Physical chemical characterization of percoll. II. Size and interaction of colloidal particles

Physical Chemical Characterization of Percoll I1. Size and Interaction of Colloidal Particles

TORVARD C. LAURENT, ALEXANDER G: OGSTON, HAKAN PERTOFT, AND BIRGIT CARLSSON Institute of Medical and Physiological Chemistry, University of Uppsala, Biomedical Center, Box 575, S-751 23 Uppsala, Sweden Received May 30, 1979; accepted October 10, 1979 The particle size of Percoll, a polyvinylpyrrolidone-coated colloidal silica, has been analyzed by various techniques. The mean diameter of the dry particle from electron microscopy and from the molecular weight and partial specific volume is 21-22 nm. Hydrodynamic measurements (viscometry and sedimentation) give values of 29-30 and 35 nm in 0.15 M NaC1 and water, respectively, indicating a layer of hydration on the particles which is more pronounced at low ionic strength. The second virial coefficient for Percoll in 0.15M NaC1, as estimated from light scattering, osmometry, and sedimentation equilibrium, was 1.3-2.9 x 10-6. Assuming excluded volume to be the origin of the second virial coefficient one can calculate an effective particle diameter of 35-46 nm which is larger than the measured hydrodynamic diameter. Electrostatic forces do apparently contribute to the exclusion. The second virial coefficient in water is 100-fold larger than in 0.15 M NaC1. In the course of this investigation a technique was designed to measure sedimentation equilibria of concentrated Percoll solutions in a preparative zonal rotor. The colloid was concentrated in the bottom fraction of one run to 0.58 g/ml which is close to tight packing of the hydrated spheres. The material was still a clear fluid and no precipitate or gel was noticed during a long observation period. INTRODUCTION

The preceding paper (1) describes the determination by light scattering of the particle weight of Percoll, a colloidal silica coated with polyvinylpyrrolidone (PVP). The molecular weight was determined under ideal conditions, i.e., at high dilution of the colloid. Percoll is, however, used in concentrated solutions when it is applied to biological work (2) and it is important to understand the interaction that occurs between the individual colloidal particles and how this interaction expresses itself in the properties of the solutions. The present paper deals with the hydrodynamic volume of the Percoll particles and the nonidealffy en ~ countered when the solutions are concentrated. In the process of this work we designed a technique to study sedimentation

equilibria of Percoll in a zonal rotor at low speed. MATERIALS AND METHODS

Materials. Percoll (Batch 96544D) and a pure silica colloid, Ludox HS, were the same as described in the preceding paper (1). The pH of the solutions was adjusted to 8 with hydrochloric acid. The material to be used for determinations of partial specific volume and viscosity was then thoroughly dialyzed against distilled water and when appropriate dialyzed against 0.15 M NaC1 as described (1). Percoll solutions to be used for ultracentrifugations were adjusted to their respective solvents by addition of concentrated sodium chloride or sucrose.

133 0021-9797/80/070133-09502.00/0 Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980

Copyright © 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.

134

LAURENT ET AL.

The concentration of colloid was determined by dry weight analysis (1) or from the density of the solutions (see below). Density. Density was determined in a Precision density meter DMA 02C (Anton Parr, Graz, Austria) at 20 ___ 0.01°C. Organic density gradient columns (3) were also used at room temperature in the analyses of some of the sedimentation equilibrium runs. Viscosity. Viscosity was measured at 25 ___ 0.01°C in several Ostwald viscometers with outflow times for water of approximately 30 or 280 sec (Cannon Instrument Co., State College, Penn.). Outflow times were measured with a precision of _0.1 sec. The relative viscosity was determined from the expression T~rel (01"tl)/(P2"t2), where pl and p2 are the densities and tl and t2 the outflow times of the solution and solvent, respectively. Sedimentation. Sedimentation runs were performed at 20°C and 12,000 rpm in a Spinco Model E ultracentrifuge equipped with RTIC unit and electronic speed control. Single sector cells (30 mm) were employed and the sedimentation was followed by schlieren optic. Sedimentation equilibrium. Sedimentation equilibrium runs were made in a Beckman Model L4 preparative ultracentrifuge and a Type B-IX continuous-flow rotor at speeds between 2000 and 10,000 rpm. The cylindrical rotor has an annular space for the sample with an outer radius of 5.1 cm and an inner radius of 4.1 cm. The total height of the space is 25.6 cm and the volume 750 ml. The rotor was filled with a pump from the outer edge with 250 ml of Percoll at a rotor speed of 2000 rpm, followed by 100 ml (+ the volume of connecting tubings) of a dense liquid which does not mix with Percoll (Fluorinert FC-70, density 1.93 g/ml; 3M Co., Saint Paul, Minn.). Percoll was preceded by solvent so that the total sample space was filled when the rotor was run at 10,000 rpm. The total width of the Percoll layer at the start was 0.32 cm. The runs were continued at the desired speed for 50 hr and =

Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980

the temperature of the rotor was kept at 4°C. The average speed was calculated from the odometer readings at specified times in the beginning and the end of the run. The experiment was completed by pumping Fluorinert to the outer edge and collecting approximately 5-ml fractions from the inner edge. The volume of each fraction and its density were measured. The concentration, c (g/ml), of Percoll in the fractions was calculated from the density of the fraction (ps), the density of the medium (P0), and the partial specific volume (fzp) according to the equation c -

Ps - Po 1 -

[1]

V~p'Po

The interphase between Percoll and Fluorinert (at a distance from the rotor center of 4.98 cm) was used as reference point when the total volume pumped out of the rotor was translated into distance from the rotor center during the run. Nitrogen analyses. Nitrogen analyses were kindly performed by Pharmacia Fine Chemicals AB using the Kjeldahl technique. RESULTS

Partial Specific Volume Density measurements were performed on Percoll and L u d o x HS in distilled water or 0.15 M NaC1. Four to five measurements at concentrations of 0.06 and 0.43 g/ml were made on each colloid and in each solvent. The partial specific volume was calculated according to Eq. [1] and the results are summarized in Table I.

Viscosity The relative viscosities of Percoll and L u d o x HS solutions were measured in a large concentration range in water (Fig. 1) and 0.15 M NaC1 (Figs. 2a and b). The bottom fraction from sedimentation equilibrium run No. 3 (see below) was used to obtain Percoll with a concentration above 0.25

PHYSICAL CHEMICAL CHARACTERIZATION

TABLE I The Partial Specific Volume (r~sp) (ml/g) of Percoll and Ludox HS Colloids Measured in 0.15 M NaCI and in Distilled Water Solvent

Percoll Ludox HS

0.15 M NaC1

H20

0.497 -+ 0.001 0.449 +- 0.001

0.496 -+ 0.001 0.454 -- 0.005

Note. Each value is the average of four or five measurements. g/ml. This fraction gave the same viscosity as the original Percoll when diluted to the same concentration. The viscosity values of L u d o x HS did closely follow the Percoll values (Figs. 1 and 2a). It was, however, notable that while the viscosity of Percoll was constant with time, L u d o x showed a time-dependent decrease in viscosity when allowed to stand in distilled water. The L u d o x HS values in Fig. 1, which were obtained immediately after the dialysis, should therefore only be regarded as approximate. The relative viscosity of a suspension of spheres should theoretically (4) follow the equation q%el =

-

Sedimentation The sedimentation values of Percoll in 0.15 M NaC1 and water in the concentration interval 1-10 mg/ml are shown in Fig. 3. Earlier sedimentation analyses (5) were carried out at much higher concentrations and allowed only an approximate extrapolation of the values to infinite dilution. The use of 30-ram cells in the present experiments made it possible to observe sedimenting peaks in the schlieren optics below 1 mg/ml. The extrapolated values from Fig. 3 o f 0s20,w are given in Table II.

8

[2]

I -f- [T/]'C q- 1.60['012"C 2

+ A exp(0.40 -B . [r/] .c).

/

9

where 6 is the volume fraction occupied by the solute and A and B are empirical constants. The last term is included to take account of effects appearing in the shearing of highly concentrated solutions (4). The effective hydrodynamic volume fraction of Percoll is equal to e "gsp'k, where c and 9~p have been defined above and k is the ratio by which solvation increases the unhydrated volume. Knowing that the intrinsic viscosity, [,/], is equal to 2.5 .f'~p'k and the value of f % is 0.495 (Table I), Eq. [2] can be rewritten T/rel =

The values of the intrinsic viscosity can be obtained either by extrapolation of (q~rel 1)/c to infinite dilution or as the initial slopes of the curves in Fig. 1 and 2a and are given in Table II. The values of A and B (Table II) were calculated by trial and error for best fit of the experimental data to the theoretical relationship at high concentration. Figures 1 and 2 demonstrate that Eq. [2] describes the viscosity behavior of Percoll both in 0.15 M NaCI and in water.

1 + 2.56 + 10.054)2 + A exp(B6),

[2a]

135

OF PERCOLL

7 6

I ~reL

S!

2 0

0.1

0.2

0.3

CONCENTRATION [g/mr)

Fro. 1. The relative viscosity (~re~) of Percofi (©)

and Ludox HS (•) in water as a function of concentration. The values obtained for Ludox HS were not stable and decreased upon standing. The line is drawn according to Eq. [2a] using the values of [~], A, and B given in Table II. Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980

136

LAURENT ET AL.

3,0'

a

•

~00

80 2.5-

60

~rel ~r~l.

2.0

~0

1.5" 20

1

011 CONCENTRATION

012

(g/~l}

I

I

I

03

0.4

05

CONCENTRATION (g/ml)

FIG. 2. The relative viscosity (~re]) of PercoU (e) and Ludox HS (,t) in 0.15 M NaC1 as a function of concentration. The line is drawn according to Eq. [2a] using the values of [~], A, and B given in Table II.

Sedimentation Equilibrium of Percoll

in 0. 15M sodium chloride than in the absence of salt. The dip in concentration close to the bottom is presumably an artifact due to turbulence when the rotor was emptied.

Three runs were performed in 0.15 M sodium chloride, one in water, and one in 0.25 M sucrose (salt free): the last solvent is commonly used i n separation o f cell - " It is remarkable that colloid concentraorganelles in Percoll. The experimental tions in the order of 0.58 g/ml were reconditions for the five runs are given in covered at the bottom of the gradients in Table III. Figures 4a and b show the disruns 2 and 3 (Fig. 4). The bottom fraction tribution of Percoll after sedimentation was a clear solution without any signs of equilibrium. Steeper gradients were formed gel formation such as is usually encountered when Ludox HS is centrifuged at high TABLE II speed. The Percoll solution recovered from Viscometrie Parameters and Sedimentation the Bottom of run No. 3 was used for Constants of Percoll osm0metry (1) and viscometry (see above). Solvent The pH was followed along the gradients in sodium chloride and water (runs Nos. 3 0.15 M NaC1 H20 and 4). It varied from 7.7 at the low conIntrinsic viscosity (ml/g) 2.5 4.8 centration end to 8.0 and 8.4, respectively, A (Eq. [2]) 6.0 × 10-4 1.9 x 10-2 in the bottom fraction. B (Eq. [2]) 22.2 10.7 Nitrogen analyses on fractions taken from The ratio of the volume of run No. 3 showed that the top fraction (colthe hydrodynamic unit from viscosity and the M i d concentration 0.013 g/ml) contained unhydrated particle, k 2.0 3.9 19% PVP per dry weight, a middle fraction Sedimentation constant, (colloid = 0.25 g/ml) contained 11.9% PVP, s°0,w (sec x I01~) 192 165 and the bottom fraction (colloid = 0.58 Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980

PHYSICAL CHEMICAL CHARACTERIZATION OF PERCOLL

137

X

,,, loo-

so

"~-.--_.__.~.~._~

,

~

,

~

,

~

,

CONCENTRATIONIg/=d.xlO 31 FIG. 3. The sedimentation constant of Percoll in 0.15 M NaG1 (0) and in water (©) as a function of concentration.

g/ml) contained 11.1% PVP. This is in agreem e n t with the earlier suggestion that the Percoll solution contains small amounts of free P V P and that the relative proportion of this decreases towards the b o t t o m of the gradient (1). In the evaluation of the data we will follow the t r e a t m e n t of Nichol et al. (6). The equilibrium sedimentation of a nonideal h o m o g e n o u s solute (7) is d In c

oJZM •dp/dc -

dr 2

.....

2RT(1 + d In 7/d In c)

,

[3]

where c is the concentration in grams per milliliter, r the radius of rotation, co the angular velocity, M the molecular weight, 3' the activity coefficient of the solute (as-

s u m e d to be independent of pressure), and dp/dc its density increment (assumed to be

independent of concentration). The latter is related to the partial specific volume f'sp and the density of the solvent (Oo): do~de = (1 - fzso'Po). I f the solutions were ideal (3' = 1), then In c should be linearly related to r 2 and the slope should be proportional to the molecular weight. Such plots are shown in Fig. 5 and it is quite evident that the solutions behave nonideally, especially at low ionic strength. F r o m the initial slopes of the runs p e r f o r m e d in 0.15 M NaC1 one can estimate a molecular weight of 1.6 x l0 s but this estimate is impaired b y errors both due to polydispersity and nonideality.

TABLE III Experimental Data on Equilibrium Sedimentation Runs of Percoll Run No. I 2 3 4 5

Solvent 0.15 0.15 0.15 H20 0.25

M NaC1 M NaC1 M NaC1 M sucrose

Original colloid . . . . concentration (g/ml) 21.5 18.3 23.8 12.1 21.7

Rotor speed (rpm)

Time (hr)

1,980 4,216 4,103 10,355 4,084

48 49 44 48 49

..... ....

A2 (mol g-2 cm3) 2 2.8 2.9 1.7 2.4

x x x x x

10 -6 10 -6 10 -6 10 -4 10 -5

A3 (mol g-Z cm~)

-5.4 x 10-4

Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980

138

LAURENT ET AL. b

--0,6

h

~

0.2

u- 0,4

it

~

0.1

o,2

i

4.7 Distance

i

&.8 &9 from rotor center, r (cm)

5.0

4.6 Distance

4.7 k.8 from rotor center, r (c m)

t~.9

5.0

FIG. 4. Sedimentation equilibrium of PercoU under different conditions. Run Nos. 1 (0) and 3 (A) were made in 0.15 M NaCI at 1908 and 4103 rpm, respectively. Run No. 4 (A) was made in water at 10,355 rpm and run No. 5 (O) in 0.25 M sucrose at 4084 rpm. For further experimental details see text and Table lII. N i c h o l et al. (6) e x p r e s s t h e a c t i v i t y c o e f ficient as a virial e x p a n s i o n In y = M ( 2 A 2 c + 3/zA8c2 + 4[3A4c3 q- "" "),

in t h e o s m o t i c p r e s s u r e e q u a t i o n . T h e authors also proposed a practical way of estimating the nonideality coefficients. They showed that

[4]

w h e r e A2, A s , e t c . , a r e t h e virial c o e f f i c i e n t s

Z -

t°2dp/dc r 2 2RT

cln_ M

_ 2A2c

+ 3/2A3c 2 - I , -1

-2

-3

i

-4

i

22

23 r2(cm 21

i

2/*

25

FIG. 5. Plots ofln c vs r ~for the sedimenti~tion equilibrium runs described in Fig. 4. Run Nos. 1 (0) and 3 (A) in 0.15 M NaC1, 4 (/x) in water, and 5 (O) in 0.25 M sucrose. All curves deviate from linearity as expected for strongly nonideal systems and do not allow a calculation of molecular weights. Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980

[5]

w h e r e t e r m s c o n t a i n i n g A4, A s , e t c . , h a v e b e e n n e g l e c t e d . I is an i n t e g r a t i o n c o n s t a n t . Z can be calculated, using an independent estimate of M, and plotted versus c and the coefficients can be calculated by leasts q u a r e s fitting. W e h a v e m a d e s u c h p l o t s (Figs. 6a a n d b) u s i n g t h e v a l u e o f M = 6.5 × 106, o b t a i n e d f r o m light s c a t t e r i n g (1). I n 0.15 M N a C I t h e r e w a s a g o o d l i n e a r c o r r e l a t i o n in r u n N o . 1. I n r u n N o s . 2 a n d 3, p e r f o r m e d at higher speed, there was a deviation from l i n e a r i t y at l o w c o n c e n t r a t i o n s ( < 0 . 0 5 g/ml), w h e r e e r r o r s in t h e c o n c e n t r a t i o n d e t e r m i n a t i o n s will b e m a g n i f i e d . T h e s c a t t e r e d p o i n t s a t high c o n c e n t r a t i o n s a r e d u e to t h e a p p a r e n t a r t i f a c t c r e a t e d in e m p t y i n g t h e r o t o r . F r o m t h e s l o p e s in Fig. 6a a v a l u e o f A2 in 0.15 M N a C I o f 2 - 2 . 9 x 10 -6 c a n b e e s t i m a t e d (see T a b l e III). S i m i l a r p l o t s m a d e f o r P e r e o l l at l o w ionic s t r e n g t h a r e n o t as e a s i l y i n t e r p r e t e d . F r o m t h e s u c r o s e

139

PHYSICAL CHEMICAL CHARACTERIZATION OF PERCOLL

33

50 32

49 31

/*8 30 47

29

~o /*6 )¢ N

=o x N

/*5

28 27

S

5

11 /* 1

i

0

i

i

i

,

0.1 0.2 0.3 0.6, 0,5 CONCENTRATION ( g / m r )

i

0.6

3 i 0.1 CONCENTRATION

I

0.2 ( g,/mt )

I

0.3

FIG. 6. Plots according to Eq. [5] to evaluate the virial coefficients for Percoll from sedimentation equilibrium run Nos. 1 (0) and 3 (A) in 0.15 M NaCI, 4 (A) in water, and 5 (©) in 0.25 M sucrose. The slope of the linear z vs c plot in runs Nos. 1, 3, and 5 gives the second virial coefficient (A2) (Table III). The values from run No. 4 were fitted in a computer by regression analysis to a quadratic equation giving both A2 and Aa. The calculated line is drawn in the figure.

run one can obtain a value of Az one magnitude larger than that in NaC1 (Table III). The data in water give a curve the slope of which varies 10-fold. The shape of the curve indicates that A3 in Eq. [5] must be significant and negative. The experimental values at c < 0.2 were fitted by the least-squares approximation in a computer to a secondorder equation and the values of A2 and A3 obtained are given in Table III. DISCUSSION

The present results show that Percoll can exist as a stable solution containing 0.15 M sodium chloride over a large concentration interval--even at as high concentrations of the colloid as 0.55-0.60 g/ml. The properties of the Percoll solutions at high dilution are determined by the parameters of the individual particles and at high concentrations by the interaction between the particles. The data presented in this report provide a basis for discussion both of the individual particle size and of the interaction.

The Size of the Nonhydrated Particle We can calculate the mean diameter (d) of Percoll from the molecular weight (M = 6.5 × 106 (1)) and the partial specific volume with the relationship / 3MI;" \1/3 d = 2[ s___~) , [6] where N is Avogadro's number. This gives a value for d of 21.7 nm (Table IV). Similarly, we can calculate d from electron microscopy. The weight-average diameter of the silica nucleus was estimated to be 19.3 nm (1). If the coat of PVP corresponding to 10% of the particle weight is tightly packed on the surface and if the polymer is assumed to have a density of 1.2, then the PVP layer is 0.6 nm thick and the total diameter 20.5 n m , in good agreement with the above estimate,

The Hydrodynamic Unit Size of Percoll The size of the particle in solution may be larger than the dry weight particle due to Journal of Colloid and Interface Science, V o l . 76, N o . I , J u l y 1980

140

LAURENT ET AL. TABLE IV

of the solvent, respectively. Using the molecular weight of 6.5 × 106 (1) and the partial specific volumes and sedimentation constants from Tables I and II we find diameters of PercoU in 0.15 M NaC1 and water of 30 and 35 n m , in good agreement with viscometric data.

Mean Diameter of Percoll Colloidal Particles Diameter (nm)

Dry particle From molecular weight and partial specific volume From electron microscopy Hydrated particle From viscometry From sedimentation and molecular weight Diameter of excluding unit from A2 Light scattering Osmometry Sedimentation equilibrium

22

The Size o f the I n t e r a c t i n g P e r c o l l Particle

21 In 0.15 M NaC1

In H20

28

35

30

35

35 39

260

41-46

180

hydration. An estimate of the size of the hydrated particle is obtained by hydrodynamic techniques, e.g., viscosity and sedimentation. The intrinsic viscosity of a compact spherical particle without hydration should be 2.5. l?sp (4; 8, pp. 333-341). A larger value can be due either to asymmetry of the particle or to hydration. As the former alternative can be excluded on grounds of electron microscopy we can assume that the deviation is due to a hydrated volume larger than the dry volume by the factor k (Table II). If the dry diameter is 22 nm then the hydrated diameter is 22 .k v3, which gives d values o f 28 a n d 35 nm in 0.15 M sodium chloride and water, respectively. A combination of Svedberg's and Stokes' equations (8, pp. 349, 359, 380) gives a relationship between the diameter of a sphere and its sedimentation constant M.(1 - 9sp'po) d

=

,

[7]

N "3 " rr "Oo "S

here po and ~/oare the density and viscosity Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980

Independent estimates of the second virial coefficient in 0.15 M sodium chloride were arrived at from light scattering (1.3 × 10-6 ) and o s m o m e t r y (1.8 × 10 -6 ) by Laurent e t al. (1) and from sedimentation equilibrium (2-2.9 x 10 -6) in this investigation. A2 can be regarded as a parameter describing the excluded volume in a macromolecular system (9). N'u A2

-

-

-

2"M 2

N'2"rr'd 3 -

3 "M 2

'

[8]

where u is the volume excluded around each individual molecule and which is eight times the real volume of the spherical particle. The apparent molecular diameters of Percoll from the second virial coefficients are shown in Table IV. They are larger than the estimates of the hydrodynamic units, indicating that other forces than sterical exclusion, e.g., electrostatic repulsions, also contribute to the interaction. This is especially marked when water is the solvent. Further terms in the virial expansion than the one containing A2 are needed to describe the behavior of very concentrated systems. This is apparent from the sedimentation equilibrium run in water (No. 4, Table III). A negative value was observed forA3. Negative values of higher virial coefficients have recently been reported in polymer solutions by Preston and Wik (10) and can be ascribed to a decrease in effective molecular volume with increasing concentration (l 1). In the present case we may interpret the negative A3 as a decrease in the repulsive forces between the particles at increasing concentration due to an increased concentration of

PHYSICAL CHEMICAL CHARACTERIZATION OF PERCOLL

counterions which screen the surface charge of the colloid. Very similar effects were encountered in the preceding studies by lightscattering (1). The experiment in 0.25 M sucrose gave a lower value of A2 than that registered in water. Only tentative explanations of this result can be given. The presence of even small amounts of electrolyte impurities in the sucrose could give such an effect. Sucrose could also osmotically dehydrate the polymer coat of the Percoll particles and thereby affect the degree of dissociation and the surface charge of the colloid. The concentration observed at the bottom of sedimentation equilibrium runs Nos. 2 and 3, 0.58 g/ml, is very close to closest packing of hydrated spheres. The concentration corresponds to a volume fraction of dry spheres of 0.29 ml/ml, of hydrated spheres (d = 29 nm) of 0.69 ml/ml, and of "interaction spheres" from A2 (d = 40 nm) of 1.7 ml/ml. The closest packing of equally sized spheres which is possible corresponds to a volume fraction of 0.74 ml/ml.

141

ACKNOWLEDGMENTS This project was supported by grants from the Swedish Medical Research Council (project No. 4) and Pharmacia Fine Chemicals AB. We are grateful to Dr. L. K~tgedalfor valuable discussions. REFERENCES 1~ Laurent, T. C., Pertoft, H., and Nordli, O., J. Colloid Interface Sci. 76, 124 (1980). 2. Pertoft, H., and Laurent, T. C., in "Methods of Cell Separation" (N. Catsimpoolas, Ed.), Vol. 1, p. 25. Plenum, New York, 1977. 3. Oster, G., Sci. Amer. 213, 70 (1965). 4. Thomas, D. G., J. Colloid Sci. 20, 267 (1965). 5. Pertoft, H., Laurent, T. C., L~t/ls,T., and K~gedal, L., Anal. Biochem. 88, 271 (1978). 6. Nichol, L. W., Ogston, A. G., and Preston, B. N., Biochem. J. 102, 407 (1967). 7. Casassa, E. F., and Eisenberg, H.,Advan. Protein Chem. 19, 287 (1964). 8. Tanford, C., "Physical Chemistry of Macromolecules," Wiley, New York, 1961. 9. Flory, P. J., "Principles of Polymer Chemistry," Chap. XII-2. CorneUUniv. Press, Ithaca, N. Y., 1953. 10. Preston, B. N., and Wik, O., in press (1980). 11. Ogston, A. G., and Preston, B. N., Biochem. J. 183, 1 (1979).

Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980