J. Phys. Chem. 1882, 86,4073-4077
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and 5% depending on the precise values of Gruneisen molar compressibilities of the two benzenes at infinite constants employed to define the volume dependence of dilution in C6H12,APc/(XBXc) = AVO(1 - PmHVmH/ the lattice frequencies, and the relations defining the ~ " D V ~ DThe ) . limiting partial molar compressibility of benzene at infinite dilution in cyclohexane, OmH, is more isotope dependence of the Gruneisen constants. In other than 10% larger than PH(the compressibility of pure words, the excess volume from effect I1 as estimated in eq 5 is consistent in both sign and magnitude with the analysis benzene)22and it follows from model calculations2 that of the data in Figure 2. An experimental value for OH O m ~ / / 3 "is~ larger than, but still of the same order of OD would be of considerable value in delineating the This is a consequence of the larger magnitude as, PH/PD cavity which benzene enjoys at infinite dilution in C6H12 physical origins for the excess volume effect. We are presently engaged in the construction of experimental as compared to benzene itself. The change in the volume apparatus to determine the isotope dependence of 0 in ratio, while of the same sense, is not as marked. The order to further test the thermodynamic framework.24 analysis leads to the conclusion that the excess volume isotope effect should be negative and display a significant Effects in C&12/c& and c&12/c&6Solutions. The temperature dependence in agreement with the present isotope effects for excess volumes on this system, data (Figure 1). More quantitative comparisons await Pmd, - PvdF = A P are shown in Figure 1. The effects are negatwe. o asymmetry due to effect I is found experimental measurements of the partial molar comwithin the experimental precision, which is about *20% pressibility isotope effect which are planned for the near future (vide infra). of the measured effects. However, it must be emphasized O C ~ ~ ~ that, because the differences, vOcd12- VOc and ~ Acknowledgment. This material is based upon work - VOcsD6, are about 100 times larger than - vOcsD6, supported by the National Science Foundation under extremely precise data defining the limiting curvatures of the eXCeSS Volumes Of the C6H12/C& and C ~ H ~ Z / C ~ DGrant ~ CHE 81-12965. solutions at high benzene concentration are required to define the asymmetry. At high C6H12 to the extent that (22) Kiyohara, 0.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodyn. this contribution is defined by the host lattice alone, effect 1978, 10, 721. (23) Zwolinski, B., et. al. "SelectedValues Propertie of Hydrocarbons"; I vanishes. Within the present experimental uncertainty American Petroleum Institute: College Station, TX, 1971. it is therefore useful to ascribe A P as arising only from (24) Note Added in Proof. Preliminary measurements of the comeffect 11, and the A P difference can therefore be appressibility isotope effect give (& - &)/& = 0.010f 0.002 at 15,25, and proximated (eq 5 ) in terms of AVO and the limiting partial 40 OC.
Pc,
Translational Dlffuslon Coefficlent of a Macropartlculate Probe Species in Salt-Free Poly(acrylic acid)-Water Thy-Hou Lln and George D. J. Phlllles" Department of Chemistry and Micromolecular Research Center, The University of Mlchigen, Ann Arbor, Michigan 48 109 (Received: February 26, 1982; In Final Form: March 9, 1982)
Quasi-elastic light-scattering (QELS) spectroscopy was used to measure the mutual diffusion coefficient D, of 0.038-pm diameter carboxylate-modified polystyrene latex spheres dissolved in poly(acry1ic acid) (mol wt 300000)-water solutions of concentration 0.0-171 g/L. D exhibits a complex dependence on the polymer concentration, viscosity, and temperature. The apparent hydrodynamic radius of the polystyrene spheres, as obtained from the Stokes-Einstein equation, increases markedly with increasing polymer concentration.
I. Introduction In a previous paper, one of us1 reported on the diffusion of polystyrene latex spheres and bovine serum albumin in solutions of water-glycerol and water-sorbitol of various temperatures and compositions, over a range of solvent viscosities q of 0.8-1000 cP. While at low viscosity the electrophoretic mobility A,, and diffusion coefficient D are proportional to q-l, for q 5 10 CPthe q dependence of A,, and D can be more complex. If the solute molecules are small, but comparable in size or larger than the solvent, one2y3finds D q-l. If the solute molecules are smaller than the solvent molecules, D and X arek7 -qQ, for 0.63
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(1) Phillies, G. D. J. J. Phys. Chem. 1981, 85, 2838. (2) Singh,K. P.; Mullen, J. G. Phys. Rev. A. 1972,6, 2354. (3) Bemer, B.; Kivelson, D. J.Phys. Chem. 1979,83, 1401. (4) Green, W. Heber, J. Chem. SOC.1910,98, 2023. (5) Stokes, Jean M.; Stokes, R. H. J. Phys. Chem. 1966,60, 217. 0022-3654/02/2006-4073$01.25/0
< a < 0.7. For the systems studied in ref 1, it was found that D depends on q and temperature in accordance with the Stokes-Einstein equation regardless of whether q was changed by varying the temperature or the solvent composition. Here ro is the hydrodynamic radius of the probe species. This difference between the results of ref 1-3 and 4-7 suggests that it would be interesting to use light-scattering spectroscopy to study diffusion in highly viscous systems as solute and solvent sizes are gradually made more similar. Making the sizes more similar by using smaller macroparticulate probes would lead to significant experimental (6) Stokes, Jean M.; Stokes, R. H. J. Phys. Chem. 1958, 62, 497.
(7) Hiss, T. G.; Cussler, E. L. MChE J. 1973, 14, 698.
0 1902 American Chemical Society
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difficulties due to the ensuing reductions in the lightscattering cross sections and the spectral correlation times of the probes. We instead elected to make solute and solvent of similar size by making the solvent molecules bigger, namely, by replacing the small-molecule solvents with an approximately chosen water-polymer solution. In this new mixed solvent, some though not all of the solvent molecules have the same dimensions as the probe solute. We are also motivated by the recent revival of theoreticals and experimental@interest in polymer diffusion. While some of this worklo has recently been subject to criticism," it is clear that additional experimental data on polymer solutions using modern techniques may aid in demonstrating or rejecting particular theoretical models. The polymers systems studied here correspond to the three-component solutions whose diffusion'* and lightscattering spectral3have previously been analyzed. In such three-component solutions, it will, in general, be true that a concentration gradient in either solute will lead to a diffusion current in the other solute species, potentially resulting in complex scattering spectra. Here we have taken advantage of two physical simplifications. First, the "solvent" polymer scatters very little light, so that the observed spectra are due almost entirely to scattering by the probe species. Second, the probe concentration in the experiments below is so small that probe-probe interactions are virtually nonexistent, so that probe diffusion may be attributed entirely to interactions between the probe particles and the medium. In this way, in a system containing several Brownian species, the motions of the particles of one species may be used to reveal the physical nature of the medium around them. Our experimental procedures are discussed in section 11. Section I11 of this paper presents the experimental findings. A discussion and comparison with related literature results will be found in section IV. 11. Experimental Methods The experimental methods used here are largely identical with those of ref 1. Solvent viscosities were measured with a series of calibrated Ubbelohde and Cannon-Fenske viscometers, determinations of 7 being reproducible to better than 1% Measurements of the viscosity were made at 5 "C intervals over the temperature range 0-50 "C and graphically interpolated to the precise temperatures at which the diffusion measurements were performed. Poly(acry1ic acid) (mol wt 300000) was obtained (Polysciences, Inc.) as the 25 wt % solution and progressively diluted with 18MQdeionized water to make the individual solvent systems studied. After some initial experiments (not reported in detail here), the 0.091-pm polystyrene latex spheres and black-anodized aluminum (light-scattering) cells of the previous study' were replaced with 0.038 I.rm carboxylate-modified polystyrene latex spheres (Dow Chemical) and all-glass fluorimeter cuvettes (four sides polished). The sphere concentration was uniformly held at -5 X lo4 by volume. The 0.091-pm polystyrene spheres and the aluminum cells were replaced because the spheres showed rapid and irregular aggregation, even after substantial amounts of sodium dodecyl sulfate were added to the solvent. The carboxylate-modified spheres in the
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(8)Freed, K. F.; Edwards, S. F. J . Chem. Phys. 1974,61,3626. (9)Adam, M.;Delsanti, M. Macromolecules 1977,10,1229. (10)deGennes, P. G. "Scaling Concepts in Polymer Physics"; Comell University Press: Ithaca, NY,1979. (11)Patterson, G. D.; Jarry, J.-P.; Lindsey, C. P. Macromolecules 1980,13,668. (12)Kirkwood, J. G.; Baldwin, R. L.; Dunlap, P. J.; Gosting, L. J.; Kegeles, G. J. Chem. Phys. 1960,33, 1505. (13) Phillies, G. D. J. J. Chem. Phys. 1974,60,983.
Lin and Phillies
glass cells showed no sign of aggregation over the periods of time used in these experiments. Diffusion coefficients were determined by means of quasi-elastic light-scattering spectroscopy, using a 25-mW He-Ne laser and 64-channel Langley-Ford Instruments digital correlator. In some cases, the incident laser beam was attenuated with a neutral density filter. The correlator time base was adjusted so that the correlation time ( l / e time) corresponded to 7-25 correlator channels. The actual time per channel varied from 10 (for a poly(acry1ic acid) concentration of 0.37 g/L) to loo00 (for a poly(acry1ic acid) concentration of 171 g/L) ps. Spectra were analyzed by means of the method of cumulants, the spectra being fitted to the form n
In [ ( I ( t ) l ( t + ~-)B) ] = 2C (-1)Xi?/i!
(2)
is0
For our data, the best fits were obtained with n = 3 or sometimes n = 2. The root-mean-square error in the fit was usually 51% of the total spectral amplitude The assumed scattering vector k was consistently corrected for the refractive index n of the solvent, n being measured for each solvent by means of a Bausch and Lomb Abbe-56 refractometer. For each solvent system, a determination was made of the intensity of the spectrum due to the polymer-containing solvent in the absence of polystyrene spheres. While the poly(acry1ic acid)-water solutions do have a measurable QELS spectrum, which could be studied by making sufficiently prolonged (2-3-h integration time) measurements, with the iris settings and laser power that we used the poly(acry1ic acid)-water scattering was negligible (51%) by comparison with the scattering due to the polystyrene spheres. The actual poly(acry1ic acid) concentrations were determined by titration with sodium hydroxide. The pH of the yet-to-be titrated, salt-free solutions was also determined. The observed pH ranged from 4.41 at a concentration of 0.18 g/L polymer to 2.60 at 23.5 g/L and 2.35 at a poly(acry1ic acid) concentration of 171 g/L. 111. Results The temperature dependence of the viscosity was studied for solutions of poly(acry1ic acid) (mol w t 300000) at polymer concentrations c of 0.37, 1.5, 5.9, 6.1, 11.7, 17.4, 17.9,34.1,35.2,52.5,86.3,117,and 171 g/L over the range of 5-50 "C. Measurements were also made of the viscosity of solutions of concentration 0.18,0.64,0.88, 1.2, 2.4, 23.5, 69.9 g/L poly(acrylic acid) at the fiied temperature 15 "C. The viscosity of these solutions ranges from 1.36 (for the 6.1 g/L solution at 50 "C) to 984 CPfor the 171 g/L solution at 5 "C. Figure l shows the temperature dependence of the viscosity for each of the solvents. The viscosity increases with increasing polymer concentration and with decreasing temperature. It is interesting to note that the temperature coefficient of the viscosity depends noticeably on the polymer concentration, ds/dT being substantially larger for pure water and for the most concentrated polymer solutions than for the intermediate (10-50 g/L) polymer concentrations. In Figure 2, the concentration dependence of the viscosity at 23.1 "C is shown. Over the range 0-1 g/L, the viscosity rises extremely sharply with increasing concentration; dv/dc then falls dramatically, so that in the range 1-10 g/L of polymer the viscosity increases by less than 50%. Increases in the polymer concentration above 10 g/L lead to disproportionately large increases in the viscosity, so that a plot of log 9 against log c is not linear in the higher concentration ranges. Patterson et al.ll have emphasized
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a Macroparticulate Probe
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T ("C) Flgm 1. Temperatwe dependence of the vlscoslty of pdy(aayllc add) (mol wt 300000) In water at concentrations of (a) 0.0, (b) 5.9, (c) 6.1, (d) 11.7, (e) 17.4, (f) 34.1, (9) 35.2, (h) 52.5, (I) 86.3, 0) 117, and (k) 171 g/L. IOOC
f IOC
1
'I (CP) IC
I
Figue 2. pbt of the concentraw dependence of the vlscoslty q of mol wt 300 000 pdy(acrylic acid) In pure water. The inset shows (in a linear-linear plot to the same unlts) the low-concentratlon dependence of ?.
that apparent breaks in log-log plots may result from the numerical propertiea of the Naperian logarithm rather than from a change in the putative scaling relation for the property under consideration. Figure 2 therefore includes (inset) a linear plot of 7 against c, showing that the apparent discontinuity in the slope near 1 g/L is real. In order to calculate the viscosity, it was necessary to find the density p of each of our poly(acry1ic acid)-water solutions. The partial volume of poly(acry1ic acid) de-
Flgure 3. Dlffusion coefflcient D (In units lo-' cm2/s) of 0.038-~m diameter carboxyiate-modmed polystyrene spheres in water-pdy(acrylic acid) solutions of various concentrations and temperatures, plotted as a function of the Inverse viscosity T / ? (in units K/cP).
termined from p is d = 0.61 cm3/g; within experimental error, d is independent of the solute concentration. The diffusion coefficient of the 0.038-pm polystyrene spheres was found in pure water and in each of the above-mentioned poly(acrylic acid)-water solutions in the temperature range over which each solution's viscosity had been determined. The measured diffusion coefficient ranges from 6.81 X 10-8 cm2/s in 0.37 g/L poly(acrylic acid) to 7.1 X 16" cmz/s in 171 g/L poly(acrylic acid). For each solvent composition and temperature, three measurements of D were made on at least two different samples. For a given sample, the spread of measurements of D, around their averaged value, remained near *3% for poly(acrylic acid) concentrations of less than 45 g/L. At higher polymer concentrations, the fluctuations in D increased, reaching &5% in the 86.3 g/L solution and f10% in the 171 g/L solution. This increase appears in significant part to be due to the difficulties in studying spectra with very large correlation times (up to about 0.1 s for the 171 g/L solution at 5 "C). The variation in D between different but nominally identical preparations was generally similar in size or smaller than the variation between repeated determinations of D in the same preparation. The cumulants fitting procedure which we use also provides the variance u = K2t2/Kl,which is a measure of the deviation of the spectrum from a single exponential. When pure water was used as the solvent, we observed u 0.25 f 0.05. u increases gradually with increasing polymer concentration. Our complete results are indicated in Figure 3, which shows the measured diffusion coefficients as a function of T/q.The straight line indicates the diffusion coefficient predicted by the Stokes-Einstein equation, using the observed diffusion coefficient of the spheres suspended in pure water to fix r,. Unlike the results found in ref 1, it is clear that D is not simply a linear function of TI?. In our previous work on polystyrene spheres in waterglycerol and watemrbitol, for a frxed solvent composition
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Lin and Phillies
5.9 g/l poly-acrylic ocid-:J
r
.06 0.6
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.04 0.4
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20\ I
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ob
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Figure 5. Apparent hydrodynamic radius r o of 190-A radius polysty~ene spheres as a function of the poly(acrylic acid) concentration in the solvent.
Figure 4. Mutual diffuslon coefflclent D (units of lo-' cm2/s) of 0.038-pm polystyrene spheres In 5.9 (open clrcles) and 52.5 g/L pdy(aaylic add), as a functbn of TI7 (units of K/cP) showing the linear relatlon of these varlables. The straight lines represent least mean square fits to eq 3, the fits Indicating hydrodynamlc radii r of 290 and 321 A, respectively. Note the use of separate scales for the two solubions, and the nonreproducble low-temperature anomaly wlth the 52.5 g/L solvent.
D was found to be a linear function of Tlq. In those solvents, fitting D for each solvent composition to the equation D = kBT/(6.1rqro)+ Do (3) gave a constant value of the hydrodynamic radius ro,which was the same in all solvents. Values for the intercept Do were scattered randomly around Do = 0. This simplicity is not entirely preserved in the present observations. While data on each system (except perhaps the 117 g/L solution) fib well to eq 3, with Do E! 0, for each solvent system a different value of ro is obtained. Figure 4 shows the complete measurement on two representative solvent systems. As data were consistently obtained at 5 "C intervals between 5 and 50 "C, inclusive, the approximate temperature at which each group of data points was taken may be obtained by inspection, high temperatures corresponding to large values of T/q. The low-temperature anomaly noted in ref 1, consisting of excessively large values of D at 5-15 "C, was also occasionally seen in this study. Figure 5 shows the values of r, obtained by applying eq 3 to our data on each solvent system. Some of the points, corresponding to solvents whose full 7-T plots were not obtained, only incorporate data obtained at 15 "C. One sees that the apparent hydrodynamic radius of the spheres depends strongly on polymer concentration. Between 0.0 and 10 g/L, ro increases by roughly 50%. ro then remains roughly constant as solutions of 10-90 g/L are compared. Above 90 g/L, r, again increases, reaching in the 171 g/L solution nearly double its pure-water value. Several experimental controls indicate that our data do not include the more obvious physical artifach. As noted above, the poly(acry1ic acid)-water solvents used in this study scatter extremely little light, so the observed spectra
are directly indicative only of the motion of the polystyrene spheres. Two lines of indirect evidence indicate that the polystyrene latex does not form aggregates under our conditions. First, the diffusion coefficient of the spheres in the mixed solvents was observed at fmed T &s a function of time, data being taken as little as 40 min and as much as 9 h after suspending the polystyrene spheres in the solvent. Over this time interval, there was no indication of a systematic change in D. As a further control, temperature series on different preparations of the same solvent were made by ascending from 5 "C or by descending from 50 "C,so that the initial point of one series and the final point of another series (on a fresh preparation) would be a t the same temperature. No systematic difference between ascending and descending series was found, which further confirms that we do not have slow aggregation phenomena in our systems. Multiple scattering could lead to an apparent dependence of D on the poly(acry1ic acid) concentration c, the mechanism being as follows: (i) the index of refraction of the solvent depends on c; (ii) the sphere-scattering cross section a (and hence the amount of multiple scattering) depends on the index of refraction mismatch 6, between spheres and solvent; (iii) the amount of multiple scattering depends on a. Changing c thus leads to changes in the double-scattering terms in the spectrum, thereby changing the apparent average diffusion coefficient. This effect is unlikely to be significant here. We have previously shownl4 by means of homodyne coincidence spectroscopy that, for 0.091-pm spheres in water, multiple scattering is not important for concentrations 4 5 0.1 % . In this work the use of smaller (0.038)-fim)spheres, a lower index of refraction difference between spheres and solvent (due to the poly(acrylic acid)), and a sphere concentration # lo4 would all tend to make multiple scattering negligible.
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IV. Discussion The viscosity of poly(acry1ic acid)-water has previously been studied by a number of ~ 0 r k e r s . l This ~ work largely (14)Phillies, G . D.J. Phys. Reu. A 1981, 24, 1939.
Diffusion Coefficient of a Macroparticulate Probe
The Journal of Physical Chemistry, Vol. 86, No. 20, 1982 4077
solution might be thermodynamically immiscible, the spheres are so dilute that the polymer is unlikely to create a significant sphere-sphere potential of average force. On the hypothesis that the sphere motions do reflect properties of the medium, over distances ro and times -ro2/D, the most direct implication of our results is that the solvent microviscosity experienced by the spheres is unequal to the macroscopic shear viscosity 7 . A "cage" model in which the spheres must hunt for openings in a polymer mesh work would predict this effect. The cage image suggests that changes in dro/dc correspond to critical ratios of the mesh size to the sphere diameter, rather than to changes in the properties of the polymer solution, so the lack of correspondence between the breaks in ro-c and T C plots might not be extremely important. Crossley et al.16 have recently reported on the diffusion coefficients of low-density lipoprotein in buffer solutions and polystyrene latex spheres (Polaron Equipment Ltd., U.K., 0.109-pm diameter) in pure water over the temperature range 20-45 OC, finding that D20,w,the diffusion coefficient corrected for the temperature and solvent viscosity in the experiment, has a temperature coefficient of about -0.6% K-l. That is, their results indicated that D9/ T decreases substantially with increasing temperature. We do not observe this effect with out carboxylate-modified spheres, nor was it observed in our previous study of D for 0.091-pm spheres in H20-glycerol-sodium dodecyl sulfate. As these authors emphasize, temperature coefficients vary substantially (even as to sign) from substance to substance. Presumably, this is a surface effect, so polystyrene latex spheres from different sources under different conditions might show different coefficients, as is observed. However, we see no significant indication of a nonzero temperature coefficient for our polystyrene spheres in pure water. In conclusion, the diffusion coefficient of 0.038-pm diameter polystyrene spheres in water-poly(acry1ic acid) has been studied. D depends in a complicated way on the polymer concentration. Unlike our earlier work on the diffusion of polystyrene spheres in water-glycerol and water-sorbitol, the variations in D as temperature and polymer concentration change are not simply determined by the macroscopic shear viscosity of the solvent, so the apparent hydrodynamic radius ro of the sphere depends on the solvent. The apparent crossover effects in T C and ro-c plots do not occur at the same polymer concentration.
emphasizes observations on the partially neutralized polymer in the presence of a background electrolyte and is thus not trivially to be compared with our own, in which the nonneutralized (though presumably partially charged, due to ionization) polymer was dissolved in deionized water. A comparison of our viscosity data with modern theories based on renormalization group techniques is also not necessarily reasonable. As has previously been emphasized," the easy scaling approaches presume that only a single (physical) length is significant, rather than the three (Debye electrostatic, effective hydrodynamic, and polymer radius of gyration) which arise in the treatment of a polyelectrolyte. There is a clear break in 7-c plots near c = 1.0 g/L, as well as a sharp increase in 9 at larger concentrations. In the language of some modern work, c = 1.0 g/L might be taken to be a transition concentration c* at which ones sees a crossover from one type to solution behavior to another. If one assumes that the lowest-concentration data takes the form 9 = ?jo(l + 2.5d1*C) (4) dl* being an effective hydrodynamic volume, Figure 2 shows dl* = 300 cm3/g or c* = 3 g/L. This is far larger than the thermodynamic d which was obtained from density measurements. An effort to refine Ul* by fitting 9 to a higher-order polynomial in #IJ = dl* c did not give convincing results. The ro-c relation shows substantial changes in slope, but at very different concentrations(viz.,10,90 g/L) from those at which changes in (dq/ac) are evident. In the absence of extensive theoretical work on the diffusion of macroparticles through a background polymer, one cannot readily make comparison with the literature. If spheresphere interactions are not important, the concentration dependence of ro would be due to the sphere-solvent interaction. The spheres would then serve as a probe of the properties of the medium on the distance and time scales on which they move. It may readly be argued that sphere-sphere interactions are indeed not significant in our systems. At a volume fraction 4 1X in pure water, the polystyrene spheres do not interact significantly with each other; the measured D is essentially that of a single isolated sphere. Adding to the pure water some amount of polymer should not increase dynamic interactions between the spheres; indeed, the hydrodynamic screening hypothesis suggests that the polymer should reduce the long-range hydrodynamic interactions between the spheres. While a concentrated sphere suspension and a concentrated polymer
Acknowledgment. This work was supported in part by Grant CHE-79-20389 from the National Science Foundation.
(15) Osterheld, J. E.; Flory, P. J. J. Phys. Chem. 1954,58, 653; J. Chem. Phys. 1957,26, 114.
(16) Crossley, J. M.;Spragg, S. P.; Creeth, J. M.; Noble, N.; Slack, J. Biopolymers 1982,21,233.
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