2838
J. Phys. Chem. 1981, 85, 2838-2843
3000
2600 T, K
2200
1800
0
.2
.4
.6
.8
Xcao
Flgure 3. Calculated CaO-AIO1,S phase diagram.
or free energy function (Planck function) values for the solid calcium aluminates to match the known liquidus for CaA1407must be increased curves. For example, Ylooo from 302.5 to over 310 J mol-’ K-’ to show it as a stable solid in the phase diagram. 313.418 J mol-’ K-’ gives the best fit. The Planck function at 1000 K for Ca3A1206was
decreased by 1.181 to 326.687 J mol-l K-’ to obtain the liquidus curve plotted in Figure 3, but the adjustment required for CaA1204was negligible. The final thermodynamic values accepted for the solids are listed in Table I. The agreement of some of the curves could be improved by allowing for some nonstoichiometry in the solid phases as suggested by Edmund and Taylor7for CaA1407,but this refinement does not seem to be required. &Aluminas generally show variable composition, but we have used stoichiometric CaAllzOlgand a number of estimates in obtaining the values shown for this solid in Table I and in the calculations for Figure 3. We estimated Hlm (Hlooo - H2Qs),(Slooo - SzQs), and C, as the sum of the corresponding values for C d 4 0 7 and 8 A101.6(c,gamma). The phase diagram calculation using these estimated values yielded Ylooo= 753.3, and thus SZgs= 424 J mol-’ K-’ for CaAll2Olg. The calculated liquidus curve for CaAllzOIQ leaves CaA1407with a congruent melting point, but a small correction for nonstoichiometry in CaAll2Olg would correct this. It would be nice to treat calcium /3-alumina (CaAllzOlg)with Redlich-Kister coefficients as we have done for the K00,6-A101.6system,26but without further data on the range of variability in the composition this refinement does not seem warranted. Acknowledgment. This work was supported by the U.S. Department of Energy, Office of Magnetohydrodynamics, under Contract No. ET-78-C-01-3087(SUB-9-AOOl-C). (25)I. Eliezer and R.A. Howald, High Temp. Sci., 10,l (1979).
Translational Diffusion Coefficient of Macroparticles in Solvents of High Viscosity George D. J. Phiilles DePelrmnt of Chemlstty, Unlversm of Mlchlgan, Ann Arbor, Mkhigan 48 109 (Received: March 10, 198 1; In Final Form: June 8, lQ81)
Quasi-elastic light-scattering spectroscopy was used to measure the mutual diffusion coefficient D of bovine serum albumin and of 0.091-pm polystyrene latex spheres in water-glycerol and water-sorbitol as a function of solvent composition and temperature. In a given solvent mixture, D-’ is linear in the viscosity/temperature ratio q/T (as varied by changing T ) , even for viscosities 2100 cP. While comparison of D for macroparticles in solvents of differing composition is less accurate than comparison of D for the same sample at different temperatures, to within 10% D-l/ (q/T ) is independent of solvent composition.
Introduction A quantitative connection between the translational diffusion coeffcient D of a spherical Brownian particle and the viscosity q of the solvent suspending the particle was suggested by Einstein,’ who proposed D = KBT/(6aqr,) (1) KBT being the thermal energy and rs the particle’s hydrodynamic radius. The quantity 6a is the numerical factor from Stokes’ law which assumes “stick” boundary conditions; i.e., the fluid at the surface of the Brownian particle is assumed to be at rest with respect to the particle. Einstein’s application of Stokes’ law-originally obtained for a macroscopic particle moving with constant velocity
in a continuum-to the thermal motions of a Brownian particle is open to some reservation. Equation 1 and its counterpart
D R = KBT/(8aqr,3)
as proposed by Debye2 for the rotational diffusion of a sphere, have been the subject of extensive experimental examination. Even with pure water as the solvent, the classical boundary spreading technique takes many hours to obtain the mutual diffusion coefficient of a protein or other macromolecule. In solution of high viscosity, weeks or months might be required to measure D for a large molecule by this technique; not surprisingly, classical mea~~
(1)A. Einstein, Ann. Phys. ( L e i p i g ) , 17, 549 (1905).
(2)
~
(2)
P.Debye, Ber. Dtsch. Phys. Ges., 15,777 (1913).
0022-365418112085-2838$01.25/0 0 1981 American Chemical Society
The Journal of Physical Chemistry, Vol. 85, No. 19, 198 1 2830
Diffusion Coefficient of Macroparticles
surements of D in viscous solvents are limited to studies of the motion of ions and small molecule^.^ By using Mossbauer spectroscopy, Singh and Mullen4 measured D for macroparticles in highly viscous solvents. However, most Mossbauer studies@have emphasized measurements on s m d (r < 4 A) species in very viscous (7= 1o2-1ol3cP) liquids. With the advent of quasi-elastic light-scattering spectroscopy, it has become practical to measure diffusion coefficients of as little as W0cm2 s-l, permitting one to study systems which are not accessible to boundary spreading methods. While the success of the StokesEinstein formula for D,at least for macromolecules in solvents of low viscosity, suggests what values of D will be found in more viscous solutions, the paucity of experimental data motivates further experimental study of D for macromolecules diffusing in solvents of viscosity 1-500 cP. Here the mutual diffusion coefficient for several species of macroparticles dissolved in highly viscous, essentially continuous (i.e., solvent molecules much smaller than solute molecules) solvents is reported. Experimental Section The macromolecular probe species used here were bovine serum albumin (BSA) (Sigma Chemicals, essentially fatty acid free) and 0.091-pm polystyrene latex spheres (Dow). To prevent slow aggregation of the polystyrene latex, 0.4 g L-* of sodium lauryl sulfate was included in the solvent mixtures used for the spheres. Charge effects in BSA solutions were suppressed by the addition of NaCl to the solvent mixtures used for BSA, to a final concentration of 0.15 M. Measurements were performed on polystyrene-sphere solutions of concentrations 4 X by volume and on BSA a t a concentration of 25 g L-'. Our light-scattering spectrometer is of conventional design. The sample was illuminated by a 25-mW He-Ne laser; in some cases, neutral density filters were used to attenuate the incident beam. The light scattered through 90° was isolated by a pair of irises and a collecting lens which focused parallel rays of scattered light at the entrance pinhole of the detector, which was an RCA 7265 photomultiplier tube. Solutions being studied were held in black-anodized aluminum microcells of 0.2-mL capacity. Temperature was controlled to -0.1 OC by means of a regulated bath/circulator pumping coolant through a massive brass cell-mounting block. The detector was always operated in the homodyne mode; the laser intensity and the iris diameters were adjusted to keep the count rate below 700 000 photocounts s-l. The intensity-intensity autocorrelation function of the scattered light was determined by means of a 64-channel Langley-Ford Instruments digital correlator. The correlator time base (channel width) was adjusted so that the correlation time ( l / e time) was -10-25 channels. The actual channel width varied from 2.0 p S (for serum albumin in water) to 10000 p S (for the 0.091-pm polystyrene spheres in 65% sorbitol-water). Data analysis was done on a Data General Nova 3 computer by means of the method of c u m u l a n t ~ . The ~ data (including a base line B measured from a 64-channel delay placed between channels 56 and 57) were assumed to be of the form
+
C ( 2 ) ( ~ ) B = S d I ' A(I')e-rr
+B
(3) E.g., T. G. Hiss and E. L. Cussler, AZChE J., 14, 698 (1973). (4) K. P. Singh and J. G. Mullen, Phys. Rev. A, 6 , 2354 (1972). (5) A. Abras and J. G. Mullen, Phys. Rev. A, 6 , 2343 (1972). (6) P. P. Craig and N. Sutin, Phys. Rev. Lett., 11, 460 (1963). (7) D. E. Koppel, J. Chern. Phys., 57, 4814 (1972).
(3)
and were each fit to the cumulant expansion n
02)(7)
= exp( C [ (-l)nK,,~n/n!]) is0
(4)
for values of n ranging from 1to 4. For our sample, the best overall quality of fit was generally obtained for n = 2. The root-mean-square error in the fit was usually 51% of 0 2 ) ( 7 ) for the first channel. A substantial limit on the accuracy of light-scattering spectroscopic measurements of diffusion arises from the presence in the sample of dust or gas bubbles. To control these factors, the same dust-removing procedure was used for all mixtures. The polystyrene sphere-water-glycerol samples were cleaned by first passing the sphere-free solvent through a 1.0-pm Nucleopore polycarbonate membrane, serially diluting the stock samples of spheres by 104:1with filtered solvent, and then loading the diluted spheres directly into a clean sample cell. For the watersorbitol mixtures, 0.2-pm Nucleopore polycarbonate filters were used to clean the solvent. The bovine serum albumin samples were prepared by dissolving BSA in unfiltered solvent and passing the resultant solutions through a 1.0-pm filter into a clean sample cell. The sample cells had themselves been cleaned by passing through them -50 volumes of filtered (0.2 pm) distilled water and then drying with filtered N2gas. With a solitary exception noted below, the observed light-scattering spectra showed no indication of a second, very slow decay, as might have been expected if dust had entered the system. In all cases, the assumed scattering vector lkl in the experiment was corrected for the index of refraction n of the solvent, using 4m 0 Ikl = - sin x 2 A being the wavelength in a vacuum of the incident light and 0 being the scattering angle. Indexes of refraction were determined as a function of temperature with a Bausch and Lomb Abbe-56 refractometer. The temperature correction for n over the range 0-50 "C, due substantially to the thermal expansion of the solvent, was never as great as 1%. Viscosities were obtained by the use of a series of calibrated viscometers which were mounted in a water bath stabilized to 0.1 "C. Measurements of the elapsed time in a viscosity determination were reproducible to better than 1% The viscosities of solutions containing sodium lauryl sulfate or NaCl were determined separately before adding the corresponding macromolecular solutes. An extensive search was made for a background spectrum arising from scattering by any of the solvent mixtures used. Samples were prepared in the usual way except that no BSA or polystyrene-sphere suspension was added. With our laser power and collecting angles, the light scattered from these solvent blanks showed (with one exception) no appreciable quasi-elastic scattering spectrum.
.
Results Macromolecular Probes in Aqueous Glycerol. The mutual diffusion coefficient of 0.091-pm polystyrene spheres suspended in water-glycerol-0.4 g/L sodium lauryl sulfate was determined at glycerol concentrations of 0,19, 38,57,67,70,86, and 91 w t %. D was measured over the range 0-50 OC at -5 OC intervals. For each sample at each temperature, three spectra were taken. For each glycerol concentration (except 86%), at least two samples were prepared on different days, so our results on D include more than 500 separate measurements. To search for possible time-dependent slow aggregation effects, we made
2840
The Journal of Physical Chemistry, Vol. 85, No. 19, 1981
Phlllles 19% f6 86%
0
1 9 % Glycerol 86 % Glycerol
t 4
2
D-'
400
I
200
I
0
t 0,l
0
.XI2
.4
10
20
30
40
50
60
T ("C) Flgure 1. Viscosity of water-glycerol-sodium lauryl sulfate as a function of temperature and glycerol concentration, XG. The curve for X, = 0.80 is interpolated from Herz and Wegner (ref 8 and 9).
alternate series of measurements beginning near 0 and 50 "C. No significant repeatable differences were observed between data taken at successively higher temperatures from 0 to 50 "C and at successively lower temperatures from 50 to 0 "C. For each water-glycerol-sodium lauryl sulfate solvent system, viscosities were measured at 5 "C intervals over the temperature range 0-50 "C and graphically interpolated to obtain q for each temperature at which D was measured (Figure 1). Herz and Wegner8v9report values of q for water-glycerol mixtures for 10 IT I 80 "C at glycerol concentrations as high as 92 w t %. Our results agree with theirs to within 2%. This disagreement may reasonably be explained in terms of a 0-2% systematic error in estimating the amount of water in each solution. Since it is notoriously difficult to prepare anhydrous glycerol, such a variation is not unreasonable. A t each nominal concentration we compared D with q for the same batch of solvent, so this systematic uncertainty does not enter into our results. D for 0.091-pm spheres in the 0% glycerol system ranges from 0.27 (at 1.9 "C)to 0.87 (at 50 "C). (All values of D are in units of lo-' cm2/s.) By increasing the glycerol concentration, one may greatly reduce D. At 21 "C,D values of 0.45,0.108,0.024, and 0.0019 were found for the spheres in the 0, 38, 77, and 91 w t % glycerol systems, respectively. As expected, viscosity decreases with increasing temperature and increases with increasing glycerol composition. Over the temperature range 10-50 O C , viscosity falls from 1.38 to 0.58 cP for the 0 wt % glycerol solution, from 4.6 to 1.6 cP for the 38 wt % glycerol solution, and from 150 to 36 cP for the 91 wt % glycerol solution. Figure 2 exhibits representative data on the diffusion of 0.091-pm spheres in particular solvent mixtures. D-' has been plotted against TIT. For each example, above (8)W. Herz and A. Wegner, Z. Dtsch. Oel- Fett-lnd., 45, no. 5, 1 (1926). (9)W.Herz and A. Wegner, Z.Dtsch. Oel- Fett-lnd., 45,401 (1925).
I
,004
VT
86%
.6
I
19%
,006
(dK)
Figure 2. D-' (In units of I O 7 s cm-*) as a function of v/T(ln CP K-') for 0.091-pm polystyrene spheres in aqueous glycerol solutions, at glycerol concentrations of 19 and 86 wt % . The sphere concentratlon Is by volume. Note the different scales for the two sets of data. The two systems showing the greatest difference In slopes d(D-')/d(q/ T) were chosen for this figure. 1
1
1
1
I
I
I
Figure 3. log (17') (D-' in units of I O 7 s cm-*) against log ( q / T )(q/ T in units of CP K-l) for 0.OQl-pmpolystyrene spheres In water-glycerol. The straight line is the theoretical prediction (no free parameters) of the Stokes-Einstein equation (eq 1). Points are averages of two to four measurements.
15 "C, D-l is linear in q/T. Below 15 "C,there is more scatter in the data. This scatter appears to have been an experimental artifact which did not arise during the studies on water-sorbitol. Figure 3 shows the complete measurements of D-l for polystyrene spheres in aqueous glycerol as a function of q / T. To compress all of the results into a single figure, we have used a log-log plot. Each symbol is an average of the two to four (usually three) measurements of D on a given sample at a given temperature. The straight line, slope of unity, is the theoretical
The Journal of Physical Chemistry, Vol. 85, No. 19, 1981 2841
Diffusion Coefficient of Macroparticles I
I
I
I
----
I
-
100.0
-
10.0-
7) (CP) LO-
c
Flgure 5. Viscosity of aqueous sorbffol as a functlon of temperature and weight fraction X s of sorbitol. .06
.02
1.0
I
I
I
I
I
I
I
I
~
I
Fi ure 4. D-' (in units of IO7 s cm-') as a function of qlTfor 25 g L- bovine serum albumin in aqueous glycerol solution at a glycerol concentration of 60 w-i %. Comparison with other data indlcates that the nonzero Intercept is due to noise.
B
prediction of eq 1 for 0.091-pm spheres. The results for D of BSA are qualitatively the same as those for D of the 0.091-pm spheres. q was measured for water-glycerol-0.15 M NaCl and D of BSA dissolved in this mixed solvent was obtained for glycerol concentrations of 0, 20, 40, 60, 80, and 95 wt %. Good agreement was found between our q values and those of Herz and Wegner, which were therefore used to interpolate between our measurements. Because BSA denatures at high temperature, D was measured in ascending and not in descending temperature series. Apparent denaturation of BSA (as evidenced by a fall of D below its predicted value D = DoTqo/(qTo)) was seen near 45 "C; observations on apparently denatured BSA were not included in our quantitative analysis. The scattering intensity from the serum albumin samples was substantially less than that from the polystyrene spheres. For each sample at each temperature, two spectra were obtained with a 20-min averaging time for each. These results, which are based on more than 150 measurements of D, are represented by Figure 4. The spectrum of BSA in 95 wt % glycerol at low temperatures exhibits a substantial slow decay in addition to the rapid decay which we studied. Samples at lower glycerol concentrations did not show this effect. The same procedures were used to remove dust in different solvents, so sample contamination seems unlikely to be responsible for the slow exponential. Relaxations due to the interdiffusion of water, glycerol, and NaCl ought to occur at higher frequencies than the solvent-polystyrene interdiffusion, but the extra spectral term noted here is seen at lower frequencies. For each BSA sample at temperatures below the denaturation temperature, D-' is linear in q/T. Macromolecular Probes in Aqueous Sorbitol. The mutual diffusion coefficient of 0.091-pm polystyrene latex spheres dissolved in aqueous sorbitol was measured over the temperature range 5-50 "C at sorbitol concentrations of 5, 12, 42, 55, 62, and 65 wt %. Figure 5 gives the viscosities of each of these solutions over the 5.0-50.0 OC range; q changes from 0.63 cP for the 5 wt % solution at 50 "C to 176 cP for the 65 wt % solution at 5.5 "C.
0.1
-
77T
.o I
i
Figure 6. log (D-') (D-' in units of IO7 s cm-2) against log (q/T ] (q/T in units of CP K-') for 0.091-pm polystyrene spheres in water-sorbitol. The straight line is the prediction of the Stokes-Einstein equation. Points are averages of three measurements.
Under the experimental conditions used to study the polystyrene suspensions, of the sphere-free mixed solvents only the 65 wt % sorbitol solution exhibits an appreciable light-scattering spectrum. At 15 "C the 65 wt % solution has a very weak spectrum (less than 1% of the scattering intensity of the sphere preparations), implying an apparent mutual diffusion coefficient of this solvent of 4.5 X 10-lo cm2s-l at 15 "C and 5.6 X 10-l' cm2 at 5 "C. This weak scattering could not have affected measurements of D appreciably but might have enhanced the second spectral cumulant at low temperatures. Data on sphere diffusion were obtained in both ascending and descending temperature series. Series of measurement taken on one sample starting at 5 "C sometimes showed a substantially broader scatter than those series obtained by descending from 50 O C . In a few cases, a search was made for hysteresis in D vs. T during a cooling-heating cycle; no hysteresis effects were observed. Our data are given in Figure 6, which graphs log D-' against log (q/Z'). The points are averages of three
2842
The Journal of Physical Chemistw, Vol. 85, No. 19, 1981
Phiiiies
(sometimes four) measurements; the straight line is the theoretical curve generated by eq 1. The scatter of data around the line largely reflects sample-to-sample variation. If one compares D-' with q/T for a single sample, the scatter around a linear fit is much smaller than the scatter seen in Figure 6.
SPHERES
Discussion Here we have studied the mutual diffusion coefficient of macroparticles (radii of -35 and 455 A) while changing q/T by more than 3 orders of magnitude. As seen in Figures 4 and 6, to high accuracy D-l is linear in q / T over a wide range of temperatures and solvent compositions. For each sample the temperature dependence of D-' is given by
7
D-1
=qq/T)
+
(7)
The corresponding form for the rotational diffusion time T is widely used.1G13 A random scatter of intercepts Do-' around D0-l = 0 was found. While Do-' is more often negative than positive, the magnitude of the intercepts is comparable to the scatter in the measured values of D-' so we find no clear evidence that Do-' # 0. For a series of nominally identical preparations, the variation in r around its average is from several to 10%. This sampleto-sample variation, which probably stems from the difficulty in clarifying highly viscous solutions, is substantially greater than the other sources of noise in our results. Figure 7 plots I' for BSA or spheres in water-glycerol against the composition of the solvent. For the polystyrene spheres, Figures 3,5, and 7 represent the same data. While the scatter is appreciable, for either solvent r is substantially independent of the solvent composition. The dashed line in Figure 7 indicates r, for a 0.091-pm sphere as calculated from eq 1. At XG = 0, the data show r = 660, (10)G. R. Alms, D. R. Bauer, J. I. Brauman, and R. Pecora, J. Chern. Phys., 58, 5570 (1973). (11) G. R. Alms, D. R. Bauer, J. I. Brauman, and R. Pecora, J. Chern. 59. 6310 -Phvs.. I - . - - ~ (1973). - - 7
(12)G. D. J. Ph-aeS and D. Kivelson, J. Chem. Phys., 71,2575(1979). (13) W.Lempert and C. H. Wang, J. Chern. Phys., 72,3490 (1980). (14)B. Berner and D. Kivelson, J. Chem. Phys., 83, 1401 (1979).
BSA I
103
I
I
1 too
5001
o+o
400
2o 40
0 ,091,~ spheres in water 0 BSA in water glycerol
(6)
to within 5%. Comparison of D-l in solvents of different composition indicates that I' is independent of composition to within 11%(outer limit of error) in water-glycerol and to within 7% in water-sorbitol. The literature results most closely related to those reported here are those of Singh and Mullen: who studied the motion of colloidal CoSn(OH)6 (radius, 650 f 150 A) in glycerol, using Mossbauer line broadening to determine D of the colloid particles. Their study was in substantial part intended to test theoretical calculations of the diffusive contribution to Mossbauer line broadening; for light scattering the relation between mutual diffusion and line width is not presently in doubt. For 0.5 < q/T < 25 CPK-l, Singh and Mullen found D a (T/q),l a result consistent with our own. The results here contrast with those of Singh and Mullen in that this paper covers a different viscosity range (2 X < q/T < 0.6 CPK-l), uses a different physical technique (lightscattering spectroscopy) to measure D, and studies a wider range of solvent compositions and particle sizes. A quantitative analysis of the dependence of D-l on q/T was also made by least-squares fitting the data taken on each sample to the form D-' = r(q/T) Do-1
I
loo
00
glycerol
Q ,091,~ spheres in water:sorbitol
02
04
X
06
0,8
20
I O
Figure 7. Results obtained by fitting D-' for each sample to D-' = rq/ T Do-', and Do-' being independent free parameters. XIS the weight fraction of glycerol or sorbitol In the solvent. The points represent averages over as many as four distinct samples.
+
r
which is slightly larger than the theoretical value Fa = 621. For the spheres, values of I? at XG > 0 are scattered around the theoretical value but tend to be slightly larger. I t is weli-known that the hydrodynamic radius for polystyrene latex spheres, as obtained by light-scattering spectroscopy, is slightly larger than the spheres' nominal radius as found by other techniques, so the difference between r and rth is not unexpected. This disagreement is usually attributed to the fact that most techniques report a number- or molecular-weight-averaged value for the particle size, while D from light scattering is (crudely speaking) weighted by the square of the molecular weight (i.e., by r:). However, to my knowledge, no detailed quantitative study has been made to confirm this widespread attribution. Recent studies of the viscosity dependence of the rotational diffusion time 7 R of small molecules in solvents of varying temperature and composition have obtained disparate results. Alms et a1.loJ1 find that rR for several different probes (nitrobenzene, toluene) dissolved in a series of nonpolar solvents at a fixed temperature satisfies 7~ = rT/T + 70 (8)
-
where r and r0 depend on the probe concentration. 70,the 0, is comparable to the limiting value of TR as T/T thermal rotation time (KBT/l)-1/2 of the probe, but the physical significance of this coincidence is not obvious. Phillies and Kivelson12report 7R as a function of temperature for nitrobenzene in several polar solvents. When q/T was varied by changing T in systems of fixed composition, 7 R fitted eq 8 with 70 = 0 to within experimental error. Phillies and Kivelson also found that r depends on the solvent and on the probe concentration. Lempert and Wang13 report rR for pure nitrobenzene and nitrobenzene-CC4 mixtures between -10 and 40 OC, finding that, as T is changed in each system, 70 = 0, with r dependent on the probe concentration. For translational motion in our polar solvents, we find that I' is nearly the same for all solvent compositions. Graphs (Figures 2 and 4) of D-l against q/T, as obtained by varying T at constant X G or Xs,show no sign of an q/T 0 intercept for D-l. Since is not systematicallydependent upon composition, a graph of D-' against q/T at fixed T, by analogy with the
-
J. PhyS. Chem. 1981, 85, 2843-2851
-
graphs of Alms et al., will give a vanishing intercept for D-I as q/T 0. The viscosity dependence of the motion of small ions and intermediate-size molecules has been studied by measuring both their diffusion coefficients and their limiting conductances A,, at zero concentration. Berner and Kivelson4used paramagnetic enhancement of nuclear spin relaxation to determine the translational diffusion coefficient of di-tert-butyl nitroxide in ethylene glycol, water, and other solvents, confirming the Stokes-Einstein equation (eq 1)to within a factor of 4 as q varied over 4 orders of magnitude. For simple ions and small molecules moving in a solvent whose molecules are bigger than those of the solute, at high viscosity a relation for the drag coefficient f of the form f a qa for 0.63 < a < 0.70 has long been s~ggested.'~This form is found both for A,, of simple electrolytes in sucrose16and mannitol1' solutions and for the diffusion coefficient of n-hexane and naphthalene in compound oils with q > 10 c P . ~We have studied the motion of macromolecules, finding for q > 10 cP that a = 1.0 to within experimental error. The Stokes-Einstein equation (eq 1) is properly obtained for a single Brownian particle, not for a solution in which interactions between Brownian particles are significant. Conventional theories18indicate that corrections where for hard to eq 1 take the form D(c) = D ( l + aC#J), spheres a is a constant of order 1 and C#J is the volume fraction of the solute. For the polystyrene suspensions, (15)W.Heber Green, J. Chem. Soc,, 98,2023 (1910). (16)Jean M. Stokes and R. H. Stokes, J. Phys. Chem., 60,217(1956). (17)Jean M. Stokes and R. H. Stokes, J. Phys. Chem., 62,497(1958). (18)B. U.Felderhof, Physica A (Amsterdam),89,373 (1977);B. U. Felderhof, J. Phys. A , 11, 929 (1978).
2843
on which the bulk of our discussion rests, 9 = 4 X W ,so D for these systems should be indistinguishable from its infinite dilution limit. The concentration dependence of
D for polystyrene spheres in plain water solution has been examined experimentally; if multiple scattering is suppressed by means of homodyne coincidence techniques,l8 D is independent of concentration for 4 as large as 0.01.20 The diffusion coefficient of isoionic serum albumin in solutions of moderate ionic strength is only weakly dependent on concentration.21 While it is sometimes said that glycerol tends to stabilize protein conformations, there do not appear to be any data on conformational changes of serum albumin in concentrated glycerol. Such changes, if they exist, would obscure any interpretation of the results presented here on BSA diffusion. Our discussion therefore emphasizes measurements made on the polystyrene-sphere suspensions, since no polar, hydrogenbonding solvent seems likely to affect the size of the spheres. In conclusion, it seems unlikely that interactions between the macromolecular probe molecules have a significant effect on the results presented here. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. Additional support was provided by Grant Che-79-20389from the National Science Foundation. We thank Mr. Richard Finegold, Mr. Larry Chin, Miss Katie MacNamara. and Mr. Todd Werner for their technical assistance a t various steps in this research. (19)G. D. J. Phillies, J. Chern. Phys., 74, 260 (1981). (20)G.D. J. Phillies, Phys. Rev. A, in press. (21)G. D. J. Phillies, G. B. Benedek, and N. M. Mazer, J. Chern. Phys., 65,1883 (1976).
Small Ionic Clusters of LiC104 in Low Dielectric Constant Media from Conductivlty Data at 293 K M. Nlcolas" and R. Relch Laboratoire de Physique des Solhles. Universlt6 Paris-Sod, Centre d'Orsay, 9 1405 Orsay, France (Recelved: March 11, 1980; In Final Form: February 13, 1981)
The molar conductivity of LiC104,AM,is measured in various solvent systems, over a wide range of concentration (lov55 c 5 saturation) and dielectric constant (2.20 5 e 5 33). The influence of the solvent mixtures (POlar-nonpolar, protic-aprotic) on ion pair and triple ion formation is investigated. The KT variations are found to be linear functions of ( c - ~ ) ~for , the four studied systems, provided e R 3.00. The sharp rise of AM d c 5 1M) is attributed to small ionic cluster formation. For a given solvent system, all tangents to the log Awlog c curves converge to the same point, W. The model suggested by the authors leads to the conclusion that the maximum number, n,of LiC104-neutral pairs, bound to a free ion Li' or ClO, to give a cluster, is determined by the dielectric constant of the medium, for a given solvent system.
Introduction In the last few years, there has been renewed interest in the nature of ionic species present in low dielectric constant solvents. The use of various organic mixtures allows the effect of the dielectric constant on the solute behavior to be studied. In particular, for dilute solutions of salts in high dielectric constant solvents, ion-solvent interactions take place, which depend strongly on the 'Laboratoire associ6 au CNRS. 0022-3654/81/2085-2843$01.25/0
nature of the solvent: polar or nonpolar, protic or aprotic; the cations and/or the anions may be more or less solvated depending on the case. If the dielectric constant of the medium is sufficiently low, complex ions may be formed. New studies of cluster formation have recently been carried out, especially for very small charged1p2 or uncharged3t4 (1)T.P. Martin, J. Chem. Phys., 69,2036 (1978). (2)T.P. Martin, J. Chem. Phys., 72,3506 (1980). (3)J. W.Brady, J. D. Doll, and D. L. Thompson, J. Chem. Phys., 71, 2467 (1979).
@ 1981 American Chemical Society
