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J. Phys. Chem. 1985,89, 692-700
furcation appears to occur. The phase portrait as a function of temperature should then be as shown in Figure 7. Note that the progression of phase portraits in Figure 7 may be thought of as coinciding with a horizontal or diagonal (along a line bisecting the oscillatory region) cut through the phase diagram of Figure 3, while the corresponding progression in Figure 2 corresponds to a vertical cut.
Summary and Conclusions The present study extends earlier work5,6*9~’o on the Br03-Br--Mn2+ system at room temperature to the entire temperature range over which the system oscillates. The results obtained lend support to the contention6 that the NlT mechanism provides an accurate description of this reaction. By studying the range of flow rates in which the system oscillates as well as the period and amplitude of oscillation as
functions of temperature, we have been able to assign reasonable activation energies for those mechanistic steps to which the oscillatory behavior is most sensitive and to infer the phase portrait of the system. The remaining discrepancies between theory and experiment, while quantitative rather than qualitative, should nevertheless be explored in order to improve still further our knowledge of this “best understoodn6among chemical oscillators and of the many other derived from it.
Acknowledgment. This work was supported by National Science Foundation Grant C H E 8204085 and by BRSG SO7 RR07044 awarded by the Biomedical Research Support Program, Division of Research Resources, National Institutes of Health. We thank Mohamed Alamgir, Donald Boyd, Ofra Citri, Kenneth Kustin, and Robert Olsen for helpful discussions and suggestions. Registry No. Mn, 7439-96-5; Br03-, 15541-45-4; Br-, 24959-67-9.
Probe Dlffusion of Polystyrene Latex Spheres in Poly(ethy1ene oxide)-Water Gregory S. Ullmann, Kathleen UUmann, Robert M. Lindner, and George D. J. Phillies* Department of Chemistry, The University of Michigan, Ann Arbor, Michigan 481 09 (Received: June 4 , 1984)
The diffusion coefficient D of polystyrene spheres in poly(ethy1ene oxide)-water was studied as a function of sphere radius R, polymer concentration c, and polymer molecular weight M . The Stokes-Einstein equation for D fails badly, D being larger than predicted from R and the macrarcopic shear viscosity 7. The failure increases with increasing polymer concentration. We obtain scaling law relations between D,7,and c for high molecular weight ( M > 100000) polymers. However, at large c the scaling law coefficients depend on M and not R, contrary to theoretical expectations. D can be interpreted quantitatively in terms of non-Stokes-Einstein hydrodynamics and the Langmuir adsorption isotherm for the polymer by the spheres.
I. Introduction In this paper we report a study of the diffusion of spherical probe particles through solutions of a neutral polymer. The diffusion of globular particles arises in several scientifically important contexts, such as the transport of macromolecules within and between cells and tissues. Furthermore, modern techniques for biopolymer fractionation, such as gel electrophoresis and gel chromatography, are influenced by the diffusion rate of biopolymers through a background of long-chain molecules. This work is a continuation of our previous studies on probe diffusion in viscous liquids,’ suspensions of charged colloidal particles,* and poly(acry1ic acid) solution^.^ Our original interest was the range of validity of the Stokes-Einstein equation
Classical measurementse of diffusion and electrophoretic mobility find that eq 1 fails badly for small-molecule probes in smallmolecule solvents with viscosity > 10 cP. W e found that eq 1 is valid for large molecules (bovine serum albumin, polystyrene spheres) in small-molecule solvents of high viscosity (100-1000 cP), such as water-glycerol and water-sorbitol. Lin and Philliesj studied probe diffusion in water-poly(acry1ic acid), measuring D for different probe sizes and polymer molecular weights. For probes in a system of fixed composition, eq 1 does predict the temperature dependence of D. However, at fixed temperature eq 1 does not predict how D depends on 7; in concentrated polymer solutions D can be much larger than expected ( 1 ) Phillies, G. D. J. J . Phys. Chem. 1981, 85, 2838. (2) Phillies, G . D. J. J . Chem. Phys. 1983, 79, 2325. Phillies, G. D. J. J . Phys. Chem. 1982,86,4073. J . Colloid (3) Lin, T.-H.; Interface Sci. 1984, 100, 82. Macromolecules 1984, 17, 1686. (4) Heber-Green, W. J . Chem. SOC.1910, 98, 2023. (5) Stokes, Jean M.; Stokes, R. H. J . Phys. Chem. 1956, 60, 217 ( 6 ) Stokes, Jean M.; Stokes, R. J. J . Phys. Chem. 1958, 62, 497.
0022-3654/85/2089-0692$01.50/0
from 7 and eq 1. This non-Stokes-Einsteinian behavior depends on the probe diameter and increases with increasing c and M . Studies of probe diffusion in solutions of poly(ethy1ene oxide) are here reported. The major experimental variables were the probe radius R and the concentration c and molecular weight M of the polyethylene oxide. Experimental methods are treated in section 11. Section I11 of this paper presents the measurements of D and 7. Section IV gives a phenomenological interpretation. Section V compares our findings with some modern theories of polymer solutions. Section VI considers other experiments on probe diffusion. Conclusions are found in section VII.
11. Experimental Methods Solutions of polyethylene oxide (PEO) of molecular weights 7500, 18 500, 1 X lo5,and 3 X lo5 (all from Polysciences, Inc.) were prepared in 14 M a deionized water at the concentrations seen in Figures 1-3. For the 1 X lo5 and 3 X lo5 amu polymers, 3 g/L is rather close to being a saturated solution. Langevin and Rondelez’ note that the highly linear PEO prepared by anionic polymerization is appreciably more soluble than is commercially available material, such as that studied here. Studies were also performed on probe particles in poly(ethy1ene oxide)-water solutions which contained small amounts (except as otherwise noted, 0.1 wt %) of Triton X-100 (polyethylene glycol p-isooctylphenyl ether, Aldrich). The 1 X lo5 amu polymer was also studied in solutions containing 0.005 M pH 9 CHES (cyclohexylaminoethanesulfonic acid, Calbiochem) buffer and in aqueous solutions containing 17 wt % diglyme (bis(methoxyethy1) ether, Sigma). The probe particles were carboxylate-modified polystyrene latex spheres. Extensive measurements were made on spheres of nominal diameter 0.038 pm (Dow Pharmaceuticals, surface charge 0.296 mequiv/g of polymer), 0.12 pm (Polysciences, Inc., nominal surface charge 0.92 mequiv/g of polymer), 0.70 pm (Polysciences, Inc., nominal surface charge 0.46 mequiv/g of polymer), and 1.28 (7) Langevin, D.; Rondelez, F. Polymer 1978, 14, 875
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 4 , 1985 693
Probe Diffusion in Poly(ethy1ene oxide)-Water pm (Polysciences, Inc. nominal surface charge 0.12 mequiv/g of polymer). More limited measurements were made on 0.15-, 0.30-, and 3.0-pm spheres (Polysciences, Inc.). According to the manufacturer the sphere sizes other than 0.038 pm are nominal; while monodisperse, actual and nominal sizes may vary from batch to batch. In pure water, the nominal 0.038-, 0.12-, 0.70-, and 1.28-pm spheres have diffusion coefficients of 1.15 X lo-’, 4.64 X 7.44 X and 3.66 X cm2/s, respectively, corresponding to hydrodynamic radii of 208 A, 517 A, 0.322 pm, and 0.655 pm, respectively. Spheres were added to the polymer-water mixtures to concentrations of 0.05% (w/v) for the 0.038-pm spheres, 0.006% (w/v) for the 0.120-pm spheres, and 0.0025% (w/v) for the larger spheres. At these concentrations, multiple scattering and sphere-sphere interactions are insignificant. Light scattering by the poly(ethy1eneoxide) is negligible by comparison with scattering by the spheres. As justified in section IV, the diffusion coefficient D inferred from the light scattering spectra is hereinafter referred to as the diffusion coefficient of the spheres. Our working temperature was 24.9 “C. Viscosities were obtained with Cannon-Fenske and Ubbelohde viscometers. Diffusion coefficients were determined with quasi-elastic light scattering spectroscopy. Samples were illuminated with 20-30 mW of 6328-A laser light; the scattering angle was 90“. Spectra were acquired on a 64-channel (56 data channels, 8 delay channels) Langley-Ford Instruments digital autocorrelator and on a 144channel (128 data channels, 16 delay channels) Langley-Ford Instruments Model CM64 digital correlator. The sample time (the time width of an individual correlator channel) was chosen by beginning with a sample time which was too large, and then reducing the sample time until the measured spectrum decayed to 50% of its initial value in 8-20 sample times. Spectra were fit to the cumulant expansion
‘/z In
4
n
[S(k,r) - E] = CKj(-tY/j! j=O
(2)
where n is the order of the fit, Kj is the j t h cumulant, and B is the base line. Base lines were obtained from the correlator delay channels. Spectra were examined with a variety of delay times to ensure that the base line channels actually contained the base line, Fits with n = 1 to 5 were made; the best fit was identified from the root-mean-square error and the quality parameter Q
+P) i
N-1
Q = iC(di - Ci)(di+i - Ci+i) =l
(3)
where dj is the measured spectrum at time i, Ci is the calculated spectrum at time i, and N is the number of correlator data channels. As the best fit is approached, the magnitude of Q tends to a minimum. The optimum value of n was 2 or 3. At least three measurements of D were made on each sample; Figures show the average and spread of these measurements. Signal-to-noise ratios [the ratio of S(k,O) to the rms error in the fit] were generally in the range 250-750. D was obtained from the first cumulant
D = K,/2k2
(4)
Here k = lkl,k being the scattering vector. Scattering vectors were corrected for the index of refraction of the solution, as obtained with a Bausch & Lomb Abbe-56 refractometer. A significant concern was the extent to which adsorption of poly(ethy1ene oxide) by the probe particles might change their hydrodynamic radii. Efforts were made to control this effect by (i) rendering the solutions more basic (pH 9), thereby ionizing the carboxylic acid groups on the spheres and preventing hydrogen bond donation; (ii) adding to the solutions 17 wt % (1.2 M) diglyme, in order to saturate ether binding sites on the spheres with this small-molecule triether; (iii) adding to the solutions 0.1 wt % of the nonionic detergent Triton X-100 to passivate the sphere surfaces. As seen below, methods i and ii were apparently ineffective; control experiments demonstrate the effectuality of method iii.
C
c(g/L)
Figure 1. (a) Concentration dependence of the viscosity of water-poly(ethylene oxide) of molecular weight 3 X los,1 X lo5,18 500, and 7500 amu. The lines represent linear least-squares fits of the data. (b) Concentration dependence of the viscosity of poly(ethy1ene oxide) of molecular weight 18500 and 7500 amu. The curves represent fits of the data to eq 5 .
111. Experimental Results Figure 1 shows the viscosity of poly(ethy1ene oxide)-water. TJ increases with increasing polymer concentration c and polymer molecular weight M . Below 3 g/L, TJ is essentially linear in c. At the higher concentrations attained with the 7500- and 18 500-amu polymers, TJ is described well by the form (solid lines, Figure lb)
TJ/TJ,, = exp(-a”P)
(5)
a’’ and Y” being arbitrary parameters discussed in section IV. Separate measurements were made on water-polymer-Triton X-100 solutions; the effect of the Triton X-100 on TJ is small. Measurements of D of the 208-%r, 517-A, 0.322-pm, and 0.655-pm polystyrene spheres are presented in Figure 2. The horizontal scale is logarithmic in c; dashed lines are guides for the eye. Each point represents the average of at least three spectra.
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The Journal of Physical Chemistry, Vol. 89, No. 4, 1985
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!a
d
I
I
I
' 0 .
I
1
I
IO
I
"".9,
,
100
c (g/L)
Figure 2. Concentration dependence of the diffusion coefficient D (in units lo-' cm2s-I) of (a) 208-A, (b) 517-A, (c) 0.322-pm, and (d) 0.655-pm diameter carboxylate modified polystyrene spheres in (0) 7500, ( 0 )18500,(+) 1 X lo5,and (X) 3 X lo5 amu poly(ethy1ene oxide)-water solutions. The curves are drawn to guide the eye. Insets show the D of the spheres in concentrated 7500-amu poly(ethy1ene oxide)-water solutions.
Error bars indicate the dispersion in measurements of D on a single sample. The behaviors exhibited by the 208-%r,517-A, and 0.322-pm spheres are largely similar. In each water-polymer system, the diffusion coefficients decrease substantially with increasing polymer concentration. D of the 208-A spheres in pure water is 1.15 X cm2/s and falls to 1.02 X 0.66 X 0.60 X and 0.4 X cmz/s in 3 g/L solutions of the 7 500, 18 500, 1 X lo5, and 3 X lo5 amu polymers, respectively. D of the 208-A cm2/s in 50 g/L of the spheres falls further to 1.7 X cm2/s in 350 g/L of the 18 500-amu polymer and to 1.4 X 7500-amu polymer. At fixed pol mer concentration, the diffusion coefficients of the 208-A, 5 17- , and 0.322-pm probe particles almost always fall with increasing M . The dependence of D upon M is largest when the polymer concentration is high. The behavior of the 0.655-rm spheres in the presence of poly(ethy1ene oxide) is surprising. In solutions of the 7500-amu polymer, D falls monotonically with c. However, in solutions of the 18 500 and 3 X lo5amu polymers, D increases with increasing c. This effect, which is appreciably larger than the experimental scatter of the data points, is also seen if Triton X-100 is present in solution. Figure 3 shows D for polystyrene spheres in water-polymerTriton X-100. D for spheres in water-Triton X-100 in the absence of polymer is smaller than d for spheres in pure water, as expected if the detergent had coated the sphere surfaces. Figure 3, a and b, shows results on the 208- and 517-A spheres, for which D
1
decreases smoothly with increasing polymer concentration and with increasing polymer molecular weight. D is linear in c at low polymer concentration; this linearity is obscured by the semilogarithmic scale for c. Figure 3c gives D against c for the 0.322-pm spheres. For c < 30 g/L, D is larger for spheres in the 18 500-amu polymer than for the same spheres in the 7 500-amu polymer. This effect might be a precursor of the effects seen in Figure 3d, which gives D for the 0.655-pm spheres. In the 18 500-amu polymer solutions, D of the 0.655-pm spheres increases substantially with increasing c. Each data point was obtained from a different sample; the data show a smooth, reproducible curve, as would be expected if this non-monotonic relation between D and c were a physical effect rather than a heterodyning artifact due to dust combination of a few samples. Scaling arguments suggest that D has a form like D / D o = exp(-cyc") (6) Do being D in the limit of low polymer concentration, and cy and v being arbitrary parameters. The solid lines of Figure 3 show fits to this form; the corresponding a and v are given in Table I. Figure 4 shows tests of methods of preventing polymer binding by the spheres. The a parent hydrodynamic radius rH (eq 10) of the 208- and 517- spheres is plotted as a function of the concentration of 1 X lo5 amu poly(ethy1ene oxide). Neither raising the p H to 9 nor adding 17 wt % diglyme has an effect on the concentration dependence of rH.
kt
The Journal of Physical Chemistry, Vol. 89, No. 4, 1985 695
Probe Diffusion in Poly(ethy1ene oxide)-Water 1.2
I
I
I
I
I
I
~
c (g/L) 0.5I,,
,
I
1
O6
I
1
d
x’
05 -
BI
1
5.
3 \a \
04-
\
c
D 0302.01-
0
c (g/L)
c(g/L) Figure 3. Concentration dependence of the diffusion coefficient D (in units lo-’ cmz s-I) of (a) 208-A, (b) 517-A, (c) 0.322-pm, and (d) 0.655-pm diameter carboxylate modified polystyrene spheres in (0)7500, ( 0 )18500, (+) 1 X lo5, and (X) 3 X lo5 amu poly(ethy1ene oxide)-water-O.l wt % Triton X-100 solutions. The solid curves represent fits of the data to eq 6. The dashed line is a guide for the eye. r
750
T
T
0*
E-
0.250
4{
o.2
w t.%
20
15
0.1
,
,
,
,
Water I 2 M diglyme
^;a
lo
7,
005 M CHES, pH9
500 2
I
3
c(g/L)
Figure 4. Effects of the presence of 1.2 M diglyme or 0.005 M CHES on the concentration dependence of D of the 208- and 5 17-A spheres in poly(ethy1ene oxide)-water. Neither species apparently hinders polymer binding.
Figure 5 shows r H of probe particles in 1 X 10’ amu polymer solutions as a function of Triton X-100 concentration. Even at high polymer concentration (1 .O g/L), r H changes substantially
Figure 5. Effect of Triton X-100 on rH of 0.322-pm spheres in 1.0 g/L of polymer solution. Note that rH for solution containing Triton X-100 and no polymer is larger than rH for solutions containing neither polymer nor Triton X-100.
on first addition of Triton X-100, and then remains constant over a range of Triton concentrations cT. This dependence of rH on CT has the form expected if the polymer adsorption were described by the mechanics of Langmuir adsorption isotherms, in which data were only obtained a t the two endpoints of the titration. If the Triton were unable to displace the polymer, rH would not change on first addition of detergent. If there were effective competition between polymer and detergent binding, r H would depend on cT for cT > 0. Neither of these effects is observed. At moderate polymer concentrations, the Triton X- 100 prevents polymer binding by the spheres. The following sections analyze the above at increasing levels of abstraction. In section IV, we interpret D in terms of nonStokes-Einsteinian hydrodynamics and polymer adsorption; these factors are separated with a plausible phenomenological analysis.
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TABLE I: Nonlinear Least-Squares-Optimized Parameters for Representing D / D o of Polystyrene Spheres of Various Molecular Weights in Water-Poly(ethy1ene oxide)-Triton X-100by the Form D / D o = exp(-ac’)
1 x 105amu
18500 amu
7500 amu sphere size 208 8, 517 8, 0.322 pm 0.655 pm
ff
V
ff
V
ff
V
0.035 0.034 0.036 0.052
0.81 0.83 0.82 0.75
0.076 0.061 (0.0045)*
0.78 0.84 (1 .45)b
a
a
(0.013)b 0.046 0.064 0.094
(1.95)b 1.35 1.oo 0.98
3 x io5 amu v 0.14 0.88 0.14 0.98 0.12 0.82 ff
a
a
‘The data are not consistent with this analytical form. bParentheses indicate fits of marginal quality.
Section V analyzes the data by using scaling models for polymer solutions. Section VI compares our findings with other experimental studies of probe diffusion.
IV. Discussion and Phenomenological Interpretation In this section we first justify the identification of D from the scattering spectrum with D of the probe particles. In a ternary system, this is not a trivial problem. We then introduce a parameter K to characterize the failure of the Stokes-Einstein equation. K can be obtained from data on the Triton X-100 coated spheres. D for spheres in solutions containing no Triton X-100 can rationally interpreted in terms of the previously determined dependence of K on c, together with Langmuir adsorption of the polymer by the spheres. I. Interpretation of Spectra of Multimacrocomponent Systems. One of us8 has demonstrated that the spectrum of a solution of two interacting Brownian species contains two exponentials, even ifonly one species scatters any light. The decay constants of the exponentials are determined by all four mutual and cross diffusion coefficients; neither decay constant describes the motion of a single Brownian species. In our system, the scattering species is dilute. In this special case, the spectrum is a single e~ponential~-’~ with decay constant r = 2Dk2 (7) This species is dilute, so its mutual and self-diffusion coefficients have the common form D = kBT/f Here f is the drag coefficient of the scatterers in the ternary system. The spectrum thus gives the diffusion coefficient of the probe particles. The unseen polymer hinders the probe motions but does not contribute an additional mode to the spectrum. 2. K, an Empirical Parametric Modification of the StokesEinstein Equation. Equation 1 is readily compared with experiment if the probe size is known, as is the case in water-poly(ethylene oxide)-Triton X- 100. To parametrize failures of eq 1, we write (9)
K has previously been employed to describe similar effects in rotational diffusion.I5J6 We may alternatively introduce an “effective hydrodynamic radius” rH defined as
Simple polymer adsorption would increase R, leading to the apparent result K > 1. In mixtures containing Triton X-100, we (8) Phillies, G. D. J. J . Chem. Phys. 1974, 60, 983. (9) Phillies, G. D. J. Biopolymers 1975, 14, 499. (10) b e y , P. N.; Fijnaut, H. M.; Vrij, A. J . Chem. Phys. 1982, 77,4270. (11) Kops-Werkhofen, M. M.; Fijnaut, H. M. J. Chem. Phys. 1981, 74, 1618. (12) Kops-Werkhofen, M. M.; Pusey, P. N.; Fijnaut, H. M. Chem. Phys. Lett. 1981, 81, 365. (13) Kops-Werkhofen, M. M.; Jijnaut, H. M. J . Chem. Phys. 1982, 77, 2242. (14) Kops-Werkhofen, M. M.; Pathmamanoharan, C.; Vrij, A,; Fijnaut, H.M. J . Chem. Phys. 1982, 77, 5913. (15) Kowert, B.; Kivelson, D. J . Chem. Phys. 1976, 64, 5206. (16) Phillies, G. D. J.; Kivelson, D. J . Chem. Phys. 1979, 71, 2575.
TABLE 11: Nonlinear Least-Squares-OptimizedParameters for Representing K for Polystyrene Spheres of Various Molecular Weights in Water-Poly(ethy1ene oxide)-Triton X-100by the Form K = exp(-a’&) 1 x io5 amu 3 x io5 amu
sphere size 208 A 517 8, 0.322 pm 0.655 um
a’ 0.10 0.062 0.045 0.014
V‘
0.48 0.32 0.62 -0.21
a’ 0.1 1 0.11 0.12 0.29
‘V
0.78 0.61 0.86 1.01
almost always find K C 1; Le., rH C R. This result may stem from molecular hydrodynamics but is difficult to reconcile with polymer adsorption effects. 3. Measurements of K. Figure 6 plots K against c for probes in water-polymer-Triton X- 100; data are arranged by polymer molecular weight. Do values are for spheres in water-Triton X-100 at c = 0; the corresponding probe radii (233 A, 530 A, 0.336 pm, and 0.663 pm, respectively) are slightly larger than are probe radii in pure water. The increase in R may reasonably be interpreted as arising from a 15-25-A layer of detergent adsorbed to the probe surfaces. K for probes in the 7500-amu polymer is given in Figure 6a. For the 0.665-pm spheres, K = 1 at all polymer concentrations. For the 517-Aand 0.322-pm spheres and c C 300 g/L, 0.9 C K < 1.1. K for the 208-A spheres falls monotonically with increasing C.
Figure 6b gives K for spheres in the 18 500-amu polymer. The dependences of K on c are qualitative1 similar to those noted in Figure 6a. K of the 208- and 5 1 7 - i spheres is in the range 1.0-1.1 5. K of the 0.322- and 0.655-pm spheres falls substantially with increasing c. Figure 6, c and d, gives K for probes in the 1 X lo5 and 3 X lo5 amu polymer solutions. Again, K falls monotonically with increasing polymer concentration and depends substantially on the size of the spheres and the polymer molecular weight. The dependence of K on R is not the same for these two polymers. This lack of consistency was also seen in previous studies3 of water-poly(acry1ic acid). For the 1 X lo5 and 3 X lo5 amu polymer solutions, K is described well by K = exp(-a’cy’)
(1 1)
Values for the fitting parameters a’ and Y’ are presented in Table 11. Curves generated with eq 1 are the solid lines in Figure 6. 4. Effect of Adsorption on D in the Absence of a Surfactant. The behavior of the spheres in the absence of Triton X-100can be interpreted in terms of non-Stokes-Einsteinian hydrodynamics and polymer adsorption. In the above, the behavior of polystyrene spheres in the presence of Triton X-100 was parametrized by K. To determine K , probe particles of known size are needed; the detergent ensured that R is constant. K only depends weakly on R, so, since Triton-coated and uncoated spheres have approximately the same radius, they should have nearly the same value of K. We now use K of the surfactant-coated spheres to infer the sphere radius and amount of bound polymer in solutions containing no surfactant. Preliminary results for probe particles in the 3 X 1O5 amu polymer have previously been published.17 (17) Ullmann, G.; Phillies, G. D. J. Macromolecules 1983, 16, 1947.
The Journal of Physical Chemistry, Vol. 89, No. 4, 1985 697
Probe Diffusion in Poly(ethy1ene oxide)-Water
1.2
'
! I
I I
K
0
3 ~
1
~
1
0 30
50
0.31 0
I
I
1.0
I
I
2.0
I
I
3.0
I
C(q/P)
c(g/L)
Figure 6. Concentration dependence of K of (0)208-8,, (A)517-8,, (0) 0.322-pm, and ( 0 )0.655-pm diameter spheres in (a) 7500, (b) 18 500, (c) 1 X lo5, and (d) 3 X lo5 amu poly(ethy1ene oxide)-water-O.1 wt % Triton X-100. The curves in parts a and b are drawn to guide the eye, while those of parts c and d are fits to eq 12. TABLE III: Langmuir Adsorption Isotherm Parameters 1 x io5 amu 3 x io5 amu
sphere size 208 8, 517 A 0.322 pm 0.655 pm
a
b
a
b
4.57 X lo4 4.37 x 10-3 0.166 0.215
0.555 1.21
1.69 X lo4 1.79 x 10-3
2.49 4.21
0.384
2.46
1.84 13.5
Figure 7 shows the radii r of uncoated spheres, as calculated from eq 9. Solid lines are obtained from Langmuir adsorption isotherms, determined by a least-mean-squares fit of the amount x / m of adsorbed polymer
to'*
c - -c+ - 1 -x / m a ab Here R is the bare radius of the spheres, a gives the limiting thickness of bound polymer in the limit of binding saturation (c m), and b measures the strength of binding. Our determinations of a and b are in Table 111. The constant a cannot be determined directly from the sphere radius at high polymer concentration because at the solubility limit for PEO in our system the spheres are not saturated with polymer. As seen in Figure 7, c and d, changes in r in 1 X lo5 and 3 X lo5 amu PEO are well described
-
as consequences of Langmuir adsorption isotherms. We emphasize that K was measured in solutions containing surfactant and was not obtained from a multiparameter fit to Figure 7 . To obtain b accurately, D is needed at polymer concentrations at which the probe particles are not saturated with polymer. We did not always happen to do this, so some values of b are of limited accuracy. As our primary interest is solution hydrodynamics, this point was not pursued further. Probe particles in 18 500-amu polymer solutions exhibit a variety of concentration dependences. K may be either larger or smaller than unity. Furthermore, K is not monotonic in c. If r is computed from D,7, and K,one obtains Figure 7b. We find that r is larger when polymer is present than when polymer is absent, even though D may either increase or decrease as polymer is added. If our data were analyzed with the assumption that eq 1 is always valid, one would gain an incorrect impression of the behavior of our system. For the 208- and 517-A probe particles in the 18 500-amu polymer, the apparent sphere radius has a maximum for c = 1-2 g/L polymer and decreases for c > 3 g/L. If data had only been taken for 0 6 c 6 3 g/L, as was done for the high-molecularweight polymers, this behavior of r would have been far less apparent. The Stokes-Einstein eq 1 is apparently valid for solutions of low-molecular-weight (7500 amu) polymer. In the range 0-3 g/L, r (eq 9, Figure 7a) of the detergent-free spheres increases smoothly from its value in pure water to a limiting value found at large c. For 30 < c < 350 g/L, in all cases studied (including measurements not presented in detail here on 0.08- and 1.5-pm probe
-8.0
Ullmann et al.
The Journal of Physical Chemistry, Vol. 89, No. 4, 1985
698
0
2
2.5
5 IO
I
50
I
I
I
L
200
100
,
1
300
c(g/L)
c(g/L) d
I
I
I
I
I
I
I
I
I
I
I
I
-------t 08
IO
20
31)
IO
30
c(g/Ll I
Figure 7. Concentration dependence of r / R of (0)208-A, (A) 517-A,(0)0.322-”, and ( 0 ) 0.655-pm diameter spheres in (a) 7500, (b) 18 500, (c) 1 X los, and (d) 3 X 10’ poly(ethy1ene oxide)-water. The curves of parts a and b are drawn to guide the eye, while those of parts c and d are the calculated Langmuir adsorption isotherms of eq 14.
particles) r is independent of c. That is, in the absence of the detergent, r has the form expected if polymer binding was described by Langmuir adsorption with saturation. Data on surfactant-coated 517-A, 0.326-pm, and 0.655-pm spheres also shows that K = 1.0 for solutions of the 7500-amu polymer. For the 208-A probe particles, K in the presence of Triton X-100 falls from 1.O in pure water to 0.8 at c = 300 g/L. At these very high polymer concentrations, the detergent might desorb from the probes, or it might perturb the solution hydrodynamics. Figure 8 shows how the limiting (as c m) thickness 6R of adsorbed polymer depends on R and M. 6R varies with molecular weight as 6R M‘/*for the 1 X los and 3 X lo5amu polymers.
-
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V. Comparison with Fundamental Theory This section treats relevant theoretical work on probe. diffusion. The parametrization introduced in eq 14-16 will be used in the
section VI to describe other experimental studies. In the deGennes of polymer solutions, solution properties were asserted to depend qualitatively on the polymer concentration c, with properties changing their nature as the polymer concentration passes from one regime to the next. Of particular interest is the concentration c*, which is said to divide dilute solutions (in which polymer chains are appreciably separated from each other) from semidilute solutions (in which polymer chains necessarily overlap). A semidilute polymer solution is viewed as a transient mesh of polymer chains, the distance between (18) Adamson, A. W. “Physical Chemistry of Surfaces”; Interscience: New York, 1960. (19) deGennes, P. G. Macromolecules 1976, 9, 587. (20) deGennes, P. G. Macromolecules 1976, 9, 594. (21) deGennes, P. G.; Pincus, P.; Velasco, R. M.; Brochard, F. J . Phys. (Orsay, Fr.) 1976, 37, 1461.
The Journal of Physical Chemistry, Vol. 89, No. 4, 1985 699
Probe Diffusion in Poly(ethy1ene oxide)-Water I2001
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5 17-A probes thus have R/[ 5 1, while in these same solutions the 0.62-pm probe particles have R / [ 15-30. While much of our data is for R/[ > 10, our measurements should span both eq 14 and 16. The above theoretical concepts were used to analyze our data. Section 111 gives cases in which q, D, and K were fit to the analytical scaling form of eq 5, 6, and 11. Parameters were obtained by a nonlinear least-squares search procedure, the quantity which was minimized being
Figure 8. Dependence of the high-concentration limiting thickness bR of bound polymer on the radius R of the binding particle for the (0)3 X
lo5 and (A) 1 X lo5 amu polymers.
contact (“entanglement”) points of the chains giving a scaling length [. For polymers of high molecular weight, [ is much less than the total chain length; chain ends are rare. The entanglement points are then claimed to dominate polymer dynamics. In a semidilute solution, local properties are thus predicted to be independent of M . As long as entanglement points dominate the dynamics, the exact location of the small number of chain ends should be unimportant; this physical model thus also appears to predict for the semidilute regime that the exact form of the molecular weight distribution is not important. It is further predicted that polymer solution properties follow scaling laws (e.g., eq 5,6, and 1l), in which the logarithmic dependence of a property is proportional to a power of the polymer concentration. deGennes20 predicts that c* scales as M-”.*. For poly(ethy1ene oxide) of molecular weight 3 X lo5, Destor et aLz2found c = 0.3 g/L, so for solutions of the 1 X lo5, 18500, and 7500 amu poly(ethy1ene oxide), c* is 0.7,2.5, and 5.6 g/L, respectively. Our data thus span both the dilute and semidilute ranges. Figures 2-4 do not suggest that probe diffusion is particularly sensitive to a transition near c*. The most extensive theoretical analysis of probe motion in semidilute and other solutions, in terms of this modern model, is that of Langevin and Rondelez.’ They argue that if R / [ >> 1, the polymer solution will appear to the probe particle as a continuum, so that the drag coefficient will be determined by the macroscopic viscosity the subscript “0” referring to pure solvent. On the other hand, if R / [ 5 1, the drag coefficient should following a scaling law
In our case, since fo/f
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fo/f= W / E )
(15)
(c*/c)fi, one might have
= exp(-ac’) = exp(-a*R6cu’)
(16)
Here v, v * , p, a,a*,and 6 are coefficients determined by the solution being studied; v* = p& Scaling theories predict p = for polyelectrolyte moleculesz2and p = 3/4 for neutral polymers.20 The rather different physical models of Ogston et al.z3and CukierZ4 predict 6 = 0.5 and 6 = 1, respectively. Destor et a1.22report E = 31 A for a 10 g/L polyethylene oxide solution. From the scaling-theory value for p of a neutral polymer, a t c = c* the 3 X lo5, 1 X lo5, 18 500, and 7 500 amu polymers have of 430, 230, 80, and 50 A, respectively. In the most dilute solutions of the highest-molecular-weight polymer studied here, the 208- and (22) Destor, C.; Langevin, D.; Rondelez, F. Polym. Lett. 1978, J6, 229. (23) Ogston, A. G.; Preston, P. N.; Wells, J. D. Proc. R. Soc. London, Ser. A 1973, 333, 291. (24) Cukier, R. I. Macromolecules 1983, 17, 252.
where Qi is the value of the variable being fit a t concentration c, Qois the value of Q at c = 0, a and v are adjustable parameters, and the sum is over all data points. As seen in Figure 3, when eq 16 is at all correct, it is valid both at high and a t low polymer concentrations. With the 7500-pm polymer solutions, a nd v were in the ranges 0.034-0.052 and 0.75-0.83, respectively. qo/q and DIDo of the 7500-amu polymer solutions have the same concentration dependence, so the Stokes-Einstein equation predicts D accurately. If M is increased, a increases substantially, reaching 0.12-0.14 in the presence of the 3 X lo5 amu polymer. A log-log plot of a against M is the though the possible error in the M consistent with a exponent is substantial. Y depends at most slightly on polymer molecular weight. Neither a nor v depends appreciably on R. The diffusion of the 0.655-pm spheres in the 18 500, 3 X lo5, and perhaps 1 X lo5 amu polymer solutions is not described by eq 14 or 16. For 0 < c < 3 g/L, D of these spheres increases with increasing c. At the larger concentrations accessible with the 18 500-amu polymer, D reaches a maximum and then decreases again. K is also described by the power-law form eq 11. With the 7500-amu polymer, K = 1; a’ = 0. With the 1 X lo5 and 3 X lo5 polymers, K is described accurately by the parameters of Table I1 (solid lines, Figure 6). The adjustable parameters a’ and v’ both depend on the sphere size and on the polymer molecular weight. K can be a well-behaved function of c even when D/Do is not. D of the 0.655-pm spheres in the 3 X lo5 amu polymer solution increases with increasing c, but K looks well-behaved. While dq/dc > 0 in these solutions, the polymer is more effective at causing a failure of the Stokes-Einstein equation than at increasing q , so D and q both increase with increasing c.
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VI. Other Experimental Work Studies of probe diffusion have also been reported for macromolecules in water-dextran, water-hyaluronic acid,z5-29waterpoly(acry1icacid): and water-poly(ethy1ene oxide).’ Substantially disparate values of a,v , and their dependences on M and R have been reported. In water-dextran and water-hyaluronic acid, Laurent and c o - w o r k e r ~used ~ ~ *ultracentrifugation ~~ and boundary spreading to measure the drag coefficientfof proteins and larger particles 25 A-0.365 pm). They found v = a increased with (R increasing R, so that a R’. The M dependence of a and v were not studied. Probe motion in water-dextran was also studied by Brown and Stilbs,28 who used pulsed field gradient N M R to measure the self-diffusion of a poly(ethy1ene oxide) probe species, and by Turner and Hallet:9 who used light scattering spectroscopy to study D of polystyrene spheres. These latter two studies found u = 1, not v = 0.5. Contrary to Laurent et al., for R / l 2 1 Turner and Hallet found that a is independent of probe size, but increases appreciably with increasing dextran molecular weight. Lin and Phillies3 used methods similar to ours to study probe diffusion in water-poly(acry1ic acid). In 50 000-amu poly(acry1ic acid), u = 0.9 f 0.1; at higher molecular weight (3 X lo5, 1 X
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(25) Laurent, T. C.; Persson, H. Biochim. Biophys. Acta 1964,83, 141. (26) Laurent, T. C.; Bjork, I.; Pietruszkiewicz, A,; Persson, H. Biochim. Biophys. Acta 1963, 78, 351. (27) Laurent, T. C.; Pietruszkiewicz, A. Biochim. Biophys. Acta 1961.49, 258. (28) Brown, W.; Stilbs, P. Polymer 1983, 24, 188. (29) Turner, D. N.;Hallett, F. R. Biochem. Biophys. Acta 1976, 451, 305.
700 The Journal of Physical Chemistry, Vol. 89, No. 4, 1985
lo6 amu), Y = 2 / 3 f 0.1. a increased with increasing polymer molecular weight. da/dR was not quite zero, but changed, even as to sign, as M was changed. Finally, Langevin and Rondelez7 made sedimentation studies of a series of probes of radii 17-460 A in water-polyethylene oxide, finding u = 2/3. They concluded that their data were consistent with a R’,and that neither a nor Y depends on M . Our findings do not agree completely with those of Langevin and Rondelez. We find u in the range 0.8-1.0, not 2/3. Y is independent of M and R . Furthermore, we find that a is nearly independent of R but depends substantially (Table I) on M , contrary to ref 7. Langevin and Rondelez show that artifacts due to probe-probe interactions and polymer adsorption can be important. We used very low probe concentrations, so probe-probe interactions were insigniicant. There are two potential artifacts due to polymer adsorption. First, polymer adsorption by the probe particles lowers the effective polymer concentration in the solution. With an infinitesimal probe concentration, this effect cannot be significant. Second, adsorbed polymer molecules change R , confusing interpretation of D. Our success in comparing surfactant-coated and uncoated spheres indicates that we can control this effect. Most studies of probe diffusion have not compared D and 7 . Jamieson et aL30 report D and 7 for a xantham gum-waterpolystyrene sphere system. Above 0.2 wt %, the xanthan gum causes a substantial increase in 7 , while D falls substantially. For water-dextran, Laurent and P e r ~ s o nreport ~ ~ solution viscosities and sedimentation coefficients in graphical form, noting that sols depends on concentration less strongly than does 7/qo. Their data thus anticipated ours in showing that non-Stokes-Einsteinian effects in polymer solutions cause rHto fall with increasing polymer concentration. Furthermore, Laurent and Persson emphasize that the difference between sols and q/qo, Le., the deviation of K from unity, is largest for solutions of linear polyelectrolytes. Here we have found K 0.4 to 1.1, while in poly(acry1ic acid)-water Lin and Phillies3 report K < low2. Langevin and Rondelez discuss r H ,noting a theoretical expectation that very large probes ( R / [ >> 1) should respond to or the macroscopic shear viscosity, so that rH R as R / l R a. Furthermore, because E decreases with increasing c, differences between rH and R are predicted to decrease as c is increased. We find just the opposite of these predictions. As seen in Figure 6, r H / Rdoes not approach unity as large c. The Stokes-Einstein equation is no more successful with large probes than with small probes.
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VII. Conclusions and Remarks We studied the diffusion of polystyrene latex probe spheres in aqueous solutions of poly(ethy1ene oxide). The probe diameter and the concentration and molecular weight of the polymer were each separately varied over one to three orders of magnitude. The Stokes-Einstein equation fails in polymer solutions with M > 7000. An empirical correction factor K, which allows determination of the true size of a Brownian particle in a polymer solution from its diffusion coefficient, was obtained. By using K , changes in D and rHcould be interpreted in terms of changes in 7 and in the amount of bound polymer. We find that large (up to micron-size) probe particles move through polymer solutions more rapidly than expected from the viscosity, at least for particle motions on the distance scale fixed by our scattering vector. A standard intuitive explanation for non-Stokes-Einsteinian behavior is that Stokes’ law is obtained (30) Jamieson, A. M.;Southwick, J. G.; Blackwell, J. J . Polym. Sci.: Polym. Phys. 1982,20, 1513. (31) Phillies, G.D.J. J . Chem. Phys. 1974,60, 976.
Ullmann et al. for a continuum fluid and therefore does not apply when solute and solvent molecules are similar in size. However, we find K < 1 for huge (R > 0.6 pm) particles, so the intuitive explanation for K # 1 is not relevant. As one of us31 has previously emphasized, light scattering spectroscopy and the Stokes-Einstein equation cannot be used to infer the hydrodynamic radius of a thermodynamically nonideal (Le., nondilute) species of macromolecule. From the above, light scattering spectroscopy does not yield the size of particles in a dilute suspension, unless the solvent is well-behaved. Highly viscous solutions (1000 cP) of small molecules (glycerol, sorbitol) are well-behaved in this sense. Solutions of neutral or charged3 polymers are not well-behaved, in this sense. Our results were compared with scaling theories of polymer solutions and with related studies on probe motion. We obtained scaling-law forms, such as eq 16, which describe most of our data. However, the parameter a depends on M but is independent of R, contrary to theoretical predictions that a R 1 and a w. Our values for Y do not agree with those of Langevin and Ronde le^,^ who used ultracentrifugation on the same system. An exactly parallel set of discrepancies exist between the results of Brown and Stilbs,28and Turner and Hallet,29on one hand, and Laurent and c o - ~ o r k e r s >on~the ~ ~other hand, on water-dextran. The long-term methods (sedimentation, boundary-spreading diffusiometry) used in ref 7 and 25-27 consistently find smaller values of v than do the short-time methods (spin-echo NMR, light scattering spectroscopy) used here and in ref 28 and 29. Furthermore, the long-time value of a is proportional to R, but is independent of M,while the short-time value of a depends on M , but not R. These differences between the short- and long-time methods could reflect as a residual frequency dependence of the transport coefficients. Iffwere significantly frequency dependent, extreme caution would be indicated before using the sedimentation coefficient, D from a light-scattering spectrum, and the Svedberg equation to infer the molecular weight of a probe particle in a polymer solution. A polymer solution is sometimes described as a transient net or gel. This description leads to the expectation that particles which can slip between the mesh (R/[< 1) will be retarded less by a polymer solution than particles which must push the mesh aside before they can move. We studied both the regime R / [ < 1 and the regime R / [ >> 1 ; the concentration dependence of D for small and large particles is nearly the same. On our time and distance scales, the notion that a polymer looks like a transient gel is therefore false. Physically, our results suggest that (on our time and distance scales) polymer entanglements do not dominate other (e.g., hydrodynamic) effects in polymer dynamics. Our findings may have practical application in the preparation of biopolymers. Most modern electrophoretic procedures rely on supported systems to avoid convection. However, the pore size of many supports hinders operations on macroparticles and biological cells. The finding K < 1 indicates that a polymer solution may be more effective at increasing 7 , thereby hindering convection, than at preventing diffusion. Since the drag coefficients for single-particle diffusion and for electrophoresis are closely related, the addition of a high-molecular-weight polymeric component to a free (Tiselius) electrophoresis system would stabilize against convection while perhaps not preventing the electrophoretic motion of large biological assemblies.
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Acknowledgment. The support of this work by the National Science Foundation under Grants CHE-7920389 and CHE821 3941 is gratefully acknowledged. Registry No. Polystyrene (homopolymer),9003-53-6; poly(ethy1ene oxide) (SRU),25322-68-3.
