Sedimentation of molecular solutions in the ultracentrifuge

Sedimentation of Molecular Solutions in the Ultracentrifuge I. Equilibrium Phase Behavior W I L L I A M R. ROSSEN, 1 H. T E D DAVIS, AND L. E. SCRIVEN Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455

Received September 16, 1985; accepted November 18, 1985 A theory is presented for multicomponent, multiphase chemical equilibrium in a gravitational or centrifugalfield. Bulk fluid in the fieldis stratified; the degreeof stratificationdepends on how free energy and density of the material depend on composition and pressure. Nonideal solutions of low-molecularweight species can have the sharp composition gradients in the field that are characteristic of solutions containing dense salts and colloids. Becauseeach phase in the field is not homogeneous, but is stratified in pressure and composition, the number and compositionof phasesin a strong fieldcan differstrikingly from those in Earth's gravity. The phase rule does not apply to a system so stratified. Metastable states are possible in the field as well. All of these features are illustrated by the behavior of simple model nonideal solutionsand, in two cases,by experiments.Implicationsfor interpreting ultracentrifugestudies of phase behavior and fluid structure are discussed. © 1986AcademicPress,Inc. 1. I N T R O D U C T I O N

The ultracentrifuge is widely used to separate interspersed phases (1-10) and to determine colloidal particle mass from field-induced equilibrium composition gradients within fluid phases (4, 11-20). Both applications assume that equilibrium stratification in the field depends primarily on the mass and buoyant density of the species present. When this is true, macroscopic particles of dispersed phase sediment in a field too weak to induce stratification within the phases. Likewise, a solution containing colloidal particles can stratify in composition in a field too weak to affect smaller molecules, and the mass of the colloidal particles can be inferred from their equilibrium composition gradient. If a molecular solution is sufficiently nonideal, however, sharp concentration gradients like those in colloidal systems can develop even if none o f the species in solution is large or of high molecular weight. The nature of the 1Presentaddress: Chevron Oil Field ResearchCo., P.O. Box 446, La Habra, Calif. 90631.

nonideality has to be that large differences in composition cause but small changes in the chemical potentials of all of the components in the solution. In such cases, interpretation of these gradients in terms solely of colloidal particle mass leads to false conclusions about solution structure. Field-induced stratification in nonideal solutions can be so great, in fact, that the sample in the centrifugal field contains interfaces not present in the same sample at l g: then identifying of the n u m b e r of equilibrium phases in the field with that at lg leads to false conclusions about phase behavior. It has been noted previously that gravity induces stratification and distortions in phase behavior (21, 22), especially sufficiently near a binary plait point (23-43), and that there is potential for enormous distortion in the ultracentrifuge (26, 33, 34, 41). There has been no general study o f m u l t i c o m p o n e n t , multiphase equilibrium in external fields such as gravitational and centrifugal fields, however. Toward such a study we investigate here the therm o d y n a m i c theory of stratification and phase behavior of multicomponent mixtures in an

248

0021-9797/86 $3.00 Copyright © 1986 by Academic Press, Inc. All fights of reproduction in any form reserved.

Journal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

PHASE

BEHAVIOR

external field. We illustrate the predictions of theory with simple solution models and, in two cases, confirm these predictions by experiment. 2. T H E R M O D Y N A M I C

[f°(T, n(O) + N ni(OMi¢(Old35 [ll i=1

H e r e f ° is the field-free Helmholtz free energy density of homogeneous bulk fluid at the local temperature T and composition n, n(r) is the set of m local component molar densities, Mi is molecular weight, and ~(r) is the local value of the external field potential. In a uniform gravitational field, ¢ = ¢o + gh where g is gravitational acceleration, h is height, and ~0 is an arbitrary datum. In a centrifugal field, ¢ = ¢0 - ~02r2/2, where o~ is angular velocity and r is the distance from the axis of rotation. An external field can also be produced by linear acceleration of the system. Equation [1 ] is valid if the free energy arising from component density gradients is negligible. Except in interfacial zones this is generally the case in realizable centrifugal fields. In an interfacial zone Eq. [ 1] must be modified (4449). Very near a plait point the free energy depends as well on coupling between the external field and the correlation function of the fluid (40, 48, 49). These complications are discussed in Appendix lB. We further assume that the external field is much stronger than the capillary forces which act to minimize interfacial free energy. The mass distribution at equilibrium is that which minimizes the free energy while satisfying mass-conservation constraints

Ni= f ni(Odar, i= 1. . . . ,rn. ,J v

The condition of stationarity (50), i.e., that the first variation of F vanish, subject to Eq. [2], yields the equilibrium condition, that the chemical potentials tz; be uniform throughout the system (21):

THEORY

The Helmholtz free energy of a volume of stratified bulk fluid in an external field consists of two contributions. One is determined solely by local values of density and temperature and is independent of the field; the other is the aggregate of the local potential energy density of the fluid in the field: f =

249

IN ULTRACENTRIFUGE

[2]

#i = #°(T, n(0) + MN'(O, i = 1. . . . .

m.

[3]

Here as in Eq. [ 1] there are two contributions. One is the field-free chemical potential #/0 = (Of°/Oni),j and the other is the field potential energy. The field-free equation of state f 0 thus governs the equilibrium composition profile n(0, and the composition profile as a function of ~ can be computed if f °(T, n) and appropriate boundary conditions are specified. Conversely, an experimentally determined equilibrium composition profile reveals the field-free chemical potentials if f °(T, n) is not known, and the equation of state can be fit to these data. The stability of a composition profile which satisfies Eq. [3] is determined by the second variation (50) of F. Because the external-field contribution to the free energy in Eq. [1] is linear in the component molar densities, thermodynamic stability is governed by the fieldfree function f°(T, n). Stability therefore requires that the Helmholtz stability matrix be positive semidefinite at every local composition in the system (21):

/ 02fo\ ~nrt~-~0nffn >~ 0.

[4]

Here the i, j elements of the stability matrix

02f°/OnOn is 02f°/OniOnj and 6n is any perturbation in n. If Eq. [4] is violated then at equilibrium there must be phase separation. A system at equilibrium in an external field may consist of distinct phases separated by interfaces that can be modeled as discontinuities in composition. In this case of piecewise continuous n(O , minimization (50) of the free energy leads again to Eq. [3]. In addition, because external potential changes negligibly across an interface of negligible width, the condition of equilibrium at an interface in the field is exJournal of Colloid and Interface Science, Vol. I 13, No. 1, September 1986

250

ROSSEN, DAVIS, AND SCRIVEN

actly the condition in the field's absence: the field-free chemical potentials of each component on either side must be equal. Thus the compositions on either side of an interface correspond to a field-free pair of phases and are represented by a tieline in the field-free phase diagram. We define the number of phases here as it would conventionally be determined: the number of interfaces minus one. A phase is a region unbroken by interfaces. Thus, in contrast to the field-free case (51, 52) (and to some usage (22), including Gibbs (21)) a phase as defined here is not necessarily homogeneous. A stratified sample at metastable equilibrium in the field is stable to sufficiently small fluctuations in density, but not to nucleation of an entirely new phase. It may include stable and metastable, but not unstable, local compositions. Although a sample at metastable equilibrium can have stable local compositions, no sample at globally stable equilibrium can contain metastable local compositions because then the free energy could be lowered by splitting the local metastable compositions into separate phases. In a stable equilibrium, density p is a monotonically decreasing function of field potential ~. Otherwise potential energy could be reduced without changing the field-free free energy by rearranging strata in order of decreasing density. The composition of any particular component (e.g., the densest or the lightest) need not be monotonic in ~, however. Often it is convenient to replace intensive variables T a n d n in Eq. [3] with temperature T, pressure p, and m - 1 mole-fraction compositions xi, Ui = I*°(T, p(r), x(r)) + Mi~(r), i= 1,...,m,

[5]

or temperature, pressure and m - 1 volume fractions ~i u~ = ~,°(T, p(r), 4,(0) + M ~ ( O , i = 1. . . . .

[6]

113, No. 1, September 1986

[7]

imply the condition of hydrostatic equilibrium (22) dp = -pd~P. [8] The set of pressures and compositions dictated by Eq. [5] or [6] trace a continuous path on a pressure-composition phase diagram like Fig. 1. (Similar paths could be traced on molardensity or volume-fraction composition phase diagrams.) Interfaces in the centrifugal field correspond to tielines on the field-free phase diagram. Any path which lies entirely in the stable region of the phase diagram is globally stable as shown in Appendix 1C; paths which enter the metastable region are metastable; and those which enter the unstable region are unstable. The phase diagram of a compressible system depends on pressure, but that of an incompressible system with zero volume of mixing is independent of pressure. The pressure-explicit form of Eq. [5], #i = #°(T, p0, x(0) p(r) + O,(T, p'(r), x(O)dp' + M,~, jpo

f

i= 1,...,m,

[9]

is, for such a system, #i = #°(T, pO, x(0) + vi(P(O - pO) + M tp(O ' i= 1,...,m,

[10]

where p0 is a datum and Oi is partial molar volume. (What Kwon et al. (39) call the "equation of state" in an external field is Eq. [9] recast as a relationship between p and x on a single composition path.) Pressure is eliminated from Eq. [10] by subtracting ~i/Umtimes the mth equation from the rest to obtain, for given T, a set of m - 1 equations for x(0: [ . t i - (~i/~m)[.tm

m.

Uniformity of chemical potentials #j in t h e field and the Gibbs-Duhem relation, Journal of Colloid and Interface Science, Vol.

~, ni(d#°)T = - d p

=

~0(X)

--

(~i/Om)[,tOm(X)

+ (34,- - (~3iP3m)Mm)~l'(O, i= 1,...,m-1.

[111

251

PHASE BEHAVIOR IN ULTRACENTRIFUGE COMPOSITION ~ PATH

1

~

FIELD

APPEARANCE OF SAMPLE

REPRESENTATION IN (p,x) SPACE

INCOMPRESSIBLESYSTEM: PHASEDIAGRAM INDEPENDENT OF PRESSURE

INTERMEDIATE

LIGHT

~I

:OMPOSITION PROFILE OF SAMPLE

ENSE

INCOMPRESSIBLE SYSTEM: REPRESENTATIONIN x-SPACE

FIG. 1. Equilibrium distribution of mass in the field and representation on the phase diagram.

The composition path is projected from pressure-composition space to a single composition phase diagram as in Fig. 1. Plotted on a single phase diagram the stable paths of different samples may overlap but they do not cross, as shown in Appendix 1D. Paths of real mixtures can cross but tend to be nearly parallel. Thus while experimentally determined equilibrium composition profiles reveal the changes in field-free chemical potentials #0 along the corresponding paths on the phase diagram (see Discussion below), they cannot reveal the differences in #0 between these paths. Those must be inferred from the fit of an equation of state to the data. For an incompressible, dilute, ideal solution Eq. [ 10] is

#i

=

+ n T l n x i Jr Mi(1 - rio)f,

[121

where vi 17i/Mi is partial specific volume. Equation [12] can be used to determine the molar mass of colloidal particles in ideal solution: R T ln[xi( f")/xi( f') ] Mg = [13] =

~zp)(¢' -

mi =

R T ln['Yi(Xi( fn) )xi( fn)/'Y i(Xi( f ' ) )Xi( f ') ] (1 -

~io)(¢'

¢")

-

if")

[14] Differentiating the equilibrium conditions (Eq. [3], [5], or [6]) with respect to ¢ yields an equivalent set of equations which relate the composition gradient to the volume and freeenergy properties of the mixture. With the natural logarithms of the rn - 1 mole fractions as composition variables the result is 0t~° 0 In x - - - - + M * 01nx 0f

u°(T, pO, xi = 1)

(1 -

Small deviations from ideality are accounted for with the activity coefficient 3% defined by g°(T, p, x) = #°(T, P, xi = 1) + RTln(3,ixi):

01nx_ 0~--

=0

( 0]./.0 ~-1 \0~-~nx} M*,

[151

where the i, j element of the (m - 1) × ( m - 1) matrix 0#°/0 In x is 0g°/0 In xj; the vector 0 In x/0ff contains elements 0 In xi/O¢; the (m - 1)-length vector M* contains the product Journal of Colloid and Interface Science, Vol. 113,No. 1, September1986

252

ROSSEN, DAVIS, AND SCRIVEN

of molecular weight and buoyant density of species in solution M* = Mi(1 - ~p).

[16]

In general, solution properties ~i and p depend upon pressure and composition. In an ideal solution 0t~°/0 In x is R T times the identity matrix. Equations [15] and [ 16] can be applied to equilibrium aggregates like micelles rather than their molecular constituents if it is more convenient to do so. In that case additional equilibrium conditions relate the local concentrations of monomer and aggregate. Equations [15] and [16] reveal the causes of sharp equilibrium composition gradients: large solute or particle mass Mi, large (positive or negative) buoyant density (1 - v i p ) , and solution nonideality that causes the matrix 0u°/0 In x to be nearly singular (i.e., to have an eigenvalue near zero). At a plait point or critical point 0g°/0 In x is singular: there, according to Eq. [15], the composition gradient is infinite and the composition path is tangent to the phase boundary on the phase diagram. When the gradient in a real mixture becomes extreme, Eq. [15] must be modified (44-49). Sharp equilibrium composition gradients in ideal solutions of colloids indicate that colloidal particle mass is large. But it is plain that nonideal molecular solutions can have sharp

gradients too, especially near a plait point. On the other hand, if the buoyant densities of all components are zero, the external field can induce no stratification in the sample. Analogs to Eq. [3], [4]-[6], and [ 15] for different forms of the field-free equation of state are listed in Table I. 3. SEDIMENTATION EQUILIBRIUM IN MODEL NONIDEAL SOLUTIONS

Equilibrium behavior can be complex even in simple nonideal molecular solutions. Here we select for illustration binary and ternary incompressible strictly regular solutions (52, 53), in which mixing is ideal and enthalpy is the sum of pairwise interactions. Model parameters employed are listed in Table II. The equilibrium and mass-conservation conditions, Eqs. [2] and [11 ], determine equilibrium in the external field. For mass-conservation purposes, we assume the sample is contained in a sector-shaped cell as in an analytical ultracentrifuge. The extent of stratification and thus of distortions in phase behavior depend on the difference in field potential/xff ~ gAh across the sample and not on field strength g per se. If deep enough, a sample can stratify significantly at equilibrium in a field as weak as lg. Typical values of Aft fn ultracentrifuge experiments are listed in Table III.

TABLE I Equilibrium and Stability Conditions in an External Field

Intensive variables

Field-frec equation of state

Necessary condition for equilibrium

Necessary condition for stability

o2fox

(o2io)

Molar densities

f°(T, n)

~ = #°(T, n) + Mi~

~n~[~--~) ~n >~ 0

Pressure and molefraction composition

g°(T, p, x)

/zi : #°(T, p, x) + Mi~P"

6x~o--~x)rX ~ 0

Pressure and volumefraction composition

g*(T, p, gp)b ~ = frO(T, p, 4~) + MAb c

[ 02g O~

\OnOn] d ~ + M = O [ 02g O~ d x , ~ Op = 0

[O-~xl -~ ~- v ~x

2.

\OdpOCb]

Equation for composition gradient

o

( O2g* ] d4~ Op + - : \0-~1 ~ 04J

o

Note. The i, j, elements o f cgf°]OnOnare c92f°/OniOnj, etc. M is the set of molecular weights and ~ is the molar volume of the solution. a Incompressible system: t~j(x) 0 - (~i/~m)/Z°m(X)+ ( M j - (~)i/~m)M,,)~P= constant. b If4~i ~- Ni~i/(Z~l Ni~j) with the ~j a set of standard-state molar volumes, g* = g°/(~7=l xj~j). c Incompressible system: (Og*/O(ai) + (p~ - p,,)¢, = constant. Journal of Colloidand InterfaceScience, Vol. 113, No. 1, September 1986

PHASE BEHAVIOR IN ULTRACENTRIFUGE TABLE II METASTABLE

I.O

Solution Models and Parameters Binary strictly regular solution

T

253 PLAIT POINT

METASTABLE 2.0

0.8

2.5

g°/RT = xlln xl + x21nx2 + (A/RT)xlx2

R-T

Parameter values ~ = ~z = 200 cm3 gmol -~ Mt = 250 M2 = 150ggmo1-1 A[RT as specified

I 0,6

3.33 r

~

SPINODAL ~ l I

0

Ternary strictly regular solution

0.5

1.0

MOLE FRACTION, xl

g°/RT = xlln xl + x21nx2 + x31nx3 + (A/RT)xlx2

FIG. 2. Binary regular solution: field-freephase diagram.

+ (B/RT)xIx3 + (C/RT)x2x3

Parameter values Vl = v2 = v3 = 200 cm3 gmol -l MI= 150 M2=250 M3=200ggmol A]RT, B/RT, C/RT as specified

Binary regular solution: two phases from one. T h e binary strictly regular solution has a miscibility gap if A exceeds 2R T. Its tempera t u r e - c o m p o s i t i o n phase diagram (Fig. 2) illustrates features typical o f binary solutions with an upper critical solution point: the binodal o f coexisting phases, the spinodal envelope enclosing unstable compositions, the region o f metastable compositions between binodal and spinodal curves, and the plait point, or critical point, where they are tangent. (The phase diagrams o f real mixtures are o f course very rarely symmetrical.) The effect o f the external field is to stratify the sample, i.e., to produce a radial concentration gradient. O n the phase diagram the composition profile is

a horizontal line; several examples are shown in Fig. 3. In the first case (a) T > Tc; there is stratification in the field but no interface. In the second case (b) a mixture that would display one interface and thus two phases outside the field shows stratification as well as an interface in the field. In the third case (c) a mixture that consists o f a single phase outside the field stratifies in the field a n d forms a new phase f r o m a supersaturated layer. The fourth case (d) is like the third except that the supersaturated layer remains metastable as a single phase. Centrifugation distorts the field-free diagram as shown in Fig. 4. Here the phase envelope in the external field encloses c o m p o sitions which have two phases or one interface in a given field. Compositions between the phase envelope a n d the dashed line are metastable as one phase in the field. T h e phase envelope in an external field differs strikingly

TABLE III Typical Values of External-Field Potential Difference in Sedimentation Experiments A~p/RT(gmol g-~)a

Experimental conditionsb

0.001 0.01 0.04

Analytical ultracentrifuge (AN-D rotor), 17,000 rpm Analytical ultracentrifuge (AN-D rotor), 55,000 rpm Preparative ultracentrifuge (SW 50.1 rotor), 50,000 rpm

a Assuming T = 300 K. b Sample rotors are manufactured by Beckman Instruments, Inc., Palo Alto, Calif. For a given rotor, A~p/RTis proportional to rotational velocity squared. Journal of Colloid andlnterface Science, Vol. 113, No. 1, September 1986

254

ROSSEN, DAVIS, AND SCRIVEN a) OVERALL CELL COMPOSITION SAMPLE:

I

~

T/To

SAMPLE:

T/T¢

11

X1

d)

T/T¢

;/~'N~

~

T/To~

(mef~s~lable)

Xl

X1

FIG. 3. Binaryregular solution: sample compositionprofileson the phase diagram.

from that outside the field, increasingly so in stronger fields. Thus can a centrifugal field make two phases from a one-phase binary mixture.

Tables IV and V describe a hypothetical determination of phase compositions in the binary regular solution by equilibrium centrifugation. As shown there, phase compositions

-•-=0.01

'~" = 0.001 1.0

T

0.8

//,,'

0.8

~!I/~

~1

0.1~

x

2.5

~

i/

,

'/ ,'

ii/J'"'

~ ~ll ~

/

\/

0.5

/ 2.0

o,,

i tll I

1.0

0.5

MOLE FRACTION, x,

MOLE FRACTION, x, ~ T = 0.04

1.0

2.0 /

T___

0.8

//

Tc

/ ; /

0.6

\ ~~W - 0_

/ / I

\\

2.5 \

A RT

I 3.33

I

0

0.5

1.0

MOLE FRACTION, x,

FIG. 4. Binaryregular solution: phase envelopein external field. Journal of Colloid and InterfaceScience,

Vol.113,No.I,September 1986

1.0

255

PHASE BEHAVIOR IN ULTRACENTRIFUGE

determined by ultracentrifugation, like phase counts, differ from those outside the field and depend on field strength (Table IV) and initial composition (Table V). This dependence of inferred phase composition on external field strength has been reported (5, 8, 15), but not always understood, in the literature. Phase compositions in Tables IV and V are inferred from (1) mass balance, given interface position and one field-free coexisting composition, and (2) removal of each phase and chemical analysis of the separate phases. The coexisting compositions at an interface do not vary (except as they would under compression), but overall phase composition, which reflects stratification within the phase, does vary with field strength. Figure 5 shows the difference between topend and bottom-end compositions in a sample at equilibrium in the field as a function of field strength and nearness to critical temperature. Sufficiently near the critical point even weak fields cause noticeable stratification; strong fields generate steep concentration gradients, characteristic of high-molecular-weight solutes, even far from the critical point. Here we have assumed moderate molar volumes for all components. The Flory-Huggins equation of

TABLE V Coexistence Curve Determination by Ultracentrifugation Inferred coexisting compositions Radial position of Initial cell composition

interface (cm)

>0.88 0.846 0.773 0.697 0.619 0.540 0.500

None 5.966 6.095 6.221 6.346 6.467 6.528

Mass balance x]'

Chemical analysis

x~

. . . 0.315 0.873 0.315 0.854 0.315 0.824 0.315 0.783 0.315 0.724 0.315 0.685

x7

x~

. 0.267 0.236 0.210 0.190 0.174 0.168

0.875 0.868 0.859 0.850 0.839 0.832

Note. Conditions and parameters: Binary strictly regular solution has A/RT = 2.10. True coexisting compositions are x~ = 0.315 and x~ = 0.685. Sector-shaped analytical ultracentrifuge cell has inner radius of 5.9 cm and outer radius of 7.1 cm. Rotor speed is 52,500 rpm.

state (54) can accommodate high-molecularweight solutes in nonideal solution; in that equation particle mass and solution nonideality can conspire together to generate sharp field-induced gradients. Near the plait point very steep gradients ap4.0

0.5

TABLE IV

1.0

2.0

Coexistence Curve Determination by Ultracentrifugation

~ BOTTOM-END~1

Inferred coexisting compositions Radial position of Rotor speed (rpm)

interface (cm)

<23,500 23,500 28,200 34,100 42,200 48,300 53,600

None 5.900 5.946 5.994 6.043 6.071 6.089

Mass balance

Chemical analysis

~o.oool

T x7

xf

. . . 0.315 0.778 0.315 0.795 0.315 0.814 0.315 0.835 0.315 0.848 0.315 0.857

x7

xq

. 0.315 0.304 0.289 0.266 0.250 0.236

0.778 0.796 0.816 0.841 0.858 0.870

Note. Conditions and parameters: Binary strictly regular solution has A/RT = 2.10. True coexisting compositions are x]' = 0.315 and xq = 0.685. Sector-shaped analytical ultracentrifuge cell has inner radius of 5.9 cm and outer radius of 7.1 cm. Overall cell composition is x~ = 0.778.

~).00001 I\

1.0

~ X ~

, L:"/ 0.67

\

I

~ I

/1

2.0

II JJ 3.0 ".'Jn

J

015

1.0

MOLEFRACTION,xl FIG. 5. Binary regular solution: end-to-end composition extremes when overall mole-fraction composition is 0.5. Journal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

256

ROSSEN, DAVIS, AND SCRIVEN

pear in ultracentrifuge optics as discontinuities in composition, further confusing the interpretation of data. The schlieren optical system translates the concentration gradient into the pattern y(r) (55):

by further increasing 0. Thus in a strong field, sharp gradients can make the presence of an interface and its position in the cell uncertain. Ternary regular solution with plait point: one phase from two. The phase diagram of the ternary regular solution of Fig. 7, like the bidXl dno y = LmlmEa cot 0 - - - [17] nary diagram of Fig. 2, has a plait point and dr dxl binodal curve enclosing metastable and unHere L, rn~, and m2 are machine constants; stable regions. Unlike the binary diagram, optical path length a and phase plate angle 0 however, composition profiles on the ternary are adjustable parameters; xl is mole fraction; diagrams are curved. This further complicates r is radial position; and no is refractive index. the relation between field-free and field-inBecause photographic plates are finite in size, duced phase behavior. For instance, the comlarge values ofy(r) go off-scale, rendering sharp position profile of a field-stratified sample may concentration gradients indistinguishable from lie entirely in the one-phase region while its the practically infinite gradients at an interface. overall composition falls within the field-free The schlieren pattern from Eq. [17] then has two-phase region (Fig. 8). In this way a cena band of missing gradient in which an inter- trifugal field can make one continuous phase face may or may not be present (Fig. 6). Using from a sample that is two phases in the absence a cell of shorter path length a, if available, helps of field. Again metastable states are possible to retrieve the missing pattem, as does in- (Fig. 9). In a ternary mixture a stable comcreasing the phase plate angle 0 to 90 °. In ex- position in the absence of field may be stable, treme cases, however, a portion of the gradient metastable or unstable in the field. Likewise, pattern is deflected completely out of the op- metastable or unstable compositions on the tical system and adjusting the phase plate angle field-free phase diagram may lie in stable, is ineffective. In our experience, for instance, metastable or unstable regions of the field-inany gradient pattern which is missing when 0 duced phase diagram. The stronger the field, = 80 ° is so sharp that it cannot be retrieved the greater is the distortion of the phase en-

A RT 2.2

2.0

1.8

1.6

I

I

I

10OO 1.2 80 °

o

E

11313

Z m

"F I.a

~

1.2

10

[

a=O.~crn 87"5°

1

[

= . o ~875°

1.0

I

I

1.1

1.2

T T~ FIG. 6. Binary regular solution: region of band of missing concentration gradient in the schlieren pattern. Parameter values employed are L = 60 cm, m~rn2= 2.15, dnD/dxt = 0.1, field strength = 300,000g. Journal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

PHASE BEHAVIOR IN ULTRACENTRIFUGE

257 3

3

FIELDPOTENTIADI LFFERENCE:

METASTABLE:

MAYSPLIT INFIELD

I

2

FIG. 7. Ternary regular solution with plait point: fieldfree phase diagram. Parameter values employed are A / R T = 3, B / R T = C / R T = O.

velope (Fig. 10). As in the binary regular solution, sharp equilibrium composition gradients like those of colloids yield large end-toend composition differences even far from the plait point. There are no interfaces in the profiles represented in Fig. 11. Ternary regular solution with isopycnic tieline. three phases from one or two. If the fieldfree binodal contains a pair of coexisting phases of equal density as in Fig. 12 (56, 57), there are three-phase, i.e., two-interface, profiles in the field. The composition profile of

3

/

~

I~ CoOMVCpE(~i~,0ALN

FIG. 9. Ternary regular solution with plait point: fieldinduced phase envelope and field-free phase diagram.

Fig. 13 touches an isopycnic or equal-density tieline, i.e., one connecting phases of equal density; the field-stratified sample has two interfaces and a small amount of phase rich in component 3. Thus can a centrifugal field make three phases from one--even if there is no three-phase region in the field-free phase diagram. The field-induced phase envelope for this regular solution is shown in Fig. 14. At a binary isopycnic tieline (56, 57), buoyant density changes sign as pressure varies. Thus in a compressible binary mixture, the composition profile can reverse direction on

3

APPEARANCOF E

FIELD-FREE COEXI STENCE CURVE

,

1:1(3. 8. Ternary regular solution with plait point: sample composition profile on the phase diagram.

0,04

1

2

FIG. 10. Ternary regular solution with plait point: effect of increasing field strength on phase envelope. Journal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

258

ROSSEN, DAVIS, AND SCRIVEN 5

5

APPEARANCE OF

SAMPLE:

HORIZONTALTIE LINES CONNECTEND-TO-END COMPOSITIONEXTREMES WHENMOLERATIOOF COMPONENTS1 AND 2 IS UNITY

TOP-END

®

/

~ BO'ITOM-END

/

/ /

CURVE

~N

/

~..~_,oL.,'7"-.~ "'~E~L ~L ~COMF~,T,ON\ PROFILE, \

~

~

I

2

I~

~

2

FIG. 11. Ternary regular solution with plait point: endto-end composition extremes without an interface.

FIG. 13. Ternary regular solution with isopycnic tieline: sample composition profile on the phase diagram.

the phase diagram (Eq. [11]) and cross the binodal twice, contrary to a remark by Block et al. (34). Ternary mixtures with more than one isopyc (56, 57) when stratified in the field can cross the binodal three times and show three interfaces, or four phases, in an external field. Ternary regular solution with multiple coexistence curves and plait points: one, two, three, four, or five phases from one, two, or three. The ternary system of Fig. 15 has numerous multiphase regions and plait points

and shows steep composition gradients in an external field. In even a moderate field a composition profile may intersect all four upper central coexistence curves, with the result that there are four interfaces in the field (Fig. 16). Thus can a centrifugal field make an arbitrary number of phases out of mixtures with complex field-free phase diagrams. The field-induced phase envelope of this regular solution shows as many as five phases (Fig. 17) over a finite range of overall compositions as well as temperature and ambient pressure (not

3

A

3

FIELDPOTENTIALDIFFERENCE:/X

c~,~g~CE

FIG. 12. Ternary regular solution with isopycnic tieline: field-free phase diagram. Parameter values employed are A / R T = O, B / R T = C / R T = 2.1. Journal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

II

'2

FIG. 14. Ternary regular solution with isopycnic tieline: field-induced phase envelope and field-flee phase diagram.

259

PHASE BEHAVIOR IN ULTRACENTRIFUGE 5

3

//~

NUMBERS ARE OUNTS INTC FIELD-NOF UMPHASES BER OFIN ERFACES

FIELDPOTENTIADILFFERENCE

i

2

I

2

FIG. 15. Ternary regular solution with multiple coexistence curves and plait points: field-free phase diagram. Parameter values employed are A / R T = B / R T = C / R T = 2.66.

shown). This violation of the field-free phase rule depends not on symmetry in parameter values but on field-induced stratification within the phases; the phase rule is discussed further in Appendix 1A. In principle, only the complexity of the field-free phase diagram limits the number of interfaces and thus the number of phases that can be present in a strong external field. For instance, the binary surfactant-water phase diagram of Laughlin (58) (Fig. 18) has m a n y

....

2

I

FIG. 17. Ternary regular solution with multiple coexistence curves and plait points: field-induced phase envelope.

multiphase regions, and a horizontal composition profile at 100°C can include as many as four interfaces, or five phases. 4. EXPERIMENTAL

Carbon tetrachloride, methanol, n-hexane, and hexadecane were obtained from Fisher

300 250 ~===;~(Neat Subneat fl)

~

~APPEAR~SAMPLE

200

Nigre

Soap I)

?

m

150 /

/ ~

\FOUR 7 INTERFACES -

."N

I,-

' Middle Soap

100 50

20 FIG. 16. Ternary regular solution with multiple coexistence curves and plait points: sample composition profile on the phase diagram.

Waxy

Neat (Neat

40 60 %SodiumPalmitate

80

100

FIG. 18. The binary system sodium palmitate-water: field-free phase diagram, from Ref. (58). Used with permission. Journal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

260

ROSSEN, DAVIS, A N D SCRIVEN

Scientific Company, Aldrich Chemical Company, and Phillips Petroleum Company, either certified A.C.S. or stated 99% pure. Water was doubly distilled. All were used without further purification. Sedimentation experiments were conducted on a Beckman Instruments, Inc., Model E analytical ultracentrifuge equipped with Model An-D rotor and schlieren optics. The cells used had an optical path length of 1.2 cm. Methanol and n-hexane have a large miscibility gap at room temperature (59). A one-

phase, 30.4 wt% solution ofn-hexane in methanol was centrifuged at 29,500 rpm to see i f a second phase would segregate in the field. An interface appears as a vertical line in the schlieren pattern of Fig. 19. At 37 and 61 rain there was no sign of phase separation but the refractive index gradient indicated accumulation of hexane at the top of the cell. Careful examination of the schlieren pattern at 96 rain (not shown) revealed that a hexane-rich phase had formed; by 128 min (Fig. 19c) it was quite noticeable, and it continued to grow in time.

FIG. 19. Schlieren patterns during sedimentation of a mixture of 30.4 wt% n-hexane in methanol. Phase plate angle 0 = 60°; speed = 29,500 rpm. (a) Time = 37 rain (shortly after attaining full speed), T = 23.6°C; (b) 61 rain, T = 23.6°C. Thereafter, T was maintained above 24.0°C. (c) 128 rain, (d) 454 min, (e) 1096 min, (f) 5654 min. R = reference edge; V = vapor phase; M = methanol-rich phase; H = hexane-rich phase. Journal of Colloid and Interface Science, Vol. 113,No. 1, September1986

PHASE BEHAVIOR IN U L T R A C E N T R I F U G E

By 5650 min, when the schlieren pattern was constant to within measurement uncertainty, which indicated that equilibrium had been established, there was a substantial hexane-rich phase. Carbon tetrachloride and hexadecane are miscible at room temperature and almost completely immiscible with water, whose density is intermediate between the other two. Thus the phase diagram mimics Fig. 12 except that the two-phase region covers almost the entire ternary diagram. Enough hexadecane with a trace of Sudan black dye was dissolved in carbon tetrachloride to reduce its density to slightly below that of water; this mixture was layered on water and centrifuged at approximately 60,000 rpm to see ifa third phase would form. The schlieren photographs are shown in Fig. 20. At first only two liquid phases were present, although sedimentation of carbon tetrachloride within the oil phase was evident even before full speed had been attained. The bottom of the oil phase soon grew rich in carbon tetrachloride and denser than the water below it; dense oil there traversed the water phase, presumably by drop

261

formation and recoalescence, and formed a new phase. Within this lower phase sedimentation continued, yielding at the top light hexadecane-rich fluid which again traversed the water phase. The third phase was clearly established within 45 min. 5. DISCUSSION

The ultracentrifuge is used for three purposes: (1) to separate interspersed phases; (2) to determine, by means of Eq. [13] or [14], molecular weight or particle mass from equilibrium composition gradients; (3) to determine, by means of Eq. [9], the change in fieldfree chemical potentials along the equilibrium composition profile. In only the third application does equilibrium behavior necessarily yield unambiguous results. The danger in ultracentrifuging to separate interspersed phases is that the centrifugal field may not merely accelerate the settling of phases but also alter their composition or number. Field-induced distortions in phase behavior are greatest in near-critical and colloidal mixtures, in which stratification is especially great. Moreover, once they are stratified

FIG. 20. Schlieren patterns during sedimentation o f a mixture of carbon tetrachloride, hexadecane, and water near the isopycnic tieline. Oleic phases appear pale due to a trace of Sudan black dye. Temperature varies between 17.3 and 18.4°C. (a) T i m e = 5 min, speed = 13,200 rpm, phase plate angle O = 85 °. Thereafter, speed = 59,6400 rpm and 0 = 80 °. (b) 13 min, (c) 45 min. R = reference edge; V = vapor phase; O = oleic phase; W = water phase. Loss of pattern at bottom of cell is characteristic of a sharp refractive index gradient. Journal of Colloid and Interface Science, Vol. 113,No. 1, September 1986

262

ROSSEN, DAVIS, AND

these systems are slow to return to field-free equilibrium because the thermodynamic incentive for diffusion is so small (60-62). Thus field-induced behavior is easily mistaken for field-free equilibrium. If the rate of sedimentation of large particles of dispersed phase is much larger than that of molecular species, then it may be possible to separate dispersed phases before significant stratification occurs within the phases. As we show in a companion paper (62), however, this is not always possible: for instance, stratification occurs rapidly in molecular solutions near a plait point. Therefore the extent of stratification must always be checked, either from the schlieren pattern (1, 6) or from fractionation and analysis of the sample, before the possibility of field-induced distortion in phase behavior can be ruled out. Solute particle mass can be deduced from equilibrium sedimentation patterns by means of Eq. [13] or [14] if the solution equation of state is known. Otherwise, sorting out the effects of particle mass and solution nonideality can be tricky. For instance, critical phenomena (63, 64) can distort apparent micelle sizes determined by sedimentation of surfactant solutions near the cloud point, which is a critical point (13, 20, 65, 66). Rosen and Li (67) found significant composition gradients at lg in a three-phase microemulsion sample evidently only 50 cm tall. This represents a field potential difference A~/ R T of only 2 × 10-6. The gradients disappeared when the upper phase was removed. Centrifugation of the middle phase at speeds below 18,000 rpm (probably near 1000g) produced no composition gradients, however. This suggests that the gradients were caused not by the gravitational field but by small shifts in phase equilibrium due to temperature fluctuations or gradients within the stated temperature control precision (_0.1 °C). Temperature control is crucial in determining composition gradients caused by Earth's gravity. The effect of temperature gradients is made plain by the nonisothermal form of Eq.

[15]: Journal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

SCRIVEN

Olnx

o~ Ot.t 0

-1

*

+

[18] 0T d~(r)J "

In general, therefore, systems sensitive to gravity ((0~°/0 In x) nearly singular) are also very sensitive to temperature gradients (68). Moreover, multiphase systems for which (0~°/ 0 In x) is nearly singular in the one-hase regions are especially sensitive to temperature fluctuations (61). This may also explain the observations of Good et al. (69). The sedimentation-equilibrium composition profile can be used with Eq. [9] to determine the field-free equation of state of a mixture. The sharper the field-induced composition gradients in these mixtures, the wider the range of compositions over which the free energy is probed in a single experiment. Previous studies have focused on massive solutes like polymers (70-76) or dense solutes like salts (77-79) or carbon tetrachloride (80) or on binary mixtures ((81); this study involved two binary mixtures in equilibrium, with a common solute and immiscible solvents), especially near a plait point (36-38, 41, 42); Eq. [ 11] introduces a broad new class of multicomponent near-critical mixtures (56, 82, 83) to this technique as well. Our experimental study of these mixtures (84) focused on one microemulsion system and two mixtures of oil and water with protosurfactant such as alcohol. Equilibrium ultracentrifugation revealed that the microemulsion system is near-critical along a path on the phase diagram from nearly pure oil to nearly pure water; that is, along this path the field-free chemical potentials ~0 change very little. This property of microemulsions had been conjectured (61, 82, 83) based on phasebehavior and interfacial tension patterns and is thought to govern microemulsion properties (61). On the other hand, equilibrium ultracentrifugation revealed that the field-free chemical potentials of the two protosurfactant systems depend more strongly on composition except near critical points. This may help ex-

PHASE BEHAVIOR IN ULTRACENTRIFUGE

plain the similarities and differences between these systems, sometimes called "detergentless microemulsions" (7, 10) and true microemulsions made with surfactant. APPENDIX 1

A. The Phase Rule The question of a phase rule in an external field turns not on the existence of an external potential ((85) is incorrect on this point) but on its variation across a sample. If the field potential is uniform, the phase rule is retained in its usual form because for a given component the field's contribution to #i (Eq. [3]) is the same in each phase. Phase-behavior experiments at lg usually approach this limit because the difference in external potential across the sample is so small. The variation of external potential across a finite sample invalidates the conventional phase rule, however. The required equality of chemical potentials, Eq. [3], implies the condition of hydrostatic equilibrium, Eq. [8], which replaces pressure equality in the tally of constraints. Therefore for each additional position r there are m unknowns (P(O, x(O) and m independent constraints (Eq. [3])--regardless of the number of intervening interfaces. There is in principle no limit to the number of interfaces and phases which may be present at equilibrium in an external field.

B. Free Energy in an External Field Equation [ 1] implies that the external field exerts no influence over the microstructures that govern the local field-free free-energy d e n s i t y f °. The field alters macroscopic phase behavior solely through hydrostatic pressure and redistribution of mass; the stability of the local fluid is not directly affected by the field (Eq. [4]). Differing rates of mass transfer during the approach to equilibrium can cause formation of phases not present at equilibrium (86) as well as the illusion of phase separation due to density inversion and convection (33, 34, 87).

263

To explain field-induced changes in phase behavior, some (7, 8, 88-90) have suggested that the field can alter the stability condition (Eq. [4]) and thus induce phase separation. Three arguments have been put forward why Eq. [ 1] and the theory based on it do not hold very near a critical point. From intuitive arguments Alder et al. (89) derived a formula for the increase in the critical temperature in homogeneous fluid suddenly placed in an external field. This increase stems from coupling between the external field and composition gradients between opposing fluctuations near the critical point. The computed increase in the critical temperature is 0.1 K for a binary mixture whose components differ in density by 1 g cm-3; for most mixtures it is presumably less. The experimental observations which led to this theory (88) can also be explained as convective instability (33, 34). Second, very near a critical point, field-induced equilibrium concentration gradients inhibit the divergence of the correlation length which governs many properties near the critical point (40, 43, 47, 48). For instance, in a 300,000g field, the fieldfree equations of state of pure zenon and water at the critical density do not apply within 0.3 and 0.14 K of their critical temperatures. Third, the free energy of material at an interface must be augmented by gradient free-energy terms (44-48). It is usually adequate for the study of bulk phase behavior to ignore these terms and treat interfaces as discontinuities. This approximation is even more accurate in the presence of field than in its absence because the field always reduces interfacial width (45). According to Eq. [15], the equilibrium composition gradients in bulk fluid can be so sharp that gradient terms in the free energy become significant, but this occurs only very near a plait point. For instance, in the binary regular solution of Section 3 the composition difference over a distance of 100 ~t is less than 1 mole% within even 0.01 °C of the plait point. Except within a fraction of 1 K of the plait point, then, the free energy is that given in Eq. [1]. Journal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

264

ROSSEN, DAVIS, A N D S C R I V E N

C. Global Stability in an External Field In Section 2 we show that any sample composed of local globally stable compositions is at least locally stable. Here we establish that it is globally stable because no two such samples have the same overall composition. The proof depends on the convexity of the freeenergy surface at stable compositions. Gibbs called the free energy surface composed of stable compositions and stable tie-lines, tie-triangles, etc. the "derived" free-energy surface (91): here we consider the Helmholtz derived surface f°(n) and the paths on it representing the local field-free free energies of two nonidentical samples in identical containers in an external field. The composition profiles of these samples are n"(~) and he(if), ~b° ~<~b~< ~b1. Because the derived surface is convex, for any two points a and b on it m

f°(n~) + Z /~i(n o a )(nib -- n a) ~
[A1]

i=1

The equilibrium condition [3] for the two paths n~(ff) and ne0k) requires

free phase equilibrium. A mass balance on samples a and/3 yields

N'~ - N~ = f (n'~ - n~i)d3r = f [nT(ff) - n~i(tp)]A(~b)/g(~b)d~P,

where N? are total moles o f / i n sample a, A(~p) is the area of the constant-potential surface in the sample and g(~) -= d~p/dr is gravitational acceleration. Premultiplication by ki and summation over all components gives (cf. Eq. [A4]) m

X ki(N'~ - N~) i=1 f

=

The equality holds only if g°(n~) = g°(ne) everywhere in the two samples; that is, only if the two samples are in phase equilibrium with each other throughout all their local compositions. Aside from this anomalous case,

ki(N~ - N~) > 0

[A71

i=l

= #°(nff~))

i = 1 .....

-

u0(n~(~°)),

m,

[A21

or

#O(n"(~b)) - g°(n~Op)) -- ~t°(n'~(~°)) - g°(n~(~°)) ------ki, ~0<~
m

Z ki(n'~ - n~i)A(~P)lg(~k)d~P >10. [A61 i=1

#O(n-(~)) - gO(n-(~o))

~0<~
[A5]

i = 1,...,m,

[A31

with constants ki not all equal to zero. Adding Eq. [A1] with n a = n"(~) and n b = n~(~k)to that with n a = nt~(~) and n b = n"(xb) yields

ki(nT(~b) - n~iOP)) >1 O, i=1

~o~<~b~<~b I.

[A4]

The equality in Eqs. [A1] and [A4] holds only where ~t°(n"(ff)) = g°(n~(~b)), i = 1. . . . . m; that is, where compositions n" and n a are in fieldJournal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

or

N '~ # N ~.

[A8]

Two field-induced equilibria composed of globally stable local compositions cannot have the same overall composition. Therefore each is the globally stable profile for its overall composition.

D. Composition Profiles of Incompressible Mixtures The composition profile in the field of an incompressible mixture with zero volume of mixing can be plotted on a single composition phase diagram, as illustrated in Fig. 1. Any two globally stable profiles on such a diagram may overlap but cannot cross. The proof is in two parts. First, given field-free phases a and b, if tz°(T, p, x a) < #°(T, p, xb), i = 1. . . . .

m,

[A9]

PHASE BEHAVIOR IN ULTRACENTRIFUGE

then phase b is unstable to nucleation of a a new phase; specifically, its free energy can be reduced by the nucleation of a small amount F fin a of phase a. The free energy of the new phase is ~'~i = 1 ~t°(T, p, xa)t~Na, and that of the f o remaining phase b is, by Taylor series, G(T, p, N b) _ ~mi = 1 #0(T ' p, xb)rN~. The free energy rn change upon nucleation is ~i=~ [#°(T,P' (02f°/OnOn) g(~) Xa) - - #°(T, p, x b ) ] r N a < 0. Phase b is not globally stable. Second, if two composition profiles, a and gO /3 share x"(ff ') = xa(ff') at one position but differ at if", one of them is not globally stable. Sub- g* tracting Eq. [9] for xa(ff ") from that for x~(~b") yields ki t~°(T, x"(~b")) - #°(T, xa(~b")) + 0i(P"(ff'9 L, ml, m2 -/~(ff"))=O, i = 1. . . . . m. [AIO] M Ifp~(~ ") 4:/fl(~b"), let p"(~b") >/~(~b"); then M* #°(T, x~(~")) < #°(T, xe(~b")), m i = 1, . . . . m; [All] local composition x~(~b") and therefore profile /3 are not globally stable. If p" = ~ at each point for which x" ~ x e, then

u°(T, x~(~)) =

n

u°(T, xff~)), i= l,...,m,

[A12]

and, from the condition of hydrostatic equilibrium (Eq. [8]), p"(~) = p~(~).

N

n a, n b n"(~b), n~(~)

[Al3]

Thus compositions x"(~) and xa(~b) are in isopycnic field-free phase equilibrium over a finite range of compositions. Aside from this anomalous exception, two stable composition paths of an incompressible system do not cross on the phase diagram. Two paths may split if one enters a metastable region.

nD fin p p0 P"0k), Pe(~b)

APPENDIX 2: NOMENCLATURE

A(~)

A, B, C

area of constant-potential surface through sample (Eq. [A5]) parameters of strictly regular solution model (Table II)

R r r T

265

optical path length of sample in ultracentrifuge (Eq. [ 17]) Helmholtz free energy of stratified system (Eq. [1]) Helmholtz free energy density of homogeneous bulk fluid (Eq. [l]) matrix of elements oEf°/OniOnj gravitational acceleration, d~p/

dr field-free Gibbs free energy per mole (Table I) field-free intensive Gibbs free energy defined in note to Table I constants defined by Eq. [A3] machine constants in ultracentrifuge optics (Eq. [ 17]) molecular weight vector of elements M~(1 - 0io) (Eq. [16]) number of components in system set of total number of moles of components in system (Eq. [21) set of component molar densities (Eq. [21) two arbitrary compositions (Eq. [A1]) composition profiles in two samples a and/3 as functions of external potential, or, equivalently, of position, in the samples (~p = ~b(0) (Appendix 1C) refractive index of solution a perturbation in n pressure reference pressure such as 1 atm pressure as a function of external potential in two samples a and/3 (Appendix 1D) gas constant distance from point in rotating sample to axis of rotation position in system temperature

Journal of Colloid and Interface Science, Vol. 113, No. 1, September 1986

266 V

x

xT, xf

x~(~), x~(~)

0 In x/&k

y(r) "iti

0 #i

(Ot~°/0 In x) P p°(~), 0~(~)

ROSSEN, DAVIS, AND SCRIVEN system v o l u m e m o l a r v o l u m e o f solution, E x f i i (Table I) partial m o l a r v o l u m e o f c o m p o n e n t i in solution (Eq. [9]) partial specific v o l u m e o f c o m p o n e n t i in solution, 6l/Mi (Eq. [12]) set o f mole-fraction compositions (Eq. [5]) mole fraction compositions o f phases a and 13inferred f r o m equilibrium ultracentrifugation experiment (Tables IV and V) composition profiles o f two samples a and/3 as functions o f external potential, or equivalently, o f position, in the samples (~b = ~b(0) (Appendix 1D) vector o f elements 0 In xi/&P (Eq. [15]) pattern p r o d u c e d by schlieren optical system in ultracentrifuge (Eq. [ 17]) " activity coefficient o f c o m p o nent i in solution (Eq. [14]) phase-plate angle in ultracentrifuge optics (Eq. [ 17]) (total) chemical potential o f c o m p o n e n t i in external field (Eq. [3]) field-free chemical potential o f c o m p o n e n t i, (Of ° / Oni)nj matrix o f elements 0#°/0 In xj (Eq. [15]) density o f solution density as a function o f external potential in two samples a and/3 (Appendix 1D) set o f v o l u m e fraction c o m p o sitions (Eq. [6]) external-field potential, equal to ~o - ¢o2r2/2 for centrifugal field an arbitrary d a t u m

JournalofColloidandInterfaceScience,Vol.113,NO.1,September1986

A¢

external potential at opposite ends o f a sample external-field potential at two positions in system (Eq. [ 13]) difference in ff across sample ACKNOWLEDGMENTS

Our thanks to Gerald Bratt for his expert help with the experiments and to the Department of Medical Biochemistry of the University of Minnesota and to Beckman Instruments, Inc., for the use of equipment. W.R.R. received support from the University of Minnesota Corporate Associate and Dissertation Fellowship programs. Research was partially funded by the University of Minnesota Computing Center and by the Department of Energy as part of a Universityof Minnesota program on fundamental aspects of enhancing oil recovery. REFERENCES 1. Scholte, Th. G., and Konigsveld, R., Kolloid Z. Z. Polym. 218, 58 (1967). 2. Friedman, H. A., J. Sci. Instrum. 44, 454 (1967). 3. Papamichael, S., and Conophagos, C., C. R. Acad. Sci. (Paris) 275, 785 (1972). 4. Shinoda, K., and Kunieda, H., J. Colloid Interface Sci. 42, 381 (1973). 5. Fontell, K., J. Colloidlnterface Sci. 44, 318 (1973). 6. Rietvold, B. J., Brit. Polym. J. 6, 181 (1974). 7. Smith, G. D., Donelan, C. E., and Barden, R. E., J. Colloid Interface Sci. 60, 488 (1977); Keiser, B. A., Varie, D., Barden, R. E., and Holt, S. L., J. Phys. Chem. 83, 1276 (1979); Lund, G., and Holt, S. L., J. Amer. Oil Chem. Soc., 264 (1980); Smith, G. D., and Barden, R. E., in "Solution Behavior of Surfaetants" (K. L. Mittal and E. J. Fendler, Eds.), Vol. 2, p. 1225. Plenum, New York, 1982. 8. Lang, J. C., and Morgan, R. D., J. Chem. Phys. 73, 5849 (1980). 9. Chart, K. S., and Shah, D. O., in "Surface Phenomena in Enhanced Oil Recovery" (D. O. Shah, Ed.), p. 53. Plenum, New York, 1981. 10. Schwab, A. W., Nielson, H. C., Brooks, D. D., and Pryde, E. H., J. Dispersion Sci. Teehnol. 4, 1 (1983). 11. Waugh, D. F., J. Phys. Chem. 65, 1793 (1961). 12. Anacker, E. W., Rush, R. M., and Johnson, J. S., J. Phys. Chem. 68, 81 (1964). 13. Ottewill, R. H., Storer, C. C., and Walker, T., Trans. Faraday Soc. 63, 2796 (1967). 14. Doughty, D. A., J. Phys. Chem. 83, 2621 (1979). 15. Jorpes-Friman, M., Finn. Chem. Lett. No. 7-8, 240 (1979). 16. Kirschbaum, J., J. Pharm. Sci. 63, 981 (1981). 17. Keh, E., C. R. Acad. Sci. (Paris) C272, 1441 (1971). 18. Richard, A. J., J. Pharm. Sci. 64, 873 (1975). 19. Kratohvil,J. P., J. Colloid Interface Sci. 75, 271 (1980).

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