J . Phys. Chem. 1989, 93, 5029-5039
t
0
I
A-
j
II
I” I
I
rrFigure 4. Profile of the potential energy as a function of radial coordinate for TEMPOCA anion (left) and TEMPOCA acid (right). These potential profiles are drawn only on a qualitative basis. Solid curves are for larger micelles and dotted curves are for smaller micelles. Because this molecule does not dissolve in the solvent (isooctane),the potential is very high outside of the micelle. Minima occur at the center of water pool ( r = 0) and also at the interface between the water pool and the surfactant layer (represented qualitatively as a potential well).
translational motion) for A- in the core of water pool decreases drastically because it is roughly proportional to the volume of core water. On the other hand, that of A H trapped at the interface as Figure 31V may not decrease so sharply and is essentially proportional to the surface area of the water pool. (3) A part of the apparent enthalpy change of 3.2 kcal/mol may be assigned to the potential difference between A H at the interface and Ain the core of the water pool. These may be the mechanism for the increase of A H type at high temperatures and low water contents.
5029
From the above discussion, we can depict the potential curves schematically for A H and A- solubilized in the reversed micelle as shown in the left and the right diagrams of Figure 4, respectively. The solid curve is for a larger micelle, and the dotted one is for a smaller micelle. For both A- and AH, there are potential minima at both the center of the water pool and the interface between the water pool and the apolar layer of the reversed micelle. Because A- is mainly located in the water pool, the potential at the interface for A- is considerably higher than that at the center of the pool. On the other hand, the stable form of A H of Figure 31V corresponds to the potential well. The minimum at the center of the pool becomes deeper with increasing the diameter of water pool, since the dielectric constant becomes larger with the diameter.2 Considering the phenomenon from a microscopic point of view, we can regard the phenomenon as switching of the system, the reversed micelle dissolving one amphiphilic carboxylic acid in the water pool, to the acidic form (Figure 3IV) from the basic form (Figure 31) by a single proton perception. It is interesting to point out that physiologists suggest carboxyl groups as the specific receptor sites for the taste response to salts and acids.22 Acknowledgment. Interesting discussion with Professor Nobuji Maeda, Department of Medicine, Ehime University, is gratefully acknowledged. Technical assistance from Keichi Nunome and Kazuko Yamaguchi is also acknowledged. (22) Beidler, L. M. In Handbook of Sensory Physiology; Vol. I K ChemicalSenses 2. Taste; Beidler, L. M., Ed.; Springer Verlag: Berlin, 1971; Chapter 11.
FEATURE ARTICLE The Hydrodynamic Scaling Model for Polymer Self-Diffusion? George D. J. Phillies Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01605 (Received: December 15, 1988)
The hydrodynamic scaling model for polymer dynamics is described. Phenomenological results on probe diffusion in polymer solutions, on polymer and protein self-diffusion coefficients, and on related macromolecule transport properties (viscosity, sedimentation) are considered. A universal scaling equation is obtained for the concentration and molecular weight dependences of these transport coefficients. Implications are noted for the nature of polymer dynamics. Experimentally,polymer solutions are fundamentally unlike gels, even on short time scales. A physical model for transport through polymer solutions, based on the primacy of hydrodynamic over topological forces at large concentration, is given. Using self-similarity arguments to bootstrap from small to large c, the model accounts quantitatively for the concentration dependence of 0,and is consistent with measurements of 7 and s in nondilute polymer solutions.
Introduction The motion of macromolecules in solutions and melts is a topic of substantial current interest, a wide variety of experimental techniques,’ numerical simulations,2and algebraic computations3 being applied to improve understanding of polymer dynamics. Experimentally, many transport coefficients can be measured in polymer solutions. Most transport coefficients depend strongly ‘Preliminary versions presented at the American Chemical Society Denver National Meeting (1987) and at the American Physical Society New York Meeting (1987).
0022-3654/89/2093-5029$01.50/0
on the polymer concentration and molecular weight. A fundamental problem is to understand how the forces between polymer chains affect transport coefficients, the objective being to explain the coefficients’ behaviors in terms of fundamental polymerpolymer and polymer-solvent interactions. (1) Tirrell, M. Rubber Chem. Technof. 1984, 57, 523 and references therein. (2) Skolnick, J.; Yaris, R.; Kolinski, A. J . Chem. Phys. 1988, 88, 1407, and references therein. (3) Freed, K H. Renormalization Group Theory of Macromolecules; Wiley-Interscience: New York, 1987.
0 1989 American Chemical Society
5030
The Journal of Physical Chemistry, Vol. 93, No. 13, 1989
The field is presently in the unusual position of having several incompatible paradigms for polymer motion, each of which in the eyes of its supporters provides an adequate characterization of polymer transport. The objective here is to present one of these paradigms-the hydrodynamic scaling method-for polymer solutions, showing how the picture arose from experimental phenomenology and how the picture gives a quantitative description of polymer transport. Important physical data are studies of probe diffusion through polymer solutions and measurements of polymer and biopolymer transport coefficients, notably the self-diffusion coefficient D,. A simple phenomenological equation will be shown to describe probe and self-diffusion coefficients; conclusions that could be drawn from the equation are discussed. A physical model for polymer solutions is presented, yielding the phenomenological equation, complete with numerical values for scaling coefficients and scaling prefactors. The discussion emphasizes one mathematical form, the stretched exponent ia I D, = Do exp(-ac”)
(1)
Here Do is D, at concentration c = 0, the scaling parameters being a prefactor N and an exponent u . Exponential and stretched exponential forms are not novel in polymer science. The Martin equation for the viscosity
[ q ] being the intrinsic viscosity and b a constant, was obtained empirically and derived by Adler and Freed4 from effective-medium arguments. Stretched exponentials also appear in models for probe d i f f u s i ~ n . ~ - ’ ~ The following sections treat (i) phenomenology of probe diffusion. (ii) theories of probe diffusion, (iii) polymer self-diffusion, (iv) comparison with models for self-diffusion, (v) other polymer transport coefficients, (vi) derivation of eq 1 from hydrodynamic scaling, (vii) tests of the hydrodynamic scaling model, and (viii) conclusions.
Phenomenology of Probe Diffusion
The diffusion coefficient D of large, dilute spheres in lowviscosity (7 < 10 cP”) solvents obeys the Stokes-Einstein equation (3) where k , is Boltzmann’s constant, T i s the absolute temperature, and R is the sphere radius. Our original intent in studying probe diffusion was to verify light scattering spectroscopy as an experimental technique, by testing eq 3 in aqueous solutions made viscous by adding saccharides or hydrophilic polymers. It soon became apparent that eq 3 can fail badly in some polymer solutions, even when R of the probe was >>R, of the polymer. We elected to test eq 3 by studying ternary probe:polymer: solvent mixtures. The light scattering spectra of such mixtures (see are in general quite complex. As I showed a decade (4) Adler, R. S.; Freed, K. F. J . Chem. Phys. 1980, 72, 4186. (5) Ogston, A . G.; Preston, P. N.; Wells, J . D. Proc. R. Soc. London, A 1973, 333, 297. ( 6 ) Cukier, R . I . Macromolecules 1983, 17, 252. (7) Altenberger, A. R.; Tirrell, M . J . Chem. Phys. 1984, 80, 2208. (8) Langevin, D.; Rondelez, F. Polymer 1978, 14, 875. (9) Altenberger. A. R.; Tirrell, M.: Dahler, J . S. J . Chem. Phys. 1986.84. 5 122. ( I O ) Phillies, G.D. J.; Ullmarn, G . S.; Ullmann, K.; Lin. T.-H. J . Chem. Phys. 1985, 82, 5242. (II ) Above 10 cP. small-particle drag coefficients often follow/’- ?*I3 or so (Heber-Greene. W. J . Chem. SOC.1910. 98. 2023). For modern references corroborating Heber-Greene. see Phillies, G . D.J . J . P h y s . Chent. 1981, 85, 2838. (12) Phillies, G . D.J . J . Chem. Phyc. 1974, 60, 983 ( 1 3 ) Phillies. G. D.J . Biopul.ymers 1975. 1 4 , 499.
Phillies
Figure 1. Diffusion coefficients from Phillies et aL30 of 208-.& carboxylate-modified polystyrene latex spheres through 64% neutralized poly(acrylic acid), M = 450000. Solid lines represent fits to eq I .
also ref 14 and 15), under certain conditions the spectra simplify. Namely, if the po1ymer:solvent pair is nonscattering (e.g., if polymer and solvent are i s o r e f r a ~ t i v e lor ~ *if~ ~a fluorescently labeled and unlabeled polymer are combinedI3), and ifthe probe species is dilute, the spectrum shows a single relaxation. (With quasielastic light scattering, the relaxation is exponential; with fluorescence correlation spectroscopy, the relaxation has an inverse binomial form.) The relaxation time is determined by the single-particle (tracer) diffusion coefficient D of the probe in the po1ymer:solvent mixture. The nonscattering polymer component is sometimes referred to as the “background” or “matrix” polymer. Optical probe methods, which yield tracer diffusion coefficients of probes in ternary mixtures, are fundamentally unlike light scattering spectroscopic studies of binary po1ymer:solvent mixtures, which yield polymer mutual diffusion coefficients. (In very dilute systems the self-diffusion and mutual diffusion coefficients are almost equal.) D obtained while using globular probes (proteins, polystyrene latices) is here denoted D,; D obtained with random-coil polymers as probes is denoted D,. Early experiments on probe motion in polymer solutions include work by Laurent and co-workers,1618 Turner and Hallett,19 Brown and Stilbs,20and Langevin and Rondelez,8 all of whom compared their data with exponential and stretched-exponential forms. Laurent et al.i6-’8 used ultracentrifugation to obtain the sedimentation coefficient s of proteins and small particles (2.5 nm < R < 356 nm) in water:dextran and water:hyaluronic acid. In terms of s = so exp(-ac”)
(4)
-
they found v = 0.5 and a R 1 . Laurent and PerssonI6 report s depends on dextran concentration less strongly than does 7; Stokes’ law therefore fails in dextran solutions for small particles. Turner and HalletL9(using polystyrene sphere probes in dextran:water) and Brown and StilbsZ0(using pulsed-field-gradient N M R to observe D, of poly(ethy1ene oxide) (PEO)) both saw u = 1, not v = 0.5. Turner and HallettI9 found a to be independent of R, contrary to Laurent et al.’s16-18 a R. Langevin and Rondelez6 measured s of probes (1.7 nm < R < 46 nm) in PEO:water, finding u = 2/3. My laboratory has observed a variety of probe:polymer combinations, using quasielastic light scattering to measure D,. Polystyrene ~ p h e r e s ~ l and - ’ ~ bovine serum albumin28were the
-
B. Physica 1979, 97A, 113. ( 1 5 ) Lodge, T. P. Macromolecules 1983, 16, 1393. (14) Jones, R.
(16) Laurent, T. C.; Persson, H. Biochim. Biophys. Acra 1964, 83, 141. ( 1 7 ) Laurent, T. C.; Bjork, I.; Pietruszkiewicz, A,: Persson, H. Biochim. Biophys. Acta 1963, 78, 351. (18) Laurent, T . C.; Pietruszkiewicz, A. Biochim. Biophys. Acta 1961, 49, 258.
(19) Turner, D. N.; Hallet, F. R. Biochim. Biophys. Acfa 1976, 451, 305. (20) Brown, W.; Stilbs. P. Polymer 1983. 24. 188.
The Journal of Physical Chemistry, Vol. 93, No. 13, 1989 5031
Feature Article
I
08-
0.6
-
’
I
I
I
I
I
I
I
I
\y ,+I\‘ ‘\
‘y
I
‘. ‘\
0.4 -
‘.“.. L ‘.‘-. -\
I
I
I
I
I
I
I
I
I
elevated protein concentrations, qo/? is clearly smaller than D,/Do. The macroscopic viscosity determined with capillary viscometers is larger than expected from the probe diffusion coefficient, even though the probes are nearly 100-fold larger than the matrix polymers. Protein adsorption by the spheres would yield q o / q > Dp/Do, not vice versa, so the difference between 7-l and D is not due to protein adsorption. Indeed, by diluting concentrated protein:probe solutions, we were ablez7to quantitate the protein binding by the probes, showing that adsorption leads only to small corrections on these measurements. Dp/Do and q o / q (Figure 2) follow stretched exponentials with u in the range 0.96-0.99. cy is much smaller in BSA solutions (4.4 X Icy I8.0 X than in poly(acry1ic acid) solutions (0.27 Ia I0.99); the phenomenologies of probe diffusion in BSA and poly(acry1ic acid) solutions are otherwise very similar. With both matrix polymers, De has the same functional form, and eq 3 fails. Some modern treatments of polymer dynamics emphasize the observation that polymer chains are not ghosts and cannot pass through each other. In these treatments (discussed further below), loops, knots, and other topological constraints on polymer motion, collectively described as “chain entanglements”, are presumed to dominate polymer dynamics. Since globular proteins cannot entangle, because they cannot forms loops around each other, ref 27 concluded that the non-Stokes-Einsteinian behavior of large probe particles in macromolecule solutions does not arise from polymer entanglement. Systematics of Probe Diffusion Figures 1 and 2 represent a large l i t e r a t ~ r e ~on l - ~probe ~ diffusion. This section presents generalizations from these experiments, relating D, to c, 7, R, and M . In almost all systems, the functional dependence of D, on polymer concentration c is a stretched exponential De = Do exp(-ac’)
(6)
with 0.5 Iu I1. Most deviations from eq 6 arise from polymer adsorption and polymer-induced probe aggregation. In these systems 17 obeys of the probes fell to 112 to 117 of their actual radius R . There are artifacts (polymer binding by the probes, probe aggregation) leading to apparent failures of eq 3. These artifacts cause probes to diffuse too slowly-not too quickly-relative to 9 and R , thus having the wrong sign to explain the observationg0 Rh < R . Solid lines in Figure 1 were obtained from nonlinear least-square fits to eq 1. For each I, eq 1 clearly represents the data well. The scaling prefactor a falls with increasing Z,while u increases with increasing I . A full discussion appears in Phillies et al.;30*31 see also Gorti and Ware.33 At a contrast, Figure 2 shows2’ D,/Do of 0.655-pm carboxylate-modified polystyrene latex probes diffusing through bovine serum albumin: 0.15 M NaCI, pH 7. Also indicated is the normalized fluidity Bo/’) of each solution, the subscript “0” referring to a solution containing no protein. If the Stokes-Einstein equation worked in this system, Dp/Doand vo/17 would agree. At (21) Lin, T.-H.; Phillies, G. D. J. J . Phys. Chem. 1982, 86, 4073. (22) Lin, T.-H.; Phillies, G.D. J. J . Colloid fnrerface Sci. 1984, 100, 82. (23) Lin, T.-H.; Phillies, G. D. J. Macromolecules 1984, 17, 1686. (24) Fernandez, A. C.; Phillies, G. D. J. Biopolymers 1983, 22, 593. (25) Ullmann, G. S.; Phillies, G. D. J. Macromolecules 1983, 16, 1947. (26) Ullmann, G. S.; Ullmann, K.; Lindner, R. M.; Phillies, G. D. J. J . Phys. Chem. 1985, 89, 692. (27) Ullmann, K.; Ullmann, G. S . ; Phillies, G.D. J. J . Colloid Interface Sci. 1985, 105, 315. (28) Phillies, G. D. J. Biopolymers 1985, 24, 379. (29) Phillies. G.D. J. J . Phys. Chem. 1984. 81. 1487. (30) Phillies, G. D. J.; Malone, C.; Ullmann, K.; Ullmann, G. S.; Rollings, J.; Yu,L.-P. Macromolecules 1987, 20, 2280. (31) Phillies, G.D. J.; Pirnat, T.; Kiss, M.; Teasdale, N.; Maclung, D.; Inglefield, H.; Malone, C.; Rau, A.; Yu, L.-P.; Rollings, J. Macromolecules, in press. (32) Phillies, G.D. J.; Gong, J.; Li, L.; Rau, A,; Zhang, K.; Yu,L.-P.; Rollings, J. J . Phys. Chem., in press. (33) Gorti, S.; Ware, B. R. J. Chem. Phys. 1985.83, 6449.
17 = 90 exp(ac’)
-
(7)
At very large M and c, where 9 > 104 P or so, 7 gains a power-law behavior 9 CX, while the light scattering spectrum acquires a second, fast relaxation. In this high-viscosity regime, which has not been studied extensively with probe diffusion, D continues to follow eq 6, resulting in very large deviations from eq 3. D and 9 are independent parameters. Optical probe studies using light scattering spectroscopyare not just expensive substitutes for cheap capillary viscometers. The Stokes-Einstein equation is adequate for probes in aqueous solutions of small polymers (7000 amu PE0,z6 5 X lo4 amu nonneutralized PAA2z). In solutions of larger polymers eq 3 fails, probe particles diffusing faster than expected from the macroscopic solution viscosity. Most obvious artifacts have the wrong sign to explain this deviation from D, ?-I. In at least one system,z6 the microscopic viscosity qP inferred from De, R, and eq 3 is substantially less than the solvent viscosity vo. In some cases eq 3 is valid below a limiting concentration c’. Increasing R (up to 1-3 pm) does not usually improve the validity of eq 3. The significances of the matrix molecular weight and probe radius are clarified by considering computations which might be said to treat Dp, including those of Langevin and Rondelez,8 Altenberger et a1.,7.9Cukier,6 and Ogston et al.5 Some models7 only apply in limited concentration ranges. Others appear to refer to probes in cross-linked gelsS or treat background polymers as fixed sources of frictional i n t e r a c t i ~ n . ~Approximating !~.~ matrix polymers as being held in fixed positions seems more appropriate for gels than for polymer solutions.
-
(34) Brown, W.; Rymden, R. Macromolecules 1986, 19, 2942. (35) Brown, W.; Rymden, R. Macromolecules 1987, 20, 2867. (36) Brown, W.; Rymden, R. Macromolecules 1986, 21, 840.
5032
The Journal of Physical Chemistry, Vol. 93, No. 13, 1989
Phillies True gels are size filters that selectively block the motion of large particles and allow small particles to pass. In contrast, polymer solutions are not size filters; they hinder large and small particles to roughly the same extent. This fundamental difference between gels and polymer solutions suggests that polymer solutions might in some cases have interesting properties as electrophoretic support media. Polymer solutions are sometimes described as pseudogels, containing a transient gel structure at short times. If polymer solutions were in fact pseudogels, at short timesf, from probe diffusion in polymer solutions and f, from electrophoresis in cross-linked gels should have the same R dependence, viz., eq 13. Experimentally, probes in polymer solutions actually follow
-0
a
f,/f, = exp(aR0*0.2)
-0.0
,
I
io5
io4
I
,
,
,
,
106
M O L W T (Mw) Figure 3. Parameter a from DJD, = exp(-ac") from probes in dextran:water, plotted against M of the matrix polymer (from Phillies et a1,32).
From theory and experiment, Langevin and Rondelez8conclude for large particles in semidilute (c > e*) solutions s
-
exP(-(R/E)6)
(8)
(though they note S
- = exp(-(R/E)&) so
+7 710
-
(11)
-
-
where a is independent of R but satisfies a W for y = 0.6-0.75. Altenberger et aL9 indicate without explicit proof that y 0 as M m . In contrast, the hydrodynamic treatment of Cukier6 predicts that D p / D odepends upon R. Ogston et aL5 treat sphere motion through randomly placed mechanical obstacles, a sphere being able to move only if it faces a hole large to permit its passage, so that D , / D , depends on R in semidilute solution. How accurate are these predictions? Figure 3 compares a with M , for polystyrene spheres in dextrans of measured molecular weight d i ~ t r i b u t i o n .These ~ ~ data, which are fully consistent with earlier studiesI0 of a and M , find cy
-
(12)
M?
-
where y = 0.82. Figure 3 is totally inconsistent with any model @ in semidilute solution. If a probe particle that predicts a in semidilute solution effectively interacted only with the portion of each neighboring polymer chain that extended between pairs of entanglement points, a , like E, would be independent of M . Since y is substantially nonzero, diffusing probe particles interact with the entirety of each adjoining polymer molecule, not only with polymer segments extending from the probe to the first entanglement point. The recent model of Altenberger et aL9 is consistent with eq I2 and Figure 3, at least if 5 X lo5 amu is below m limit in which ref 9 predicts D , to be independent the M of M . D, depends on R through Do. Results from my l a b ~ r a t o r y * l - ~ ~ R6 with 6 indicate a depends but weakly upon R, namely, a = 0 f 0.2. Experiments on true cross-linked gels find a very different value for 6. From electrophoresis, Rodbard and Chrambach3' find for the drag coefficientf, of particles in gels
-
-
fJf0
t2
2 0 ,= ~ Rg2 ~
(10)
where Langevin and Rondelez found 6 = 1 and u = 0.62. Neither R nor depends on the matrix molecular weight M . Treating matrix polymers as lines of fixed frictional points, Altenberger et al.9 predict for c > T ~ ) ,the pseudogel lattice would relax before probes encountered it. T, is the time in which a probe moves the typical distance between entanglement points
(9)
as a hypothetical alternative.) Here
E
(14)
= exp(aR2)
(13)
E -
For a polymer with critical overlap concentration c* Mu38 and R,
Since
(16)
- M/R:
= R,(c*/c)"
(17)
m=- U 31,- 1
(18)
@ for c > c*
and TP = -Ds-
7,
(02
Dp R:
or -2v/ ( 3 r I )
2 = 3(L) D , c* T~
(20)
where u is 0.5 or 0.6. As shown below, D, and D , have nearly the same dependences on c and M , so D J D , is nearly independent of c. By comparison with other polymer systems, Lin and Phillies' data21-23on polystyrene spheres in nonneutralized poly(acry1ic acid):water extends to c > 3c* with D , / D , = 2 . For this system Tp/Tr
0.2
(21)
Dp thus has been measured on the short time scale T, < T~ on which polymer pseudogel structures are said to exist. Polymer solutions retard all probes to the same extent, but true gels hinder the motion of large particles while letting small probes pass. From eq 13-21 polymer solutions and true cross-linked gels are fundamentally different on a short time scale. Polymer solutions therefore do not contain transient pseudogel lattices, at least as structures that are strong enough to retard particle diffusion. ( 3 7 ) Rodbard, D.; Chrambach, A. Anal. Biochem. 1971, 40, 95. (38) deGennes, P . G . Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.
The Journal of Physical Chemistry, Vol. 93, No. 13, 1989 5033
Feature Article IO
I
M
c
I
4 01
001
D,of polystyrenes of various M (and D of 598-kDa polystyrene in a 1800-kDa polystyrene matrix) in tetrahydrofuran,based on Leger et al.“ and (lines) eq 22. Concentration x in g/g. Figure 4.
Polymer Self-Diffusion From Figures 1 and 2, the phenomenologies for diffusion of globular probes through (i) long-chain random-coil polymer^^'-^^ and (ii) globular proteinsz7 are roughly the same. In both sorts of systems, Dp follows eq 6, while for specified R and 9 eq 3 often underestimates D,. Because globular and random-coil matrix polymers can be interchanged without changing the phenomenology for probe diffusion, it appeared intuitively plausible39that interchanging globular probes with random-coil probes would also not change the phenomenology. The intuitive extrapolation makes sense if probe diffusion and polymer self-diffusion proceed by the same mechanisms, but is unlikely to be correct if the modes of motion open to random polymer coils and to globular bodies are very different. The motion of a random-coil polymer probe through a random-coil matrix is described by the probe’s tracer (self-) diffusion coefficient. If the intuition were true, D, and D, would have the same concentration dependence, so
D, = Do exp(-acv)
-
3
C Figure 5. D,of polystyrenes of various M in poly(viny1 methyl ether): toluene, from Martin.” M p v M=~ 110000 amu. Lines are fits to eq 6 . c is weight fraction. rms errors in the fits ranged from 3 to 7%.
0-
2000
0 110,000
1
X
233.000 350,000
D
(22)
-
Do and a should depend on M , namely, Do and a M6 for 6 of order unity. Indeed, Turner and Halletig (using poly0 styrene sphere probes) and Brown and Stilbs20(using poly(ethy1ene oxide) chains as probes) found that very different probes behave quite similarly in water:dextran. The similarity between D, and Dp was examined in ref 39, which reviewed much of the large literature on polymer self-diffusion. The following emphasizes self-diffusion measurements not treated in ref 39. Figure 4 shows Leger et al.’s40 pioneering study on D, OS of polystyrene in tetrahydrofuran. Do was taken from the literature. The solid lines represent nonlinear least-squares fits to stretched exponentials. There is reasonable agreement between most data and eq 22. For 245 000 amu polymer, with increasing I I I I \ I c, D, first increases by nearly 50% and then decreases. The 50 100 150 200 250 2 )O inflexible stretched exponential cannot represent a nonmonotonic C D,(c); however, other experimental studies on p01ystyrene:THP~~~ Figure 6. D,for polystyrene in CC14 from Callaghan and Pinder.4”9 find D, to be monotonic on c, with no increase in D, at small c. Solid lines are fits to stretched expnentials; vertical arrows indicate ref The probe and matrix polymers can be chemically distinct, in 49’s estimates for c*. The smallest-M polymer has M < M eand hence which case D, is the single-particle interdiffusion coefficient. no c*. Figure from ref 39. c in g/L. (39) Phillies, G. D. J. Macromolecules 1986, 19. 2367. (40) Leger, L.; Hervet, H.; Rondelez, F. Macromolecules 1981, 14, 1732. (41) Wesson, J. A.; Noh, I.; Kitano, T.; Yu,H. Macromolecules 1984, 17, 782. (42) von Meerwali, E. D.; Amis, E. J.; Ferry, J. D. Macromolecules 1983, 16, 1715. (43) von Meerwall, E. D.; Amis, E. J.; Ferry, J. D. Macromolecules 1985, 18, 260.
Representative of this class of study is Figure 5 , which shows Martin’s44 observations on D, of polystyrene in 110 000 amu (nominal) poly(viny1 methyl ether) (PVME):toluene. Here Do was extrapolated from Hanley et al.45 Denoting the matrix and (44) Martin, J. E. Macromolecules 1986, 19, 922.
5034
The Journal of Physical Chemistry, Vol. 93, No. 13, 1989
TABLE I: Nonlinear Least-Squares Fits of the Universal Scaling Law to Results of Callaghan and Pinderm9 on Polystyrene:CCI, and Smith et al.” on Poly(propy1ene oxide) in Poly(propy1ene oxide)“ M C a Y 7% rmse I IO-kDa PS all 0.0775 0.75 6.2
Phillies is interpreted as the D that a PS chain would have, in pure solvent, if it were transferred from PVME solution to pure solvent without changing its R,.
probe molecular weights M and P, respectively, the stretched exponential clearly gives a good representation of experiment, both for M > P and for M c*. From Figure 6, eq 22 describes the data well at all c, ranging from a few up to 300 g/L. While no one would reasonably expect the dilute (c < c*) and semidilute (c > c*) regimes of D, to be separated by a 90’ bend, Figure 6 shows that there is no change in the behavior of D, at c*. One curve with fixed parameters passes through all data points both below and above c*. The assertion that the system’s behavior does not change between c < c* and c > c* has a stronger form. Table I shows parameters gained by fitting eq 22 to c < c* measurements. For 233 000 and 350 000 amu polystyrene, a and v obtained from the c < c* values of D, are quite close to a and v obtained by fitting eq 22 all the data. Polymer dynamics in concentrated solutions is apparently effectively prefigured by solution behavior at low (c < c*) matrix concentrations. Does eq 22 become inaccurate at very large c? Most solution studies are limited to c < 500 g/L at temperatures and pressures too low to continue to a melt at large c. Smith et aL50obtained D, of entangled (M = 32000 > Me) poly(propy1ene oxide) (PPO) in a molten low-M (1000 amu) PPO,Me being the critical molecular weight for entanglement. The mass fraction of the large PPO ranged from near zero to 1 .O; the concentration of labeled large chains (not all large chains were labeled) was small. Equation 22 is found (Figure 4 of ref 39) to describe D,(c) from dilute solution up to the melt. The five to seven measurements at smallest c yield a and v that are within 10% (Table I) of a and Y gained by analyzing all the data. Error bars on the low-c a and v were obtained by robust statistical analysis, by changing which of the small-c data points were included in the fit to eq 22. Equation 22 is only a two-parameter form. With extremely precise data on polystyrene in PVME:toluene, Lodge and Wheelers1ss2find small but systematic deviations of eq 14 from eq 22. Do from a three-parameter fit of the universal scaling equation to their measurements is larger than the c 0 limit of D,. Lodges3 has shown that these deviations are eliminated by reinterpreting Do. Namely, the radius of gyration, R, of PS in at PVME solutions falls markedly (50%) with increasing cpVME (
