Measurement and Interpretation of Diffusion Coefficients of Proteins*

Measurement and Interpretation of Diffusion Coefficients of Proteins* BY LOUIS J . GOSTING University of Wisconsin. Madison. Wisconsin

CONTENTS I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. The Diffusion Coefficient. D. an d I t s Interpretation . . . . . . . . . . . . . . . . . . . . .

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430 434 1. Flow Equations and Fick’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 a . Fick’s First Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 b . Frames of Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 e . The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 d . Fick’s Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 442 2 . Klinetic Interpretation of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Thermodynamic Interpretation of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 4. Relation of D to Molecular Size, Shape, an d Weight . . . . . . . . . . . . . . . . . . 449 a. Diffusion Coefficients of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 b. Diffusion Coefficients of Ellipsoids of Revolution . . . . . . . . . . . . . . . . . 450 e . Molecular Weights from Sedimentation and Diffusion . . . . . . . . . . . . 451 d . Diffusion Coefficients of Salts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 e. Application to Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 I11 Methods of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Steady State Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a . The Diaphragm Cell., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

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.

g . Schlieren Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . h. Gouy Interference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i. Rayleigh Interference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . j Jamin Interference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

475 476 480 484

* Recent experiments indicate t h at the subject of interacting flows considered in Section V may be of greater importance in the diffusion of proteins than was supposed from t h e limited evidence available a t the time this review was written . Accordingly a brief appendix has been added in proof to consider the present relation of Sections IV. 2 and V 429

.

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LOUIS J

.

GOSTING

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3 . Restricted Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 . a . 1‘:quations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 6 . Ilnrned’s Coiitluctance Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 487 4 . Other Init.ial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a . The Opcn-l?,nded Capillary Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 5 . Determination of 1) from t.he Spreading of a Boundary in the Ultracentrifuge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 I V . Interpretation of Experirriental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 1. Conversion to Standard Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 a . The Solvent Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 6. Ilependeiice of D on Temperature., . . . . . . . . . . . . . . . . . . . . . . . . 495 c. Dependence of I> on Concentration., . . . . . . . . . . . . . . . . . . . . . . . . 497 2 . The l’robleni of Solute P u r i t y . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 a . Invcstigation of Heterogeneity; Fringe Deviation Graphs . . . . . . . . . 502 b . Determination of D,for the Main Solute . . 3 . Some Data for Representative Solutes . . . . . . . . . a . Amino Acids and Other Materials of Low Molecular Weight . . . . . . . 511 b . l’rotcins and Other Materials of High Molecular Weight . . . . . . . . . . 514 . . . . . . . . . 525 4 . Calculation of Molecular Weights . . . . . . . . . . . . . a . Approximate Method Using Stokes’ Law . . . . . . . . . . . . . . . . . . . . . . . . 526 6. Combination of Sedimentation and Diffusion Data . . . . . . . . . . . . . . . 527 V . Interacting Flows in Systems of Three or More Components . . . . . . . . . . . . 530 . . . . . . . . . . . . . . . . . 532 1. Flow ICquations . . . . . . . . . a . F;xperirncntal Flow . . . . . . . . . . . . . . . . . 532 B . Theoretical Flow Equations . . . . . . . . . . . . . . . . . . . . 535 2 . The Concentration Distributions . . . . . . . . . . . . . . . . . . 539 3 . Determination of the Four Diffusion Coefficients for a Three-Component System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 4 . Some AppliciLtions to Biological Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 546 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

I . INTRODUCTION If pure water is layered upon an aqueous solution of sucrose in a test tube or othcr diffusion cell. there occurs a mass migration of sucrose molecules upward into the water . Simultaneously. water molecules flow downward into the solution . This process. whereby concentration differences in a solution spontaneously decrease until the solution finally beComes homogeneous. is called diffusion . Diffusion occurs because the molewles iri a fluid are in constant motion. this continual wandering about being a direct consequence of their thermal energy . It should be noted. however. that the term diffusion is applied only t o the macroscopic flow of components due to concentration differences . The microscopic random movements through the liquid of the individual molecules. which continue after the solution has reached macroscopic homogeneity. are not

MEASUREMENT AND INTERPRETATION O F DIFFUSION

43 1

referred t o as diffusion. Nevertheless, as will be seen in Section 11,2, the process of diffusion has been mathematically related to the average individual molecular movements. Diffusion is encountered frequently in investigations of biological systems. For example, a knowledge of the laws governing diffusion is basic to a n understanding of transport through the walls of living cells and within the cell itself. Also, quantitative measurements of diffusion continue t o contribute greatly to our knowledge of the molecular weight and characteristic properties of proteins and ot,her molecules of biological interest. Such measurements began over a century ago when Thomas Graham (1850) found that egg albumin diffuses much more slowly than many other common materials such as salt and sugar. By 1861 he had used diffusion and dialysis to separate mixtures of slowly diffusing and rapidly diffusing solutes and had made quantitative measurements on a great many substances. On the basis of this work he proposed (Graham, 1861) the terms colloids and crystalloids to denote classifications of matter. Unfortunately, Graham could not report his data in terms of the simple but exceedingly important diffusioncoeficient, D, defined by Fick (1855). The routine measurement of this coefficient in the equations now known as Fick’s first and second laws had t o wait for mathematical integrations of the second law by later workers, such as Stefan (1879) and Boltzmann (1894). These mathematical achievements paved the way for the development of several different experimental procedures for measuring diffusion coefficients. Descriptions of early forms of diffusion apparatus will not be given here since they have been reviewed in detail elsewhere (Longsworth, 1945; Geddes, 1949). Much of the work prior to about 1925 was concerned with the diffusion of salts and other materials of low molecular weight. These studies provided the basis for subsequent investigations of diffusion of biological materials by establishing the validity of Fick’s laws for simple (two-component) systems and by encouraging the development of improved experimental methods for measuring 1). Furthermore, out of this period came contributions of outstanding importance to all branches of the subject of diffusion: the mathematical relations between the diffusion coefficient, molecular size (more precisely, the frictional coefficient), and Brownian motion were derived about 1905 b y Sutherland (1904, 1905), Einstein (1905, 1906), and von Smoluchowski (1906). It is of interest that as early as 1905 Sutherland, using a value of D calculated by Stefan from Graham’s data, obtained a value of 33,000 for the molecular weight of egg albumin. A renewed interest in the applications of diffusion measurements t o the study of biological materials occurred about 1925 with the development of the ultracentrifuge (Svedberg and Nichols, 1923; Svedberg and

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LOUIS J. GOSTING

Pedersen, 1940). Studies made with this instrument by Svedberg and others established the fact that many proteins are relatively homogeneous substances, instead of consisting of a wide distribution of molecular sizes as believed by a number of workers at that time. Such proteins therefore have characteristic molecular weights which, by using Svedberg’s equation (Svedberg, 1925; Svedberg and Pedersen, 1940, p. 5 ) , may be calculated from measurements of their sedimentation coefficients, partial specific volumes, and diffusion coefficients. This need for values of D in order to obtain molecular weights of different proteins has not only provided motivation for studying the diffusion of new biological substances as they were recognized and purified; it has also served to stimulate the development of new and improved methods and apparatus for studying the diffusion process itself. These developments are continuing a t the present time. An important advance during the last decade has been the development of interferometric optical systems for measuring the refractive index distribution, or refractive index gradient distribution (and hence the concentration or concentration gradient distribution), in a diffusion cell. These optical systems have provided a tenfold increase in the accuracy of diffusion coefficients. As the measurement of sedimentation coefficients is also being improved, we may soon expect more accurate values for the molecular weights of a number of proteins. This increased precision of measurement is of value in studies of proteins, even when more reliable values of D are not required. One use of the high accuracy of interferometric data is to analyze the form of the distribution of refractive index (or refractive index gradient) in order to obtain information about the degree of purity of the protein preparation. It appears that a lack of homogeneity of the preparation, rather than errors contributed by the apparatus, may be the limiting factor in measuring diffusion coefficients and molecular weights of proteins. Interferometric methods are also useful for studying diffusion in systems in which two or more solute flows interact. Such coupling occurs when ions diffuse, and it is probably also present to some degree in all nonelectrolyte systems. To describe such flows the simple form of Fick’s first law is inadequate and must be extended. Very accurate refractometric measurements are required to obtain a modest accuracy for the several diffusion coefficients of the system. These studies of interacting flows not only provide basic information about the process of diffusion but they also guide studies on biological materials. For example, knowledge of this aspect of diffusion is important in studying the diffusion of proteins, especially a t pH values other than their isoelectric points. I t

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also provides quantitative information about the movement of substances against their concentration gradients caused by this coupling of the flows-a phenomenon which may be of importance in the metabolism of living cells. In this review special emphasis will be given t o the basic equations of diffusion and to consideration of correct procedures for the evaluation and interpretation of diffusion coefficients. With the rapid growth of our knowledge of diffusion, due in large part to the increased accuracy of measurement made available during the past decade, it has been found that some of the classical methods of measuring and interpreting diffusion are only approximately correct. For example, the diffusion coefficient, which was once called the diffusion constant, is now known to be appreciably dependent on solute concentration in many systems. Consequently it has become important to establish a relation between the measured value of D from a given type of experiment and the concentration to which this value corresponds. For combination with sedimentation data to determine the molecular weights of proteins, the diffusion coefficient should be measured at different concentrations and extrapolated to infinite dilution. Also some difficulty is experienced in predicting the diffusion coefficient at one temperature from a knowledge of its value at another, because the classical Stokes-Einstein relation has been found inadequate to describe recent data within the error of measurement. A further deviation from the classical laws has been referred to above, i.e., the necessity for extending Fick’s laws for systems containing two or more solutes whose flows interact. Data confirming and illustrating these phenomena will be presented below. Although many of the results were obtained with well-defined systems containing purified solutes of low molecular weight, they illustrate well the problems which should be considered when studying the diffusion of proteins. Perhaps it should be emphasized that in this review we shall consider only diffusion in liquids, although many of the concepts and equations are directly applicable to either gases or solids. The subject of rotary diffusion (Edsall, 1943) will not be discussed here. Other reviews and general references on the subject of diffusion in liquids which the reader may wish to consult include Williams and Cady (1934) , Kincaid, Eyring, and Stearn (1941), Glasstone, Laidler, and Eyring (1941), Neurath (1942), Edsall and Mehl (1943), Harned (1947), Geddes (1949), Jost (1952), Longsworth (1955), Robinson and Stokes (1955), and articles by a number of workers presented a t a conference on diffusion of electrolytes and macromolecules held by the New York Academy of Sciences (Longsworth et al., 1945).

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11. THE DIFFUSION COEFFICIENT,

D, AND ITS INTERPRETATION

1. Flow Equations and Fick’s Laws Before proceeding further we will consider briefly the subject of flow equations and their basic importance to the study of diffusion. This subject is discussed in terms of the general case of diffusion in systems with any number of components to provide perspective for consideration of the simpler two-component case. Flow equations for diffusion may be divided into two classes. The first class corisists of relations expressing flows of the components as sums of the several conc~entrationgradients, each multiplied by a diffusion eoefficient. Fick’s first law for two-component systems is the simplest example of this class. Equations in this class are basic to all experimental studies of diffusion, and we will call them experimental flow equations. The second class corisists of relations expressing flows of the components as sums of forces (gradients of the several chemical potentials or, more generally, electrochemical potentials), each multiplied by an appropriate coefficient (diffusional mobility). Although such forces cannot ordinarily be measured directly in experiments, these equations are important in theoretical considerations of diffusion and in interpreting diffusion coefficients. Relations in this second elass will therefore be referred to as theoretical flow equations. Both experimental arid theoretical flow equations describe the macroscopic niovement of components a t the different points throughout the liquid. At constant temperature and pressure, the number of terms these equations must contain to represent diffusion in the general case of interacting flows (Section V) depends on (1) the number of components present, and (.2) the number of dimensions (one, two, or three) in which diffusion occurs. However, the values of diffusion coefficients (or diffusional mobilities) appearing in these relations ordinarily depend not only on th e number and nature of the components present in the system but also on their relative concentrations. They also depend on temperature arid pressure. At eonstaiit temperature and pressure, the f o r m of the flow equations for a gillen system i s independent of the geometrical shape of the system and of the initial distribution of the concentrations.‘ By combining the experi1 A qualification is necessary. The flow of a component depends on the frame of reference, i.e., whether measurements are made relative to the diffusion cell or to some system of coordinat>esnioving relative to the cell. This fact may complicate the flow equations, especially for concentrated solutions. As will be seen later, however, the above statement is correct for experimental flow equations if volume changes on mixing may be neglected. The experimental flow equations then assume a n especially simple form when referred to a coordinate system fixed relative to the cell.

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435

mental flow equations for a given system with continuity equations expressing conservation of mass of each component, basic differential equations for describing diffusion in the system are obtained. Fick’s second law is a n equation of this type. Integration of these differential equations, subject t o the initial and boundary conditions of the case considered (Section 111),yields expressions for each concentration as a function of position and of time.2 The standard procedure for making quantitative measurements of diffusion is to utilize these integrated relations for some system of simple geometrical form to determine the numerical values of the diffusion coefficients appearing in the experimental flow equations. The known coefficients may then be used t o calculate the flows or the concentration distributions as functions of time and position in other systems of the same composition but with different initial and boundary conditions. Furthermore, by utilizing the theoretical flow equations, important interpretations of the meaning of the diffusion coefficients may be made. a. Fick’s First Law. This relation (Fick, 1855), which defines the diffusion coefficient, D, is the simplest of the experimental flow equations. It commands a unique position in the study of diffusion because of its historical position and its simplicity. It was th e only experimental flow equation in common use during the past century, and, in addition to describing experimental results for two-component systems, it also describes within the error of measurement the results for a number of systems containing three or more components. Sections I1 and IV of this review will be devoted to considerations of D and its interpretation. Modern methods of measuring D will be described in Section 111, and some parts of Section I11 will also be applicable to the more complicated general case of diffusion with interacting flows, described in Section V. For diffusion a t constant temperature and pressure in two-component systems showing no volume change on mixing, Fick’s first law for onedimensional transport of the solute (denoted by subscript 1) is J1

=

-D($)

This equation expresses the fact th at a t any time, t , and position, x, the J1,of solute relative to the cell is directly proportional to the first power of the solute concentration gradient, (ac,/ax) t . I n subsequent equaIf the diffusion coefficients or the partial specific volumes of the components vary appreciably within the system, the concentration dependence of these variations should be considered in performing the integrations. Throughout this review the boldface letter J will be used to denote flows, even in equations which are not written in vector notation.

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LOUIS J. GOSTING

tions the subscript t will be dropped from the concentration gradients; it should be understood that these partial derivatives are always taken a t constant time. The flow, J1, is positive in the direction of increasing x and is defined as the amount (moles, grams, etc.) of solute crossing a unit area perpendicular to the direction of the flow per unit of time. Consistent units must be used for c1 and J1. If J1 denotes grams of solute crossing a square centimeter per second, then x should be measured in centimeters and c1 expressed as grams per cubic centimeter; the units for D then are square centimeters per second, the standard units for reporting the diffusion coefficient. For liquid diffusion in a three-dimensional system, equation (1) may be rewritten in the vector form

where boldface type indicates vector quantities with i, j, and k being unit vectors in the positive direction along coordinates x, y, and x and V operating on el denotes the gradient of el. For simplicity we will in this review restrict equations to the case of diffusion in only one dimension; such equations are adequate for consideration of most experimental methods used to measure I). The value of D is the same for solute (component 1) and solvent (component 0) in all two-component systems if there is no volume change on mixing (Onsager, 1945; Hartley and Crank, 1949). This might be expected from the fact that in diffusion there occurs an exchange of solute and solvent, the rate of the process depending on the nature of both components, not on the nature of either component alone. This equality of the diffusion coefficients may be shown by first rewriting equation (1) as follows for each component.

Then because no volume change occurs during diffusion, the volume of solute moving up (measured relative to the cell) must equal the volume of solvent moving down, or BoJo

+

BiJl

=

0

(5)

Here and 01 are the partial specific, or partial molal, volumes of th e components, depending on the units of JOand J1. Substitution of equa-

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437

tions (3) and (4) into equation (5) gives

Differentiation of the basic relation 6oco

with respect t o x yields ac0

60-

as

+

BlCl =

1

(7)

+ 01-ac1 = 0 ax

since Bo and g1 must be independent of concentration, and hence of x, when no volume change occurs on mixing. By substituting equation (8) into equation (6) we obtain

Do = Dl (9) Therefore the same symbol, D, may be used irrespective of whether

Fick’s first law is written for the solute or the solvent. Diffusion in systems of three or more components has often been described by writing Fick’s first law for each solute separately. For this formulation, equation (1) is rewritten as

(i = 1, .

. . ,q)

(10)

so t ha t there is a separate experimental flow equation and diffusion coefficient, Di,for each of the q solutes which is present. T h e flow of the solvent, i.e., the remaining component (i = 0), may be obtained if desired from the generalized form of equation (5)

when no volume change occurs. This representation of diffusion in systems containing several components is justified within the limits of current experimental error for a number of dilute solutions (see Section IV,2,a). If large volume changes occur on mixing, other terms should be included in each flow equation as indicated below; if there is measurable interaction of solute flows, more complicated experimental flow equations are required (Section V). The earth’s gravitational field seldom has a measurable effect on the usual experimental studies of diffusion, and its influence has been neglected in all the above flow equations. If one or more of the components

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LOUIS J. GOSTINCI

has an exceptionally large sedimentation coefficient, a term should be added to describe the sedimentation produced by the gravitational field. Thermal diffusion, called the Soret effect, and th e inverse phenomenon known as the Dufour effect have also been neglected. They consist of interactions between the flow of heat and the flows of matter. Discussions of this topic are available in a number of texts including Jost (1952) and de Groot (1951). The Soret effect is usually eliminated in experimental studies of diffusion by placing the cell in a constanttemperature bath and providing adequate thermal contact between the bath and cell contents so th a t temperature gradients do not arise from the heat of mixing or from the Dufour effect. b. F r a m e s of Reference. The frame of reference for any flow equation should be given careful consideration. For example, an additional term must be added t o Fick’s first law when a bulk flow of th e liquid is superimposed on the flow due to diffusion, as occurs when there is a large volume change on L, mixing. A simple system for illustrating this problem, FIG.1. Adiagram without introducing the complication of actual illustrating the effect volume changes, is shown in Fig. 1. Here a piston of a bulk flow of forms the bottom of a long vertical diffusion cell liquid on the form of Fick’s first law. The which is open a t the top, and this piston is moving x coordinate is fixed upward with a velocity b relative t o the cell wall. relative to the cell The x coordinate is fixed relative to the cell wall and wall, while the piston x increases in the downward direction; hence b is here moves the solution i n the cell upward with a negative number. It is assumed that only two components are present and that no volume change a (negative) velocity b relative to the cell occurs on mixing in this liquid. Because the solute wall. D e n s i t y of concentration is greater a t t,he bottom of the liquid shading indicates socolumn than a t the top, solute flows upward owing lute concentration. to diffusion as well as t o the movement of the piston. To describe the flow, J1,of either component relative to the cell wall equation (1) must therefore be rewritten in the more general form

1. . I . ’-.

. . . . ..

Ji

=

dc, -I&- ax

+ c,b

(i = 0,l) (12)

where the flow c,b is the amount of component i crossing a unit area (fixed relative t o the cell wall) in unit time owing to the bulk velocity, b, of the liquid. I n Onsager’s (1945) formulation for the general case of

MEASUREMENT AND INTERPRETATION OF DIFFUSION

439

+

(q 1) components with volume changes on mixing, the bulk velocity a t each level is defined by

1aiJi 9

D =

i-0

where D is measured relative t o the cell if Jo, . . . , J, are expressed relative t o the cell. As in equations (5) and (11) the subscript zero denotes solvent and 8 0 , . . . , ii, are the partial specific volumes, or partial molal 1) components. By fixing the coordinate system in volumes, of the (q Fig. 1 on the piston, rather than on the cell wall, we could make D = 0 so t h a t the form of Fick’s first law in equation (1) would again be applicable to this case. However, when appreciable volume changes occur owing t o mixing as diffusion progresses, no simple transformation of this sort is possible. If the diffusion then occurs in a similar cell with open top but fixed bottom, the level of liquid changes with time as its volume changes. Volume changes occurring below any level x cause a bulk movement of liquid at that level, with b depending on both x and t . T o define b = 0 everywhere for this case would require that the coordinate system move with different velocities in different parts of the cell. It seems simpler t o define the coordinate system as fixed relative t o the cell, even though this requires that Fick’s first law be written in the form of equation (12) rather than equation (1).The experimental utilization of equation (12) is difficult, and further theoretical work is needed t o clarify the problem of measuring D in systems with large volume changes on mixing. Some consideration has been given t o this problem by Hartley and Crank (1949). In current experimental work it is usually hoped that equation (1) will adequately describe the diffusion if relatively small concentration differences are used within the diffusion cell. Fortunately, the problem of volume changes on mixing does not seem to be of major importance in studies of the diffusion of proteins. One other frame of reference, the local center of mass, will be considered here because it is important when writing theoretical flow equations. Its velocity, v, is given by4

+

v =

2

civi

i=O

ci i=O 4 In equation (14) component i.

ci

and

Ji

2

Ji

i=O

$

c,

i=O

must be expressed in terms of mass, not moles, of

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LOUIS J. GOSTING

where v is referred to the same coordinate system used t o measure the 1) components. The velocities, ( = Ji/ci), and the flows, Ji, of the (q flow, Ji', of component i relative to the local center of mass may therefore be written Ji' = J1. - C ~ V (i = 0, . . ,q) (15)

+

.

Throughout this review the quantities J I and v (and also b) are referred to the cell (closed a t the bottom, open a t the top, and with constant cross section) as a frame of reference unless otherwise specified. If there is no change of volume on mixing in a two-component system, use of equations (5), (7), and (14) allows simplification of equation (15) to

Consequently, for subsequent use with theoretical flow equations, another diffusion coefficient, Dl', may be defined by the flow equation

The relation of this diffusion coefficient, called the intrinsic diffusion coefficient by Hartley and Crank (1949), t o the experimentally determined diffusion coefficient for two-component systems is seen (Baldwin and Ogston, 1954) to be5 n = Bo(c0 C1)Dl' (18)

+

+

If f i 0 is expressed as milliliters per gram, (co cl) becomes the solution density, p, in grams per milliliter and Dl'-+ D as c1 --+ 0. c. The Continuity Equation. Fick's second law is obtained by substitution of his first law into the continuity equation, which expresses conservation of mass (or moles) for each of the (q 1) components. For a three-dimensional system the continuity equation may be written in the vector form

+

It is applicable t o sedimentation and electrophoresis as well as diffusion and must be satisfied by each component individually provided there are n o chemical reactions. For the one-dimensional case equation (19) reduces to aci _ at

6

See footnote on p. 448.

- -Ja

dX

(i = 0,

. . . ,q)

(20)

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44 1

It should be understood that the derivative (ac,/at) is taken with x held constant, whereas (aJJaz) is taken with t held constant. Equations (19) and (20) express the fact that the change in concentration of component i

per unit time in a unit volume is equal to the difference of the flows into and out of that volume. Their validity is independent of the particular frame of reference used, and they are true regardless of the number of components present and regardless of whether or not the flows interact. When chemical reactions occur in the diffusing system certain terms, g(ca,cj, . . .), must be added t o the right-hand side of equations (19) and (20). These terms, which describe the rate of production of component i per unit volume, must equal (&,/at) for the case of no diffusion. The form of g(cz,cJ, . .) depends on the nature of the particular chemical reaction. For example, if the reaction is zero-order so that the production of c, is independent of all the concentrations, then g(c,,cl, . . .) reduces to the constants, Kt. For a simple association-dissociation reaction of the type

.

2 A e B

where B could also be written as Az, the form of g(ca,cj, . . .) to be added to the right-hand side of the continuity equation for component A is ( K B c B - K A c A ’ ) ; the negative of this quantity is added to the right-hand side of the continuity equation for component B. It should be noted that the presence of a chemical reaction influences only the continuity equations and not the flow equations. This is true because chemical reactions. do not transport components but only create or remove them from the various positions in the system. Th e problem of diffusion with simultaneous chemical reactions has been discussed by a number of workers including Rashevsky (1948), de Groot (1951), Jost (1952), Hearon (1950, 1953a,b,c, 1954), and Steiner (1954). d. Fick’s Second Law. By substituting Fick’s first law, equation (I), into equation (20) we obtain the important differential equation for one-dimensional diffusion in two-component systems with no chemical reactions, (i = 0,l)

(2l)l

If the diffusion coefficient is independent of concentration, and therefore of 2, this relation reduces directly to Fick’s second law (i

=

0 ,l)

(22)

Equations (21) and (22) are basic to several of the methods of measurement discussed in Section 111.

442

LOUIS J. OOSTING

For systems of more than two components, but with no chemical reactions, no interaction of flows, and no concentration dependence of the diffusion coefficients, equation (10) may be substituted into equation (20) to give a generalized form of equation (22).

2. Kinetic Interpretation of D

The molecular-kinetic theory of heat, which was used by Einstein (1905, 1906) and von Sinoluchowski (1906) in developing their theories for the Brownian motion, provides us with a physical picture on which to base a description of the diffusion process in dilute solutions. Starting from this viewpoint the diffusion coefficient can be related directly to the Brownian displacements of the solute molecules. Because this approach is limited to extremely dilute solutions, however, it is less helpful to today's experimentalist than is the thermodynamic theory outlined below. Consequently only a n elementary presentation of the kinetic interpretation will be given here. According t o the molecular-kinetic theory, heat is simply a manifestation of the motion of the particles in a system. I n liquids and gases the particles can move freely from place to place, and each particle in its continual wandering will eventually visit every region in the system. This motion is characteristic of all particles, whether they are free atoms, molecules, or much larger suspended particles. The average translational kinetic energy, (>5)m?, of every particle is the same, and depends only on the absolute temperature of the system; therefore a t a given temper", of heavy particles is ature the root-mean-square average velocity, (7) less than the root-mean-square velocity of light particles. As will be seen below, however, it would be incorrect to conclude from this knowledge that the diffusion coefficient of particles in liquids is related to the particle mass. The molecules in liquids are so close together that no particle can travel an appreciable distance without colliding with other molecules and changing its direction. Einstein (1907) pointed out th a t this rapid and irregular change in direction makes impossible the measurement of velocities of particles in liquids. It is possible (with the ultramicroscope), however, to measure the overall change in position of an observed particle over a relatively long period of time (seconds or minutes), even though the complete zigzag path of the particle between the two points considered is not measurable and is much longer than the straight-line distance. This concept of the relatively long-time displacements of particles in a liquid, rather than their velocities, forms the basis of the kinetic interpretation of diffusion.

MEASUREMENT AND INTERPRETATION O F DIFFUSION

443

For a simple derivation of the relation between the diffusion coefficient and molecular (or particle) displacements, we follow an elementary theory presented by Einstein (1908). The reader can find an English translation of this paper, and also of four other papers by Einstein on the subject of Brownian motion, in a single small volume (Einstein, 1956) edited b y Furth. We restrict our consideration to diffusion in one dimension in tube of unit cross section, Fig. 2. In this tube is a dilute solution or suspension of some material referred to as the solute, with the solute concentration greater on the right than on the left. During a time interval At, taken sufficiently small so th at only small changes occur in the solute concentration a t any point, each particle will move t o a new position

FIQ.2. Illustrative diffusion cell for deriving the relation between D and the average Brownian displacements, 6, of solute particles in time At. Shading indicates solute concentration.

owing t o its thermal motion. Denoting the corresponding change of the coordinate of each particle by &, we have a value a1 for particle 1, 82 for particle 2, etc., some being positive and some negative. Because the solution is so dilute, these displacements are independent of the solute concentration; hence the probability th at a particle will have some particular displacement is independent of its position in the cell, and positive and negative displacements are equally likely. To simplify the calculation we will assume t hat in time At every particle experiences the same change, 6, in its x coordinate, half of them being to the right (+S) and the other half t o the left (- 6). With these simplifications it is clear that one-half of the particles in the region A,P will move to the right in time At for a sufficient distance to cross plane P. Since the tube is of unit cross section, this number of particles is 2

na = % ~ d

(24)

where c, is the average number of particles per unit volume in the region

444

LOUIS J. GOSTING

A.P. During the same time nb particles from region PAb cross plane P from right to left. n b = >4CbS (25) The concentration change with x can be considered linear over these short distances, so c, and c b may be taken as the values a t the middle of each region, i.e., at planes &. and &b, respectively. From equations (24) and (25) we obtain the net solute flow across plane P in the +z direction in time At. J At = 72, - nb = %6((Ca - c b ) (26) This equation describes the basic fact of the kinetic view of diffusion, i.e., t ha t all like solute particles move on the average the same distance in unit time, but a net flow results because more particles are moving from regions of higher concentration to regions of lower concentration than vice versa. Since planes &, and Qb are a distance 6 apart, the solute concentration gradient a t plane P can be expressed as dc dz

cb

- ca

6

(27)

Substitution of this relation into equation (26), and combination with Fick’s first law, equation ( l ) , leads to the desired result

D

1 6 2

=--

2 At

More accurately, a2 should be replaced by the mean-square Brownian displacement, P . Thus we see th at the diffusion coefficient in dilute solutions is simply one-half of the mean-square Brownian displacement per second measured along the x axis. This basic relation cannot be subjected to the precise experimental tests it deserves because measurements of Brownian motion with the ultramicroscope are quite limited in accuracy, and materials of low molecular weight which can be highly purified cannot be observed a t all. Einstein (1906) also showed that the mean-square Brownian displacement along the x axis is given by

where R is the gas constant, T the absolute temperature, N Avogadro’s number, and f l the frictional coefficient of a solute particle. Combination of this relation with equation (28) leads to the important result for very dilute solutions,

MEASUREMENT AND INTERPRETATION OF DIFFUSION

445

This relation is basic to most interpretations of the diffusion coefficient and will be derived in a more general form in the following section.

3. Thermodynamic Interpretation of D The generalization of classical thermodynamics t o include nonequilibrium systems (see for example Onsager, 1931a,b; Prigogine, 1947; de Groot, 1951; Denbigh, 1951; Kirkwood and Crawford, 1952; Hirschfelder, Curtiss, and Bird, 1954) has led to a formulation which is sometimes called the thermodynamics of irreversible processes. Of basic importance in this comparatively recent development are Onsager’s reciprocal relations (Onsager, 1931a,b), which lead t o new relations between the coefficients in experimental flow equations for several types of interacting flows (matter, heat, and electricity). I n this article we make no attempt to review the thermodynamics of irreversible processes, b u t will merely take from that formulation some of its simpler results which are helpful to a discussion of diffusion. In particular, that theory guides us in writing theoretical flow equations, both here and in Section V, and leads us t o associate chemical potential gradients with the driving forces in diffusion. It will be recalled that gradients of potentials are also forces in the case of electrical potentials and gravitational potentials. Early derivations of the dependence of D on temperature and the frictional coefficient, including those of Nernst (1888), Sutherland (1905)) and Einstein (1905), considered the force on the solute molecules to be a gradient of osmotic pressure. Although that formulation led t o the correct expression for D in dilute two-component systems, it is hardly suitable as a guide t o writing flow equations for systems of three or more components with interacting flows. Furthermore, it may seem esthetically unsatisfactory to base derivations on the concept of an osmotic pressure when t ha t pressure has no physical reality in the system considered. These limitations are avoided by using the gradient of chemical potential (or of the partial molal free energy), rather than the osmotic pressure, as the driving force (van Laar, 1907; Taylor, 1927; Hartley, 1931; Onsager and FUOSS, 1932; Onsager, 1945; Lamm, 1947). We first define the chemical potential, pi, per mole of component i. This quantity was introduced b y Gibbs (1875), who defined it for a homogeneous part of a system, i.e., a single phase, by the four equivalent definitions6

6 Gibbs defines the chemical potential per unit mass of component i, instead of per mole.

446

LOUIS J. GOSTING

Here ni denotes the number of moles of component i in the phase and the subscripts indicate whether entropy, S , volume, V , pressure, P , or absolute temperature, T , are to be held constant during the partial differentiation. The subscript n,,, indicates that the number of moles of every component but i is held constant. These definitions follow from the four basic equations for changes in the total energy, El the enthalpy, H , the maximum work, A , and the free energy, F , of a homogeneous phase.

dE dH dA dF

+ podno + . . . + p Q d n q + + . + pQdn, + + . + pLLpdnq + + + . + pQdnq

=

TdS - PdV

= = =

TdS VdP podno -SdT - PdV podno -SdT VdP podno

+

*

*

*

(32) (33) (34) (35)

Only pressure-volume work, Pd V ,is being considered here. Gibbs showed that equation (32) may be written as a consequence of the first law of thermodynamics and the definition of entropy, and equations (33)-(35) follow from equation (32) and the definitions of H , A , and F ,

H=E+PV A=E-TS F = E PV - 1’s

+

The last definition of pLzin equations (31) shows that the partial molal free energy of a component serves as one of the four definitions of the chemical potentid of that component. Although we have defined p i using equations (32)-(35), which are strictly applicable only to homogeneous regions, or phases, the coefficients T , P , and each p 2 are intensive variables which may also be defined for every small element of volume in nonequilibrium systems, providing the systems are not too far from equilibrium. Having defined the chemical potential, we now write the theoretical flow equation for the solute in a two-component system at constant temperature and pressure as a force, X 1 = -dpl/dx, times a diffusional mobility, L1 (Onsager and FUOSS, 1932, p. 2759).

The extension of this relation to multicomponent systems will be considered in Section V. Here J1’,measured relative to the local center of mass (see equation (15)), is used rather than J1,the flow relative to the cell, in agreement with the procedure followed by a number of recent workers when negative gradients of chemical potentials are used to represent driving forces (de Groot, 1951 ; Goldberg, 1953; Hirschfelder, Curtiss,

MEASUREMENT AND INTERPRETATION O F DIFFUSION

447

and Bird, 1954, p. 712 ff.; Bird, Curtiss, and Hirschfelder, 1955). This flow may also be expressed in terms of the average velocity, vl’, of solute relative t o the local center of mass,

and the last equality simply states the familiar relation that velocity equals force divided by frictional resistance. From equations (39) and (40) it is seen t hat Ll/cl = l/(Njl). Here the frictional coefficient, f l , per molecule of solute is multiplied by Avogadro’s number, N , to give the frictional coefficient per mole because p1 is commonly expressed as the chemical potential per mole. Units of moles per square centimeter per second are assumed for J;, and units of moles per cubic centimeter for c1. I n rewriting equation-(40) in a form analogous to Fick’s first law, we introduce the more conventional concentration scale, C1, denoting moles per liter, or molarity. (41) Here dpl/dC1 may be written as a total derivative because T and P are held constant and only two components are present. Comparison of this relation with equation ( l ) , after substituting equation (16), shows that

+

in which p = co c1 is the solution density in grams per milliliter, the partial specific volume of the solvent in milliliters per gram, and the units of concentration are seen to cancel out. To evaluate d p l / d C l we utilize the conventional relation between p l and solute concentration expressed in molarity, P I = pl0 RT In ylCl (43)

+

where the solute activity coefficient, yl, is in general a function of C1, depending somewhat on T and P ;the reference potential, plo, is independent of C1, depending only on T , P , and the reference state for y1. Differentiation of equation (43) with respect t o C1 and substitution into equation (42) leads to the desired expression for the diffusion coefficient

The term [l

+ Cl(d In yl/dC1)] is usually called the thermodynamic term;

448

LOUIS J . GOSTINQ

the term (Bop) (Baldwin and Ogston, 1954)’ appears because equations (39) and (40) were referred to the local center of mass, while equation (1) used the cell for a frame of reference. In most derivations of this equation the term (Bop), which deviates but little from unity in dilute solutions, is either neglected or combined with f l giving a different frictional coefficient. As C1-+ 0 in equation (44), D + R T / ( N f l ) in agreement with the kinetic approach, equation (30). Hartley and Crank (1949) have considered equations for the diffusion coefficient in connection with several different frames of reference; Lamm (1954) has discussed some aspects of this topic. Equation (44) can be influenced not only by the frame of reference but also by the choice of components. If the driving force is taken as the gradient of chemical potential of component l’, where 1 mole of I‘ contains a moles of component 0 associated with 1 mole of component 1, a term in a appears in equation (44) if the activity coefficient is expressed in terms of component 1 rather than 1‘ (Stokes, 1950b; Robinson and Stokes, 1955, p. 314 ff.). Some problems relating to the choice of components have been considered by Lamm (1955). Accurate experimental tests of equation (44) have been possible only with solutions of single, ionized, salts, for which the variation of D with C in dilute solutions is due largely to the thermodynamic term and the variation of f l with concentration is given b y the Onsager-Fuoss theory (Onsager and FUOSS, 1932). Although general theories of the frictional coefficient in noiielectrolyte solutions have been developed (Kirkwood, 1946), numerical values are not easily obtained and prediction of the variation of the frictional coefficient with concentration, to permit experimental tests of equation (44) for nonelectrolyte solutions, does not yet appear practical. It should be noted that the thermodynamic term in equation (44) does not necessarily equal unity for all concentrations of a n “ideal” solution. Suppose, for example, that we define an “ideal” solution as one for which the solute activity coefficient, f l , on the mole fraction scale is unity for all values of the solute mole fraction, N1. If we adopt the customary conventions that yl -+ 1 as C, -+ 0 and f l -+ 1 as N1+ 0, it is possible to derive the relation y1 =

[

lOOOp0

lO0Op - ( A l l

- hf0)Cl

1

f1

(45)

7 The equations given a t the bottom of p. 753 of the Baldwin and Ogston article (1954) should be corrected by including brackets, i.e.,

s = [M*(1 - P2P)/j%”IPIP

in their notation.

MEASUREMENT AND INTERPRETATION O F D IFFU SIO N

449

where p is the density of the solution, M 1 the molecular weight of the solute, and PO and Mo the density and molecular weight of pure solvent. If f l remains unity at all concentrations, then it is evident that in general y1 must vary with concentration and consequently the thermodynamic term will vary with concentration. 4. Relation of D to Molecular Size, Shape, and Weight T o aid in visualizing the dependence of the diffusion coefficient on particle size and shape for simple idealized models, we first summarize below in Parts a and b of this section the classical expressions for the diffusion coefficients of spheres and ellipsoids of revolution. Unfortunately, these relations, unlike Svedberg’s equation for obtaining the molecular weight from sedimentation and diffusion data (Part c), cannot lead directly to exact interpretations of experimental data in practice; protein molecules are not, in general, either spheres or ellipsoids. Even if they were, uncertainties exist because of the unknown “hydration” (or, for the general case, “solvation”) of the molecules in solution, and because of the fact that such hydration is not necessarily of uniform thickness over the entire ellipsoid. The discussions of diffusion and sedimentation (Part c ) , and of diffusion and electrical conductance in ionic solutions (Part d ) avoid completely any consideration of particle shape and hydration; instead they relate properties of the solutions measured in different types of experiments. Consideration of the application of these several relations to studies of proteins is in general left for Part e of this section. a. D i f u s i o n Coeficients of Spheres. According to Stokes (1851) the frictional coefficient, fl, of a rigid sphere of radius r1 moving through a medium of viscosity 7 is given by fi

=

67~1

(46)

This simple relation is strictly valid only if the fluid medium is continuous and if no slip occurs between it and the sphere.8 Therefore Stokes’ equation may be applied rigorously only to very dilute solutions of spherical molecules which are large compared to the size of the solvent molecules. Substitution of equation (46) into equation (30), or into equation (44)with C1 = 0, leads to the desired expression for the diffusion coefficient,

8 I n addition, Stokes pointed out t h a t the velocity of the sphere must be small so that terms in the square of the velocity may be dropped in the force-velocity equation.

450

LOUIS J . GOSTINQ

This relation is considered in more detail in Part e of this section, and it will be helpful in Section IV for converting measured diffusion coefficients to standard conditions. The range of validity of this equation ha5 been discussed in considerable detail by Longsworth (1955) in the light of recent data. It is of historical interest to note that Sutherland (1905) in deriving equation (47) used a more general form of Stokes’ law for which some slip occurs between the sphere and the medium, (4th)

where /3 is the coefficient of sliding friction. For the case of no slippage P -+ 0~ and he obtained equation (47) ; for large slippage p -+ 0 and instead of equation (47) he obtained

He interpreted the failure of equation (47) to describe diffusion of molecules with dimensions comparable to those of the solvent molecules as being due t o a form of slippage. b. D i f u s i o n Coeficients of Ellipsoids of Re lotiction. Perrin (1936) derived formulas analogous to Stokes’ law which allow calculation of the diffusion coefficients of both elongated (prolate) and flattened (oblate) ellipsoids of revolution. For a general ellipsoidal particle, with each principal semiaxis a, b, and c of a different length, there are three separate frictional coefficients, fa, f b , and fc, for movement along each of these three axes. Because of Brownian motion the particle will constantly change orientation during diffusion, and Perrin showed that its diffusion is described by a n average frictional coefficient, l/fl = ($.S)(l/fa l/fb l/fc).By considering only ellipsoids of revolution, he obtained relatively simple expressions for the ratio of D to D,,the diffusion coefficient of a sphere of the same volume as the ellipsoid. Denoting by pe the axial ratio, b / a , of the ellipsoid, where b is the equatorial radius and a the semiaxis of revolution, he obtained for elongated ellipsoids of revolution ( p e < 1)

+

+

(48)

and for flattened ellipsoids of revolution

(pe

> 1)

MEASUREMENT AND INTERPRETATION OF DIFFUSION

45 1

Here 11is the frictional coefficient of the ellipsoid and fs the frictional coefficient of a sphere of the same volume. Tables of the frictional ratio, fl/fa, as a function of the axial ratio, l/pe, for these two types of ellipsoids are available in Svedberg and Pedersen (1940, p. 41) and in Cohn and Edsall (1943, p. 406). Boyd (1955) has tabulated the inverse relation, 1/pe as a function of fl/f8, for the case of elongated ellipsoids of revolution. c. Molecular Weights from Sedimentation and Diffusion. The problem of molecular size and shape is bypassed, and molecular weights obtained directly, by combining data from diffusion and sedimentation experiments in Svedberg’s equation (Svedberg, 1925; Svedberg and Pedersen, 1940, p. 5). This is one of the best methods for determining the molecular weights of proteins. Therefore, some of the refinements, and the limitations, of Svedberg’s equation will be briefly considered. The sedimentation coefficient, sl, of component 1 is defined as its velocity of sedimentation, 01, per unit centrifugal (or gravitational) field, provided there is no volume change on mixing; the cell wall is used as the frame of reference. For an ultracentrifuge this becomes s1 =

211 -

w 2r

where w is the angular velocity of the rotor, r is the distance from the axis of rotation to the position of the sedimenting particles, and v1 is obtained from measurements of the sedimenting boundary. The subscript 1 is used on sl because even in a simple two-component system the sedimentation coefficient, unlike the diffusion coefficient, is in general different for solute and solvent. By using the negative gradient of the total potential, p 1 = pl - M1w2r2/2, as the sedimenting force for an unionized solute (Lamm, 1953; Williams, 1954), a theoretical flow equation may be written for sedimentation a s was done for diffusion. Thus an expression for s1 in a two-component system can be derived in a manner analogous to the derivation of equation (44) for D.

Here ijl is the partial specific volume, and .Ml the molecular weight, of the solute, and p is the density of the solution. The term (ijop) was shown by Baldwin and Ogston (1954)9t o appear when the velocity of sedimentation in the theoretical flow equation is referred to the local center of mass.

* See footnote on p. 448.

452

LOUIS J . GOSTINQ

A generalized form of Svedberg’s equation for M l is obtained by eliminating f l from equations (44) and (51). slRT

M 1 = D(1 - B1p)

[I

+GI-&-

In

yll

This is permissible because the frictional coefficient of a molecule is the same for sedimentation and diffusion, unless the particle is so extremely large and asymmetric that it is oriented during sedimentation. When c1 0 the thermodynamic term in brackets becomes equal to unity and equation (52) reduces to Svedberg’s equation, ---f

If every molecule of the solute dissociates into v+ cations, each of mass M+ (per mole of ions), and v- anions of mass M-, so that v+M+ v X -= M 1 , equation (52a) is no longer applicable (Svedberg and Pedersen, 1940, pp. 13-14). One then obtains instead of equation (52a) the relation,

+

where 51 denotes the partial specific volume of the neutral salt. Equation (526) shows that any attempt to calculate the molecular weight of a dissociated salt from equation (52a) leads not to M 1 but to the arithmetic mean molecular weight of the ions,

M1 (v+

+ .-I

-

(v+M+ !v+

+ v-M-). + v-)

Precise experimental tests of equation (52) are difficult because the only solutes which inay be obtained in high purity are those of low molecular weight; only recently are measurements of s1 for these slowly sedimenting solutes becoming available. However, until precise experimental tests are possible, equation (52) may reasonably be accepted as valid at all solute concentrations in two-component systems of nonelectrolytes. Because yl is seldom known for proteins, however, it is usually necessary to extrapolate data for sl, D, and al to zero concentration, where the thermodynamic term becomes unity, in order t o determine M l accurately. The choice of components should also be considered when determining molecular weights of proteins. Equations (52) and (52a) are valid for any choice of components in a two-component system, irrespective of hydration in solution, but the calculated value of M1 will correspond to the

MEASUREMENT AND INTERPRETATION OF DIFFUSION

453

component for which ih (and yl) are actually measured (Baldwin and Ogston, 1954). If, for example, a crystallized protein is assumed to be dry but actually contains some water, the value obtained for D 1 a t c1 = 0 will depend on the moisture content. The values of s1 and D a t c1 = 0 are independent of moisture in the sample. As a result the value of M 1 obtained from equation (5%) will refer to the same “hydrated” solute for which 01 was measured. If the sample was completely dry, the anhydrous molecular weight is obtained. The problem of hydration of the solute has also been considered by Cheng (1955), who obtained an equation which may be reduced to equation (52). d. Digusion Coeficients of Salts. A general consideration of the interpretation of diffusion data for proteins should include consideration of the diffusion of a weak electrolyte in a system containing three or more kinds of ions. Further research is needed, however, t o develop and test a proper formulation of this general case. In summarizing some of our present knowledge we include here a brief consideration of the interpretation of diffusion data for a single ionized salt; the case of diffusion in systems containing three kinds of ions is considered in Section V. The theory of electrolyte diffusion is in a more highly developed state than the corresponding theory for nonelectrolytes. At infinite dilution, the diffusion coefficient of a salt is given in terms of the limiting ionic conductances by Nernst’s equation (1888). Onsager and Fuoss (1932) have extensively treated the subject of electrolyte diffusion, conductance, and viscosity in dilute solutions and have obtained a n expression for the concentration dependence of D. Diffusion in electrolyte solutions has been thoroughly reviewed by Harned (1947), Stokes (1955), and Robinson and Stokes (1955). Harned and his associates have obtained a considerable amount of accurate data for testing the Onsager-Fuoss theory for many electrolytes in very dilute solutions (Harned and Nuttall, 1947; also see several other articles by Harned and co-workers, J . Am. Chem. SOC.,1949 to present). T o illustrate the type of expression obtained for D, for comparison with equation (44), consider a simple system consisting of a dilute single salt. Suppose t hat one molecule on dissolving dissociates into v+ positive ions and v- negative ions. Because of the condition of electrical neutrality the two kinds of ions must move together in diffusion, rather than diffuse at rates corresponding to the individual ionic frictional coefficients, f+ andf-. As a result the salt may be considered t o diffuse with a n average frictional coefficient, f l = (v+f+ v_f-)/(v+ v-). These ionic frictional coefficients may be replaced by the limiting ionic equivalent conductances, A+ and A_, by using the relation fi = K’lx,(/Xiin which (zil denotes the magnitude of the valence of the i t h ion, By expressing the constant K‘ in

+

+

454

LOUIS J. GOSTING

terms of the Faraday, F , and utilizing the relation v+lz+[ = v-Ix-/, one obtains the first, or Nernst, term in the following series expression for D derived by Onsager and Fuoss for dilute solutions of a single, completely ionized, salt.

D

=

(v+

+ v J R T [’”-’(---) A+AF2v+Iz+l A+ + A-

+K

+.. (1

.]x

+ CIX) d In yl

(53)

It will be noted th at no approximation of ionic shape by spheres or ellips-

oids is required; the diffusion coefficient a t infinite dilution is expressed directly in terms of other measurable quantities, just as equation (52) may be thought of as expressing D for a nonelectrolyte in terms of sl, fill yl,etc. The limiting ionic equivalent conductances are obtained from measurements of specific conductivity and transference numbers. Since Cl(d In yl/dC1) for electrolytes is proportional to at high dilution, it is seen that D is linear in fifor dilute salts; for dilute nonelectrolytes it is linear in C1.The Onsager-Fuoss theory expresses the coefficient K , arising from the “electrophoretic effect,” in terms of known constants for the system, and it also predicts the coefficient of the next term of the series. Because these coefficients are quite complicated (Harned, 1947), expressions for them will not be included here. The “time of relaxation” effect is absent in diffusion of systems with only two kinds of ions because the two ions must move with the same velocity. e. Application to Proteins. The correct molecular weight of an unionized solute in a two-component system may be obtained from equation (52), provided that D,sl, and 61 are extrapolated to c1 = 0 when y1 is unknown. However, studies of proteins are complicated by the fact that they are often ions, rather than uncharged molecules. If no electrolyte is added, so that only the protein ions and their counter-ions are present in solution, all the ions are forced by the condition of electroneutrality t o move with the same velocity. The average molecular weight obtained for this case (equation (52b)) is much less than that of the protein ion, and depends on the valence of the protein ion and on the mass and valence of the counter-ions. T o free the protein ions and allow them t o move with a rate characteristic of their own frictional coefficient, it is customary to add a relatively large amount of “swamping electrolyte” (Svedberg and Pedersen, 1940, p. 23 ff.). This procedure has a t least some inconsistency, in principle, because the partial specific volume can be measured only for neutral salts and not for ions. Errors arising from this difficulty should be small, but it is hoped that future theoretical

dc

MEASUREMENT AND INTERPRETATION O F DIFFUSION

455

work will clarify the determination of “molecular” weights of large ions in three-ion systems. I n the discussion immediately following we consider only uncharged solutes in two-component systems. The classical procedure for calculating molecular shape using Perrin’s formulas (equations (48) and (49)) is limited not only by the fact th a t most molecules are undoubtedly not ellipsoids but also by the fact t h a t a precise value for the volume cannot be determined for calculating the diffusion coefficient, D,, of a sphere of equal volume. T o approximate D, it has been customary to assume that the volume of the solute molecule in solution is given by DIM1/N,so that (54) The axial ratio, pe = b/a, of the ellipsoid is then obtained from tables of the functions in equations (48) and (49). One reason this procedure cannot be made exact is that the volume & M 1 / N is not the actual volume of the solute molecule, but a combination of this volume and the volume change produced by electrostriction in the surrounding solvent. Furthermore, the required volume should include any bound water of hydration which moves with the solute molecule. The large axial ratios which may be obtained for spheres through neglect of the binding of solvent has been discussed by Kraemer (1940). Scheraga and Mandelkern (1953) have proposed a method for calculating the size and axial ratio of an “effective” hydrodynamic ellipsoid whose properties represent the kinetic unit in solution. Instead of assuming t ha t the volume of the solute molecule is DIM1/N,they consider the effective molecular volume, V,, to be unknown and evaluate it by utilizing, in addition t o other experimental data such as D and s1, the intrinsic viscosity [q] =

[;(

- 1)

$1

= C,+O

NV,

100&!1

y

(55)

Here q is the viscosity of a solution of concentration c1 (dry weight), expressed in grams per 100 ml., and q o is the viscosity of pure solvent. The shape factor Y has been tabulated for elongated and flattened ellipsoids of revolution (Mehl, Oncley, and Simha, 1‘340; Cohn and Edsall, 1943, p. 519) using the equations developed by Simha (1940). Scheraga and Mandelkern define a new shape factor,

456

LOUIS J. QOSTING

and compute its value for a number of proteins. They give tables relating fl t o the reciprocal of the axial ratio, pe, of the effective ellipsoid. Because p changes negligibly with pe for flattened ellipsoids it may be possible t o distinguish between flattened and elongated effective ellipsoids. This approach appears t o eliminate the theoretical limitation associated with using elM1/N to approximate the effective volume when determining axial ratios. However, inspection of their tables shows th a t /3 changes very slowly with pe even for elongated ellipsoids, for which only a 2% change in p occurs as l / p e changes from 1 to 3. Consequently very accurate measurements of the quantities used to calculate will be required in order to utilize this method. When Tanford and Buzzell (1954) tested this method by combining their data for the intrinsic viscosity of bovine serum albumin with results of other workers for sl, D,and el, they obtained values for p of (2.04 k 0.06) X 106and (2.01 +_ 0.12) X lo6.They point out that since these values are less than the theoretical lower limit, p = 2.12 X lo6, the theory may be inapplicable for particles which are not rigid ellipsoids (see also Tanford and Buzzell, 1956). The problem of determining molecular shape has been considered in detail by a number of workers including Ogston (1953), Sadron (1953), and Edsall (1953). Several approaches t o this problem are possible, but none seems to be entirely satisfactory. The simplicity of the approximate relation for the diffusion coefficient of a sphere, equation (54), has attracted attention t o the possibility of using it for direct determination of rough values of molecular weights. This approach, which was used as early as the time of Sutherland (1905), is now of special interest in connection with substances having molecular weights of several hundred to a few thousand, for which measurement of s1 is diiKcult. When end-group titration or some other method of chemical analysis is used t o obtain M1 for these substances, it is generally possible to obtain only a minimum molecular weight. If diffusion data can then be used t o determine MI within 10% or 20%, the correct molecular weight can be established. Pedersen and Synge (1948) used this approach in studies of gramicidin S, tyrocidine, and gramicidin. Polson (1950), by assuming ij1 to be approximately the same for all substances, wrote equation (54) as

However, the values of K that he then computed for a number of proteins and other compounds with molecular weights ranging from 20 to 294,000 were found t o increase markedly for molecular weights below 1000, indicating appreciable breakdown of Stokes' law, equation (46), as the size of

MEASUREMENT AND INTERPRETATION O F DIFFUSION

457

the solute molecules approached that of the water molecules. He then tried a n empirical extension (Polson and van der Reyden, 1950) of equation (57), using three empirical constants, K1, K z , and KB,

and found this relation to provide a better fit of the data for the compounds. Longsworth (1952) simplified this approach by using the empirical relation

D

=

A'/(M,'.' - B' 1

(59)

which has the advantages that a plot of DMI5S versus D should be linear and t ha t only two constants must be determined. He tested this relation and also the empirical relation

n = a ' / ( P p - b')

(60)

vl

where is the partial molal volume of the solute, using precise data from interferometric measurements of free diffusion. The results of these tests showing the degree of validity of equations (59) and (60) will be described in Section IV,4,a. 111. METHODS OF MEASUREMENT Although Fick's first law, equation (l), defines the diffusion coefficient in terms of the flow of solute and its concentration gradient, a straightforward calculation of D by direct measurement of these two quantities is not possible. Only the concentration gradient term may be measured directly; the flow of solute (unlike the flow of electricity in a wire) is not susceptible t o direct measurement. The closest approach t o a direct determination of the flow is provided by the steady-state method, wherein J 1 is computed from measurement of the concentration changes which i t produces in two homogeneous solutions. All the other methods for determining D utilize integrated forms of Fick's second law, equation (22), (or equation (21) if D varies appreciably with concentration). It will be seen below that, owing to certain experimental difficulties, absolute values of D can be obtained only from the latter methods. T h e steady-state apparatus, as illustrated by the diaphragm cell, requires calibration with a material of known diffusion coefficient. To simplify the notation for the solute concentration in two-component systems, we hereafter drop the subscript 1, writing simply c ; c1 will be used only when three or more components are present.

458

-

LOUIS J. GOSTING

1. Steady-State Diffusion

I n principle this is probably the simplest type of experimental arrangement for determining D. In practice, as is true of all the methods, careful attention must be paid to a number of theoretical and experimental details in order t o obtain accurate results. Figure 3 illustrates the essential characteristics of the method. Diffusion occurs in the narrow vertical tube of uniform cross-sectional area a and length h between two reservoirs of solution, A and R, in which the solute concentration in B, C B , is greater than that in A , c A . The volumes of the two reservoirs are known and assumed to be equal for this elementary consideration, so V A = Vo = V . After the apparatus has stood for sufficient time the concentration gradient becomes constant throughout the tube and equal t o ( C B - cA)/h, provided D is independent of concentraFIG.3. Illus- tion. The solute flow, J, may then be computed by trative diffusion cell for steady- measuring the concentration change, Ac, in C A occurring state measure- in some small time At. For this simple hypothetical apparatus D is given by the relation ments.

I n practice solutions A and B must be thoroughly stirred so that the diffusion process is restricted t o the vertical tube. The stirring, however, causes convection in the ends of the tube, creating uncertainty in the value of the tube length h. Furthermore, if the experiment is continued long enough to produce sufficiently large concentration changes for accurate measurement of Ac, the values of C A and C B change enough to alter appreciably the concentration gradient in the tube. The experiment is then no longer true steady-state diffusion, but a quasi-steady state, and equation (61) must be revised accordingly. a. The Diaphragm Cell. A practical form of steady-state diffusion apparatus for studying proteins was developed by Northrop and Anson (1929). It can be used to study impure samples, provided that a method is available for determining relative concentrations of the solute in question in the homogeneous solutions taken from the cell at the end of the experiment. I n order t o obtain measurements in a reasonably short time, they

MEASUREMENT AND INTERPRETATION OF DIFFUSION

459

replaced the tube of Fig. 3 by a porous diaphragm of sintered glass or alundum. This diaphragm may be considered to approximate a large number of small, parallel tubes. The effective tube cross-sectional area, a, was thus greatly increased while a t the same time the length, h, could be made small without introducing appreciable convection into the tube ends. In addition to these advantages the diaphragm provided considerable stabilization against thermal or mechanical disturbances. Northrop and Anson’s original cell was a bell-shaped glass vessel of about 20 C.C. capacity fitted with a stopcock at the top and the diaphragm at the bottom. After filling the cell, by sucking solution up into it through the diaphragm, the cell was clamped so that its diaphragm was level and immersed in a container of solvent. Since diffusion took place downward into the solvent, convection stirred the solution inside the cell as well as the solvent below. To establish a uniform gradient in the diaphragm, the solvent was changed a t several successive intervals of equal time and the diffusion coefficient was calculated after the quantity of solute passing through the membrane became the same for each time interval. If good accuracy is to be obtained with this apparatus it is usually necessary to correct for the change of concentration with time inside the cell. Calibration of the cell using a material of known diffusion coefficient is required, because the effective pore size and length in the diaphragm cannot be measured directly. Refinements introduced by a number of workers have increased considerably the accuracy of the diaphragm cell. Gordon (1945) has critically reviewed the problems involved in diaphragm cell measurements. His choice of subjects for discussion illustrates well the problems which should be considered: ( I ) the question of cell volume, (2) homogeneity of the inner and outer solutions, ( 3 ) the assumption of a steady state in the diaphragm, ( 4 ) the mechanism of transport in the diaphragm, (5) calibration of the diaphragm cell, and (8) the relation between the diaphragm cell integral coefficient and the differential coefficient. The last problem is of special importance because the diffusion coefficients of salts, which have been used as standard calibration substances, often vary markedly with solute concentration. To minimize this difficulty Anson and Northrop (1937) proposed the use of sodium chloride as a calibration substance. Its diffusion coefficient is constant within 1%in the range of 0.1-1.0 mole per liter (Stokes, 1950b); hence a simple calibration yields results of corresponding accuracy if the concentrations of both initial solutions, A and B, lie within this range. Stokes (1950a,b) designed a diaphragm cell with improved stirring. After carefully testing several of the experimental variables and developing an improved calculation procedure to take into account the concen-

460

tration dependence of as a function of C for accuracy of the order two compartments, A

LOUIS J. GOSTlNG

the diffusion coefficient, he obtained values of D a number of salts. His results appear t o have a n of 0.1%. As shown in Fig. 4, his cell consists of and B, separated by a No. 4 sintered glass diaphragm, D, about 40 mm. in diameter and 2-3 mm. thick. The dashed line indicates the level of water in the constant-temperature water bath, The small glass tubes, E and F , contain iron wires and are designed so th a t E will sink to the diaphragm while F rises u p to it. As the magnet M is rotated these tubes wipe the surface of the diaphragm and provide thorough stirring throughout compartments A and B , each of which has a volume of about 50 ml. Stokes found th at reproducible results were obtained with stirring rates of from 25 to 80 r.p.m. ; lower rates gave markedly lower values of the cell constant. The stirrers slowly cause wearing of the diaphragm over a series of experiments, so the cell constant, & defined below, should be redetermined from time t o time. Its FIG.4. The diaphragm cell developed by Stokes, Value for a given experiment may then be read from a graph of cell constant or a / h versus time. He also found that to prevent bulk streaming of liquid the diaphragm must be level within a very few degrees when large concentration differences exist across it. When filling the cell, care must be exercised to eliminate all air bubbles and t o establish an initially constant concentration gradient through the diaphragm. First, Stokes filled the entire cell with solution, using suction at the top t o remove air bubbles. After reaching temperature equilibrium in the bath, the solution in A was sucked out through a tube and replaced by pure solvent. Following a preliminary period of diffusion of a few hours t o establish the uniform concentration gradient, compartment A was again emptied, rinsed, and filled a second time with pure solvent. The time of this filling was taken as the start of the experiment. It will be noted that the liquid in A is thus known to be pure solvent with a concentration C A O = 0 a t 2 = 0; the concentration cg0 in B a t this time is unknown because of the preliminary diffusion required t o establish th e gradient in the diaphragm. I n his studies of relatively concentrated salt solutions Stokes allowed diffusion to proceed for 1-4 days, these long times producing relatively large concentration changes which could be accurately measured. At the end of each experiment the liquids in A and

MEASUREMENT AND INTERPRETATION OF DIFFUSION

46 1

B were carefully removed and the final concentrations, denoted simply by C A and C B , were determined. An average, or integral, diffusion coefficient was then calculated from the equation for the diaphragm cell (Gordon, 1945; Stokes, 1950b)

+

The cell constant p = (a/h)(l/VA l/VB) must be determined by Calibrating the cell with a material of known diffusion coefficient. The unknown concentration, cso, can be calculated from the other three values using the known cell compartment volumes and the volume of liquid in the diaphragm. If D is independent of solute concentration, equation (62) gives the desired diffusion coefficient directly; otherwise a more complicated calculation procedure such as that described below must be used. A procedure for calculating D for concentration-dependent systems was outlined by Gordon (1945) and extended by Stokes (1950b). They showed that the average value, D, obtained from equation (62) corresponds within 0.02% t o CB 1 D = m (63) C B - CA dc

1:

+

+

where EB = (cBO c B ) / Z and EA = (cAO c A ) / Z . Consequently, if an experiment is allowed to progress for only an infinitesimal time, the average value

Do

= Cl S o f D d c

would be obtained provided cAO = 0. Denoting the value of DQ corresponding to concentration EB by D O ( c B ) , and the value for concentration h by I ) O ( E A ) , Stokes derived the expression

Then using data for D versus ceO from several experiments over a wide range of concentration, he showed how to solve equation (65) for D o ( E ~ ) versus EB by a method of successive approximations. If we denote this curve by simply Do versus c, the desired values of D versus c are finally obtained with the use of one of the differentiated forms of equation (64), i.e., D = D O + c - doo -(66) dC

462

LOUIS J. GOSTING

for nonelectrolytes or

for electrolytes. If the salt concentration was decreased below 0.05 mole per liter, Stokes (1950a) found that erroneously high values of D were obtained. These errors, which sometimes amounted to 5%, he attributed to the increased importance in dilute solutions of surface transport effects associated with the large surface area (about 1 m.z) of the diaphragm. This problem, together with the problem of adsorption on the diaphragm, probably prevents the diaphragm cell from yielding diffusion coefficients for proteins as accurate as those for salts. 2. Free Diffusion Special attention is given t o this case because it is particularly suited for accurate studies of proteins and for investigation of solute purity. It has also been used to measure the four diffusion coefficients in threecomponent systems with interacting flows (Section V). t=O

FIG. 6. (a) Free diffusion. The initial arrangement of solutions in the cell (t = 0 ) , and illustrative concentration and concentration gradient curves at some later time, t l . (b) Restricted diffusion. Illustrative concentration curve for a time t z after the concentrations have changed appreciably a t the ends of the cell.

The physical arrangement a t the start of a simple free diffusion experiment with only one solute is shown by the left-hand diagram in Fig. 5a, which illustrates the typical rectangular diffusion cell of uniform cross section. At time t = 0 a sharp initial boundary is formed at level x = 0 between the lower, more concentrated, solution B and the upper, more,dilute, solution A . Hence C E > C A , where these concentrations of solute, component 1, are expressed as grams or moles per unit volume of solution. After diffusion has proceeded for a time t l , the boundary region

MEASUREMENT AND INTERPRETATION O F DIFFUSION

463

becomes blurred and the curve of c versus x has a shape illustrated by the center diagram of Fig. 5a. The right-hand diagram of Fig. 5a shows the corresponding concentration gradient curve, &/ax versus x, a t that time. This gradient curve continues to flatten and spread with time, while keeping the same area, throughout the stage of free diffusion, i.e., until the concentrations begin to change measurably at the ends of the cell. The experimental problem is t o measure either the concentration distribution, or the concentration gradient distribution, as a function of time. From the shape of either curve at a given time, information may be obtained concerning the homogeneity of the solute, provided there is no interaction of solute flows; if only one solute is present this shape may be used to obtain information about any concentration dependence of D. From the time dependence of either curve one may calculate the diffusion coefficient if the solute is homogeneous, or some average diffusion coefficient if it is a mixture. For mixtures with interacting flows, both the shape and the time dependence must be determined to obtain the several diffusion coefficients with the use of present methods of analysis. If a free diffusion experiment is allowed to continue so long that the solute concentration begins to change measurably a t the ends of the cell, the experiment is said to have entered the stage of “restricted diffusion.” The curve of c versus x for a representative time, t2, during restricted diffusion is shown in Fig. 5b. Such diffusion will be discussed below in Section III,3. a. Equations for the Solute Concentration and Concentration Gradient Distributions. I n order to calculate D for two-component systems from measurements of the solute concentration, c, or concentration gradient, ac/ax, in free diffusion, equations for these quantities must be derived by integration of Fiek’s second law, equations (21) or (22), subject to the initial conditions shown in Fig. 5a for t = 0,

and the boundary conditions for t

>0

T o simplify this integration, use is usually made of the fact, pointed out by Boltzmann (1894), that in free diffusion the variables x and t always occur in the ratio x / d t . This is true whether or not D depends on concentration. For the case in which D is independent of c, the integration of equation (22) has been described in a number of reviews including those by

464

LOUIS J. GOSTING

Longsworth (1945), Harned (1947), and Jost (1952). The resulting equations are

and

where E denotes the mean solute Concentration

and Ac the concentration difference

- CA

(73) across the initial boundary. For a system containing q different solutes equations of the same form are applicable, provided that there is negligible interaction of the solute flows and that each diffusion coefficient is independent of concentration. Integration of equation (23) instead of equation (22), subject to initial and boundary conditions analogous to equations (68) and (69), leads to generalized relations identical with equations (70) through (73) except that c is replaced by c; and D by Di. When D depends on concentration in a two-component system, a skewed concentration gradient curve is obtained instead of the Gaussian curve, equation (71). This is illustrated b y the dashed and solid curves, respectively, in Fig. 6. I n order to integrate equation (21) for this case, the dependence of D on c must be specified; furthermore, it is customary to utilize the fact that c depends only on x / d as indicated above. B y using a linear relation for D versus c, Stokes (1952) integrated equation (21) numerically for certain cases, and Gillis and Kedem (1953) evaluated the first three terms in a series solution which converges rapidly when the concentration dependence of D is relatively small. Stokes wrote his linear equation for D in terms of the value of D a t the mean concentration of the experiment; Gillis and Kedem wrote theirs in terms of the value of D a t infinite dilution. Fujita (1954a) obtained a n exact solution for the solute concentration distribution when 1/D is a linear function of c ; this expression was written in terms of the value of D a t c = 0. Using a cubic relation to represent the concentration dependence of the diffusion coefficient, A C = Cg

D

= D(E)[l

+ kl(c - E) + LZ(C- E ) 2 + ka(c - C)’ +

*

*

*]

(74)

MEASUREMENT AND INTERPRETATION O F DIFFUSION

465

in which D(F) is the value of D a t the mean concentration c = E, and kl, k2, ks, etc., are constants for a given value of 15, Gosting and Fujita (1957) obtained c as a series expansion, C =

E

+ (Ac/2)[40 4- (Ac/2)4i + ( A ~ / 2 ) ~ 4+2 ( A ~ / 2 ) ~ 4+3

*

. -1 (75)

For simplicity we consider here only the first two terms in the brackets in equation (75), although analytic expressions for 40, CpI, and cpz in this relation and for d&/dz, . . . , d4,/dz in the corresponding relation for the

FIG.6. The effect on the concentration gradient curve for free diffusion of a dependence of D on concentration. ---, normal Gaussian curve for D independent of c ; ............, skewed curve for the same time of diffusion and same value of D a t concentration C, but with D decreasing linearly by about 50% from C A to C B .

reduced concentration gradient were derived. Here 40 is the Gaussian error function 40 = where

w,

and z =

The next term is 41 =

X

2

v%@p

- (ki/4) {2[@(2)]' .f 2 2 @ ' ( 2 ) @ ( 2 ) f

[@'(Z)lZ

466

LOUIS J. QOSTIHfa

in which W(z) denotes the first derivative of @(z) with respect to 2 . It should be observed that dl contains kl as a multiplying factor, and the effects of a linear concentration dependence of D on the shape of the curve of c versus x diminish as either k1 or Ac approaches zero. Similar considerations apply to the higher terms of the series, because $2 consists of k12 times a function of z plus kz times another function of z, etc. The concentration gradient curve in reduced coordinates is obtained by differentiation of equation (75) with respect to z .

Figure 7 presents graphs of the two dimensionless functions d$o/dz and (l/kl) dd,/dz t o help show how the skewed curve in Fig. 6 arises from -3

-

Z

3 0

I

0.4

I

0.8

dMdz

I

1.2

- 0.4

r 0

I

0.4

(l/ki)d+i/dz

FIG.7. Graphs of the functions appearing in the first two terms of the series expression for the concentration gradient curve in reduced coordinates, equation (80).

the sum of these two functions, after the latter is multiplied by ( k l Ac/2). The next term in equation (80), associated with ( A c / ~ ) ~is, an even function of x and contributes only symmetrical deviations from Gaussian form t o the concentration gradient curve. Consideration of this term associated with ( A c / ~ )is~ necessary for correct determination of diffusion coefficients using the classical “maximum height-area ” method for analyzing free diffusion experiments (see Section 1 1 4 2 , ~ ) .For comparison with equation (71), equation (80) may also be written

MEASUREMENT AND INTERPRETATION OF DIFFUSION

467

b. Equations for the Refractive Index and Refractive Index Gradient Distributions. These relations are required because free diffusion is commonly studied by using some optical method to measure either the refractive index or refractive index gradient of the liquid in the cell as a function of height. I n this way very accurate indirect measurements of the concentration or concentration gradient can be made a t successive times during an experiment, without disturbing the diffusion process. If the refractive index, n, of a two-component system is a linear function of the solute concentration over the concentration range encountered in the cell, we may write

n = n(E)

+ R(c - E)

(82) where n(E)is the refractive index a t concentration F and the constant R, which we call the differential refractive increment, is the change in refractive index corresponding t o unit change of solute concentration. Substitution of equation (70) into equation (82) yields the refractive index distribution for free diffusion for the case of D independent of c,

Here A n denotes R Ac, the total difference in refractive index across the initial boundary. An expression for the refractive index gradient analogous t o equation (71) is obtained by differentiating equation (83),

If n does not depend linearly on c in a two-component system, this nonlinearity will cause deviations from equations (83) and (84) somewhat similar to the deviations produced by a concentration dependence of D . When both effects are present, either a partial cancellation or an addition of deviations arising from these two causes may occur, depending on the forms of the concentration dependence of D and the nonlinearity of n with c (Longsworth, 1953). I n considering the problem of analyzing such curves to obtain correct values of D,Gosting and Fujita (1957) used a series to represent the refractive index as a function of the solute concentration n

=

n(E)

+ R ( c - E)[1 + al(c - E ) + a,(c - c')2 + a3(c - E ) 3 + - - .I

(85)

where R, all a2, u3, etc., are constants corresponding to the chosen value of E . A general series expression for n as a function of position and time in

468

LOUIS J. GOSTING

the diffusion cell was then obtained by substituting equation (75) for c into equation (85). Because of its complexity this result will not be given here; only the final relations for interpreting certain experimental measurements will be considered (see Section 111,2,c). When two or more solutes are present in the cell, the situation is sufficiently complicated so th at we consider here only the case of a linear dependence of n on the concentrations of the q solutes,1°

n

=

n(E1,.

. . ,EJ

+ 2 R,(C~- ci) 9

(86)

i= 1

where n(El, . . . ,En) denotes the refractive index of a solution in which the concentrations of the q solutes are El, . . . , F,. Furthermore, for this case we assume that there is no interaction of solute flows and that each Aci is made sufficiently small that the concentration dependence of the several diffusion coefficients may be neglected. A8 pointed out earlier, equation (70) is then applicable if c and D are replaced by ci and Di, respectively. Substituting this relation into equation (86) gives

Here An is the total difference of' refractive index across the starting boundary and Ri A c ~ a, = (i = 1, . . . ,q) (88) An are solute fractions on the basis of refractive index; if the value of Ri is the same for every solute when ci is expressed as mass of i per unit volume of solution, ai reduces to the weight fraction of solute i. The refractive index gradient, which will also be needed later, is obtained by differentiation of equation (87),

c. Equations for Evaluating Diffusion Coeficients. Three basic measurable properties of a refractive index gradient curve will now be defined; their relation t o individual diffusion coefficients will then be considered for the types of systems discussed in the two previous sections. These lo

The differential refractive increments,

Ri,are defined by

MEASUREMENT AND INTERPRETATION O F DIFFUSION

469

quantities will also be used in discussing interacting flows, Section V. They are the reduced height-area ratio

where (an/ax),,, second moment

denotes the maximum value of anlax; the reduced

and the reduced fourth moment 1

Dim= 12t2 an ~

1.-

x"an/dz) dx

(92)

It will be recalled that

2 is measured from the position of the initial boundary. Higher moments may also be similarly defined, but they cannot be measured with sufficient accuracy to be of practical interest a t present. The reduced quantities defined by equations (90)-(92) are independent of time and have the units of a diffusion coefficient, except the reduced fourth moment which has the units of D2. They serve as convenient measurable quantities which may be evaluated as a step in the calculation of diffusion coefficients for the more complicated systems. Equation (90) has long served as the basis for the "maximum heightarea" method for analyzing free diffusion curves (Wiener, 1893; Lamm, 1937; Quensel, 1942), and moments such as those considered in equations (91) and (92) have been used for many years to analyze data for the diffusion of mixtures without interacting flows (Graldn, 1941, 1944). For the case of two-component systems in which D is independent of c and n linear in c, it may be shown by substitution of equation (84) into equations (90) through (92) that

DA =

=

(a:,)!,$= D

(93)

For systems containing q solutes without interacting flows, and with each Di independent of every concentration and n a linear function of the concentrations, similar substitution of equation (89) leads to the relations

n

470

LOUIS J. GOSTING

Solution of similar relations for certain reduced moments to obtain the individual diffusion coefficients and composition has been proposed by Neurath (1942) for the case of q = 2 ; unfortunately, with the accuracy then available the resulting diffusion coefficients were subject to very large errors (Bevilacqua et at., 1945). Somewhat more accurate interpretations are now possible, using the interferometric optical systems. As will be seen in Section IV,2, however, i t is often more convenient to base interpretation of interferometric data for these systems directly on fringe deviation graphs and D A , rather than to evaluate a ) Z m and a:,. Two-component systems in which D depends on c, and n is not linear with c, have been considered by Gosting and Fujita (1957), who obtained and

+ = D(E)[l - L ( A c ) ~+

LDA = D(E)[l - K ( A c ) ~

*

*

Qm

*

*

*I

(97)

*]

(98)

The next higher term in both equations is of order ( A c ) ~ .Here

K

1 2 4 ~

= -[48ai2 -

(72 - 127r)alkl

+ (18 + 9

- 9~)ki'

+ (12 - 12 4+ 47r)k2 - 12?rUz]

and

L

=

4 1 2 [ - 2 ~ i l c i - kz - (6 .\/~/T)uz]

(99) (100)

in which the several coefficients are defined by equations (74) and (85). It is seen that for this case the correct value of D(C) may be obtained by performing experiments with different values of Ac, but the same value when plotted versus ( A c ) ~ of E ; the resulting values of either 3~or SZm extrapolate linearly to D(E) at ( A C ) ~= 0. If both BA and aZm are measured in these experiments, and if al and u2 are known, equations (97)(100) may be used to calculate kl and kz from the slopes of these plots. A method for determining D as a function of c for two-component systems was devised by Boltzmann (1894), who derived the relation (Beckmann and Rosenberg, 1945),

D=--

(")-' -

2t ax

x dc

= -

(")-I

2t ax

/:-

x

(g) dx

(101)

This approach has the advantage that no functional dependence of D on c is assumed ; numerical values giving this dependence are obtained directly from equation (101) after correcting for any nonlinear dependence of n on c. In practice, however, the use of this approach with modern interferometric optical systems is laborious because involved numerical integrations are required to utilize the full accuracy available.

MEASUREMENT AND INTERPRETATION OF DIFFUSION

47 1

d. Optical Theory. An understanding of the behavior of a light ray in passing through a refractive index gradient is fundamental to any consideration of the optical systems used to study free diffusion. Derivations of the relevant equations may be found in a number of articles, including those of Lamm (1937, 1940), Svensson (1946, 1954), and Kegeles and Gosting (1947). To describe the angular deflection, and the path, of a ray passing through a rectangular cell in which the refractive index varies only in the vertical dimension, consider the column of liquid represented b y the rectangle in Fig. 8. Let the cell walls have a refractive index n2, and let Y

-a-

FIG. 8. Diagram for describing the bending of a light ray by a refractive index gradient. Here the refractive index in the cell varies in the 5 direction only. Ray angles with the horizontal are made abnormally large for illustration.

the incoming ray in the wall have an angle ye with the horizontal before entering the liquid. Here ye and the deflections in the liquid are made large for illustration; in practice angles with the horizontal are seldom more than 2 or 3 degrees. On entering the liquid, the angle of the ray with the horizontal becomes Oe according to Snell’s law

n, sin Oe

= n 2

sin

ye

(102)

where n, is the refractive index of the liquid at level z,. If the solution is considered as made up of a large number of infinitesimally thin horizontal layers with n constant within each layer, but increasing from one to another in the downward direction, Snell’s law for the ray, at any point along its path in the solution, becomes

n cos 0

=

constant

(103)

472

LOUIS J. GOSTING

The cosine term replaces the usual sine because 0 is measured relative to the horizontal, instead of the vertical. Differentiation of equation (103) with respect t o 2, and substitution of tan I9 = dx/dy,leads to the basic differential equation for ray bending by a refractjive index gradient

This relation is readily integrated subject to the approximation th a t (l/n)dn/dxis constant over the ray path in the cell. This approximation is valid provided the cell thickness, the refractive index gradient, and 8 are all small. One then obtains 0 - 0 - _y_d n - ndx Therefore the angular deflection of the ray by the refractive index gradient is seen t o be proportional to both y and dnldx. To obtain the total angular deflection in the cell y is replaced by the cell thickness, a. I n practice the deflection of a ray by a gradient is measured in air, not inside the cell. To obtain this quantity consider first the angular difference, -yf - Y ~ of , the ray’s direction in the two cell walls of refractive index n2. Because these angles are small, Snell’s law may be written for the two glass-liquid interfaces as nzy, = no, and n2yf = nof;here the small change of n in the cell between entrance level xe and exit level xj is neglected. Substitution of these relations into equation (105) leads to (loti)

It can be easily shown that this relation represents the angular deflection measured in any medium provided n2 is taken as the refractive index of the medium in question; this medium may be the cell walls, the water in the constant-temperature bath surrounding the cell, or the air outside, if it is assumed that all the interfaces between these optical components are vertical. Taking the refractive index of air as unity we obtain the important result that the angular deflection (measured in air, nz = 1) of a ray by a refractive index gradient, anlax, is equal to the cell thickness, a, times dn/dx. An expression for the light path in the cell is obtained by replacing 0 in equation (105) by its approximate value, tan B = dx/dy, and integrating to obtain y2 dn 2 - z e = eey+-2n dx

MEASUREMENT AND INTERPRETATION OF DIFFUSION

473

Thus the ray path in the cell is seen to be a quadratic function of y subject to the foregoing assumptions. Within these limitations it may be shown that tangents to the ray in the cell a t their points of entrance and emergence intersect a t the center of the cell. This provides the basis for the common practice of focusing the I I camera lens on the center of the cell in several of the optical systems. The limitations of this procedure and of the assumptions made in the above presentation have been discussed by Svensson (1954) and Forsberg and Svensson (1954). Although detailed consideration of the interference optical systems requires wave optical theory, this subject will not be treated here. It is believed that the above information, together with a knowledge that interference between two rays can occur if they are out of phase, will be adequate for the following presentations. e. Diflusion Cells and Formation of the Starting Boundary. A suitable cell for studying free diffusion must not only have optically flat, parallel windows and flat side-walls aligned t o form a rectangular parallelepiped, but it must be capable of forming a sharp initial boundary. No completely satisfactory cell has yet been designed, but a number of different designs, each with certain advantages, are available, FIG.9. The Kahn and PoIson including those of Lamm (1937, 1944), boundary sharpening technique Neurath (1941, 1942), Claesson (1946), as applied to a Tiselius electroCoulson et al. (1948), Svensson (1949), phoresis cell used for diffusion Longsworth (1950), and Hall, Wishaw, measurements. and Stokes (1953). A simple capillary sharpening procedure which produces a n excellent starting boundary for a free diffusion experiment has been described by Kahn and Polson (1947). Its application to a tall form Tiselius electrophoresis cell serving as a diffusion cell is shown in Fig. 9. This cell is filled so t ha t initially the bottom section, the left limb of the center section, and the left limb of the top section, contain solution while the right limb of the top section contains solvent. The right limb of the center section, in which diffusion occurs, may contain either solution or solvent initially.

474

LOUIS J. GOSTINQ

T o form the starting boundary, the top and bottom sections are moved into alignment with the fixed center section and a fine stainless steel capillary tube lowered into the cell to the level of the optic axis. As liquid is siphoned out through the capillary the levels of liquid in both (open) limbs of the top section move down together, and the initial sheared boundary moves from its starting position t o the tip of the capillary, where i t is sharpened by the flow of solvent down to the capillary tip and solution up to it. If the capillary end is ground flat the boundary is formed slightly below the tip (Trautman and Gofman, 1952), requiring tha t the tip be set slightly above the optic axis. Alternatively, the tip may be gound t o an angle of 45” which produces a boundary a t the mean level of the opening (Longsworth, 1953). Because the starting boundary is not perfectly sharp, use of the observed times, t’, after sharpening is stopped, for computation of the reduced height-area ratio from equation (90) leads t o slightly erroneous values, QA’. To obtain the true time of diffusion, t, a small increment, At, should be added to t’. Assuming that the boundary was essentially Gaussian when sharpening was stopped, Longsworth (1947) obtained BA using the relation a ) ~ ’=

+

9 ~ ( 1 At/t’)

(108)

A plot of 9 ~versus ’ l/t’ extrapolates linearly to BA, the desired reduced height-area ratio corresponding to the corrected starting time of the experiment. Alternatively, 9~ may be obtained as the slope of a plot of (An)a/[4~(an/as)~a,] versus t’. If the reduced second moment, DD2m, is measured instead of BA, the problem of the starting time correction may be handled in the same way. f. Lamm Scale Method. This method (Lamm, 1937, 1940) measures the refractive index gradient as a function of height in the cell. A uniformly ruled glass scale is illuminated with monochromatic light and focused on a photographic plate by means of a long focal length lens (50-100 cm.). The cell is placed between the scale and lens, a t a distance of a few centimeters from the scale. With no refractive index gradient present in the cell a simple image of the scale is obtained on the plate, giving a reference photograph. With a diffusing boundary in the central part of the cell, light rays passing through the different levels in this region are deflected through small angles proportional to the corresponding refractive index gradients, equation (106). Those scale line images are therefore displaced from their original positions. Measurement of the displacement of each line from its position on the reference photograph and multiplication by the apparatus constant give the refractive index gradient for the position in the cell corresponding to that scale line image. Further

MEASUREMENT AND INTERPRETATION O F DIFFUSION

475

description of this method is available in a number of articles and reviews, including those by Bridgman and Williams (1942) and Geddes (1949). This method has played a very important part in studies of the diffusion and sedimentation behavior of proteins. It has been used less as interferometric methods have been developed. Approximately equal labor is required to measure scale photographs as interference fringe photographs, but considerably greater accuracy is possible with the interference optical systems. The accuracy of diffusion coefficients obtained with the Lamm scale method is of the order of 1%, this limitation being due both t o the accuracy of measuring scale line displacements and to the fundamental limitation that light does not exactly follow ray optics, on which the method is based (Adler and Blanchard, 1949). g. Schlieren Methods. A graph proportional to anlax versus x is obtained directly on the photographic plate with either the cylindrical lens schlieren apparatus (Philpot, 1938; Svensson, 1939, 1940) or the schlieren scanning apparatus (Longsworth, 1939). Both forms of schlieren apparatus utilize a long focal length lens, with a diameter slightly larger than the height of the cell, to form an image of a horizontal source slit which is illuminated with monochromatic light. The cell is placed close to the schlieren lens on its image side, and the lens is masked so that only light passing through the cell reaches the image plane. Alternatively two schlieren lenses may be used with the cell placed in the collimated light between them. Rays passing through regions of homogeneous liquid in the cell focus to form a normal slit image. Rays passing through levels a t which refractive index gradients are present are bent downward according to equation (106) and form images at corresponding levels below the normal slit image; these images merge into a band of light extending down t o the position corresponding to the maximum refractive index gradient in the cell. A camera lens, of diameter sufficient to collect all of the deflected rays, is placed just beyond this image plane of the source slit and focuses the cell onto a photographic plate at the end of the apparatus. In the schlieren scanning method a horizontal knife edge is slowly moved upward in the focal plane of the slit images. It first cuts off the most deflected rays, corresponding to the maximum refractive index gradient, and then cuts off light from parts of the cell with smaller and smaller gradients until finally the normal slit image is covered, leaving the cell image on the plate dark everywhere. By gearing the plateholder so that it moves the plate sideways a t the same time that the horizontal edge is moved upward, a curve proportional to anlax versus x is produced between light and dark regions on the plate. This curve is sharpened by placing a stationary vertical slit over the cell image just in front of

47 6

LOUIS J. GOSTING

the plate to minimize the horizontal width of the band of light being photographed. A similar diagram is obtained on the plate with the cylinder lens apparatus, but no moving parts are required and the refractive index gradient curve can be observed visually. The horizontal knife edge in the slit image plane is replaced by a diagonal knife edge (or diagonal slit or bar) which cuts across the normal slit image and through the band of deflected light below. The more an image of the source slit is deflected downward by a refractive index gradient in the cell, the greater is the horizontal displacement of its point of intersection with the diagonal edge. The horizontal components of these intersections along the diagonal edge are then focused as horizontal displacements on the photographic plate by means of a cylinder lens with vertical axis, which focuses the plane of the diagonal edge onto the photographic plate in the horizontal dimension only. Since this does not disturb the focusing of the cell on the pIate in the vertical dimension, the line of demarcation between light and dark regions on the plate is proportional to dn/dx: in the horizontal direction and to the cell coordinate, z, in the vertical direction. Detailed descriptions of the schlieren methods are available elsewhere (Longsworth, 1946; Geddes, 1949). Determinations of dn/dx by these methods is limited to about 1% accuracy, as was the case for the Lamm scale method, both by experimental uncertainty and by the fact that light does not exactly follow the ray optical theory on which they are based. A further limitation of the cyliridcr lens system, which is avoided by the schlieren scanning procedure, is the difficulty of obtaining vylindrical lenses of high optical quality. h. Gouy Interference Method. The interference fringes formed by this optical system permit a precise determination of the shape of a refractive index gradient curve, provided the curve is symmetrical and has only one maximum. Therefore this method is particularly useful for the study of free diffusion, yielding values of D for two-component systems with a n accuracy of 0.1 % and providing information about heterogeneity and interacting flows for systems with more than two components. It cannot measure movement of a boundary in a cell, a s required for studies of sedimentation and electrophoresis. Although these interference fringes were described about 75 years ago by Gouy (1880), who suggested th a t they might be used to study diffusion, no further development of the method occurred until interest in it was revived by Longsworth (1945). A theory for the fringe spacing derived by Kegeles and Gosting (1947) provided the basis for interpreting the first experimental work with the method in this country (Longsworth, 1947; Gosting et aE., 1949) ; in these

MEASUREMENT AND INTERPRETATION O F DIFFUSION

477

forms of the apparatus rapidity of measurement was sacrificed in an attempt to achieve maximum accuracy by using a long optical lever arm. An independent development, in England, of theory and apparatus (Coul1948) placed emphasis on making rapid measurements with a son et d., small volume of solution. Subsequent refinements of the theory have evaluated small correction terms and considered non-Gaussian refractive index gradient curves (Gosting and Morris, 1949; Gosting and Onsager, 1952).

FIG. 10. Schematic side view of the Gouy diffusiometer, showing a t thc bottom representative Gouy fringe photographs for successive times t l , t ~ etc. ,

Figure 10 shows a schematic side view of the apparatus, where for simplicity the constant-temperature water bath surrounding the diffusion cell, C , is omitted. The horizontal source slit S is illuminated by monochromatic light from the assembly on the left containing the mercury lamp, condenser lens, and filter. This slit is focused through the cell onto the photographic plate, P , by means of the long focal length lens, L. Light traversing the upper and lower parts of the cell, where there is no refractive index gradient, forms a normal image of the slit on the photographic plate at level 1’ = 0. Rays passing through levels in the central region of the cell are deflected downward by the refractive index gradient according to equation (106) and form a band of light below the normal slit image. By making the source slit quite narrow it is found that this band resolves into the Gouy interference fringes shown at the bottom of Fig. 10. These fringes are formed because each ray of every cofocusing

478

LOUIS J. GOSTING

pair of rays traverses a region of different refractive index in the cell and a different path through air to reach the photographic plate. The intensity along the right-hand edge of each Gouy pattern in Fig. 10 was diminished by means of a step filter to facilitate observation in the intense region near the undeviated slit image. Reference photographs (Gosting et al., 1949; Gosting, 1950), which are visible to the left and right of each normal slit image, are essential for precise measurement of this important position on each photograph. The basic data obtained from each photograph are the displacements, Yj, of several fringe minima below the normal slit image, these minima beginning (positions of zero intensity) being numbered j = 0, 1, 2, with the lowest minimum. Also required is the total number of fringes,

...

J = -a An

x

where A n is the refractive index difference across the initial boundary, the wavelength of the monochromatic light," and a the cell thickness along the light path. Measurement of the Gouy fringes usually establishes J within a few tenths of a fringe; the fractional part of J is determined within about 0.01 fringe during the boundary sharpening process by measurement of Rayleigh fringes from a double slit placed so that light from the upper slit comes through solvent and that from the lower slit through solution. I n terms of these measured quantities equation (90) for the reduced height-area ratio becomes (JXb)z = 47rtCt--2~

Here b is the optical lever arm from the center of the cell to the photographic plateI2 and Ct, which i s the m a x i m u m dejlection of light predicted by ray optics for time t, is proportional to (dn/dx),,,,. To evaluate Ct for equation (110) one first computes the reduced fringe number@

11 The reference medium for X and An must be the same; if A is the wavelength in air, the refractive index difference An is relative to a refractive index of unity for air. 12 In a convergent light Gouy diffusiometer the optical lever arm is defined by 6 = Zdi/ni where di is the distance along the optic axis in each medium, i, of refractive index ni; when A is the wavelength in air, ni should be expressed relative to a refractive index of unity for air. The function f@) is defined by

MEASUREMENT AND INTERPRETATION O F DIFFUSION

479

for the fringes which were measured, where the approximation 2, to the series j 3i . . . has been tabulated (Gosting and Morris, 1949). The reduced cell coordinate^,'^ 5;, need not be determined because the ideal reduced fringe displacements, e-rJ2, corresponding to a Gaussian boundary, are obtained directly from tables of e-rr2 versus f(s;) (Kegeles and Gosting, 1947); the values of e-rj2 then are used to calculate values of the ratio YJ/e-riafor the lower Gouy fringes. This ratio is constant for all fringes of a given photograph and equal to Ct if the experimental refractive index gradient curve is Gaussian. If more than one solute is present (equations (87) and (89)), the ratio Y3/e-z7' varies somewhat with j ; by plotting this ratio versus Z354 it extrapolates linearly15 t o Ct a t ZJ7'"= 0 (Akeley and Gosting, 1953). This extrapolation is valid, and the correct value of BAis obtained, even when the solute flows interact (Fujita and Gosting, 1956). When the solute flows do not interact, the resulting value of BA is related to the individual diffusion coefficients by equation (94). More information about analyzing Gouy fringes for systems containing two or more solutes is presented in Sections IV and V. There is some experimental evidence that this extrapolation procedure for obtaining Ct, and hence a>A, for systems of three or more components may also be applied to two-component systems in which D varies appreciably with c and (or) the dependence of n on c is appreciably nonlinear (Gosting and Fujita, 1957). However, the Gouy method is quite insensitive to skewness of the refractive index gradient curve; most two-component systems do not show appreciable variation of Y3/e-c72with j , and for such systems Ct is obtained by simply averaging values of this ratio, using data for the lower fringes for which the greatest accuracy is available. Some experimental confirmation of the theory for the Gouy fringes has been provided by the constancy of Y3/e-rlz for each photograph when studying substances which would be expected to give nearly Gaussian refractive index gradient curves. In addition, Riley and Lyons (1955) have measured the distribution of light intensity in the lower fringes and found good agreement with the predicted relative values.

+ +

"For the simple two-component case with linear dependence of n on c and no concentration dependence of D , we have p , = z , = x,/(2 dot). '5This is seen from equation (31) of Akeley and Gosting (1953), which may be written _' I _ -- Ct

e-f7l

- KCtZ17s + . . .

(lllb)

for any experiment in which the refractive index gradient curve is a sum of Gaussian curves. The quantity K is constant for a given experiment and need not be evaluated when obtaining C,by extrapolation; it depends on J,the diffusion coefficients, and the relative amounts of the solutes.

480

LOUIS J. GOSTING

i. Rayleigh Interference Method. The fringes produced by this modification of the Rayleigh interferometer have a shape directly proportional to n versus z in the cell. Therefore from a single experiment on a twocomponent system one may obtain the diffusion coefficient with a n accuracy of 0.1% and also obtain information about the concentration dependence of D. T o analyze for solute heterogeneity, a procedure must be used which is not sensitive t o the symmetrical deviations from the ideal refractive index curve which are produced by concentration dependence of the diffusion coefficients or by nonlinear dependence of n on the concentrations. All analyses of data from this method must be made in terms of the n versus x curve because the curve of anlax versus x cannot be easily computed from the integral curve without appreciable loss of accuracy. However, Svensson (1949, 1950b) and Svensson, Forsberg, and Lindstrom (1953) have developed an ingenious modification of this optical system which automatically performs the equivalent of a numerical differentiation of the n versus x curve. As the optical principles of their modification are readily understood if one is familiar with the form of the Rayleigh diff usiometer described below, and the interpretation of curves of anlax versus x has been described in Section 111,2,b and 2,c, the details of their method will not be given here. Its accuracy appears to be considerably greater than that of the Lamm scale and schlieren methods but not quite as great as that of the Gouy method or the Rayleigh method giving the integral curve. A brief review of the conventional Rayleigh interferometer seems appropriate before describing its adaptation to diffusion measurements. Figure l l a will serve to illustrate the optical arrangement (viewed from above), if we neglect the cylinder lens L,. Monochromatic light from a vertical source slit S is focused by lens I; to give a slit image in plane P. Mask M containing two identical vertical slots is placed near the lens; the beam from one slot passes through cell C and th a t from the other through cell C’, both of which are filled with pure solvent. These identical cells have no effect on the shape of the Rayleigh interference pattern formed in plane P by the double slit in mask M . The appearance of this pattern is identical with the reference fringe patterns shown in Fig. 12. A Rayleigh pattern has the gross characteristics of a diffraction pattern from a single slit (produced b y covering one of the slots in the mask), but in addition it contains a fine structure of equally spaced interference fringes. When the liquids in cells C and C’ are identical the zeroth order intensity maximum of the pattern is always located a t the center of the diffraction envelope, corresponding to the center of the “slit image”; this maximum is the one for which rays (or waves) from both slots in M traverse equal optical paths. Since the light waves arriving a t either side

MEASUREMENT AND INTERPRETATION OF DIFFUSION

481

of this position have traversed unequal optical paths from the two vertical slots, they undergo destructive or constructive interference depending on whether the path difference is an integral number of wavelengths. If the refractive index of the liquid in cell C is increased the interference fringes move sideways (downward in Fig. 11a) within the central diffraction envelope, their displacement being proportional to the refractive index difference between the liquids in C and C' (equation (109)). The width of the interference fringes is directly proportional to the wavelength of the light and inversely proportional to the distance between centers of the slots in mask M ; the width of the central diffraction envelope is directly proportional t o the wavelength of the light and inversely I

c'

n

I

a

c

FIG.11. Schematic diagram of a simple form of the Rayleigh diffusiometer. (a ) Top view. ( b ) Side view.

proportional t o the width of the slots in the mask. Thus the ratio of slot separation t o width determines the number of interference fringes in the central diffraction envelope. I n the following considerations we will not be concerned with the use of compensating plates commonly present in Rayleigh interferometers or with the utilization of white light to determine the zeroth order interference fringe. A diff usiometer based on Rayleigh fringes was described by Calvet (1945a,b) and Calvet and Chevalerias (1946). Articles describing more recent forms of diffusion apparatus utilizing Rayleigh fringes include those of Philpot and Cook (1948), Svensson (1949, 1950a, 1951a), and Longsworth (1950, 1951, 1952). Schematic side and top views of a simple form of this apparatus are shown in Fig. 11. A point source, S, is considered here, although a vertical slit may also be used t o obtain greater light intensity. Source S is illuminated with monochromatic light which is focused by the lens L of long focal length to form an image on the

482

LOUIS J. OOSTING

photographic plate P . Mask M contains two vertical slots allowing one band of light to pass through the diffusion cell C and the other through an identical reference cell C' filled with solvent. Since the diffusion cell is placed in a constant-temperature bath cell C' is often eliminated, the windows of cell C being made so that they extend beyond the edge of the cell, thereby forming part of the reference path with thermostat water making up the remainder of the reference path. Cylinder lens L, with its axis horizontal has no effect in the horizontal direction, appearing as a glass plate in Fig. lla. As seen in the side view, Fig. l l b , however, it focuses each level in the cell to a corresponding level at plate P .

FIG.12. Rayleigh fringe photographs of a boundary between HzO and 0.75% levulose a t the start of diffusion, after diffusion for 900 sec., and after diffusion for 1800 sec. The adjacent reference pattern of straight fringes was photographed simultaneously with each diffusion pattern (Longsworth, 1952).

I n this way rays deflected by the gradient are brought back to the proper level to interfere with rays from the reference path. Because it is difficult to obtain high-quality cylinder lenses, some workers (Svensson, 1951a; Longsworth, 1952) have modified the apparatus slightly to permit using the cylinder lens with only a small aperture in the direction normal to its axis. Illustrative photographs of the Rayleigh fringes from a levulosewater diffusion boundary are shown in Fig. 12. The straight fringes shown a t the right of each diffusion pattern are reference fringes produced by a separate set of slits in the cell mask; they are displaced horizontally by a deflecting plate to prevent their merging with the pattern from the cell. They facilitate alignment of the plate in the comparator used to measure the diffusion patterns. Several procedures for evaluating photographs of Rayleigh fringes to obtain diffusion coefficients have been described, including those of

MEASUREMENT AND INTERPRETATION OF DIFFUSION

483

Svensson (1951a,b), Longsworth (1952, 1955), Hoch (1954), and Creeth (1955). Creeth’s procedure is presented here because not only is it applicable to simple two-component systems but i t is easily extended to systems in which D varies with c and/or the dependence of n on c is nonlinear. It may also be used to interpret data from systems containing more than two components. Two basic kinds of measurements of the Rayleigh fringes are required: first, determination of the total number of fringes, J , defined previously by equation (109) in connection with the Gouy method; second, measurement of fringe positions to obtain the shape of the n versus x curve. Determination of J is possible from a fringe photograph taken during diffusion, Fig. 12, by counting the integral number of fringes directly; the fractional part is determined as the horizontal displacement of fringes a t the top of the photograph from those a t the bottom, divided by the fringe width. Because the straight portions of the fringe system at the top and bottom of the photograph are so far apart vertically it is customary, however, to obtain the fractional part of J from a separate photograph taken during sharpening of the boundary, such as that shown a t the left in Fig. 12. Measurements of fringe photographs taken during diffusion (assuming no imperfections in the optical system) are made by aligning the plate in a comparator so that the reference fringes are parallel to the main axis of measurement and the cross-hair is initially set on a fringe minimum conjugate to pure solvent. With this fringe denoted by 0, comparator readings, X j , are taken as the cross-hair passes over successive fringes, j = 1, 2, 3, etc. To obtain readings symmetrically about J / 2 , and thus simplify computations (Creeth, 1955), the cross-axis screw is readjusted slightly on reaching the J / 2 position so that the cross-hair is set on the fringe minimum conjugate to solution rather than solvent. The remaining comparator readings then correspond to XJ-j. For example, if J = 80.4 a sequence of readings is made on fringes j = 1, 2, . . , 39, 40, 40.4, 41.4, . . . , 79.4. Interpretation of Rayleigh fringe data is based on the reduced cell coordinates, z*, defined by 2 _j_-_ J_ - 9(z*)

.

J

Evaluation of z* for each fringe is facilitated by using extensive tables16 of the probability function 9, equation (77). For two-component systems 16 “Tables of Probability Functions,” Vol. 1, Federal Works Agency, Work Projects Administration (sponsored by the National Bureau of Standards), 1941. Available from the Superintendent of Documents, Government Printing Office, Washington 25, D. C.

484

LOUIS J. GOSTING

in which D is independent of c arid n varies linearly with c, x* reduces to x/(2 Then D may be computed by averaging values of the quantity

a).

for a number of different fringe pairs. Here M is the magnification, i.e., the distance on the photographic plate divided by the corresponding distance in the cell. For two-component systems in which D varies with c and n is nonlinear in c, Creeth (1955) has shown that pairing of fringes symmetrically about the J / 2 value causes cancellation of the first-order effects from these dependences, giving with high accuracy

When more than one solute is present, equation (87), the quantity ( X J - j - X j ) / ( 2 M x * ) varies with z*. For relatively small x* it becomes (Creeth, 1957) and on extrapolation to ( z * ) ~= 0 yields linear in ( z * ) ~ the quantity 2 .d/a)At, where DAis the reduced height-area ratio, equation (90) * Deviations of the n versus II: curve from the form of the probability integral will be considered further in Section IV. j . Jamin Interference Method. This optical system also provides data for the n versus z curve. However, it yields somewhat more information than the Rayleigh fringe method because each point in the cell is focused as a point on the photographic plate, whereas in the Rayleigh method each level in the cell is focused a t a corresponding level on the plate, giving an average refractive index for each level. In some applications the Jamin method should be preferable; however, it is often convenient to have the residual optical imperfections averaged out by the Rayleigh method. Adaptations of the Jamin method have been described by Labhart and Staub (1947) and by Scheibling (1950a,b, 1951); a modified form has been used with the equilibrium ultracentrifuge by Beams et al. (1954) and Beams et al. (1955). Interference is obtained by using a beam-splitting mirror to separate collimated monochromatic light into two beams, one of which is passed through the cell containing the refractive index gradient and the other through a reference cell; these beams are then combined by a second, similar mirror. Then they pass through a camera which focuses the cell onto the photographic plate. No cylindrical lens is required, since the camera lens focuses the cell onto the plate in both the vertical and horizontal dimensions. Although this method appears to

MEASUREMENT AND INTERPRETATION O F DIFFUSION

485

have the same potential accuracy as the Gouy and Rayleigh interference methods, further development is required before its full potential is reached. 3. Restricted Diflusion

a. Equations. When a free diffusion experiment is continued until the solute concentration, c, begins t o change appreciably a t the ends of the cell (Fig. 5b), the concentration is no longer a function of the single variable x / d . For this stage of restricted diffusion we choose the origin of x

FIG.13. Schematic vertical cross section of Harned’s conductance cell for studying restricted diffusion.

at the bottom of the cell and measure 2 upward, as shown in Fig. 13. An expression for c is obtained (Harned, 1947) by integrating Fick’s second law, equation (22), subject to the initial distribution of concentration G =

(t

f(x)

=

0)

(115)

and the boundary conditions which state th a t no flow occurs through the ends of the cell a t any time

Provided t ha t D is independent of c, integration yields the desired expression m

i= 1

in which h is the cell height and the constants

Ao

=

l

1

h

f ( x )d x

486

LOUIS J. GOSTING

and

depend on the initial concentration distribution, f(x). These constants need not be evaluated in using the following method. b. Hurned's Conductance Method. I n this method (Harned and French, 1945; Harned and Nuttall, 1947) the extremely high accuracy of conductance measurements is used to determine the difference in solute concentrations at two levels in the cell at several different times during the experiment. By placing the two pairs of electrodes E and E' (Fig. 13) equidistant from the two ends of the cell, at x: = C; and x = h - t , all even terms in equation (117) cancel out. i = 1,3,5, . . .

Furthermore, by placing these electrodes so that f = h/6, the term corresponding to i = 3 vanishes, and only the first term is significant for large values of t. If we neglect terms corresponding to i = 5 , 7, etc., equation (120) can then be written r2Dt In [ c ( f ) - c(h - [)I = - - constant (121) h2

+

and D may be determined from the slope of In [c(.$) - c(h - f ) ] versus t. In practice the conductance of a salt solution is nearly proportional to the salt concentration, so the difference in conductance, (KE - K p ) , at the two pairs of electrodes may be substituted directly for the concentration difference; the proportionality constant does not influence the slope of the graph. Not only is this method capable of measuring D with an accuracy of about 0.1%) but this accuracy can be achieved in extremely dilute salt solutions having concentrations in the range 0.0014.01 molar. Therefore this method has made possible experimental tests of the Onsager-Fuoss relation for Ll versus c, equation (53), and consequently of the Nernst limiting value, for many different electrolytes (Harned and Nuttall, 1947; also see several other articles by Harned and co-workers, J . Am. Chem. SOC.,1949 to present). A variation of D with c has less influence, in general, on measurements of restricted diffusion than on measurements of free diffusion. Because the concentration in the cell approaches a uniform value near the end of the experiment, the values of D obtained a t the later times can be associ-

MEASUREMENT AND INTERPRETATION O F DIFFUSION

487

ated with the final concentration of solute a t infinite time. However, it is now possible to evaluate the magnitude of errors caused by a concentration dependence of D in studies of restricted diffusion; a theory has been developed for the c versus x curve when D depends on solute concentration (Snider and Curtiss, 1954). The primary limitation of this method for measuring D has been the requirement that the solution have an electrical conductance and that this conductance be nearly proportional t o solute concentration. I n its present form it is therefore unsuited for studies of proteins. Further studies would be required to determine whether an interferometric optical system could be adapted to measurements of restricted diffusion without serious loss of accuracy.

4. Other Initial and Boundary Conditions Solutions of Fick’s second law have been derived for a number of initial and boundary conditions other than those considered above, and for three-dimensional, as well as for one-dimensional, diffusion. Several of these cases are considered in Chapter I of Jost (1952). Because the equations for diffusion and heat flow are identical, the solutions to problems of heat flow presented by Carslaw and Jaeger (1947) are also applicable to studies of diffusion. Their book presents integrated expressions for temperature distributions subject to a variety of initial conditions in blocks, cylinders, spheres, cones, etc. Unfortunately, the case of threedimensional diffusion in liquids is not as simple, experimentally, as heat flow, because the earth’s gravitational field tends to produce convection. Most precise methods for measuring D therefore use diffusion in one dimension, arranged so that the earth’s gravitational field stabilizes the concentration gradients. The problem of diffusion when D is dependent on concentration has been treated for some cases other than those considered in the above sections. Fujita (1954b) obtained an exact solution for the concentration distribution in a semi-infinite medium when 1/D is a quadratic function of c. Hwang (1952), using a series relation t o represent D, obtained a series solution for the concentration for this case. a. T h e Open-Ended Capillary Method. This method is in a sense intermediate between free and restricted diffusion. It seems especially suitable for studies of diffusion using radioactive tracers. Anderson and Saddington (1949) used it for studying the diffusion of polytungstic acids. In practice one or more small capillaries, closed a t the bottom, open a t the top and filled with a solution of the radioactively tagged solute, are immersed in a well-stirred solvent in a flask. The solute concentration at the top of each capillary is thus maintained almost a t zero throughout

488

A

LOUIS J . GOSTING

Carslaw and Jaeger, 1947, p. 82 ff.) (2i i=O

h

+ 1)irz 2h

(122)

After diffusion for a time t the average concentration, caul which is obtained by mixing and analyzing the contents of the capillary, is given by C""

For values of Dt/h2 less than 0.2, Jost (1952, pp. 4 0 4 2 ) has shown that a series of error functions converges more rapidly than the trigonometric series in equation (122). For this case he derived the relation (125) Using open-ended capillaries about 0.5 mm. in diameter and 2-5 cm. long, Wang and co-workers have studied the diffusion of trace components in a number of different systems including radioactive water in normal water (Wang, 1951a,b; Wang, Robinson, and Edelmsn, 1953) and radioactive trace ions in the presence of large amounts of other electrolyte (Wang, 1952a,b, 1953; Wang and Miller, 1952). The latter studies were made to test the Onsager-Fuoss theory for the diffusion of trace ions in systems containing three kinds of ions (Onsager and FUOSS, 1932; Onsager, 1945; Gosting and Harned, 1951). Wang, Anfinsen, and Polestra (1954) have also used this method t o study the diffusion of tagged molecules of ovalbumin in normal ovalbumin solutions, as well as tagged water in these solutions.

489

MEASUREMENT AND INTERPRETATION OF DIFFUSION

This method has the advantage that the apparatus is simple and only very small amounts of material are required. However, it has not yet been capable of yielding accuracy comparable to Stokes’ form of the diaphragm cell method, to the free diffusion methods using interferometric optical systems, or to Harned’s conductance method. In favorable cases the open-ended capillary method can provide an accuracy of about 1%. Wang (1952a) has investigated the effects of convection at the ends of the capillary by using capillaries of different lengths. With suitable rates of stirring this effect was negligible within the error of measurement. 5. Determination of D from the Spreading of a Boundary in the Ultracentrifuge The possibility of determining the molecular weight of a protein from a single ultracentrifuge experiment, by measuring its sedimentation coefficient, s, from the rate of movement of the boundary and its diffusion coefficient, D, from the rate of spreading of the boundary, has attracted the attention of investigators for many years. Although the accuracy of such measurements of D cannot be as great as those obtained from simple free diffusion experiments using interferometric techniques, the results may be quite adequate for many purposes provided the protein is substantially homogeneous and the calculations take into account the marked boundary sharpening produced by a concentration dependence of s (Fujita, 1956; Baldwin, 1957). In earlier work the effects of a concentration dependence of s were neglected and D was calculated by using the first term in Faxhn’s (1929) series solution for the solute concentration gradient curve for a rapidly sedimenting solute; this series may be written in the form (Gosting, 1952)

where W can be written either in closed form or as a series

W =

2 D ( e 2 J ~-Z 11) SW2

1

+ sw2t + 32

(~w’t)~

-1

in which su2t is usually small compared to unity. Here r and ro are the distances from the axis of rotation to the point in question and to the meniscus, respectively; co is the solute concentration at the start of the experiment; and t is the time after the rotor is “instantaneously” brought to its constant angular velocity, w. All terms in the series in brackets, equation (126), are small compared to unity and may be neglected within

490

LOUIS J. GOSTING

the limits of the usual experimental error. Because coe--2no*t is simply the (changing) concentration difference across the sedimenting boundary at a given time, it may be denoted by Ac. Equation (126) is then seen to have the same form as equation (71); the term in the exponent simply shifts the entire curve along the r axis. Consequently any procedure for analyzing free diffusion curves, such as the maximum heightarea method, permits determination of W . From equation (127) it is seen that W is nearly equal to 4Dt a t early times (since s d t is usually less than 0.1). Lamm (1929) pointed out that neglect of the series in brackets in equation (127) could cause a measurable error in D. For even a modest variation of s with c, such as is present in nearly all protein solutions, sufficient boundary sharpening occurs so that equation (126) becomes grossly in error. Use of equations (126) and (127) to calculate a value of D for such systems may lead to errors of 50% or more (Baldwin, 1957). By considering the concentration dependence of s to be linear with c s = so(1 - kc) (128) where SO is the value of s at infinite dilution, Fujita (1956) was able to solve the differential equation for sedimentation and obtain a more general form of equation (126). He then obtained the following expression for the maximum concentration gradient

where in which

The function (and its derivative a’) are defined by equation (77). Here co is the initial solute concentration a t t = 0, and Ac is the concentration difference across the sedimenting boundary at time t. Since F(E,) + 2/ fiand h + 0 as k + 0, it is seen that, if s is independent of c, equation (129) reduces to the same expression for (aclar), that is obtained from equation (126) (neglecting higher terms in the series in brackets in equation (126)).

MEASUREMENT AND INTERPRETATION O F DIFFUSION

49 1

Baldwin (1957) used equation (129) to calculate the diffusion coefficient of bovine plasma albumin after first evaluating Ic from several sedimentation experiments. Using an approximate value for D he calculated tmandplotted [A~F(€~)/(ac/ar),]~ versus (eZsowzt- 1)(1 dT'7i)z. From the slope, 2D/(s0w2),of this graph D was obtained. If the line was not quite straight this value of D was used as a second approximation t o reevaluate Em and the calculation was repeated. The final value of D agreed with the value obtained from a free diffusion experiment within an estimated experimental uncertainty of 5 %. Because the rotor cannot be brought t o the desired operating speed instantaneously, as assumed in the above mathematical solutions, some procedure for estimating the effective starting time is necessary for determination of D by this method.

+

IV. INTERPRETATION OF EXPERIMENTAL RESULTS Probably the most important single application of the diffusion coefficient of a protein at the present time consists of its use, in combination with values of the sedimentation coefficient and partial specific volume, for obtaining the molecular weight. However, the accuracy of modern diffusion measurements can be fully utilized in this way only if careful attention is given to certain problems. One of these is the conversion of values of the diffusion coefficient t o standard conditions. Another, and in many cases more serious, problem is the presence of impurities. Both of these factors are considered in this section, together with some recent data providing information about the magnitude of errors which may arise from them. A third problem which may be of importance is th a t of interacting flows. Because our knowledge of this subject is both new and incomplete, we defer discussion of it to Section V; in the present section i t is assumed that the solute flows are adequately described by equations (10). Diffusion coefficients of a number of amino acids and other low molecular weight materials of biological interest have been determined in water with high precision during the past few years. Some of these data will be reviewed because they provide interesting information about the effects of chain length (or molecular weight) and electrostriction on the diffusion coefficient, and hence on the frictional coefficient. 1. Conversion to Standard Conditions

The diffusion and sedimentation coefficients of a protein are often measured in different laboratories using different temperatures, concentrations of buffer salts, and protein concentrations. Therefore the practice arose of converting measured values to certain standard conditions, such as zero protein concentration in pure water at 20" C. From the data re-

492

LOUIS J. GOSTING

viewed below i t is evident th at the classical procedure of reporting diffusion coefficients a t 20" C. is not valid within the error of modern interferometric measurements unless the experiments are performed at temperatures near 20" C. On the other hand, there is increased need for extrapolating diffusion and sedimentation coefficients to zero protein concentration in order to obtain accurate values of protein molecular weights. The procedure of converting diffusion coefficients measured in a buffer to values in pure water appears from the limited data available to be reasonably satisfactory. Diffusion measurements on proteins are customarily performed in the presence of a buffer salt to maintain the desired p H and t o make negligible the effects of any net charge possessed by the protein molecule if it is not a t its isoelectric point. Since the theory for determining molecular weights by sedimentation and diffusion in systems of three or more components is not fully developed, it has been common practice to treat the buffer system-i.e., the solvent (water) and low molecular weight saltsas a single component when using equation (52a). Thus any interaction of the solute flows is neglected. In the following consideration of standard conditions the equations given are for two-component systems. The use of these equations with protein experiments should be considered a semiempirical extension, in which the salt and water are treated as a single component. This procedure, although not strictly valid, seems t.o yield accurate results in a number of cases but its reliability should be considered t o be subject to additional experimental investigation. a. The Solvent Viscosity. For experiments a t a given temperature, the diffusion coefficient, Do,a t infinite dilution of solute in a solvent of viscosity 7 is commonly converted t o the corresponding value, (Do))r,in a reference solvent of viscosity qT by using the following form of the StokesEinstein relation (0"r

=

~"(11/11r>

(133)

This relation for two-component systems may be derived by letting Cl in equation (44) approach zero and assuming th a t the frictional coefficient, fl, of the solute is then proportional to the solvent viscosity (equation (46)) * Data testing the validity of equation (133) for some two-component systems of small molecules have been reported by Stokes, Dunlop, a i d Hall (1953). Using the diaphragm cell method they measured the diffusion coefficients of iodine in several organic solvents, and extrapolated the data for each system t o obtain Do,the diffusion coefficient corresponding t o zero concentration of iodine. Table I summarizes their data for D otogether with their measurements of q, the viscosity of each or-

MEASUREMENT AND INTERPRETATION OF DIFFUSION

493

ganic solvent. The values of DOT, shown in the last column, should be constant if the assumptions inherent in equation (133) are valid. Data in the first three lines are for solvents in which iodine is believed to exist in the simple molecular form, Iz.The constancy of DOT within 7%, while TABLE I Test of the Stokes-Einstein Relation for Dependence of the Diffusion Coefficient o n Solvent Viscosity: Iodine at Zero Concentration in Several Organic Solvents at ,025’ C . DO x 1 0 6 (0111.2 sec.-l)

x 103 (poises)

DO7 X l o 7

Hexane Heptane Carbon tetrachloride

4.05 3.42 1.50

3.19 3.96 9.18

1.29 1.35 1.38

Dioxane Benzene Toluene m-Xylene Mesitylene

1.07 2.13 2.13 1.89 1.49

12.16 6.05 5.59 5.89 6.81

1.30 1.29 1.19 1.11 1.01

Solvent

T varies by 300% for these three solvents, is quite remarkable; the solute and solvent molecules are of the same order of size so th a t the assumption in Stokes’ law of a sphere moving in a continuous medium is not satisfied. Evidence exists that some formation of solvent-solute complexes occurs in the other cases, the smaller values of the product Doq perhaps indicating a larger diffusing unit. A test of equation (133) with an approximately isoelectric protein has been made by studying the diffusion coefficient of bovine plasma albuminL7(BPA) as a function of the concentration of buffer salts (Gosting, unpublished data). So that the buffer solution for each experiment would have the same relative composition,’* each buffer sample was prepared b y diluting the same stock solution; this stock solution was 3.000 molar in potassium chloride, 0.200 molar in potassium acetate, a n d 0.200 molar in acetic acid (ionic strength = r / 2 = 3.2). After dialysis of each protein solution against the appropriate dilution of buffer, the mean protein concentrations, i h k , were found to be nearly the same for every experiment, being about 0.50 g./100 ml.; the pH of the outside solutions after dialysis ranged from 4.58 t o 4.70. A value of the reduced height-area ratio, a)A 1’ The author is indebted to the Research Department of The Armour Laboratories, Chicago, Illinois, for supplying this sample, Lot No. 212113. 18 These relative compositions were not exactly the same because the acid dissociation “constant ” varies somewhat with salt concentration.

494

LOUIS J. GOSTING

(equation (go)), was measured for each experiment using the Gouy diffusiometer. These values are represented by crosses in Fig. 15. Each value of DA was then multiplied by the corresponding value of v/qw, the viscosity of the buffer relative t o the viscosity of water. These reduced height-area ratios converted to water, (DA),, are plotted as circles on Fig. 15. T ha t they are nearly constant, trending upward b y no more than 0.5% from r / 2 = 0.16 to r / 2 = 2.56, provides some confirmation of the

3 X 73

!=

m

-

h

2 X

t66.3 6,i

'

a

X

,

I

I

2 .o

1.0

r/2 FIQ. 15. The reduced height-area ratio of a sample of bovine plasma albumin as a function of ionic strength, r/2, of the buffer. X, DA; 0 , values converted to water, (BA),. Data are for the temperature of the experiment, 25.00" C.; pH 'V 4.65; CBPA 'V 0.5 g./IOO ml.

validity of equation (133) for converting protein diffusion coefficients from buffer t o pure water. An exception is the markedly low value of (a,), for ionic strength 0.04. This anomalous result may be due t o some aggregation of the albumin a t low ionic strength; the fringe deviation graph (see Section IV,2,a) for this experiment was abnormally large, the maximum fringe deviation, Q2,,,, being 15 X For the experiments with I'/2 ranging from 0.16 to 1.28, QmaX ranged from 8 X to 10 X 10-4; for r / 2 = 2.56, Qmax was about 13 X An ideal test of equation (133) for protein solutions would require several experiments a t each ionic strength using different protein concentrations; after extrapolation of %A to zero protein concentration in each

MEASUREMENT AND INTERPRETATION OF DIFFUSION

495

buffer, the constancy of values of (a>,"), = a ) A O ( q / q , ) could then be investigated. Pending further experimental tests, however, it seems th a t equation (133) may be used with reasonable confidence, provided th a t the experiments are performed in a range of buffer concentrations in which DAis not affected by aggregation, dissociation, etc. b. Dependence of D on Temperature. The classical expression for converting a diffusion coefficient at infinite dilution of solute, Do, from the temperature of measurement, T', to another temperature, T , in the same solvent is

Here I] is the viscosity of pure solvent a t temperature T, and q' is the corresponding value a t T'. Symbols T and T' denote temperatures on the absolute (Kelvin) scale, while T and 7' are used as subscripts on D o t o denote temperatures on the centigrade scale. As in the case of equation (133), equation (134) is readily obtained for two-component systems from equation (44) by letting C 1 4 0 and assuming that f l for the solute is then proportional to the solvent viscosity (equation (46)). Experimental data suitable for testing the validity of equation (134) have been obtained in recent years for a number of aqueous systems using the interferometric methods. With the Gouy method Gosting and Morris (1949) determined the diffusion coefficient of sucrose in water for several concentrations a t 1" and 24.95" C . and extrapolated the results to zero sucrose concentration. The ratio of these extrapolated values is (Do)24.96/(Do)1= 2.156; with the use of data of Bingham and Jackson (1918) for the viscosity of water, the value predicted from equation (134) is 2.104. I n the same way Lyons and Thomas (1950) obtained (Do)2b/(Do)1 = 2.045 for glycine in water, and for n-butanol in water Lyons and Sandquist (1953) obtained (Do)2b/(Do)1= 2.150; equation (134) predicts a value of 2.107. Using the Rayleigh interference method Longsworth (1954) measured this ratio for heavy water in ordinary water obtaining (DO)26/(OO)1= 2.004. From these data it is clear that the frictional coefficients of solute molecules at infinite dilution cannot be considered exactly proportional to the viscosity of the medium over a wide range of temperature. More detailed information about this effect was obtained by Longsworth using the Rayleigh interference method. He measured the diffusion coefficients of a large number of amino acids and other low molecular weight solutes in water at several different temperatures (Longsworth, 1952, 1953, 1954). Although these data were not extrapolated to zero solute concentration, they were obtained for dilute solutions with mean

496

LOUIS J. GOSTING

concentrations (equation (72)) in the range 0.2-0.45 weight %. Some of the data are summarized in Fig. 16, which shows t,he ratio of the Stokes' radius (equation (46)) a t a temperature 7 to that a t I" C. as a function of temperature. Because this ratio is equal to (D,/o,)[rrll/(274.16rl)l, its deviations from unity indicate deviations from equation (134). Of the solutes shown, only alanine conforms to equation (134) with high precision. The general decrease in slope of the lines in Fig. 16 with increasing molecular weight is of interest. The dependence of D on temperature

5

1.02

1.00

4

U

B PA 0.96

Alanine

Cycloheptaarnylose I

I

I

I

1

13

25

37

1,

"C.

FIG.16. Deviations from equation (134) as measured by the temperature dependence of the ratio ( T ~ / T ~of) the Stokes' radius a t temperature T to t h a t a t 1" C. Here HDO and BPA denote heavy water and bovine plasma albumin, respectively (Longsworth, 1954).

has been discussed in some detail by Longsworth (1954, 1955)) who considered both the Stokes-Einstein approach and the reaction-rate theory (Glasstone, Laidler, and Eyring, 1941). Using the open-ended capillary method, Fishman (1955) obtained values for D v / ( k T ) for radioactively tagged n-pentane and n-heptane in normal n-pentane and n-heptaiie, respectively; here lc = R / N is the Boltzmanri constant. For both systems Dq/(lcT) was found to change appreciably over the temperature range studied, -78.5' C. to well above room temperature. From the data reviewed above it is clear th a t precise determinations of D for proteins should be reported a t the temperature of measurement and not converted to some standard temperature by means of equation (134), unless the temperature difference is only a fraction of a degree.

MEASUREMENT AND INTERPRETATION O F DIFFUSION

497

c. Dependence of D on Concentration. Here we consider some data illustrating aspects of this problem for simple two-component systems; data for the concentration dependence of the diffusion coefficients of proteins will be deferred to Section IV,3,b below. According t o equation (44), the diffusion coefficient in a two-component system depends on solute concentration because of the concentration dependence of f l , yl, p, and UO, the last two terms arising from the choice of the local center of mass as a frame of refereiice for equation (40). T o eliminate the influence of most of these factors it is common practice t o extrapolate data for D to zero solute concentration, thus obtaining the value Do which depends only on the absolute temperature, T, of the system and the frictional coefficient of the solute molecule in pure solvent. If desired, the numerical value of this frictional coefficient may then be found from equation (44) with C1 set equal to zero. The validity of equation (44) for nonelectrolytes with C1 Z 0 cannot yet be tested experimentally because the concentration dependence of f l cannot be independently calculated. I n the absence of a rigorous approach, the viscosity, q, of the solution divided by the viscosity, qo, of pure solvent has sometimes been used to approximate the variation of the frictional coefficient with concentration in dilute solutions (Gordon, 1937; James, Hollingshead, and Gordon, 1939). With this approximation equation (44) may be written

(135) where for simplicity the subscript 1 has been dropped. This approximation differs somewhat from the use of the macroscopic viscosity to describe the frictional coefficient in equations (133) and (134); there the particles considered were present at infinite dilution and the viscosity changes were due entirely to changes in the medium, while here the observed viscosity changes are due t o the particles whose frictional coefficient we wish t o consider. Some limitations of this approximation have been pointed out by Gordon (1950). It is of interest, however, to see how closely equation (135) represents data for several systems. Curves comparing this equation with experimental data and indicating the magnitude of the several terms in the equation are shown in Fig. 17. The circles in Fig. 17 indicate experimental values of D as a function of solute molarity, C, for the dilute region of some representative aqueous systems of two c o m p ~ n e i i t s Data . ~ ~ for KC1 are included t o contrast the l9 The sources of these data are: KCl, Harned and Nuttall (1949) and Gosting (1950); sucrose, Gosting and Morris (1949); glycine, Lyons and Thomas (1950); urea, Gosting and Akeley (1952); or-alanine, Gutter and Kegeles (1953); n-butanol, Lyons and Sandquist (1953); glycolamide, Dunlop and Gosting (1953).

498

LOUIS J. GOSTINQ

linear dependence of D on C < for a salt, as C + 0, with the linear dependence on C for nonelectrolytes, both nonpolar and dipolar. Other recent data for the concentration dependence of D in two-component systems of nonelectrolytes include those of English and Dole (1950) for POTASSIUM

CHLORIDE

P

0.6

IW,

,

,

I

.

,

,

0.60

1.26

01

0.40

06

0.2

0.4

FIG.17. The dependence of D on molarity, C, for several dilute aqueous solutions at 25" C. 0 , experimental data; -..----,D"[l C(d In y/dC)]; Do[l C(d In ~ / d C ) I / ( d d ;. . . . . ., Do[l C(d In y/dC)I@~p)/(drlo).

+

+

+

concentrated solutions of sucrose in water, Gladden and Dole (1953) for glucose in water, Hammond and Stokes (1953) for ethanol in water, and Sandquist and Lyons (1954) for biphenyl in benzene. Except for some of the values for KCl in water, which were obtained with Harned's conductance method, the data in Fig. 17 were obtained using the Gouy interference method. Experiments were performed a t different mean concentrations, 6, with relatively small concentration increments, AC, across the initial boundary. Then measured values of the reduced height-

MEASUREMENT AND INTERPRETATION O F DIFFUSION

499

area ratio, DA (or numbers closely approximating this quantity), were taken as equal to D ( c ) , equation (97); thus the effects on each experiment of the relatively small change of D across each diffusing boundary and of any nonlinear variations of the refractive index, n, with concentration were neglected. For some of the systems the validity of this procedure was tested by performing several experiments with different values of AC at substantially the same value of These data for urea in water (Gosting and Akeley, 1952) are shown in Table 11, where no appreciable

e.

TABLEI1 A Test for Dependence of BA on AC for Urea in Water at

6 = 0.248 mole/liter

AC (moles/liter)

D A x lo6 (cm.2 sec.-l)

0.19676 0.24998 0.34992 0.49499

1.3626 1.3628 1.3626 1.3637

C.

dependence of on AC can be observed. That this procedure for determining D ( 6 ) is not valid within experimental error for all systems is seen from the data of Lyons and Sandquist (1953) for n-butanol in ~ 3.941 X for water at 1' C. At 6 = 0.300 molar they obtained a ) = IACI = 0.1326 and for lACl = 0.2644 they obtained DA = 3.92, X lo+. Subsequent experiments a t 25" C. with this system using larger values of [AC[ show a marked dependence of DA on AC, as seen from the data TABLE I11 The Dependence of DAon A C for n-Butanol in Water at 25' C. 6 = 0.4000 mole/liter AC (moles/liter)

a>A x lo6 (cm.z sec.-l)

-0.2000 -0.3000 -0.4501 -0.6500 -0.8000

0.8432 0.8414 0.8391 0.8331 0.8268

(Gosting and Fujita, 1957) in Table 111. These values of AC are considered negative because, to maintain gravitational stability, the solution containing the higher concentration of n-butanol had to be placed above the more dilute solution in the cell. It is evident that for this case

500

LOUIS J. OOSTING

relatively small values of A C must be used if DAis to approximate closely D ( c ) . This rather large nonideality exists for n-butanol in water because the two sources of boundary skewness, the concentration dependence of D and the nonlinear variation of n with C, add rather than partially canceling, giving a significantly large value of K in equation (97). T h e Gouy fringes, which are quite insensitive t o boundary skewness, did not show the appreciable deviations of the refractive index gradient curve from Gaussian shape for this system except when lACl was greater than 0.3. The Rayleigh fringe method has been found to be very sensitive t o skewness of a refractive index gradient curve for free diffusion, and it is now possible t o use Rayleigh fringe data to obtain information about the concentration dependence of D from a single experiment. Longsworth (1953) found appreciable boundary skewness for n-butanol in water; he later (Longsworth, 1955) showed similar deviations t o exist for heavy water in ordinary water and detected a small but measurable skewness for sucrose in water. To show these deviations from the normal curve he measured distances between several pairs of fringes along the fringe system with the fringes of each pair separated by approximately J / 2 , where J denotes the total number of fringes, equation (109). This procedure is seen to be different from that described above in Section III,S,i, where the fringes of each pair were chosen symmetrically about the J/2 position. He then normalized the separation of each pair of fringes so th a t it would be unity for a Gaussian boundary, and plotted this normalized fringe separation against the corresponding mean relative concentration for each fringe pair. This graph is nearly a straight line and independent of time; therefore, data from the several photographs may be superimposed and averaged. These points form a horizontal line a t a value of unity if D is independent of c and n is linear in c ; otherwise it has a finite slope. From the slope of this graph, and Stokes’ (1952) table for the shape of the c versus x curve when D is linear in c, Longsworth showed how t o obtain a value for dD/dc = k,. By utilizing the series solution, equation (75), for the solute concentration curve when D varies with c, Creeth (1955) devised another deviation graph for determining boundary skewness from Rayleigh fringes. His graph is somewhat similar to that used by Longsworth, but it approaches a straight line with slope -kl(Ac/2) as Ac is made small. Here kl is the desired linear dependence of D on c (equation (74)). Following Longsworth’s procedure of using pairs of fringes with the fringes of each pair (denoted by j , and j,) separated by approximately J / 2 , Creeth derived the relation AZ - Az*

=

-kl(Ac/2)[AR(z*)]

+

’

*

(136)

MEASUREMENT AND INTERPRETATION O F DIFFUSION

501

where

and His deviation graph consists of the time-independent plot of the reduced fringe deviations, (Az - Ax*), versus the function AR(z*). I n equation (137) ( X , - XI) is the measured distance between fringes j~ and j , on the photograph, M is the magnification factor of the apparatus, and

80 40

-

-

I

- 80

-0.6

I

I

I

-

-I

I

-0.4

I

I

-0.2

I

0

1

1

0.2

1

1

0.4

1

0.6

AR(Z+)

FIG.18. Itayleigh fringe deviation diagram for glycine in water, c = 0.6101 g./100 ml. Crosses denote averages of experimental values determined at four different times. Solid line represents expected deviations corresponding to the value kl = -0.0245 (g./lOO m1.)-1 (Lyons and Thomas, 1950). Effects due to the nonlinear dependence of n on c have been eliminated (Creeth, 1955).

2 d D m t is determined from equation (114) by averaging data for a number of symmetrically chosen pairs. Values for xl* and z2* for fringes jl and j z of each pair considered here are obtained directly from equation (112) defining x * . Creeth has tabulated values of the function20 R(z) for a wide range of x so that AR(z*) can be evaluated readily using equation (139). A procedure for taking into account a nonlinear dependence of n on c was included in the development. He presented Rayleigh fringe deviation graphs for experiments on aqueous solutions of sucrose, glycine, and n-butanol a t 25' C., the slope of each graph being in good agreement with the known dependence of D on c. His graph for glycine is shown in Fig. 18, where z* is used in place of z* to indicate that these results have already been corrected for the nonlinear variation of n with c. 2o

R(z) = +1/[hW(2)1, where

+1

is defined by equation (79).

502

LOUIS J. GOSTINO

The analysis of data obtained using Scheibling's method (Section 111,2,j) has been considered theoretically by Daune and Benoit (1954) for the case of a concentration-dependent diffusion coefficient and also for polydisperse systems. 2. The Problem of Solute Purity

The ideal experiment for measuring the diffusion coefficient of a protein either would be insensitive to the presence of impurities, or would provide a means for detecting their presence and eliminating their effect on the resultant value. For both cases we assume that the solute flows do not interact and t ha t equations (10) are applicable. I n the first case some type of concentration measurement is required which is sensitive only to the component in question. One example is the study of a biologically active material with the diaphragm cell, using activity measurements to determine the concentrations of the solute of interest. This approach was used by Davisson et al. (1953) for part of their measurements of the diffusion of rat liver lactic dehydrogenase. Another possibility exists if the solute in question is colored, since then it may be studied either with the diaphragm cell or with free diffusion, determinations of concentration being made by measurements of optical density. The primary limitations of such approaches are that only those substances can be studied whose concentrations are readily measurable in the presence of impurities, and that the accuracy of such concentration measurements is sometimes quite low. The second case is illustrated by the studies of free diffusion which utilize some optical system to measure the shape of the resultant refractive index gradient curve, or refractive index curve. Analysis of these data requires the determination of deviations of the experimental curve from a normal probability curve, or probability integral curve. This second approach is still subject to serious limitations also, but it will be reviewed in some detail because its usefulness in studies of proteins has been greatly increased by the development during the last few years of the interferometric optical methods. Some classical procedures for analyzing refractive index gradient curves for the free diffusion of mixtures (equation (89)) have been reviewed elsewhere (Neurath, 1942; Bevilacqua et al., 1945) and were briefly described in Section 111,2,c; hence they will not be discussed further here. a. Investigation of Heterogeneity; Fringe Deviation Graphs. We will consider now the analysis of Gouy and Rayleigh interference fringe patterns for diffusion of mixtures of two or more solutes, reviewing procedures for the calculation and interpretation of fringe deviation graphs which are independent of time and which contain all the information about devi-

MEASUREMENT AND INTERPRETATION O F DIFFUSION

503

ations of the refractive index gradient curve from Gaussian form. These compact graphs are especially suited for reporting experimental results, being closely related t o the actual measurements yet providing at a glance certain information about solute purity. They also provide basic data for more detailed numerical analyses, such as the calculation of reduced moments or evaluation of the diffusion coefficient of the main solute. Before describing these graphs and their interpretation, however, it should be pointed out that interferometric data for diffusion of mixtures may also be analyzed by other procedures. Ogston (1949a) described a method for analyzing experiments with two solutes which was based on the use of r

TI

IMPURE

Gouy

fringes

SOLUTEPositions minima

of

IDEAL

PATTERN Positions o f minima

c,

FIG.19. Illustrative Gouy fringe patterns showing the physical significance of the reduced fringe deviations nj = (Yj' - Yi)/Ct = (e-
two widely separated Gouy fringes; he also devised a procedure (Ogston, 1949b), based on a numerical differentiation and integration, for obtaining the reduced second moment, 9 2 % (equations (91) and (95)). Charlwood (1953) used this procedure to calculate reduced second moments and extended it to obtain values for the reduced fourth moment, Dim(equations (92) and (96)). A method for obtaining reduced second moments from the Rayleigh interference patterns was described by Svensson (1951b). Fringe deviation graphs for the Gouy method are obtained (Akeley and Gosting, 1953) by plotting reduced fringe deviations, !2j, defined by

versus the corresponding reduced fringe numbers, f([j) , defined by equation (111). The fringe displacements from the undeviated slit image, Yj,

504

LOUIS J. GOSTING

are measured directly, e-ri' is obtained from tables of this quantity versus

f(3;.), and Ct is obtained by an extrapolation procedure, equation ( l l l b ) .

Visualization of the meaning of these reduced fringe deviations may be aided by Fig. 19, which shows on the left an illustrative Gouy fringe pattern for an impure solute and in the center the positions of fringe minima for this pattern. On the right are positions of fringe minima for the diffusion of a single solute; this ideal pattern has the same total number of fringes, J , as that for the mixture and the time is chosen so that both patterns have the same maximum displacement, Ct, and hence the

*-. 0.5

0.5

-.-. 1

F ~ Q20. . Illustrative graphs of Gouy fringe deviations for systems with a small amount of impurity, component 2, in the main solute, component 1. The solid curves show the effect of varying D1/D2 when az = 0.01. The dashed curves for aa = 0.005 are included to help illustrate the proportionality between fringe deviations and a p , when (YI is small.

same value of (dn/dz),,,,,. Since for the ideal pattern there exists the relation Y,'/Ct = e-17', the reduced fringe deviation, $, for any fringe j is simply the observed difference in downward displacement (Yj' - Y 3 ) divided by Ct. It should be noted that a,is zero for all fringes if the refractive index gradient curve is Gaussian; therefore fringe deviation graphs which are zero for all values of f({i) indicate that the solute is homogeneous within the error of measurement. Hereafter, for simplicity, we will drop the subscript j on Q j and 5;. Illustrative graphs of Gouy fringe deviations are shown in Fig. 20 for the case of a small amount of a single impurity (component 2) in the main solute (component 1). When the fraction, a2 (equation (SS)), of im-

505

MEASUREMENT AND INTERPRETATION O F DIFFUSION

purity is relatively small these graphs are described accurately by the seriesz1 Q =

a2F(C7D1/D2) - aa2G(T,D,/D2)

+.

*

(141)

*

The functions F({,D1/D2) and G({,D1/D2)have been tabulated (Akeley and Gosting, 1953) for a number of values of f({) and of B l / D 2 . For sufficiently small values of a z , such as those considered in Fig. 20, the term in may be neglected. Then the general shape of a fringe deviation graph is determined entirely by F ( { , D , / D J , while its height is proportional t o a z ; this dependence on a2 is illustrated in Fig. 20 by the dashed curves corresponding to az = 0.005. Thus, provided az is small for a mixture of two solutes, the ratio D l / D Z may in principle be obtained directly from the value [f(C)]ma. corresponding t o the maximum fringe deviation, Q,,,. Then, knowing D 1 / D z ,a2 may be obtained from QmBxusing equation (141). I n practice this procedure is limited by the difficulty of obtaining a precise value for [f({)Irnsx, so it is more satisfactory t o use equation (141) to obtain the best possible fit of the entire curve. If D , is not too close to D,, the fringe deviation graphs provide a sensitive test for th e presence of impurity. This is well illustrated by Fig. 20 and also by Table IV, which shows the minimum fraction of impurity, which may be detected for several values of the ratio D 1 / D z ; for these calculations the experimental uncertainty in D was taken t o be kO.0002, corresponding to an error of +0.0002 cm. in the measurement of Gouy fringes when Ct is 1 cm. The theoretical sensitivity shown in Table IV may not be fully realizable when 0 1 / 0 2 is very large (>9), unless J is also large, because of some difficulty in obtaining Ct by extrapolation. When D l / D z 'v 1 (equation ( 1 4 1 ~ ) the ) fringe deviations are of course small even for large amounts of impurity; this is illustrated by the data of Dunlop (1955), who found barely detectable fringe deviations for the diffusion in water of a mixture of glycinc and glycolamide (oi/oz = 0.922) with (Ygiyooismid. N 0.4. 21

Here (141a)

and (141b)

wheref(l) is defined by equation (llla). An alternative series expansion has also been derived (equation (27) of Akeley and Gosting, 1953) which converges rapidly when D I 'v Dz,regardless of the value of a ~ . 52 = az(1

- a z ) r z e - r i Z ( d / D 1 / D 2 - 1)2(1+ %[(I - 8az)

- 2r2(1 - 2az)](l/D1/Dz

- 1) -I-

*

*

*)

(141~)

506

LOUIS 3 . GOSTING

TABLE IV T h e M i n i m u m Fraction, ( L Y ~ ) ~ ~of, , Impurity , Which C a n Be Detected f r o m Gouy Fringe Deviations for Several Representative Ratios of the Diffusion Coeficients, DdDza

DilDa 100 49 25 9 4

2 1.5

(admin

DdDz

(adrnin

0.00003 0.00005 0.00009 0.0002

X O O

0.0005 0.0006 0.0007 0.0010 0.002 0.006 0.015

0.0007 0.004 0.012

%9

35 5

M

%

35

111.5

m a y be obtained from t h e a For values of D , / D s lying between 1.5 and 1/1.5, estimates of relation ( a ~ ) ~ i , J l( a ~ ) ~2i1d0 . 0 0 0 5 4 / ( ~ / ~-21)' derived from equation (141c) by setting CI = 0.0002 and t = 1.

-

If small amounts of several impurities are present the resultant fringe deviation graph is the sum of the graphs for the individual impurities. The mathematical relation describing this general case is

+

cr,F(T1D~/D,)

= i=2

*

..

(142)

where terms of order at2,a,aj, and higher, have been dropped. When desired, values of the second and fourth reduced moments for an experiment can be computed22from the fringe deviation graph and the value of the reduced height-area ratio, DA. Except for the inability of the Gouy fringes to measure skewness of the refractive index gradient curve, these computations are valid regardless of the number and amount of impurities and whether or not the solute flows interact (Section V). 22

The expression for the rth even reduced moment ( r

=

2,4,

..

.) is

in which (1426) (Equation (39) of Akeley and Gosting, 1953, or equation (47)of Dunlop and Gosting, 1955.) The integrals, which are frequently small compared t o unity, are evaluated graphically using the heights and slopes of the graph of fi versus f(r). The functions of needed for this computation have been tabulated (Table IV of Akeley and Gosting, 1953) for r = 2 and r = 4.

r

MEASUREMENT AND INTERPRETATION O F DIFFUSION

507

Fringe deviation graphs for some samples of plasma albumin have been reported by Baldwin et al. (1955). Two of these graphs for bovine plasma albumin (BPA) and one for normal human albumin (NHA) are shown in Fig. 21 to illustrate typical fringe deviations. Assuming that the same impurity is present in both BPA samples, the greater deviations for BPA No. R370,295A indicate that it contains more impurity than does BPA No. 284-8. Both samples of BPA, which were crystallized samples from Armour and Co., appear to be more nearly homogeneous than the NHA sample. The absence of large values of il near f(() = 1 provided proof that no appreciable buffer gradients were present to contribute to the reduced height-area ratio, D.4 (if no interaction existed between the solute

2ol+

NHA

i

NO

f

*

f(f)

I25

BPA No. R 3 7 0 2 9 5 A

f(f)

BPA NO.

em-a

fCf)

FIG.21. Graphs of Gouy fringe deviations for two samples of bovine plasma albumin (BPA) and one sample of normal human albumin (NHA), all approximately isoelectric. Each initial diffusion boundary was formed between the protein solution and the buffer against which that solution had been dialyzed. Dots a t a given value of f(r) represent values of 1;1 measured a t different times; averages of these values are represented by a cross.

flows). Analysis of Gouy fringe deviation graphs for proteins will be considered in more detail below. Experimental evidence for the independence of solute flows for some quite dilute solutions, and hence for the validity of equation (10) and relations derived from it for those solutions, has been obtained with the help of fringe deviation graphs. It was found (Akeley and Gosting, 1953) that dilute aqueous solutions containing sucrose with potassium chloride, or urea with potassium chloride, gave experimental graphs that agreed with those predicted from the known diffusion coefficients for the solutes diffusing alone in water. Similar agreement was obtained by Dunlop (1955) for dilute aqueous solutions containing sucrose with glycolamide or glycine with glycolamide. The reduced height-area ratios, D A , from these experiments were also in quite good agreement with the predicted values. However, when two different salts are present together in adilute aqueous solution the flows are not independent (Section V). The situation for concentrated solutions of norielectrolytes has not yet been clarified. Rayleigh fringe data for mixed solutes have also been interpreted by means of fringe deviation graphs. Using deviation graphs of the type

508

LOUIS J. GOSTING

suggested by Longsworth (1953) for reporting boundary skewness (see Section IV,l,c), Hoch (1954) reported data for normal bovine plasma albumin, for albumin from heated lyophilized plasma, and for mixtures of bovine plasma albumin and dextran. H e also presented theoretical curves designed t o assist in interpreting the experimental data. Longsworth (1955) reported a deviation graph from a diffusion experiment performed on a mixture of raffinose and glucose in water; agreement between the expected and observed deviations was excellent. While skewness of the refractive index gradient, curve produces fringe deviations which form a,=o.oi O,/D,=

L

n

c3

4

4

2okLc3 0.5

OO

I

a,=o.oi

Q/D2 = 1/4 n

oo

0.5

I

p(z)13

FIG. 22. Illustrativc graphs of Itayleigh fringe deviations, O R , for systems with a small fraction of impurity, 012 = 0.01.

a sloping line on these graphs, an impurity produces a symmetrical deviation curve somewhat resembling a parabola. It is therefore easy to distinguish between these two causes of non-Gaussian boundaries. Using the Rayleigh interference method, Creeth (1957) studied the diffusion of some known solute mixtures and reported his results using fringe deviation graphs of the type shown in Fig. 22. The first-order effects of boundary skewness were eliminated by pairing of fringes symmetrically about the J / 2 value, as described above in Section 1142,i. Then the Itayleigh fringe deviations defined by Q2R

= @(ZL3)

- @(Z*)

(143)

were plotted against [@(2*)13;by using [@(z*)I3 rather than @(z*) for the abscissa the graphs approach @ ( x * ) = 0 with a finite, nonzero, slope. Here @ ( z * ) is defined by equation (112) while

MEASUREMENT AND INTERPRETATION OF DIFFUSION

509

in which X J , and X j are the comparator readings for fringes J - j and j , and M is the magnification factor of the apparatus. The quantity 2 & for this calculation is obtained by the extrapolation procedure mentioned in Section 111,2,i. For small values of az in two-component systems the fringe deviations, Q E , are proportional to a 2 ;therefore the basic shape of each fringe deviation graph (Fig. 22) is determined by the ratio D l / D z as was the case for fringe deviation graphs from the Gouy method. The procedure for analyzing these graphs is similar to that described above for the Gouy method. It is of interest to note that the Gouy and Rayleigh fringe methods complement each other in providing values for the fringe deviations : most of the integral fringes arise from the central part of the boundary, whereas, owing to the contribution of the air paths to the Gouy pattern, the “tails” of the boundary produce a significant number of the Gouy fringes. This is illustrated by considering a 100 fringe pattern from a Gaussian refractive index gradient curve. The lowest Gouy fringe minimum ( j = 0) corresponds to (2/&) e-b’dp = 0.24, or to Rayleigh fringes 38 and 62 in the central region of the pattern. The 85th Gouy e-b’dp = 0.98, or to Rayleigh fringe minimum corresponds to (2/&) fringes 1 and 99. b. Determination of D lfor the Main Solute, Calculation of D1, the diffusion coefficient of the main solute, from refractometric data for the free diffusion of mixed solutes is practical only if i t can be assumed th a t a single impurity is present. Subject to this limitation, however, the interferometric optical systems can provide a value for D1 which, although subject to more error than the value of D A , is much closer to the desired diffusion coefficient than is DA. Ogston’s (1940a) procedure for obtaining D , from 3 A and the positions of two Gouy fringes has been applied to a number of materials of biological interest including lactoglobulin (Cecil and Ogston, 1949; Ogston, 1949a), dextro- and levorotatory polysaccharides prepared from PeniciEZiu~Euteum (Ogston, 1949b), hyaluronic acid complex from synovial fluid (Ogston and Stanier, 1950), and fumarase (Cecil and Ogston, 1952). Hoch (1954) has analyzed Rayleigh fringes t o obtain some information about the degree of homogeneity of plasma albumin samples. Values of D 1for both bovine and human plasma albumins have been determined by utilizing the fringe deviation graphs from the Gouy diffusiometer together with the values of (Akeley and Gosting, 1953; Baldwin et al., 1955). Using fringe deviation graphs and values of a ) ~ measured with the Rayleigh interference method, Creeth (1957) obtained information concerning D1 for bovine plasma albumin, ribonuclease, and p-lac toglobulin.

[

[

510

LOUIS J. GOSTING

The use of Gouy fringe deviation graphs in obtaining D,is illustrated by Fig. 23. Here the crosses represent average values of the reduced fringe deviations, 52, f6r the sample of normal human albumin shown

y,

c

........ , ........ , D,/D, , =,

----

DJD, D,/D,

OO

2 ,

= 1.75 = 1.5

,

'\ '. \

'..\

' \

0.5

I

frn

FIQ.23. Analysis of the graph of Gouy fringe deviations for a sample of normal human albumin (NHA No. 125, Fig. 21). Here equation (141) was fitted to the crosses, representing experimental data, by adjustment of 012 for each of three different values of DI/Dz.

earlier in Fig. 21. The dotted, solid, and dashed lines represent values of Q obtained from equation (141) for D l / D ~equal to 2, 1.75, and 1.5, respectively. In each case a value for a2 (Table V) was chosen t o give TABLEV Data Illustrating the Calculation of D I for Normal H u m a n Alhumin from BAand the Fringe Deviation Graph in Fig. 83 ENHA = 0.519 g./lOO ml. ( a ) ~ ) 2 ~= , ~6.460 X lo-' cm.2 see.-' pH = 4.60

1.5 1.75 2

0.152 0.068s 0.0414

6.91 6.75 6.68

the correct maximum deviation, Qmsx. Of the three curves, that for D 1 / D z = 1.75 appears to fit the data best, but all the curves fit within the error of measurement. For each value of D1/Dz the corresponding value of D 1was computed from equation (94) after writing it in the form

DI

=

+(

BA[~

d m- 1)%l2

(145)

~ shown in the last column of These values are denoted by ( D J z s ,and Table V. The subscript 25 on D1 and a ) ~denotes 25" C. (the temper-

MEASUREMENT AND INTERPRETATION OF DIFFUSION

511

ature of measurement) while w indicates th at the values have been converted t o those for water using equation (133). The values of ( D 1 ) % 6 , w are seen t o be appreciably different from th a t for the value ( D 1 ) 2 6 ,= ~ 6.76 X em.' sec.-l may be tentatively taken as the best estimate from these data of the diffusion coefficient of human serum albumin a t p H 4.6,, and C = 0.519 g./100 ml. If a n independent determination of a2 is available from some other type of experiment, such as an electrophoretic or sedimentation analysis, the precision with which D,may be determined is considerably increased. An analysis of this sort has been performed for a sample of bovine plasma albumin by Baldwin (1957). Interpretations of deviation graphs for Rayleigh fringes, of the type shown in Fig. 22, may be made in a manner quite analogous t o th a t described above for interpreting Gouy fringe deviations. If there is no main solute, but instead the sample is polydisperse with a distribution of diffusion coefficients, the best procedure for analysis appears t o be the determination of the reduced moments, equations (91) and (92). These may be obtained from the Gouy fringes by a procedure utilizing a numerical differentiation and integration (Ogston, 194913; Charlwood, 1953) or by a procedure utilizing %A and the fringe deviation . may also be obtained from Rayleigh fringe graph (equation ( 1 4 2 ~ ) )They measurements by performing a numerical integration (Svensson, 1951b). These moments give directly certain simple average diffusion coefficients, equations (95) and (96), or they may be used t o obtain the standard deviation of the diffusion coefficient distribution in the mixture (Charlwood, 1953).

3. Some Data for Representative Solutes

a. A m i n o Acids and Other Materials of Low Molecular Weight. Studies of aqueous solutions of a number of substances have provided information about the effect on the diffusion coefficient of chain length, chain branching, and polarity. For a detailed consideration of these problems in relation t o molecular volume, as approximated by the partial molal volume, the reader is referred to a comprehensive review by Longsworth (1955). Here we will simpIy summarize a few of the available d a ta which illustrate the magnitude of the effects. Information about the relation of the diffusion coefficient to chain length is provided by the data of Longsworth (1952, 1953), some of which are summarized in Table VI. These data were obtained with the Rayleigh interference method. Since the average concentration in the cell was quite small, these values of D may be assumed t o have nearly the same relation t o each other as the corresponding values a t infinite dilution. It is

512

LOUIS J. GOSTING

TABLE VI E$ecl of Chain Length on the Diffwsion Coe&ient Temperature: 25" C. Solvent: HtO

-

(wt. %)

D X lo8 (cm.2 sec.-l)

a-Aminobutyric acid a-Aminovaleric acid a-Aminocaproic acid

0.300 0.319 0.313 0.318 0.321

10.554 9.097 8.288 7.682 7.249

Glycine Diglycine Triglycine

0.300 0.288 0.289

10.554 7.909 6.652

Compound Glycine

A 1anin e

Here 6 =

(WA

+ w r ) / Z . where w denotes weight

WQ

$6 of solute. For these experiments

WA

= 0.

seen that the effect on D of a CH2 group becomes progressively smaller with increasing chain length, as might be expected from its decreasing contribution to the molecular volume and radius, equations (47) or (54). The influence of chain branching on the diffusion coefficient has also been studied by Longsworth (1953). Some of these data are shown in Table VII. For the five and six carbon acids the branching results in a TABLEV I I Effect of Chain Branching on the DiffusionCoeficient Temperature: 25" C . Solvent: H 2 0 (wt. %)

D X lo8 (cm.2 sec.-l)

a-Aminobutyric acid a-Aminoisobutyric acid

0.313 0.320

8.288 8.130

a-Aininovaleric acid a-Aiiiinoisovaleric acid

0.318 0.308

7.882 7.725

a-Aniinocaproic acid a-Arnirioisocaproic acid

0.321 0.326

7.249 7.255

Compound

W

higher diff usioii coefficient indicating a more compact molecule, whereas the opposite effect occurs for the four carbon acids. These effects are consistent (Longsworth, 1955) with data for the partial niolal volumes of each pair of isomers.

MEASUREMENT AND I N T E R P R E T A T I O N O F D I F F U S I O N

513

The effect of polarity on the diffusion coefficient is illustrated a t the top of Table V III by data for two sets of isomers. It is seen th a t the diffusion coefficient of glycine (Lyons and Thomas, 1950), which exists as a dipolar ion in solutio?, is appreciably less than the value (Dunlop TABLE VIII Eflects of Polarity and of Group Sequence on the Diflusion Coeficient Temperature: 25" C. Solvent: H20

-

(wt. %)

D X 106 (cm.z sec.-l)

0 0

10.635 11.428

o-Aminobenzoic acid rn-Aminobenzoic acid p-hminobenzoic acid

0.244 0.236 0.227

8.40 7.741 8.425

Glutamine GI y cylalanine Alanylglycine

0.338 0.302 0.300

7.623 7.221 7.207

Glycylleucine Leucylglycine

0,290 0.308

6.231 6.129

Compound Glycine Glycolamide

W

and Gosting, 1953) for its nonpolar isomer, glycolamide. This is consistent with the concept th at the charged groups on glycine strongly attract water molecules, producing a kinetic unit of larger size. T h a t solvent molecules are a t t r x t e d sufficiently to produce electrostriction (i.e., to make the neighboring solvent have an abnormally high density) is indicated by data for the partial molal volumes, P. The value of P for glycine a t infinite dilution at 25" C. is 43.199 ml. per mole, which is considerably less than the corresponding value for glycolamide, 56.156 ml. per mole (Gucker, Ford, and Moser, 1939). Longsworth's (1953) data for diffusion of the aminobenzoic acids, Table VIII, also illustrate the effect of polarity. It is seen th a t the ortho and para compounds have essentially the same value of D, while th a t for the meta compound is appreciably lower. This correlates with other evidence that the ortho and para coinpounds are present in the uncharged state, while the meta compound exists as a dipolar ion. Data for the partial molal volumes of these compounds indicate that electrostriction occurs with the meta isomer (Cohn and Edsall, 1943, p. 159).

514

LOUIS J. GOSTING

The remaining data in Table VIII (Longsworth, 1953) are difficult t o interpret but they provide interesting comparisons. Thus while glycylalanine and alanylglycine have nearly the same diffusion coefficient, the value for the third isomer of this group, glutamine, is appreciably larger. Furthermore, there appears to be a greater difference in the diffusion coefficient of the two isomers gly cylleucine and leucylglycine than between glycylalanine and alanylglycine. Optical isomers of a few amino acids have been studied but no significant difference was found in the diffusion coefficients of any given pair (Gutter and Kegeles, 1953; Longsworth, 1953). b. Proteins and Other Materials of High Molecular Weight. Extensive tables of diffusion coefficients of proteins, together with values for their sedimentation coefficients and molecular weights, have been published by Svedberg and Pedersen (1940, p. 406) and by Cohn and Edsall (1943, p. 428). Subsequent summaries of data for a large number of proteins and other materials of high molecular weight have been prepared by Lundgren and Ward (1951), Pedersen (1953), and Dandliker (see Table VIII of Edsall, 1953). With these compilations available, the present review is devoted for the most part to results published since 1952. It is recognized that even for this restricted period the following summary is incomplete, but the data shown illustrate the diversity of materials studied, the range of diffusion coefficients encountered, and the variety of methods available for measuring diffusion. These data also indicate what information about experimental conditions should be reported with the diffusion coefficient, and some of them illustrate the problems encountered because of the impurities usually present in protein samples. Tables IX and X contain most of the data for proteins and other substances of high molecular weight which are presented here; however, a few other results will be considered in the text below to permit discussion or inclusion of information which is not easily tabulated. Data obtained with interferometric methods are grouped together in Table X t o facilitate reporting information about solute purity in the last five columns, one of which lists some values for D1,the diffusion coefficient of the main solute component (on the assumption that equations (10) are applicable). Values in these columns could be obtained with moderate precision because the interferometric optical systems are about ten times more accurate than the optical methods based on ray optics. Some data published prior to 1953 are included in Table X because interpretations concerning solute purity were made. The results shown in Table X, together with those considered in the text below, are thought to include most studies of the diffusion of high molecular weight materials of biological interest which have been made using the modern interferometric optical methods.

MEASUREMENT AND INTERPRETATION OF DIFFUSION

515

Unfortunately certain inportant pieces of information, such as some of the conditions in the experiment or the method of analyzing the data, are sometimes omitted when diffusion coefficients for proteins are reported in the literature. The seriousness of such omissions becomes greater as the precision of measurement is increased. Consider, for example, the protein concentration: frequently the value reported is the concentration, C E , of the protein in the lower initial solution rather than the mean concentration for the experiment, F, even though F corresponds much more closely t o the vaIue of the diffusion coefficient obtained b y several standard methods (Sections III,Z,c, III,Z,i, and IV,l,c). Furthermore, it may be impossible t o determine whether the value reported is CB or F. Concentrations are sometimes expressed as “per cent,” without further clarification; usually the quant.ty meant is grams of protein per 100 ml. of solution, but it can be confused with per cent by weight. The composition and pH of the buffer solution should be given, as well as the temperature of measurement. I n reporting accurate values of diffusion coefficients, such as those obtained int erferometrically, the temperature of measurement is important because a significant error may be introduced by the classical procedure for converting to 20’ C. (Section IV,l,b). T o obtain the diffusion coefficient a t zero concentration of protein it is usually necessary to perform experiments a t several values of F (Section IV,l,c) and then extrapohte these results t o C = 0. Adequate data for this extrapolation were not obtained for all the substances considered in Tables IX and X. It is of interest t o note th at Creeth (ref. 6 of Table X) was able t o use Rayleigh hinge data from a single experiment to determine the diffusion coefficient of bovine plasma albumin a t zero protein concentration besides obtaining information about the purity of the sample. Using a modified Boltzmann type of calculation, Thompson (ref. 18 of Table X) also analyzed individual experiments performed with the Rayleigh method to obtain information about the concentration dependence of the diffusion coeffi&ent. The presence of impurities in protein samples constitutes a serious limitation on the accuracy of measured diffusion coefficients. With the development of interferometric methods, a fairly reliable determination of D,for the main solute component may be made (Table X) if only a single impurity is present in small amount (Section IV,2,6). Otherwise refractometric data must ’38 reported as averages, such as SA,a ) 2 m , and equations (94) through (96). It should be noted that a,,,and (T$,)$4 lie between D 1and DA, as illustrated by the data for bovine plasma albumins in Table X; therefore the range of these averages must not be the desired diffusion coefficient taken as indicating possible limits for D1, of the main component. Elecause of the problem of impurities, it is im-

. N

L

2

m

10

R

i

W

W

m

D

0

0 N

-

h

$ 0

N

0

0

3

0

-

h

h

m

(0

.

m

h

h

h

.

m

___

0

0

3

0 N

0

0

n 0

8

e r n n~

m.

m h

d

R

*

N

m.

R

(

LOUIS J. GOSTING

(0

N

N 0

N

10

Fetuin

11

Fibrinogen

-

0.4 M NaCl Na phosphates ___-.

3.6 7

0.2 0.2

6.2

0.45

3.72 5.37 2.02

HzO

0.4

__ __

~

~-

~

__-

12

Hyaluronidase inhibitor of human blood

Borate Sulfate

8.6

13

Hypophyeeal growth hormone

0.10 M NaH2POa 0.04 M H I P O ~ 0.1 M HJBOJ 0.09 M NaOH 0.15 M NaCl 0.05 M Na&iPOc 0.034 M NaOH

2.32

0.35%

20

20

6.37

6.71

9.94

0.25 %

20

20

7.19

7.57

11.50

0.35%

20

20

7 67

7 OR

0.18 M NaCl Phosphates

14

0.1

0.56%

_-__

7.5

0.20

16

__ 17

-1s

Several 0.0323 M KaHPO4 0.00323 M KHzPO. 0.0323 M KzHPO4 0.00323 M KHzPO'

Lactic dehydrogenase, rat liver

L-Lysine decarboxylase from Escherichia coli B Lysozyme, papaya

D

0.2 M phosphate 0.02 M cysteine 0.001 M Versene Acetate 0.02 M cysteine 0.001 M Versene Acetate

19

Myosin

0.4

M KCl

Phosnhate

10.6

_

7.8

0.1

Several 0.33

7.8

0.1

1.23

__ 6.0

20 20

0.8

20

__

_-

3.9

0.12

0.43%,

1.5

3.9

0.12

1.09%

1.5

0.6%

1

1.1

6.8

-

_

0 0.8

4

Multiple myeloma proteins

__

I

1%

I _

15

3.4

0.5

0

~ 5.72 5.42

20

H?O

9.29

20

H?O

9.41

1

20

HtO

3.7& 3.96/

II

20

HzO

i

~

5.60

13.+ 2.11e

20

1

~ _ _ 5.3

1.05g

___

-

TABLE IX (Continued)

i

I Substance

Method=

I

Ovalbumin

Papain

21

C C C

S(S)

22

Phosphatase, alkaline

ll

23

Prothrombin. bovine

S(S)

24

Rhodanese

___

25

1 Soyin Trypsinogen

26

Wool protein

i i

Colnpositionb Aqueous Aqueous Aqueous

1 1'

j

1

0.02 M cysteine

Several 0 . 1 M NaCI

Phosphate

S S

10.2 M NaCI

Protein conc.c

pH

6.9-9.

'1 1 I

S(C)

0.02 M N a acetate 0.18 M NaCI Acetic acid I

S(C)

8 Murea Bisulfite Acetate NaCl

II

Temp. of mess. r'

("C.)

4.76 4.76 4.76 3.9

Acetate

Phosphate Phosphate

S

I

27

Buffer-

i

0.12

10 G wt. 5 19.0Wt.'2 21 4 nt. Tc

10.0 10.0 10.0

I . 61-1.08 %

1.5

Ref. temp.

20

I i

1.38h 1.13h 0.979

H20

10.27

-_

5l

3.14. t

0.15

0.5%

0

0.1 0.1

0.176 % 0.338~

0.3 1.1

1%

0

4

0.9

20 20

_

-_ Hz0

20

0.9-3.5

7.5

_

H20 HnO

_

20

H20

20

Hz0

_

20

6.24

8.35

-

_

__5.5

-Diffusion coe5cientX 10' cm.' 8ec.-1

("(3.)

-_ 7.44 7.44

Ref. medium

~

7.36 7.60 5.72

~ 9.68

___ H~O

4.51

4.49

C, open-ended capillary method; D, diaphragm cell; L, Lamm scale method; S, schlieren method IS(@ denotes sohlieren scanning and S(C) the cylinder lens method]. b Concentration scales are those used by the authors. Values are expressed in grams of protein per 100 ml. of solution except where otherwise indicated. I n many cases it was not specified whether the concentration was that of the protein solution, CB, or the mean value for the experiment, 5. Values of zero indicate that the diffusion coefficient was extrapolated t o zero concentration of protein. Values other than DAand Dm are placed in this column. I n the case of the diaphragm cell and open-ended capillary measurements, these are values of D for the component of interest. Data from optical methods reported in this column were calculated by unspecified methods or by methods giving averages other than DAand Dzm. * Results from different fractions. Three patients. 9 Averaee of several methods: BA.inflection ooints. etc. . Self dkusion coefficients of ovalbumin. (1) Hayes and Velick (1954). (2) Sher and Mallette (1954b). (3) Taylor, Epstein, and Lauffer (1955). (4) Deutsch (1955). (5) Wagman and Bateman (1953). (6) Smith, Cook, and Neal (1954). (7) Sullivan et al. (1955). ( 8 ) von Hippel and Waugh (1955). (9) Whitaker, Colvin, and Cook (1954).

Deutsch (1954). Shulman (1953). Newman et aZ. (1955). Li and Pedersen (1953). Gordon and Semmett (1953). Davisson et al. (1953). Sher and Mallette (1954a). Smith et al. (1955). Pntnam and Udin (1953).

(19) Parrish and Mommaerts (1954). (20) Wang. Anfinsen, and Polestra (1954). (21) Smith, Kimmel, and Brown (1954). (22) Mathies and Goodman (1953). (23) Lamy and Waugh (1953). (24) Sorbo (1953). (25) Pallansch and Liener (1953). (26) Tietze (1953). (27) Friend and O’Donnell (1953).

TABLEX Some Data for the Daflusion of Proteins and Other Substances of High Molecular Weight. Results Obtained with Interferometric Methods Ref. Substance

I

Buffe-

Composition*

Protein c0nc.c

~1,G Adenosinetriphosphate-creatine transphosphorylase

Glycine

2,G Albumin, bovine plasma

(Several)

9.0

0.1

0.050.2

0.986

4,R Albumin, bovine plasma (Armour No. R370295B)

5,G Albumin, bovine plasma (Armour No. 284-8) Albumin. bovine plasma (Armour No. 212113) 5,G

6,R

Albumin, bovine plasma (Armour No. R370295A) Albumin, bovine plasma (Armour No. R370295A)

Phosphate

7.4

0.1

0.15 M NaCl 0.02 M acetate 0.15 M NaCl 0.02 M phosphate

0.15 M KCl 0.01 M K acetate 0.01 M acetic acid 0.15 M KCI 0.01 M K acetate 0.01 M acetic acid 0.15 M KCl 0.01 M K acetate 0.01 M acetic acid 0.15 M KCI

0.01 M K acetate 0.01 M acetic acid

--

0

B Z l %

7.30

0.67

-4.60 0.16

__4.60 0.16

_

25

20

21-25

20

25

~

~

5.78 6.14s

-

25

7.10

6.81

1.87

0

2.9380

1.85

0

3.0940

-

__-

4.60 0.16

(~A)T,w

7.05

~

0.67

--

(OC')

-__-

4.55

4.60 0.16

7'

~ _ _ _ _ _ _

__3,G Albumin, bovine plasma

mess.

_

__6.88.6

Diffusion coefficientX 10' cm.2 sec.-1

Ref. temp.

("c.)

r/2

PH

Temp. of

E= 0.498

25

i =

25

0.507

0.515 E = 0.518

0

I

A

25

6.696

~-__6.720

6.742

25

6.661

6.706

6.74s

25

6.670

-_____ 25

0.025

8

25

6.79

0.031

10

---

--____ 25

6.80

__--

6.685 6.751'

6.73,

-

6.791

6.84

0.039h

12

---

Albumin, human plasma

5,G

7,G

Blood-group A substance

8,G

I Blood-group B substance Blood-group B' substance

8,G 9,G

-

I-

1-

0.15 M KCl 0.01 iM K acetate 0.01 M acetic acid

4 . 6 0 0.16

____

0.35 0.35

0.23 0.40

_

0.4 M NaCl Na phosphates

11,G Fibrinogen, human

25 25

_

_

0.4

j

25

____

13,R Glyceraldehyde-3-phosphate Phosphate dehydrogenase

1

15,G 0-Lactoglobulink

0.1 M NaCl 0.1 M N a acetate 0.04 M acetic acid

&Lactoglobulin

0.15 M KCl 0.01 M K acetate 0.00282 M acetic acid

16,G &Lactoglobulin, goat

NaCl, phosphate NaCl. phosphate

6,R

I 1 5.14

5.20

~

O

i. = 0.47%

:::

0.2 10.2

1

6.460

1 -11

6.552

1

6.60r

-I-I1

6.75

10.068

1

21

25 25

0.067 M NaoHPO4 0.017 M KHzPOI

Insulin, oxidized Fract. A Fract. B

25

__

1.00

-

12,G Fumarase

0.519

0.11 0.22

NaC1, phosphate NaC1. phosphate

-I

25

0.42 0.81

Blood group H substance

10,G Fibrinogen, bovine

t =

i

25

~

A A

TABLEX (Continued)

Ref. and meth.

Buffer

Substance

Protein c0nc.c

Composition')

od-

17,G Licheniformins from

Bacillus licheniformis

__

A 0.2 M NaCl B 0.2 M NaCl C 0.2 M NaCl

~

18,R Mercaptalbumin. bovine plasma

Na acetate Acetic acid

19.G Myokinase

0.15 M KC1 0.01 M I(phosphatc

20,G

Peroxidase, horse-radish

21,G Polysaccharide A from hog gastric mucin

-20.17

0.388

0.2 M NaCl 0.046 M KHzPOI 0.010 M NazHPO4

22,G

&-Metal-combining protein, human plasma (metal free)"

0.28-

i

25

S

8 8

-0.267 0.267

0.25 0.50 0.635 0.665

-~

5.58 a t 0.10 23O C.

I = 0.48 0

_ I -

2o 20

6.66

-25

--

Na acetate Acetic acid 0.10 m M Versene

---

0.1000 M KCl 0.03227 M KtHPO, 7.74 0.00319 M KHzPO.

Tropomyosin

-1

25

25

0.34

Phosphate NaCl

Sheep salivary mucoid

20

-__

0.56%

6 3 Ribonuclease (Pentex No. A3301)

20 20 20

4.75 0.05 at 25' C. E =

I

~

Diffusion coe5cientdXlO7 cm.9 see.?

("CJ

-__

_ I -

-

7'

(q-) -__

7.02

~~~

Femp. Ref. of temp. mess.

20

-25 25

20 20

2.31 2.27

20 20

2.26 2.22

-__20.17

20

15

10.68

-I

--5.294."

A A

G. Gouy method; R, Rayleigh method. Concentration scales are those used by the authors; in general M denotes moles per liter. c Concentrations expressed as grams of protein per 100 ml. of solution unless otherwise indicated. Values of zero indicate either that the diffusion coe5cient was extrapolated to zero concentration of protein. or that the diffusion coefficienta t zero concentration was determined using data for the skewness of the refractive index gradient curve. d All values converted to water, w . as reference solvent unless otherwise specified. * The diffusion coe5cient of the main component was calculated from DAand the fraction of impurity, az,determined from the shape of the refractive index gradient curve; it was assumed that only one impurity was present and that the solute flows did not interact. f Here the maximum fringe deviations for the Gouy method and (Qzz)max for the Rayleigh method are defined by equations (140) and (143). respectively. The symbols S (slightly non-Gaussian) and A (appreciably non-Gaussian) are used in this column to provide qualitative information when deviations of the refractive index gradient curve from Gaussian shape were not reported in terms of Q. 0 Calculated using unsymmetrical Rayleigh fringe pairs; some of these values may not be exactly 'DA. The primary reason for the difference in these two values of as is that D t / D %was taken as 1.75 in ref. 5 and 1.5 in ref. 6. ;Calculated from 'DAa t E = 0.518 using the value for k t , equation (74). obtained by the procedure described in Section IV,l,c. i Da~a iur %A were ubiaiual; iu iLe uuuocuiiaiiuu iauge 0.2-i.0 a d e.xCrapui*Led i u zeru ~ u u ~ e u i t i l i i uegie~iiug u~~, ilu apptaoLLie iixreaee Leiuw 0.15 g./100 ml. This protein is sometimes called simply lactoglobulin. as in ref. 15. Obtained by extrapolating data from several experiments with protein concentrations in the range 0.04 I E 5 1.154; the diffusion eoe5cient wm found to increase with decreasing concentration. Concordant results were obtained when one of the experiments was analyzed using a modified Boltzmann type of calculation. m Referred to buffer instead of water. " This protein is frequently referred to as transferrin. 0 Obtained using a modified Boltzmann type of calculation. 0

b

Noda, Kuby, and Lardy (1954). Creeth (1952). Charlwood (1953). Hoch (1954). Baldwin et ol. (1955). Creeth (1957). Kekwick (1950). (8) Caspary (1954s).

(1) (2) (3) (4) (5) (6) (7)

Kekwick (1952). Shulman (1953). Caspary and Kekwick (1954). Cecil and Ogston (1952). Fox and Dandliker (1956). Gutfreund and Ogston (1949). Cecil and Ogston (1949). Caspary (1954b).

(17) (18) (19) (20) (21) (22) (23)

Ogston (1952). Thompson (1956). Noda and Kuby (1956). Cecil and Ogston (1951). Creeth and Record (1952). Caspary (1953). Bailey, Gutfreund. and Ogston (1948).

524

LOULS J. GOSTING

portant that the procedure used for analysis be stated when reporting data for diffusion coefficients of proteins. I n addition t o the data included in Tables IX and X, other results which were somewhat more complex and less susceptible to simple tabulation should be mentioned. A soluble protein obtained by the degradation of elastin with urea was studied at p H 4 and ionic strength 0.2 by Bowen (1953), using the cylindrical lens schlieren method. Because of the marked increase in diffusion coefficient (and decrease in sedimentation coefficient) with decreasing concentration, it was concluded th at there existed a reversible dissociation. Samples of dextro- and levorotatory polysaccharides prepared from Penicillium luteum were studied by Ogston (1949b) using the GOUY method. For the levorotatory sample, which was found to be polydisperse with respect to sedimentation coefficient, he obtained ( a ) 2 m ) ~ ~ , w=

8.32 X lo-' cme2sec.-'

From sedimentation studies the dextrorotatory sample was known to consist of two main fractions, a relatively homogeneous component and a faster sedimenting fraction which was heterogeneous; this sample was analyzed as a mixture of two solutes by the procedure he had applied earlier to analyze Gouy fringe data for impure samples of lactoglobulin (Ogston, 1949a). Hyaluronic acid complex in synovial fluid from cattle was also studied (Ogston and Stanier, 1950) following this procedure for analyzing mixtures. Insulin samples were studied by Creeth (1953) using the Gouy method; he not only compared the diffusion coefficients of different samples but investigated the effects of temperature, protein concentration, and pH on the results. Evidence for dissociation was obtained. Human albumins, both normal and pathological, have been studied extensively by Charlwood using the Gouy method. Samples from different normal individuals showed small differences in a)A (Charlwood, 1952a); these differences were presumably due to slight variations in the relative amounts of components, since some deviations of the refractive index gradient curves from Gaussian form were observed. The effects of albumin concentration and buffer composition on DA for three samples of material were investigated (Charlwood, 1954a) ; although the buffer (and/or pH) influenced the diffusion coefficient a t finite albumin concentrations, this effect tended to disappear on extrapolation to zero concentration of protein. Studies of albumins from cases of nephrosis indicated (Charlwood, 195213) that the serum albumins generally diffuse more slowly, and the urinary albumins more rapidly, than normal serum

MEASUREMENT AND INTERPRETATION OF DIFFUSION

525

albumin. Albumins from the sera of patients with liver disease were normal with respect to diffusion coefficient (Charlwood, 1954b). Some data for the effect of ionic strength of buffer on D~ for a sample of crystallized bovine plasma albumin have been considered earlier (Fig. 15). Crystallized bovine plasma albumin has been studied at 1' C. by Wagner and Scheraga (1956). Using protein solutions which were 0.5 molar in potassium chloride and which had a pH of 5.14, they obtained values for bDA as a function of the mean protein concentrations, E B p A , over a range of 0.25-1.25 g./100 ml. They found DA to decrease slightly with decreasing protein concentration, extrapolating to (a)Ao)l=

3.261 X lo-' em1.*set.-'

a t EBPA = 0 in this medium. The interferometric optical system described by Scheibling (1950a)b) has been used t o study several proteins. Although this apparatus did not provide data for the refractive index a t all levels in the cell, some information concerning heterogeneity was also obtained (Scheibling, 1950b). Extensive studies were made of ovalbumin (Champagne, 1950, 1951), and data were reported for blood serum (Scheibling, 1951) and serum albumin (Champagne, 1951); the size and shape of serum albumin was considered (Champagne, 1953). Subsequently more extensive results were reported (Champagne and Sadron, 1955) for crystallized bovine plasma albumin. Data for both the diffusion coefficient and intrinsic viscosity were obtained over a range of p H and protein concentration in water and also in salt (and buffer) solutions of various concentrations. Except for protein solutions in pure water, the value of the diffusion coefficient was found t o be relatively independent of protein concentration, but it did vary measurably with pH. An ellipsoid of revolution was assumed as a model, and dimensions of the molecule were calculated a t various values of the pH. A theoretical investigation has been made by Daune and Benoit (1954) of the effects of polydispersity and/or concentration dependence of the diffusion coefficients on data obtained using Scheibling's method. 4. Calculation of Molecular Weights An accurate value for the molecular weight of a solute cannot be obtained from measurements of diffusion alone; only its frictional coefficient can be determined with precision (equation (44)) and even this calculation requires that D be extrapolated to zero solute concentration. Other characteristics of the solute, such as its sedimentation coefficient and partial specific volume, must also be measured to permit accurate evalu-

526

LOUIS J. GOSTING

ation of the molecular weight, equation (52) or (52a). However, some materials of biological interest have molecular weights in the range of several hundred to a few thousand wherein measurements of sedimentation are difficult. For such materials there has been interest in the possibility of using Stokes' law for spheres to obtain rough estimates of M from values of D alone; then if the minimum molecular weight has been determined from chemical analysis the correct value of M can be established. We will therefore review some recent data indicating the degree of validity of this procedure, before considering the accurate determination of M by combination of data for sedimentation and diffusion. o Aliphatic amino acids 38 -

1D

36 -

2

8

x Aromatic and heterocyclic

amino acids

A Peptides 0 Sugars Amino benzoic acids + Sarcosine

X

3 34-

I>

n

28-

4

;

I

I

6

7

I

8 D x lo6

1

I

9

10

11

FIG.24. The product D Vx as a function of D for a number of compounds in water at 25" C. The horizontal line represents the value predicted by equation (54) (Longsworth, 1953). a. Approximate Method Using Stokes' Law. If it is assumed that the solute molecules are unhydrated spheres moving in a continuous medium and that the partial molal volume, P, of the solute provides a reliable measure of molecular volume, we see from equation (54) that DP'.4 should be identical for all solutes in the same medium at a given temperature. For water at 25" C. this constant is 33.3 X 1 P . The inadequacy of these assumptions to describe actual solutes is well illustrated by Fig. 24, which summarizes some data obtained by Longsworth (1953) for the diffusion of several compounds. These data, instead of lying on the horizontal line representing the theoretical value, are seen to be too high for low molecular weights (large D) and too low for high molecular weights. By neglecting data for the aromatic and heterocyclic amino acids, which because

MEASUREMENT AND INTERPRETATION OF DIFFUSION

527

of their different structure might be expected to have properties appreciably different from the other compounds, Longsworth was able to fit the remaining data by the sloping line with an average deviation of only 2%. This line corresponds to the empirical relation discussed previously, equation (60), with values of a’ = 24.182 X and b’ = 1.280. A test of Longsworth’s empirical equation relating M and D (equation (59)) is provided by his data for several amino acids, peptides, and sugars (Longsworth, 1952), as summarized in Fig. 25. Here the scatter is somewhat greater than in Fig. 24, presumably because equation (59) introduces, in effect, the assumption th at all compounds have the same partial 22

21

0

-

20 -

0

o

Aliphatic amino acids x Aromatic and heterocyclic amino acids A Peptides Sugars

X

s5 1 9 n

18 -

2.0

2.5

3.0

3.5

D x lo6

4.0

4.5

5.0

FIG.25. The product D M g as a function of D for a number of compounds in water at 1’ C. (Longsworth, 1952).

specific volume, 6. However, by using values of A’ = 11.66 X and B’ = 1.893 for the constants in equation (59), Longsworth obtained the straight line shown in Fig. 25 which fits the data with a n average deviation of 4.7%. Therefore by using equation (59) rather than equations (54) or (57), i t seems that M may be determined from measurements of D alone with a n average deviation of about 15%, provided the unknown material does not differ greatly in size or structure from the compounds used t o determine the empirical constants in the equation. b. Combination of Sedimentation and Diffusion Data. Some recent refinements and tests of this procedure are of interest and will be the primary subject of the present discussion. The reader interested in numerical results obtained by this method, utilizing equation (52a), can find extensive tables in the literature summarizing data for s, D,6,and M for a large number of proteins (Svedberg and Pedersen, 1940, p. 406; Cohn

528

LOUIS J . GOSTING

and Edsall, 1943, p. 428; Lundgren and Ward, 1951; Pedersen, 1953; Dandliker (see Table VIII of Edsall, 1953)). The difficulty of obtaining homogeneous protein preparations constitutes a primary limitation on the accuracy of available values of molecular weights. Usually the molecular weight of the main component, and not an average value for the sample, is desired. Therefore it is important t o examine both the sedimentation and diffusion data for evidence of impurities and to use methods of evaluation which allow the calculation of s1 and D1 for the main component whenever possible. When refractometric data for diffusion are obtained with an interferometric optical system, sufficient accuracy is available t o provide a fairly sensitive test for the presence of impurities, as in Section IV,Z,a (assuming no interaction of solute flows, see Section V). If only one impurity is present the fringe deviation graph (which summarizes the deviations of the refractive index gradient curve from Gaussian shape) may be used together with the measured average diffusion coefficient, D A (equations (90) and (94)), to obtain D 1for the main component (Section IV,2,b). The accuracy of this procedure is considerably increased if independent data for the amount of impurity are available, such as those provided by a sedimentation analysis. However, even if no impurity is detected in sedimentation experiments, i t is important to analyze the diffusion d a ta for evidence of heterogeneity because (1) a buffer gradient may be present owing to incomplete dialysis, (2) a sample may not be homogeneous with respect t o D even if it is with respect to s, and ( 3 ) the presence of an impurity with a high sedimentation coefficient may not be noticed in a n ultracentrifuge experiment if it sediments to the bottom of the cell while the rotor is reaching normal operating speed. The sedimentation coefficient, sl, of the main component in a sample for which the sedimentation coefficients are independent of concentration is readily evaluated if the observed refractive index gradient curve resolves into separate peaks corresponding to the different solutes; the theory for this case has been presented by Svedberg and Pedersen (1940). If the peak corresponding t o an impurity does not resolve entirely from tha t of the main component, but its “tail” does not overlap the maximum of the main peak, s1 may still be determined directly, provided again that the sedimentation coefficients are independent of concentration. Dependence of the sedimentation coefficients on the concentrations produces a nonideality in the movement of each peak with time which should be considered in precise determination of sedimentation coefficients (Alberty, 1954; Trautman et al., 1954; Fujita, 1956). The per cent of impurity as measured by a sedimentation experiment can be advantageously used in calculating the diffusion coefficient of the

529

MEASUREMENT AND INTERPRETATION O F DIFFUSION

main component, D1,from the measured average value BA (Section IV,2,b). This determination of the per cent of impurity is readily made from the areas of the sedimenting peaks if they resolve and if the sedimentation coefficients are independent of concentration; if they resolve only partially an analysis may still be possible (Svedberg and Pedersen, 1940, p. 296). The latter case is illustrated by Baldwin’s (1957) study of a sample of bovine plasma albumin; he used the resulting data for the per cent of impurity in obtaining a value of ( D l ) z 5 , w= 6.97 X lo-’ cm.2sec.-l a t EBpd = 0.515 g./100 ml. from measurements made earlier (ref. 5, Table X, sample No. R370295A) with the Gouy diffusiometer. When the sedimentation coefficients depend on concentration this dependence alters the relative areas of the peaks and a more complicated analysis is required (Johnston and Ogston, 1946; Trautman et al., 1954). The theory for the shape of such boundaries has not been developed; only those equations for the case of a single solute have been solved (Fujita, 1956). TABLEXI Some Physical Constants of Heteropol y Acidsa

(s0)zs.w

Acid Silicotungstic Phosphotungstic Phosphomolybdic

Anion

1013

(sec.)

S ~ W I Z O ~ O - 4.56 ~ P W 1 ~ 0 4 ~ - 7 3. 66 PM011039-7 2.04

x

Mol. weight of anion

(D”)zs,u x 106 (cm.2 sec.-l)

(c.c.g.7)

4.515 3.701 3.95

0.140s 0.142 0.244

a

+

sed. diff.

formula

2910 2875.1 2860 2909 1690 1706

a All values except u refer t o t h e anion; so a n d DO denote values at zero concentration of heteropoly acid in t h e buffer.

Besides the problem created by the presence of one or more impurities, attention should be given to problems considered above in Section IV,l, such as the extrapolation of data to zero concentration of protein. Because of errors in the classical procedure for converting data from one temperature to another, the sedimentation and diffusion experiments should in accurate work be performed at approximately the same temperature. Experimental tests of Svedberg’s relation (equation (5%)) have been made recently by Baker, Lyons, and Singer (1955a,b) using inorganic solutes with molecular weights in the range 1000-3000. They studied inorganic heteropoly acids which, in spite of their relatively low molecular weights, are sufficiently dense to have sedimentation coefficients comparable to those of many proteins. Their data, which were obtained in an aqueous sodium acetate-acetic acid buffer of pH 4.58 and ionic strength 0.2, are summarized in Table XI. It will be noted that values of s and D

530

LOUIS J. GOSTING

were extrapolated to zero concentration of heteropoly acid in the buffer and converted t o values for water a t 25' C. The measurements were performed a t a temperature near 25" C., sedimentation coefficients being measured with the Spinco model E ultracentrifuge and diffusion coefficients determined with the Gouy diffusiometer. To obtain ij, the partial specific volumes of the anhydrous acids, pycnometric measurements were made using samples which had been dried to constant weight; as these samples were still hydrated, the preliminary partial specific volumes were converted t o values for the anhydrous material by utilizing data for the composition of the hydrates. The resulting values of ij shown in Table XI are not for the heteropoly acid anions, as are the values of s and D, but they should not differ markedly from the values for the ions since only a few ionizable hydrogen atoms are involved. To test whether sufficient electrolyte was present to free the heteropoly acid anions from their counter-ions, sedimentation experiments were also performed a t ionic strengths of 0.1 and 0.3; the resulting values of ( S ) Z ~ , ~agreed within the estimated error of rt 3%.23I n diffusion experiments the silicotungstic and phosphotungstic acids gave values of Yj/e-fJ' (Section 111,2,h) which were constant within 0.2% over 8.5% of the gradient curve; however, this information is inadequate to allow computation of the maximum fringe deviation, Q,,,,. A noticeable drift of the ratio Yj/e-fJP occurred with the phosphomolybdic acid, indicating heterogeneity. Consequently, data for this anion in the last line of Table XI do not provide as satisfactory a test of Svedberg's equation as do the data for the other two anions, even though the measured and formula weights of all three anions are in agreement within the experimental error of the Sedimentation coefficients. It is hoped that the accuracy of sedimentation studies of these or other compounds of known molecular weight can eventually be improved to make possible a more precise check of Svedberg's equation and provide additional information about the effect of the supporting electrolyte. Whenever a suitable solute becomes available, equation (52) should be tested a t finite concentrations where the thermodynamic term is appreciable.

V. INTERACTING FLOWSIN SYSTEMS OF THREE OR MORECOMPONENTS Whenever diffusion occurs in systems containing more than two components, there exists the possibility of some coupling between the flows. This phenomenon is observed in solutions of salts because of electrostatic $8 It should be noted, however, that the secondary charge effect (Svedberg and l'edersen, 1940, p. 27) is not eliminated by increasing the amount of supporting electrolyte.

MEASUREMENT AND INTERPRETATION O F DIFFUSION

531

coupling between the ions, and it is probably also present t o some degree in all solutions of nonelectrolytes, partly as a result of molecular interactions. It is illustrated by the system shown in Fig. 26, which summarizes the initial conditions for a free diffusion experiment with aqueous solutions of two salts, LiCl and KC1; this system is one for which data will be reported below. Although the concentration of KCl is the same in the two initial solutions, A and B , it is found that some KC1 moves upward with the LiCl during free diffusion. Clearly this flow of KC1 cannot be described by the form of Fick’s first law given in equation (lo), which predicts no flow because the concentration gradient of KCl is initially zero. This type of coupling phenomenon has been reported for electrolyte solutions by many investigators (Arrhenius, 1892; Thovert, 1902 ;Osborne and Jackson, 1914; Walpole, 1915; McBain and Dawson, 1934; Burrage

0.2 N KCI

0.35 N LiCl

“P

FIQ. 26. Diagram of the initial conditions for a diffusion experiment exhibiting interaction of solute flows. Here the KCl concentrations (expressed as normality, N , i.e., molarity) are the same in the two solutions, A and B, used to form the initial boundary, while the LiCl concentrations are different. It is found that a flow of KC1 occurs as diffusion proceeds, in contradiction to the elementary form of Fick’s first law: Ji = -Di(aCii/dz).

and Allmand, 1937), and it has also been observed for a few cases in which one solute was an electrolyte and one a nonelectrolyte (Osborne and Jackson, 1914; McBain and Liu, 1931). Doubt concerning the validity of the classical laws for the diffusion of mixed solutes has been expressed by Bevilacqua et al. (1945). The primary purpose of this section is to consider certain experimental flow equations which are adequate for describing diffusion in the general case of a system with any number of components, and to summarize some procedures for measuring, for three-component systems, the diffusion coefficients in these flow equations. Attempts to use equation (10) and the classical equations derived from it (such as the probability integral for the case of free diffusion) to calculate a “diffusion coefficient” for a component in a system with interacting flows can lead only to erroneous or ambiguous results. The importance of using an adequate formulation for describing diffusion in systems of more than two components was stated some ten years ago by Onsager (1945) :

532

LOUIS J. GOSTING

“Very little attention has been given t o quantitative studies of diffusion in systems of more than two components, although this general case of diffusion is inevitably involved in many techniques of electrochemistry on every scale from micrograms to tons. One can understand how the experimental difficulties have deterred investigators. However, a long period of inactivity has probably added a n element of mental inhibition. From sheer weight of tradition the conclusions of relatively primitive theories tend to be accorded some of the reverence th a t is properly given t o practical knowledge. “It is a striking symptom of the cominoii ignorance in this field that not one of the phenomenological schemes which are fit to describe the general case of diffusion is widely known.” A person interested in biological problems will a t once recognize the importance of complicated diffusion phenomena in certain life processes. 1. Flow Equations

a. Experimental Flow Equations. It is important to write these equations in a form such that the diffusion coefficients defined by them do not depend on the concentration gradients of the several components, although they may depend on the concentrations. One-dimensional diffusion in a system containing q 1 nonelectrolyte components (i = 0,1, . . . ,a) may be described by the set of phenomenological flow equations proposed by Onsager (1945),24

+

2 Q

J+=

-

j=b

+

dC.

Dij”d X

(i = 0,l

...

,q)

(146)

where Dtj denote (q 1)2 diffusion coefficients. For convenience of representation later, the solvent, or component present in largest amount, is denoted by 0.25I n the general case Dij # D,%# 0 for i # j. Terms of the form cab (equation (12)) have been omitted because in this section we limit our consideration to systems showing no appreciable volume change on mixing, and all flows are measured relative to the diffusion cell. Consequently equation (13) reduces to the restriction 0

Onsager wrote his equations for the three-dimensional case. This notation is convenient because some subsequent flow equations, which do not include the solvent, may then be written simply as summations over all the solutes, 1, . . , q. It should then be remembered, however, that t h e total number of components remains equal to p 1. 24

26

.

+

MEASUREMENT AND INTERPRETATION OF DIFFUSION

533

Substitution of equations (146) into equation (147) leads t o Onsager's first set of restrictions on the values of his ( q 1)2diffusion coefficients,

2 &Dij

+

a

=

0

i=O

Furthermore, because the relation iiici = 1

(149)

i=O

can be differentiated to give

i=O

when there is no volume change on mixing, he was able t o impose a second set of restrictions 0

+

Equations (148) and (151) contain altogether 2q 1 restrictions on the (q l ) zdiffusion coefficients because they are independent except for

+

+

+

Accordingly there are ( q l ) z- (29 1) = q2 diffusion coefficients to 1 components. By be experimentally determined for the system of q utilizing his reciprocal relations (Onsager, 1931a,b) which follow from the assumption of microscopic reversibility, Onsager (1945) then derived certain additional relations between the diffusion coefficients and the thermodynamic properties of the solution. The testing of these relations constitutes one of the important problems in the field of diffusion. I n the remainder of this section we will use a modified form (Baldwin, Dunlop, and Costing, 1955; Dunlop and Gosting, 1955) of Onsager's flow equations (equations (146)) which seems especially convenient for experimental studies. Because only solute concentrations (or concentration gradients) are ordinarily measured in experimental work, terms containing the concentration gradients of the solvent, component 0, are omitted.

+

534

LOUIS J. GOSTING

The diffusion coefficients, Dij, defined by these experimental flow equations differ from those of Onsager, Dij, in equations (146), but relations between the two sets can be readily obtained. Equations (153) may be obtained from equations (146) simply by eliminating the concentration gradient of the solvent, dco/dx, using equation (150). No equation for JOis included in the set described by equations (153) because this flow is given indirectly by equation (147). This formulation is seen to contain q2 diffusion coefficients in agreement with the requirement derived by Onsager. Most of the following discussion will be limited for simplicity to systerns of three components for which only four diffusion coefficients must be measured; systems containing 4, 5, etc., components and requiring in general the measurement of 9, 16, etc., diffusion coefficients will not be considered in detail. For the three-component case equations (153) become and

in which Dll and Dzz will be called the main d i f u s i o n coeficients and DI2and Dzl the cross-term digusion coeficients. I n general D12# DZl;if, however, Dlz = Dzl = 0, equations (154) and (155) are seen to reduce to the case of noninteracting solute flows (equations (10)). Complete ionization of one or more of the q 1 components to give a total of r 1 kinetic species (molecules and ions) in solution does not necessitate any basic alteration in the above formulation. Here r is some integer greater than q. Although more flow equations, each with a greater number of terms, are in general required because of the ionization, some additional restrictions are imposed by the condition of electroneutrality. Consider, for example, a solution containing r different kinds of ions (i = 1, . . . , r ) in addition to unionized solvent, i = 0. Equations (146) may still be used to describe diffusion of the solvent and the several ions simply by replacing q by r, but in addition to the restrictions imposed by the absence of a volume change on mixing there are also the relations

+

+

and

c i=

1

z,Jz = 0

MEASUREMENT AND INTERPRETATION O F DIFFUSION

535

summariaing the requirement of electroneutrality. Here zi denotes the valence (both magnitude and sign) of ion i, and for this form of the relations ci and Je must be expressed in units proportional to the number of ions of i per unit volume of solution. When the restrictions described by equations (147), (150), (156), and (157) are all imposed on the original set of flow equations (146) (after replacing p by r in each relation), it is seen t ha t instead of equations (153) the relations r-I

Ji

dC.

D..L

= i =1

a3

ax

(i = 1, . . . ,r - 1) (158)

are adequate for describing diffusion in this system containing ions. Therefore diffusion in an aqueous solution containing three kinds of ions (two salts with a common ion, p = 2 and r = 3) can be described by equations (154) and (155), provided that the volume changes on mixing are negligible and that the ionization of the water can be neglected. b. Theoretical Flow Equations. Few experimental data are yet available for testing theoretical flow equations for diffusion in systems of more than two components, and we will not undertake a detailed consideration of these relations here. Only an elementary introduction to some of the basic formulations will be given, with special emphasis on systems containing ions because of their importance to measurements on proteins and to the general problem of diffusion in biological systems. Additional information eoncerning flow equations may be obtained from many different sources, including Onsager (1945), Prigogine (1947), Lamm (1947, 1954), de Groot (1951), Denbigh (1951), de Groot and Mazur (1954), and Hirschfelder, Curtiss, and Bird (1954). A general formulation for nonelectrolyte systems is obtained by expressing each flow as a linear function of all the forces, i.e., as a sum of the chemical potential gradients of the q -4- 1 components multiplied by appropriate coefficientsz6(Onsager, 1931a,b, 1945). J{ =

2

2)

L ~ -~ (

(i = 0,

. . . ,q)

(159)

j-0

a6 I n this elementary consideration we will not distinguish between Ji, the flow of a component relative to the cell, and Ji’, its flow relative to the local center of mass (equation (14)); they become identical in the limit of infinite dilution of all solutes. Furthermore, Onsager and Fuoss (1932, p. 2760) state that the reciprocal relations, equation (160),,apply independently of the frame of reference for the flows. Sometimes the force-flow equations are written as

536

LOUIS J. GOSTING

Restrictions such as the Gibbs-Duhem relationship may then be applied to these relations if desired. Assuming microscopic reversibility, Onsager (1931a,b) obtained the reciprocal relations

L,,

=

(160)

L,,

between the several coefficients. For electrolyte solutions, equations (159) are presumably applicable if q is replaced by r as described in Section V,l,a above, arid if the chemical potential, p i , is replaced by the electrochemical potential per mole

Here as before zj is the valence of ion species j , cp denotes the local electrostatic potential in the liquid, and E is the charge on a mole of electrons (without negative sign). I n order t o consider the predominant charge effect when three or more kinds of ions are diffusing in a solution, we assume that I J , j = 0 for i # j in equations (159). This leaves only the main termsz7

(i = 1, .

..

,r)

(162)

corresponding t o equation (39). Consequently we are neglecting the time of relaxation effect and the electrophoretic effect (Onsager and FUOSS, 1932; Onsager, 1945; Gosting and Harned, 1951), and are considering only the primary charge effects. By letting L,, = c , / ( N f , ) as was done in equations (39) and (40), and utilizing equation (lcil), the approximate equation (162) becomes

Substitution of this relation for each ion into equation (156) allows solu-

-a*i ax

=

2

RjiJi

( j = 0,

...

,q)

(159a)

i=O

This form may be obtained from equations (159) by algebraic solution of that set of relations for the chemical potential gradients in terms of the flows. I n the course of such a computation relations are obtained between the several coefficients Lij and Rii; application of the reciprocal relations, equations (lSO),then leads to

R%. I. - R 1.s $7

(160a)

The flow of solvent, Jo, is omitted from this set; it is determined by equation (147).

MEASUREMENT AND INTERPRETATION OF DIFFUSION

537

tion for the electrical potential gradient,

i=l

If variations of the ionic activity coefficients with the several concentrations are neglect,ed, differentiation of equation (43) yields

Substituting equations (164) and (165) into equation (163), and deJining

we obtain expressions for the ionic flows, which may then be further reduced by eliminating c,. and dc,/dx using equations (157).

( i = l , ..

.

The relation for J7 is omitted, this flow being determined by equation (156). It should be emphasized that f i i defined by equation (166) is not, in general, a measurable quantity; it simply denotes the hypothetical dtffusion coefficient that the ion would have if its charge were zero and if its activity coefficient, yi, were independent of every concentration. Alternatively, B, is proportional t o the ionic conductance or mobility, when the time of relaxation effect and electrophoretic effect are neglected. By using the approximate equations (163) as a starting point, relations for describing diffusion in ionic systems have been derived previously by several different workers including Taylor (1927), Hartley and Robinson (1931)) and Vinograd and McBain (1941). Taylor (1927) considered a case in which the ionic mobilities and activity coefficients were permitted certain variations with concentration. In some formulations attempts were made t o define a single diffusion coefficient for each ionic species in such a way that these “diffusion coefficients” depended on the concen-

538

LOUIS J. GOSTING

tration gradients, and this tended to obscure the linear relation between each flow and the several concentration gradients. Our purpose in considering here the approximate theoretical equations for ionic flows, equations (167), is to aid in interpreting the coefficients in the corresponding experimental flow equations (158). Both sets of relations are seen to have the same mathematical form, the flows being given as sums of r - 1 ionic concentration gradients. They differ in th a t the Coefficients in equations (167) contain the quantities, b,,which are defined theoretically rather than experimentally, whereas the coefficients in equations (158) (or (154) and (155)) were not assigned any theoretical meaning but may be measured experimentally in the same sense that the diffusion coefficient, D , in Fick’s first law for a two-component system may be measured experimentally. By equating coefficients of corresponding concentration gradients in the two sets of equations we obtain expressions for D,,in terms of the quantities in equations (167). For a system containing only three kinds of ions in a single, unionized, solvent, for which experimental flow equations (154) and (155) are applicable, we obtain Dl1 = Dl[l - Z 1 2 C 1 ( D 1 - D3)/x] (168) Diz = -Di[~izzci(D2 - D~)/x] (169) Dzi = -D,[~?zicz(Di - D ~ ) / x ] (170) n 2 2 = D2[1 - Z 2 2 C 2 ( b 2 - D3)/x] (171) where X =

21(XlD1

-

Z3D3)Cl

+

22(2Zb2

- ZQB)J)C~

(172)

D2 = D,, then D Z Z= D2 and D12 = 0 in approximate agreement with data for the LiCl-KCLH20 system reported below in Table XII. If c2 << cl, corresponding t o a very dilute solution of a charged protein (i = 2) in a n aqueous solution containing an excess of a single electrolyte, equations (168)-(172) become

If

D21

0

D22 N D2

Since Dzl N 0, the flow, Jz, of protein ions is independent of any gradient in the supporting electrolyte, as is commonly assumed. However, because D12is not zero, except for the unlikely case th a t b2 = Ds,it is seen th a t the protein gradient will produce some flow of salt, i.e., J1 # 0 when

MEASUREMENT AND INTERPRETATION O F DIFFUSION

539

acl/ax: = 0, unless acz/dz is also zero. Therefore, measured values of S A

or SZ,(equations (90) or (91)) do not necessarily equal &. Some caution should be exercised in drawing conclusions from equations (168)-(176), however, because of their approximate nature. It will be recalled that in obtaining equations (167) the coefficients Li, were arbitrarily set equal to zero for i # j , and any variations of the activity coefficients with concentrations were neglected in equation (165). The inadequacy of the former approximation for the case of trace ion diffusion with cz << c1 has been shown by Onsager’s theory (Onsager, 1945; Gosting and Harned, 1951). Although his treatment indicates, in agreement with equation (175), that Dzl should be zero, it predicts, in contradiction t o equation (176), that for small values of c1 the coefficient Dzz should vary linearly with because of the time of relaxation effect. Some experimental data supporting the validity of Onsager’s theory for trace ion diffusion have been obtained by Wang (1952a,b, 1953) and Wang and Miller (1952) using radioisotopes and the open-ended capillary method. T o test Onsager’s equations further i t is hoped th a t procedures eventually will be developed for obtaining data of this type with an accuracy of 0.1 %.

6

2. The Concentration Distributions in Free Difusion

Only three-component systems will be considered here; furthermore, it will be assumed that the diffusion coefficients may be taken as constant across the diffusing boundary and again that no appreciable volume changes occur on mixing. Expressions for the concentration distributions in more complicated systems have not yet been derived. Substitution of flow equations (154) and (155) into the continuity equations (20) leads to the following partial differential equations.

By utilizing Boltzmann’s (1894) observation th a t the variables z and t always occur in the ratio x / d for free diffusion, the new variable

may be introducedzs to reduce these partial differential equations t o a 2 B Some experimental evidence for the validity of this substitution for the case of interacting flows has been obtained (footnote 19 of Dunlop and Gosting, 1955).

540

LOUIS J. GOSTING

set of ordinary differential equations. They were solved using series expressions for the concentrations (Dunlop and Gosting, 1955) which converge rapidly when one cross-term diffusion coefficient is small. Subsequently, the following exact solutions were obtained (Fujita and Gosting, 1956) which show that each solute concentration consists of the sum of two probability integrals (plus a constant term). c1

= El

c2 =

Fz

+ Kf@(d/.-tv)+ K T @ ( 6 - y ) + K;N%Gy) + K , @ ( 6 - y )

(180) (181)

Here y is defined by equation (179) ; the mean concentrations, E1 and Cz, are givtm by equation (72) ; and @ denotes the probability integral, equaand - on u denote positive and negative tion (77). The subscripts values of the square-root term in the expressions

+

where =

(D22

+ 4D1zDz1

- Dll)z

(184)

If either D I z or Dziis zero it is seen that these expressions reduce (when D2z > Dii) t o (182a)

and g-

=

1

(183a)

--

D 2 2

The coefficients of the probability integrals are given by

.+ -

Kz

-

I<,

=

(Dli D11

(

D22

+ .fi)Ac2

(186) - 2D21Aci -

4 6 - Dzz - d 6 ) A c Z - 2DZlAcl -4 fi

(187)

(188)

It may be seen from equations (180) and (181) that interaction of solute flows does not produce skewness of the concentration gradient curves,

MEASUREMENT AND INTERPRETATION OF DIFFUSION

54 1

provided that the diffusion coefficients are independent of concentration. This has also been shown by examination of the moments of the concentration gradient curves (Baldwin, Dunlop, and Gosting, 1955). Presumably a concentration dependence of the four diffusion coefficients leads to skewness as for the case of two-component systems, equations (75) and (81). A solution giving the concentration distributions for this case would be of interest. Taylor (1927) treated the case of solutions containing three kinds of ions without assuming that the coefficients corresponding to Dll, D12, 0 2 1 , and D22 were constant. However, his equations were sufficiently complex so that only one correction term to the simple Gaussian function for free diffusion could be easily evaluated. Furthermore, approximations in his solution prevented the calculated values of the three ionic concentrations from satisfying the condition of electroneutrality. 3. Determination of the Four Diffusion Coeficients for a Three-Component System

The refractive index distribution for free diffusion in a three-component system with interacting flows is obtained (Fujita and Gosting, 1956) by substituting equations (180) and (181) for c1 and c2 into equation (86) giving n = n ( ~ & ( A n / 2 > [ r + N d & y ) r - ~ d G ) i (189) where r+ = (a/An)(RiKf 4-&Xi) (190) and r- = ( 2 / A n ) ( R l K T &K;) (191)

+

+

+

For systems containing three kinds of ions we let c1 and c2 denote concentrations of the two neutral salts, in order that equation (86) be applicable and the differential refractive increments Rl and Rz be susceptible to straightforward experimental measurement. Differentiation of equation (189) shows that anlax is simply the sum of two Gaussian curves. Therefore, provided that the diffusion coefficients and the differential refractive increments are independent of concentration, the refractive index gradient curve for diffusion in a three-component system is the sum of two Gauss error curves regardless of whether the solute flows interact or not. The two cases do differ, however: with no interaction of the flows anlax is Gaussian when either Ac, = 0 or Acz = 0 ; if the flows interact an/& is in general non-Gaussian both for Acl = 0 and for Ac2 = 0, though it may be Gaussian for some other values of Acl and Acz. It should be noted th a t from measurements of the refractive index (or refractive index gradient) distribution alone, without some knowledge of the concentration differences, it is impossible to determine whether the solute flows interact.

542

LOUIS J. GOSTING

Procedures for determining the four diffusion coefficients have thus far utilized the Gouy interference method; at least one of them should be readily applicable to the Rayleigh interference method.29Because a single experiment does not provide enough data to permit determination of all the coefficients with reasonable accuracy, the procedure has been adopted of performing two or more experiments (preferably a t least four) with different values of the refractive fraction, a1 (equation (88)). The same value of El, and also of C 2 , is used for each experiment of this group to minimize complications arising from concentration dependence of the diffusion coefficients. Data for the reduced second and fourth moments, Dzrnand Dim (equations (91) and (92)), for two or more of these experiments may be used to evaluate the four diffusion coefficients (Baldwin, Dunlop, and Gosting, 1955). Equations (180) and (181) for the concentration distributions are not required. This procedure has been applied to measurements on the systems LiC1-KCl-H20 and LiCI-NaC1-H20 (Dunlop and Gosting, 1955). However, the resulting diffusion coefficients are less accurate than those computed by the following methods because values of Dimare subject to greater experimental errors than are values of DZmand DA. Measurements of aZrn and the reduced height-area ratio, a ) A , for two or more experiments (preferably a t least four) can be used to evaluate the diffusion coefficients (Fujita and Gosting, 1956). Again the experiments are arranged so that a1 is varied while El and E 2 are kept unchanged. is linear From equations (90) and (189) it may be shown that I/* in all = IA SA~I (192)

+

Here Sd is the slope and I A denotes the intercept a t

LA

= IA

a1 =

0, while

fS A

(193)

is used to represent the intercept a t a1 = 1. From the theory for the moments of the refractive index gradient curves (Baldwin, Dunlop, and Gosting, 1955) it may be shown th at D2m is also linear in all Dzm =

I2m

where Szm is the slope, 12m

is the intercept a t al L2m

=

=

=

0 2 2

+

+

S2ma1

(194)

(R1/&)D12

(195)

0, and I2m

+ S2m = Dll + ( R ~ / R I ) D z I

(196)

29 It may be that the diaphragm cell will be found well suited for studies of interacting flows, especially for determination of cross-term diffusion coefficients.

543

MEASUREMENT AND INTERPRETATION O F DIFFUSION

denotes the intercept a t a1 = 1. T h a t straight lines are obtained in practice, in agreement with equations (192) and (194), is illustrated by data

dz

FIG.27. The linear dependence (equation (192)) of 1/ on the refractive fraction of LiCl, a1,for the two systems for which data are summarized in Table XI1 (Fujita and Gosting, 1956).

for the two systems LiC1-KC1-H20 and LiC1-NaC1-H20 (Dunlop and Gosting, 1955) shown in Figs. 27 and 28. The relation

(.\/m)3 +

[XZm

-

I A ( s 2 n ~ / * T A ) ] ( d ) 1 ) 1 1 1 )~

(SZm/SA)'

=

0

(lg7)

was derived; using the observed intercepts and slopes it may be solved for I D,, I , where (198) lDtJ1 = DiiDzz - Di2D21 With I D,I known, the main diffusion coefficients may then be computed from the relations and

Dii

=

D22

=

+L

-[/02j1

[lD,l -k

Izm

2m

\/lDJ

+

+

L2mIAS2m/SAI/S2m

I2&ASZm/SA]/SZrn

(199) (200)

544

LOUIS J. GOSTING

The cross-term diff usion coefficients are readily obtained by substituting these values into equations (195) and (196). Data for DA and the Gouy fringe deviation graphs for the experiments may also be used to determine the four diffusion coefficients (Fujita and Gosting, 1956), if up/.+ is sufficiently close to unity for equation (141c)

" In

0 X

E

Q*

a, FIG.28. The linear dependence (equation (194)) of axmon the refractive fraction of LiCl, a,,for the two systems for which data are summarized in Table XI1 (Fujita and Gosting, 1956).

to be applicable. 30 This procedure probably yields slightly more accurate values for the diffusion coefficients than the procedure utilizing and Dn, but i t is less general and the computations require a procedure of successive approximations. T o illustrate the magnitude of D,,, D12, D Z 1 and , D Z 2for one composition of each of two representative electrolyte systems, the data in Table XI1 are presented. As indicated, one set of values was computed by the 3 0 To generalize equation ( 1 4 1 ~ and ) other relations for systems without interacting flows so that they describe three-component systems with interacting flows one simply replaces a l , 012, l / D l , 1 / D % ,and D1/Dx in the former relations by r+,I?-, u+, U-, and U-/U+, respectively (when D Z Z> D l l ) .

545

MEASUREMENT AND INTERPRETATION O F DIFFUSION

method utilizing DA and Bzm;the other set was computed from the values of BAand the Gouy fringe deviations (denoted for brevity as "&graphs"). Values for the ratios R l / R 2 ,which are required for determining D l z and 0 2 1 , were obtained from the auxiliary refractometric data (using equation (109)) for experiments with a1 N 0 and 011 N l. The subscript l denotes LiCl; 2 denotes KC1 or NaC1. It will be observed th a t for a system with only three kinds of ions the subscripts on the flows and concentration gradients, equations (154) and (155), may denote either the two neutral salts or the two ions whose charges have the same algebraic sign. TABLE XI1 Values of the Four Difusion Coeficients for Each of Two Representative Systems with Interacting Flows" T = 25' C.; c.g.s. units

Method of calculation 9.4 and %,,, Daand O-graph

-------LiCl-KCl-€I~O~-------LiCl-NaCI-H2@-= C ~ i c i= 0.25d 6, = CL;ci = 0.25 = C K C l = 0.2 = C"aC1 = 0.2 Dll X D I X~ D21 X D22 X D I I X D12 X D21 X 0 105 105 105 105 105 105 105

c2

e,

1.154 -0.002 1.134 -0.001

0.19$ 1.812 0.215 1.811

1.113 O.OS1 0.185 1.098 0.100 0.198

2 2

X

106

1.369 1.351

The experimental measurements (Dunlop and Gosting, 1955) were made with the Gouv diffusiometer and results were calculated using the methods presented by Fujita and Gosting (195F). b Cross-term diffusion coefficients were computed using RI ~ R =P 0.9080. c Cross-term diffusion coefficients were computed using R I J R , = O.9OOa. d Concentrations are expressed in moles of solute per liter of solution.

I n the latter case the common ion (Cl-) is denoted by i = 3. It is seen that calculations made by the two methods yield the same diffusion coefficients within 0.02 X Both sets of results for Dij predict values of a ) and ~ 92, t ha t agree with most of the observed values within approximately 0.1 %; the results obtained by the method utilizing values of and the S2-graphs predict fringe deviation graphs which are in reasonable agreement (Fujita and Gosting, 1956) with those obtained experimentally. T ha t D12should be approximately zero for the LiC1-KC1-Hz0 system is predicted from equation (169), because the ionic conductances of K+ and C1- (and hence fi2 and D3) are approximately the same. Similarly, from equation (171) we would expect that for this system D Z ZN I%, where fi, may be approximated by the diffusion coefficient for KC1 diffusing as a single electrolyte in water; a t 25' C. and ionic strength 0.45 this value is 1.846 x (Gosting, 1950), in fair agreement with the value of DZZ in Table XII. Because equations (168)-(172) are approximations at best, no attempt will be made here to interpret the other

546

LOUIS J. GOSTING

values in Table XII. It is hoped that a more rigorous theoretical treatment will eventually make such interpretations possible.

4. Some Applications to Biological Systems An understanding of the problem of “interacting flows,” i.e., a n understanding of a general formulation for treating diffusion in systems of more than two components, is basic to measurements of the diffusion of proteins and to most problems of diffusion in biological systems. It should be recognized, however, that our knowledge of this subject is still incomplete and more research in the field is urgently needed. Studies of the diffusion of proteins should take into account the fact that coupling may occur with flows of other ions or components (equations (153)-(155)). The procedure of adding a n excess of supporting electrolyte has apparently been adequate t o suppress the primary charge effect in earlier measurements. However, further investigation is needed to establish whether added electrolyte will free the protein ions and make the cross-term diffusion coefficients negligible within the accuracy of modern measurements. It will be recalled that the standard procedure of dialyzing a protein solution against a buffer prior to diffusion ensures that the electrochemical potentials, but not necessarily the concentrations, of the dialyzable ions will be the same in both solutions. These initial salt gradients may just balance the tendency of a (charged) protein to produce flows of the other ions as mentioned above in connection with equation (174), but this should not be assumed without further evidence. Additional complications may also be introduced because proteins are weak, rather than strong, electrolytes. Pending further clarification of these problems, it is important that precise data from diffusion experiments on proteins be reported in some detail, including information about the initial conditions for diffusion, the buffer composition, pH, the type of “diffusion coefficient” reported (DA,a)2m,etc.) and all available information about deviations of the refractive index gradient curve from Gaussian form. The transport of ions and other substances through the walls of living cells is a process in which diffusion undoubtedly plays a n important part. Although no attempt will be made here to review this subject, it seems appropriate t o point out a few implications of the phenomenon of interacting flows which may relate to this problem. The form of equations (153)-(155) summarizes the fact that even in simple systems some ions can flow against their concentration gradients, provided that other ionic concentration gradients are present to provide the “driving forces.” For example, the da t a in Table XI1 indicate that for the LiCI-KC1-H20 system with = 0.25 molar and 6 K C I = 0.2 molar a concentration gradi-

e,,,:

MEASUREMENT AND INTERPRETATION O F DIFFUSION

547

ent of LiCl is a t least Mo as effective in producing a flow of K+ as is a concentration gradient of KC1. Therefore, for this case flows of Li+ and C1- can move K+ against its concentration gradient until the (negative) K+ gradient becomes a t least ).io of the Li+ gradient. It is seen from Table XI1 that this coupling effect is somewhat greater for the LiClNaCl-HZO system. This phenomenon may be of some importanpe in transport through cell membranes. Electric potentials, usually known as liquid junction potentials, are produced when ions diffuse. Consideration of this subject is also beyond the scope of this review, but its application in connection with electric potentials in cells and in nerves deserves mention. An understanding of diffusion in ionic systems is necessary for its description. Treatments of some electrochemical aspects of liquid junction potentials have been given by Taylor (1927), MacInnes (1939), and Harned and Owen (1950). ACKNOWLEDGMENTS The author is indebted to Drs. R. L. Baldwin, P. J. Dunlop, and J. W. Williams for many helpful discussions and suggcstions during the preparation of this review. He is also grateful for the continued generous support of our program in diffusion by the Research Committee of the Graduate School from funds supplied by t h e Wisconsin Alumni Research Foundation.

APPENDIX Since this review was written, a limited number of experiments have been completed which provide additional information about the phenomenon of interacting flows in the diffusion of systems containing more than two components. From these data it seems probable th a t this phenomenon is more general than previously supposed, and th a t i t exists to a measurable degree not only in solutions containing ions but also in other systems with appreciable concentrations of two or more solutes, including those composed entirely of nonelectrolytes. If this is correct, the solute flows uncouple completely only in the limit of infinite dilution of all the solutes. Until this question is resolved by further experiments, the procedures which are described in Section IV,2 for investigating solute purity and analyzing mixtures should be used with some caution. Tests of these procedures (Section IV,2,a) have shown them to be valid within experimental error for some dilute solutions, including the diffusion of a relatively dilute solute containing a small amount of impurity. However, these tests do not necessarily indicate that the procedures are valid for the diffusion of a n impure protein in a buffer solution, unless the buffer solution is quite dilute; the use of a dilute buffer may of course lead t o difficulties because of electrostatic charges on the protein. It is recommended that readers intending to utilize procedures discussed in Section

548

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IV,2 for investigating t.he purity of proteins should first consider in some detail the subject discussed in Section V.

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