Self-Diffusion and Sedimentation of Dextran in Concentrated Solutions

W. BROWN, P. STILBS, and R. M. JOHNSEN, Institute of Physical Chemistry, University of Uppsala, Box 532, ,5751 21 Uppsala, Sweden

Synopsis

Frictional coefficients for dextran in water have been evaluated from (i) self-diffusion coefficients determined by pulsed-field-gradient NMR, and (ii) sedimentation coefficients in concentrated so- lutions. The results show that these frictional coefficients are only equal at infinite dilution and that fs increases more rapidly than f ~ ' as the concentration increases.

INTRODUCTION

In describing the translational diffusion of a polymer molecule in solution, the mutual and self-diffusion coefficients are required. The mutual diffusion coefficient, which characterizes the relaxation of a concentration gradient, is usually written1

(1)

The self-diffusion coefficient, describing the Brownian or random motions

D, = ( R T / N A f D ) (1 - (6) (1 4- 2A2Mc 4- 3A3Mc2 4- * * .)

where (6 is the volume fraction of polymer.

of an individual molecule, is given by

Ds = R T / N A f D * (2)

At infinite dilution, D, = D, and ( f 0 ) O = (fb)o. It has frequently been assumed that fr> approximately equals YD even at finite concentrations (see, for example, refs. 2,3), so that the two diffusion coefficients are related through the thermo- dynamic virial coefficients. It is now becoming clear that this is not the case.4

The sedimentation coefficient is normally expressed

s = ( M / N A f s ) (1 - u2p) (3) where p is the solution density and V2 the partial specific volume of the polymer. This frictional coefficient, ( f S ) O at infinite dilution, is known to equal both ( f 0 ) O

and ( f b ) O . Calculations of the frictional factor f D by combining D , and osmotic pressure data for polystyrene in good and poor solvents in the dilute and semi- dilute regions5 apparently lead to the identity of f D and f s at finite concentrations (see Discussion). More experimental data over extended ranges of concentration are clearly required in order to clarify the interrelationships between these various frictional factors. The present article is restricted to a comparison of the frictional factors evaluated from the self-diffusion coefficients and sedi- mentation coefficients for dextran in water over a broad concentration range. The pulsed-field-gradient nuclear magnetic resonance technique is a powerful

Journal of Polymer Science: Polymer Physics Edition, Vol. 20, 1771-1780 (1982) 0 1982 John Wiley & Sons, Inc. CCC 0098-1273/82/101771-10$02.00

1772 BROWN, STILBS, AND JOHNSEN

method for determining D, in the range 10-13-10-9 m2 s-l. Recent advances6 allow determination of diffusion coefficients for the individual components in a solution. The technique, first used by James and McDonald,7 has been applied to a number of multicomponent systems, for example in the elucidation of the self-diffusion of solvents and small molecules in microemulsions? micellar sys- tems? gels,Io and polymer solutions.11J2 While application has hitherto been restricted to small solutes, it has been found possible to extend it to encompass even polymeric components12 in systems in which the polymer has a t least one NMR band which does not exhibit excessively rapid transverse (T2) relaxa- tion.

EXPERIMENTAL

Materials

Dextran 70, lot number 5162 from Pharmacia Fine Chemicals, Uppsala, Sweden, was used. The molecular weights were (M), = 64,200 and (M), = 44,000 and intrinsic viscosity [ q ] 2 5 = 0.026 m3 kg-l.

Methods

Solutions were prepared by weighing with correction for moisture content.

Partial Specific Volume Measurements

Partial specific volume measurements were made using a digital density meter (DMA 60, Anton Paar K.G., Graz, Austria) thermostated at 25°C for solutions up to 35% (w/w). The volume V of solution containing 1 kg water was calculated from the measured density p of a solution and the concentration of the solution, expressed as kilograms dextran per kilogram water and denoted c * ; see Table I. V was then plotted versus c * , as in Figure 1. The partial specific volume at any concentration is then, by definition, the tangent to this curve at that con- centration. A least-squares fit of a polynomial to these data indicated that concentration terms of order greater than one were not significant a t the 95% confidence level. Thus the partial specific volume of dextran in water a t 25OC was determined to be independent of concentration in the concentration region below 35% (w/w) and with 95% confidence to lie in the interval (0.643 f 0.002) X m3 kg-l.

TABLE I Densities of Solutions of Dextran 70 at 25OC

5.25 1016.27 10.18 1034.97 15.09 1054.61 20.35 1076.37 25.14 1096.07 30.09 11 18.01 34.86 1139.65

SELF-DIFFUSION 1773

1.4

1.2

1.0

Fig. 1. see text.

0.1 0.2 0.3 0.4 0.5

'olume-concentration relationship for dextran in concentrated aqueous solution at 25

Self-Diffusion Measurements

The Fourier-transform nuclear spin-echo measurements were made a t 99.6 MHz using improved versions of the methods described previously? where the experiments were now made at a fixed A for all 6 values. All measurements were made at 75°C in order to make TZ for the dextran signals sufficiently long. In order to leave the dextran signals unobscured by the strong averaged H20-OH signal, it was necessary to use D20 as a solvent in the present work. D20 was also needed for lock purposes. The primary data are summarized in Tables I1 and I11 and the uncertainties in the measured quantities are included.

Sedimentation coefficients were measured at 59,000 rpm in an MSE analytical ultracentrifuge using the Schlieren optical system. Sedimentation coefficients were evaluated using both second-moment and position of maximum height methods (Table IV).

TABLE I1 Values of the Self-Diffusion Coefficient for Dextran 70 in D20 a t 75OC

D75 x 10" c ("lo, w/w) (m2s-1) ( f O / / c D

2.72 7.3 f 0.4 0.74 4.67 6.8 f 0.3 0.69 7.49 5.1 f 0.3 0.52 9.09 4.2 f 0.2 0.42

10.15 4.4 f 0.5 0.44 14.3 2.5 f 0.2 0.25 19.8 1.9 f 0.3 0.19 24.8 1.2 f 0.3 0.1 2 28.6 0.84 f 0.4 0.09

1774 BROWN, STILBS, AND JOHNSEN

TABLE I11 Values of the Self-Diffusion Coefficient for Water in Dextran 70/Water (D2O) at 25'C

DZ5 X 109 Dg$l X lo9 c (%, w/w) (m2 s-1) (m2 s-l)a

2.72 1.77 f 0.05 1.78 10.15 1.57 f 0.04 1.60 24.8 1.02 f 0.04 1.07 28.6 0.95 f 0.03 1.01 33.3 0.61 f 0.04 0.66

a Corrected for exchange processes by means of eq. (5); see text.

RESULTS AND DISCUSSION

Self-Diffusion of Dextran

The results obtained are shown in Figure 2 and 3 and summarized in the Ta- bles. Within experimental uncertainty, logD decreases linearly with increasing concentration [Fig. 2(a)] as was found for self-diffusion data for short-chain polystyrenes by MoseleyI2 and for serum albumin by Keller et al.13 and Kitchen et al.3 This decrease is considered to result in part from the so-called obstruction effect14 and in part from the enhanced frictional interactions between polymer molecules. The dashed line in Figure 3 denotes the magnitude of the obstruction effect as calculated using the equation given by Mackie and Meares.15 The frictional interactions will be a function of the polymer conformation: thus, as anticipated, the ratio DID0 at a given concentration (where DO refers to the value at infinite dilution) is smaller for dextran 70 than for serum albumin of nearly

TABLE IV Values of the Sedimentation Coefficient for Dextran 70 at 25°C

s 2 5 x 1013 c (%, w/w) (S) ( f o l f c )s

0.36 3.24 f 0.2 0.98 0.74 2.91 f 0.1 0.88 0.99 2.54 0.76 1.12 2.50 f 0.1 0.77 1.50 2.23 f 0.1 0.69 1.96 2.03 0.61 2.27 1.46 f 0.07 0.45 2.91 1.57 0.47 3.85 1.25 0.38 4.76 1.05 0.31 5.25 1.00 f 0.06 0.28 9.09 0.57 0.17

10.2 0.46 f 0.02 0.14 13.0 0.36 0.11 15.1 0.24 f 0.02 0.09 16.7 0.24 0.07 20.35 0.16 f 0.01 0.06 25.14 0.10 0.04 30.09 0.07 0.03

1.0

0 . 8

0.6

0.4

0.2

C

- 0.; I 1 1

10 20 30

0.4

0.2

0

- 0.2 101

1775

% (w/w)

I 1 1

0 10 20 30

Fig. 2. (a) Plot of log self-diffusion coefficient for dextran 70 in water (DzO) solution at 75°C. obtained from pulsed-field-gradient NMR measurements, as a function of concentration. (b) Logarithmic plot analogous to (a) but referring to self-diffusion coefficients for water protons at 25°C in concentrated dextran-water (DzO) solutions.

1776

1

0.8

0.6

0.4

0.2

0

BROWN, STILBS, AND JOHNSEN

I 1

I I I

10 20 30

'I. P / w )

Fig. 3. Radio D/Do as a function of concentration for (a) proton diffusion, (b) dextran 70 diffusion. The dashed line is calculated for the obstruction theory according to Mackie and Meares.'5

the same molecular weight, owing presumably to the more compact form of the latter molecule.

Using scaling arguments, de Gennes derived16J7 a power law which predicts that the self-diffusion coefficient should decrease with increasing concentration within the semi-dilute regime according to

(4)

This exponent is supported by data for polystyrene in benzenels cited by de Gennes717 those for polystyrene in carbon tetrachloride given by Callaghan and Pinder,lg and for polystyrene in toluene.20 The present results are presented in Figure 4. The broken line represents the slope predicted by de Gennes.17 Since such curvature appears to be a common feature of plots of this type (see refs. 17,19,20), unequivocal support for this exponent cannot be said to exist.

D - C-1.75

Self-Diffusion of Water

Results of water (HDO) self-diffusion are included in Figures 2(b) and 3 and Table 111. Owing to fast exchange of protons between potential sites on water molecules and dextran hydroxyl groups, it is pertinent to correct the observed diffusion coefficients using a relationship21 of the form

(5)

where p is the mole fraction of exchangeable sites on the dextran molecules.

Dobs = (1 - p)Dh" bound H D O -k PDdex

I0

0 8

0.6

0 4

0.2

0

-0 2

SELF-DIFFUSION

I ! 1

1777

0.5 1 0 1 5 Fig. 4. Scaling of diffusion coefficient with concentration according to de G e n n e ~ ' ~ , ' ~ for dextran

70 in water (D20) solution. The dashed line represents D -C-*,75.

Dbound will be several magnitudes smaller than Dfree so that the second term may be neglected. It can be concluded from Figure 3 that the reduction in water mobility due to the presence of dextran a t a given concentration is due entirely to the obstruction effect, although the exact agreement between the experimental points and the theoretical line is most likely fortuitous. This finding agrees with previous onesz2 regarding the translational diffusion of small molecules in polymer solutions and gels of moderate concentration.

Concentration Dependence of the Sedimentation Coefficient

Scaling theory for sedimentation leads to23

S - C-0.54 (good solvents) (6) and

S - C-1.0 (8-solvents) (7) where these exponents apply in the limit of infinite molecular weight. Experi- mental support for these values exists24 but there are marked divergences in both good and 8-solvents a t low molecular weights.

1778 BROWN, STILBS, AND JOHNSEN

0.5

0

-05

-1.0

- 1.5

+\ +

!+ \+

0.5 1.0 1.5

Fig. 5. Scaling of sedimentation with concentration for dextran 70 in aqueous solution.

The present data (Fig. 5 ) leads to a continuous curvature over the concen- tration interval investigated; see the discussion in ref. 24.

Concentration Dependence of Frictional Coefficients

From the experimental self-diffusion data, the frictional ratio was evaluated using the relationship

where the subscript c refers to a given concentration and DO was obtained from a least-squares treatment of logD vs. concentration.

For sedimentation data the following expression was used:

The frictional ratios from eqs. (8) and (9) are shown as a function of concen- tration in Figure 6; it is found that the frictional ratio from sedimentation data decreases more rapidly than that obtained from self-diffusion coefficients. This finding is qualitatively the same as that of Kitchen et al.3 for serum albumin in water and of Nystrom and Bergman25 for hydroxypropyl cellulose in water. It is possible, however, that polydispersity may differently influence these frictional ratios and to an extent which is difficult to predict. This factor will be considered in a forthcoming contribution. Roots et a15 calculated the frictional ratio ( f 0 / f c ) ~

SELF-DIFFUSION 1779

1.0

0.8

0.6

0.L

0.2

0 I0 20 30

Fig. 6. Concentration dependence of the frictional coefficient ratio f0/fc for the system dextran 70-water: (a) results from self-diffusion coefficients, (b) results from sedimentation coefficients.

for polystyrene in semidilute solutions by combining mutual diffusion and os- motic pressure data and found it to be equal to (f~lf,.)~ over a broad interval in concentration in both good and poor solvents. It is questionable, however, in view of the magnitude of the thermodynamic contribution at high concentrations, whether such an indirect approach can yield an unambiguous demonstration of the equality of these frictional ratios as claimed in ref. 5. Since the initial submission of this manuscript, Callaghan and PinderZ0 have presented self- diffusion data for polystyrene ( M = 110,000) in toluene determined by the FT pulsed-field-gradient NMR technique. These authors also find that self-dif- fusion coefficients do not agree with those derived indirectly from sedimentation or mutual diffusion at finite concentration. Callaghan and Pinder, also comment that the equations used by Roots et al.5 are only valid a t infinite dilution and are unsuitable for the use in the semidilute regime. It may finally be noted that conversion of D:,5ex~20 to DPe,m20 by means of the relationship

yields a value of ( f ; ) O which is identical to ( f S ) O , in accord with theoretical pre- dictions a t infinite dilution. It is concluded that the frictional coefficients in self-diffusion and sedimentation (and mutual diffusion) are only equal at infinite dilution and that they diverge with increasing concentration. As emphasized by Callaghan and Pinder,20 there are distinctly different mechanisms governing

1’780 BROWN, STILBS, AND JOHNSEN

sedimentation (or mutal diffusion) and self-diffusion prevailing under conditions of thermodynamic equilibrium in which polymer molecules exchange posi- tions.

The authors wish to thank the Swedish Natural Science Research Council and Swedish Forest Products Laboratory, Stockholm, for financial support.

References

1. H. Yamakawa, Modern Theory of Polymer Solutions, Harper and Row, New York, 1971. 2. G. D. J. Phillies, J. Chem. Phys., 67,4690 (1977). 3. H. G. Kitchen, B. N. Preston, and J. D. Wells, J . Polym. Sci. Polym. Symp., 55,39 (1976). 4. J. A. Marqusee and J. M. Deutch, J. Chem. Phys., 73,5396 (1980). 5. J. Roots, B. Nystrom, B. Porsch, and L.-0. Sundeliif, Polymer, 20,337 (1979). 6. P. Stilbs and M. E. Moseley, Chem. Scr., 15,176 (1980). 7. T. L. James and G. G. McDonald, J. Magn. Reson., 1 I , 58 (1973). 8. P. Stilbs, M. E. Moseley, and B. Lindman, J. Magn. Reson., 40,401 (1980). 9. P. Stilhs, J. Colloid Interface Sci., 80,608 (1980); 87.385 (1982).

10. B. Nystrom, M. E. Moseley, W. Brown, and J. Roots, J. Appl . Polym. Sci., 26,3385 (1981). 11. B. NystrBm, M. E. Moseley, P. Stilbs, and J. Roots, Polymer, 22,218 (1981). 12. M. E. Moseley, Polymer, 21,1479 (1980). 13. K. H. Keller, E. R. Canales, and S. 1. Yum, J. Phys. Chem., 75,379 (1971). 14. J. H. Wang, J. Am. Chem. Soc., 76,4755 (1954). 15. J. S. Mackie and P. Meares, Proc. R. Soc. London Sec. A , A232,498 (1955). 16. P. G. de Gennes, Macromolecules, 9,587,594 (1976). 17. P. G. de Gennes, Nature, 282,367 (1979). 18. H. Hervet, L. Leger, and F. Rondelez, Phys. Reo. Lett., 42,1681 (1979). 19. P. T. Callaghan and D. N. Pinder, Macromolecules, 13,1085 (1980); 14.1334 (1981). 20. P. T. Callaghan and D. N. Pinder, Polym. Bull., 5,305 (1981). 21. E. V. Goldarnrner and H. G. Hertz, J. Phys. Chem., 74,3734 (1970). 22. W. Brown and R. M. Johnsen, Polymer, 22,185 (1981). 23. F. Brochard and P. G. de Gennes, Macromolecules, 10,1157 (1977). 24. J. Roots and B. Nystrom, J. Polym. Sci. Polym. Phys. Ed., 19,479 (1981). 25. B. Nystriim and R. Bergman, Eur. Polym. J., 14,431 (1978).

Received September 10,1981 Accepted March 25,1982