Probe Diffusion in Hydroxypropylcellulose−Water: Radius and Line

Light-scattering spectroscopy was used to examine diffusion of optical probes of various radii in solutions (0 ≤ c ≤ 15 g/L) of 300 kDa hydroxypro...
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J. Phys. Chem. B 1997, 101, 39-47

39

Probe Diffusion in Hydroxypropylcellulose-Water: Radius and Line-Shape Effects in the Solutionlike Regime George D. J. Phillies* and Mickey Lacroix Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 ReceiVed: May 23, 1996; In Final Form: September 4, 1996X

Light-scattering spectroscopy was used to examine diffusion of optical probes of various radii in solutions (0 e c e 15 g/L) of 300 kDa hydroxypropylcellulose (HPC) at 25 °C. Spectra follow well a stretched-exponential g(1)(t) ) A exp(-θtβ) ≡ A exp((t/τ)β), plus at elevated c a weak, rapid initial decay. Accurate values for θ, β, and τ were obtained via multi-τ spectral splicing techniques. For 109, 189, and 760 nm diameter spheres, θη (η is the solution viscosity) is independent of c. HPC:water shows pseudo-Stokes-Einsteinian behavior for large probes. For 21 nm probes, θη increases by 10-fold over 0 e c e 15 g/L. From the η and scattering vector dependences, θ and β are inferred to be the quantities most likely to be amenable to direct theoretical calculation.

Introduction The dynamic behavior of nondilute polymer solutions has long been a vigorous topic of investigation in polymer physical chemistry. Two decades ago, substantial interest was attracted by the tube/reptation/scaling models of Doi and Edwards1 and deGennes.2 These models have substantial disagreements with experiment. For example, reptation/scaling predicts that the selfdiffusion coefficient Dc of a polymer in semidilute solution depends on polymer concentration c and molecular weight M as

Dc ∼ c-1.75M-2

(1)

In contrast, the published literature shows3,4 that Dc in dilute as well as semidilute solution follows a stretched exponential

Dc ) D0 exp(-RcνMγ)

(2)

with ν ∈ [0.5, 1.0] and γ ) 0.8 ( 0.1. Equation 2, including quantitative predictions for ν, for γ, and for R with no free parameters, follows from fundamental considerations.5 Similarly, reptation/scaling predicts that the solution viscosity η of a polymer in semidilute solution follows a scaling law η ∼ cxM3. However, the published literature4,6,7 shows that η at almost all c follows a stretched exponential:

η ) η0 exp(-R′cν′Mγ′)

(3)

η0 being the solvent viscosity, but where the exponents ν′ and γ′ and prefactor R′ are phenomenological numbers that have not yet been calculated quantitatively from first principles. Experimentally, one finds4,7,8 that in some but not all systems at very large η there is a transition from solutionlike to meltlike behavior, in the sense that eq 3 is replaced by an approximate power law in c and M. Recent reviews9-12 find “an abundance of evidence argues against the simple reptation model’s validity”,10 “...it is unlikely that reptation is significant in the semidilute regime”,11 or “The experimental results on the concentration dependence of D in polymer solutions are also not in accord with the prediction * To whom communications may be addressed. E-mail: phillies@wpi. wpi.edu. X Abstract published in AdVance ACS Abstracts, December 15, 1996.

S1089-5647(96)01511-8 CCC: $14.00

based on the reptation model.”12 It thus continues to be interesting to inquire into how polymer chains move in nondilute solution. Many experimental techniques have been used to study transport in polymer solutions. This laboratory has stressed the use of quasielastic light-scattering spectroscopy (QELSS) and optical probe methods. In an optical probe experiment, one uses QELSS to observe diffusion of a dilute, relatively intensely scattering “probe” (typically polystyrene latex) through a relatively nonscattering dilute or concentrated polymer or surfactant solution (the “matrix”). Under conditions found here, optical probe experiments obtain the single-particle (“self-”) diffusion coefficient Ds of the probes through the matrix.13 It is necessary to emphasize that a probe sphere is not a polymer chain. While Ds is the self-diffusion coefficient of a probe sphere moving through the complex solvent formed by the solvent and polymer, just as Dc is the self-diffusion coefficient of a labeled chain moving through the same complex solvent, it has never been proposed that Ds is equal to the selfdiffusion coefficient Dc of the chains. Indeed, when Brown et al.14 determined the diffusion coefficients of spherical probes and labeled probe chains of equal hydrodynamic radius in the same medium, reanalysis15 showed that matrix chains were up to 30-fold more effective in hindering chain self-diffusion than at hindering sphere self-diffusion. This paper represents an extension of our earlier work on optical probe diffusion6,18 and viscosity17 in (hydroxypropyl)cellulose (HPC):water. Viscosity measurements on HPC:water have also been reported by Russo et al.,18-20 Yang and Jamieson,21 Phillies et al.,8 and Bu and Russo22 in connection with their optical probe studies as discussed below. Our previous work obtained8 Ds for one probe in HPC solutions at four different M. For the 300 kDa HPC studied here, probe spectra were fit adequately by a cumulant expansion yielding Ds and were fit no better by a sum:

(1)

(∑ ) (∑ ) 2

Ki(-t)i

i)0

i!

g (t) ) exp -

2

Kj(-t)j

j)0

j!

+ exp -

(4)

of two-cumulant series than by a single-cumulant expansion. Here g(1)(t) is the field correlation function and the Ki and Kj are cumulants. Analysis of Ds as a function of polymer © 1997 American Chemical Society

40 J. Phys. Chem. B, Vol. 101, No. 1, 1997 concentration, molecular weight, and solution viscosity appeared in the previous paper. In the previous paper8 we observed that our spectra also fit well to a stretched exponential exp(-θtβ) in time. We measure spectra over a limited time window with finite (S/N < 1 × 103) precision, so it is not surprising that the same data are both adequately described by several functions, namely, by cumulants and by a stretched exponential in time. Stretched exponentials in time arise naturally as part of the Ngai coupling analysis of relaxations in strongly interacting systems.23 A comparison of the Ngai model with results here will appear separately.24 The primary objective here was to determine θ and β to high precision and to compare them with c, M, and other experimental quantities. The purpose of the comparison is to investigate if θ and β give perspectives about probe dynamics other than those afforded by Ds. As extensions to our previous work, we (1) used probes of several sizes, clarifying the connection between our older results and the recent excellent study of Bu and Russo,22 who largely studied smaller probes, (2) obtained multiple spectra with different time scales, thereby greatly increasing the accuracy of our line-shape determinations, and (3) focused on an intensive study of a single intermediatemolecular-weight polymer in its lower-concentration solutionlike regime. Further sections treat experimental methods, results, and their phenomenological description and a comparison with the literature. A discussion closes the paper. Experimental Methods The polymer used in these experiments was (hydroxypropyl)cellulose (HPC, Scientific Polymer Products, Ontario N.Y., lot 2, nominal molar mass 300 000, measured Mw ) 415 kDa) at concentrations 0-15 g/L. The same material (supplier, nominal mass, lot number) was studied by Bu, Russo, et al.19-22 For details of the chromatographic determination of the molecular weight distribution, see ref 8. Water was conductivity (16-18 MΩ) grade prepared by Millipore Type-RO and Milli-Q water purification units. To prevent polymer adsorption by the probe particles, all solutions also contained 0.2 wt % Triton X-100 (Aldrich); ref 8 describes control experiments used to choose this surfactant concentration. Probe particles were carboxylatemodified polystyrene latex spheres having nominal diameters of 189 and 21 nm (Interfacial Dynamics, Portland OR) and 760 and 102 nm (Seradyn). We had previously studied 67 nm carboxylate-modified spheres (Seradyn).8 Polymer samples were prepared by serial dilution from stock solutions. In sample preparation, light scattering cells (disposable plastic and glass fluorimeter cuvettes, four sides polished) were rinsed repeatedly with dust-free (0.2 µm filter) conductivity water and blown dry with dust-free (0.22 µm filter) nitrogen. Samples were prepared by mixing polystyrene latex sphere stock into the polymer matrix solution, and filtering the mixture through an 0.2-2.0 µm pore diameter microporous filter. While use of large-pore-diameter filters is good practice for study or probefree polymer solutions, the physical requirement that the pore diameter be larger than the probes (e.g., > 1 µm for 760 nm probes) ensures that the probes are the largest and dominant particulate scattering species in solution. The concentration of sphere stock required to give good spectra, without risking multiple scattering or failure to dominate the scattering intensity, varied with sphere diameter, from e0.1 µL/mL for the 760 nm spheres to 5 µL/mL for the smallest spheres. In all cases, the probe spheres were physically highly dilute with volume fraction 5 g/L) of high molecular weight (1 MDa) HPC. The use of stretched exponentials to describe polymer relaxations has a range of antecedents in polymer fluid dynamics. Carroll and Patterson28 studied quasielastic light-scattering spectra of polystyrene near its glass transition, finding after Laplace inversion that the relaxation time distribution of their spectra agreed well with the distribution function expected for a stretched exponential in time. Nystrom et al.29 and Ngai and co-workers23 have demonstrated the use of a sum of a regular exponential and a stretched exponential for fitting relaxations of complex systems. Carroll and Patterson28 emphasize that extracting reliable lineshape parameters from a fit to a stretched exponential in time is difficult unless accurate measurements of C(t) are obtained over a wide range of times. Our previous work8 found that accurate measurements at early times are especially important.

J. Phys. Chem. B, Vol. 101, No. 1, 1997 41 With our correlators, obtaining an adequate range of delay times t requires making multiple measurements on the same sample while using different sample times δt and then fitting a single function simultaneously to all spectra. This standard “splicing” procedure was previously used by Carroll and Patterson28 for precisely the reasons it was used here. The utility of taking multiple spectra of the same material, while using multiple rather than a single value of δt, may be inferred from Figures 6c and 7c in ref 8. When only one spectrum is taken, spectral fitting parameters are scattered. When multiple spectra having different values of δt are taken and analyzed simultaneously, the resultant spectral fitting parameters lie on a single smooth curve. How does the fitting procedure work? Spectra may consistently be written

C(t) ) [Ag(1)(t)]2 + B

(7)

Here g(1)(t) is the field correlation function, with fixed normalization g(1)(0) ) 1, while A is the spectral amplitude and B is the baseline. We determined B by direct measurement of C(t) for times much larger than the largest times used by the data channels. The spectral baseline may also be formally calculated from the total number of photocounts P and the total number of sample times M during the experiment as B ) P2/M. When δt was sufficiently large that g(1)(t) had decayed to zero before the baseline channels were reached, the directly measured and formally calculated values of B agreed to within photon counting statistics. We used two forms for g(1)(t). Reference 8 used

g(1)(t) ) exp(-θtβ) ≡ exp(-(t/τ)β)

(8)

with time constants θ and τ and stretching parameter β; θ ) τ-β. We also followed Nystrom et al.,29 who used

g(1)(t) ) (1 - F1) exp(-θtβ) + F1 exp(-Γt)

(9)

where F1 is the amplitude fraction of the plain exponential and Γ is the exponential’s time constant. This form was useful only at higher polymer concentrations. The exponential mode was always weak (F1 e 0.04) and fast (the exponential disappeared before the stretched exponential decayed substantially.) Use of eq 9 rather than eq 8 improved the accuracy of the fit; these two equations for g(1)(t) lead to the same values for θ and β to within 10%. We fit the above forms for g(1)(t) to a series of spectra, on the same sample, having different channel widths δt; this procedure was used previously, e.g., by Carroll et al.28 Values of δt were typically varied by a factor of 8 between spectra to ensure substantial spectral overlap. As done previously by Carroll and Patterson,28 each spectrum was truncated above its long-delay-time noise limit, typically for g(2)(t)/g(2)(0) g 0.01. Usually four (sometimes three or five) spectra were spliced in this manner. A single set of field correlation parameters (θ, β, ...) was fit to all spectra on a given sample; A was of course varied independently for each spectrum. For sufficiently small δt, the measured baseline channels lay in regions where C(t) is nonzero; such baseline channels were treated as additional data channels lying above a baseline with free parameter B. While various methods were used to determine good initial guesses, in the final step all parameters were refined simultaneously. We obtained stable fits whose plots (cf. Figure 1) agree well with measured spectra. The rms error from final fits was 1-3%. The fitting process implicitly tests the validity of the assumed field correlation function. If spectra did not have stretched-

42 J. Phys. Chem. B, Vol. 101, No. 1, 1997

Phillies and Lacroix

g2(t)

Figure 1. Representative data: spliced multitau spectra of 102 nm polystyrene latex spheres in 10 g/L HPC:water (small circles) as fit to eq 9. Largest-τ spectrum was truncated before the noise became substantial.

Figure 3. Scaling exponent β for eqs 8 and 9 as a function of HPC concentration. Solid lines are stretched exponentials in c describing the 21, 102, and 760 nm β parameter from Table 2; noise in β for 189 nm spheres prevents a good fit. Other details as in Figure 2.

TABLE 1: Fits of the Spectral Parameter θ (Eq 8) for Probes of Nominal Radius R to a Stretched Exponential θ θ0 exp(-rcν) in Concentration R (nm)

θ0

R

ν

760 189 102 21

300 860 1810 7300

0.63 0.59 0.47 0.49

0.72 0.77 0.82 0.59

TABLE 2: Fits of the Spectral Parameter β (Eq 8) for Probes of Nominal Radius R to a Stretched Exponential β ) β0 exp(-rcν) in Concentration

Figure 2. Scaling prefactor θ for eqs 8 and 9, obtained as described in the text, as a function of HPC concentration, for spheres having nominal diameters of (a) 21 nm (diamonds), (b) 102 nm (circles), (c) 189 nm (triangles), and (d) 760 nm (squares). Solid lines: stretched exponentials in c (parameters are in Table 1).

exponential decays, the process would still yield “best-fit” parameters, but the spliced spectra would not interlace properly. There would be visible discontinuities between datasets arising from normalization mismatches. Control tests in which spectra described well by eq 9 were fit to eq 8 led to poorly-spliced, badly interleaved spectra, confirming that normalization mismatches are immediately visible when an incorrect form is used for g(1)(t). Figure 1 shows representative results of our analysis procedure, using spectra of 102 nm spheres in 10 g/L of HPC. Three spectra of different δt were simultaneously fit to eq 9, the fitting parameters including θ, β, Γ, F1, and one value of A for each spectrum. The spectrum of shortest δt had B as a free parameter; other spectra used the measured B. The two regions of very large horizontal density of points are long-time ends of a single correlator output; the small, dense group of points near t ) 2.5 mS is the floated baseline channels of the shortest-δt

R (nm)

β0

R

ν

760 189 102 21

0.90 0.97 0.99 1.01

0.040 0.015 0.044 0.051

0.56 1.02 0.64 0.33

spectrum. Over 3.5 decades of time and 2+ decades of decay, C(t) - B ) g(2)(t) follows very well the functional form implied by eq 9. The form that most accurately describes g1(t) depends on c and R. With 760 nm spheres, at concentrations above or about 5 g/L eq 9 was clearly preferable to eq 8. For the 189 nm spheres, the single stretched exponential was preferable at lower concentrations, while the exponential plus stretched exponential form of eq 9 gave more stable fits at polymer concentrations of 12.5 and 15 g/L. For the 102 nm spheres eq 9 gave a better fit at concentrations g10 g/L; at polymer concentrations below roughly 5 g/L the fitting process failed to find a fast exponential distinct from the dominant stretched exponential. Finally, for the 21 nm spheres eq 9 was preferred except at the lowest few concentrations studied. The amplitude of the pure exponential consistently increased with increasing c. Results Figures 2 and 3 and Tables 1 and 2 give θ and β for the 760, 189, 102, and 21 nm nominal diameter probe particles in (hydroxypropyl)cellulose: water solutions (numerical results appear in the tables in the Supporting Information). As seen in Figure 2, over 0 e c e 15 g/L, θ falls ∼100-fold for the larger spheres, but only ∼10-fold for the smaller spheres. Solid lines in Figure 2 were obtained from nonlinear least-squares

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J. Phys. Chem. B, Vol. 101, No. 1, 1997 43

Figure 4. Relaxation time τ from eq 8 computed as τ ) θ-1/β. Solid lines are stretched exponentials using parameters in Table 3. Other details are as in Figure 2.

TABLE 3: Fits of the Spectral Parameter τ (Eq 8, As Calculated Using θ and β from Fits of Actual Spectra to Eqs 8 and 9) for Probes of Nominal Radius R to a Stretched Exponential τ ) τ0 exp(+rcν) in Concentration R (nm)

τ0 (mS)

R

ν

760 189 102 21

3.11 1.06 0.48 0.134

0.385 0.358 0.208 0.138

0.89 0.89 1.02 0.91

fits of θ for each R to

θ ) θ0 exp(-Rcν)

(10)

Table 1 gives the intercept θ0, scaling exponent ν, and scaling prefactor R for probes of each radius. The exponent ν is markedly smaller for the 21 nm probes than for the larger probes. Figure 3 gives the stretching exponent β, taken from the fits that gave the values of θ in Figure 2. We find that β falls with increasing polymer concentration, decreasing from slightly less than 1.00 to ≈0.8 for the larger spheres but to ≈0.9 for the 21 nm spheres. At concentrations below e5 g/L, β is nearly independent of probe radius; at larger concentrations, β for the 21 nm probes is clearly larger than β for the larger probes. The solid lines in Figure 3 are stretched exponentials β ) β0 exp(-Rcν) describing the concentration dependence of β. Table 2 shows the parameters used to generate these lines. The parameters in Table 2 are not especially accurate, because the data on β(c) have intrinsic limits: Over the measured concentration range, β changes only by 15%. Furthermore, as seen from Figure 3, determinations of β (especially for the 189 nm spheres) are not very accurate. The fit for the 189 nm spheres is a nearstraight line, the line’s straightness arising from accidents in the noise for the 189 nm sphere data set. Figure 4 and Table 3 show the alternative parameter τ that represents the same spectra, in the form exp(-(t/τ)β) seen in eq 8. τ and θ are related by θ ) τ-β, so Figures 2-4 reflect only two independent parameters. While θ has transcendental dimensions (time)-β, τ has dimensions (time)1. Figure 4 and Table 3 give τ for the four sphere sizes as functions of c. Over the range of c and 25-fold range of R that we studied, τ varies from a tenth of a microsecond up to 0.2 s. We find that τ is described well by τ ) τ0 exp(+Rcν), a stretched exponential in c, with ν a near-exponential 0.9-1.0. τ depends on R both

Figure 5. Fractional amplitude F1 of the very fast exponential decay of eq 9 for 189 nm spheres as a function of polymer concentration c.

through τ0 and through R. However, the variation in τ with R arises primarily from a 23-fold variation of τ0 with R, and only secondarily from a 3-fold variation of R with R. Figure 5 gives the fractional amplitude F1 of the weak fast exponential of eq 9 as a function of polymer concentration. F1 was always quite small, in the range 1-3% of the total amplitude; F1 increased monotonically with increasing polymer concentration. We infer from its concentration dependence that this very fast mode reflects weak scattering by the polymer itself or by internal polymer modes coupled to sphere motion. While a very fast relaxation might in principle arise from micelle scattering due to the added Triton X-100, the concentration dependence of F1 does not support such an interpretation. For the 189 nm spheres, for which the least scattered values of Γ were obtained, Γ is typically ∼3000-4000 s-1, with no indication of an appreciable c dependence. Γ for the other sphere sizes was badly scattered. Regardless of interpretation, the F exp(-Γt) mode does not interfere with fitting almost all of C(t) to a single stretched exponential in time. Discussion What does the motion of optical probes reveal about the motion of the surrounding matrix? Any probe perturbs the surrounding polymer solution: A probe chain creates a correlation hole. A probe sphere excludes polymer chains from its interior and perturbs the concentration of neighboring chains by creating around itself a depletion zone of characteristic thickness ∼Rg, Rg being the chain radius of gyration. The displacements of a probe sphere and its depletion zone are necessarily highly correlated; one cannot simply think of a probe as moving through unperturbed solution. A probe and its depletion zone may be envisioned as a collective object whose motions are retarded by the matrix. A complete theoretical treatment of polymer solution dynamics will describe the motions of spheres and their depletion zones, just as such a theory describes other motions in which depletion effects arise. As an example of these “other motions”, a complete treatment of polymer dynamics will describe solutions flowing in mesoscopic pores, in which depletion near the walls causes c to depend significantly on the position within the pore. While computation of Ds involves challenges, measurement of Ds as a function of c and M tests any nominally complete model of polymer dynamics, and provides constraints that guide the development of a sound theory.

44 J. Phys. Chem. B, Vol. 101, No. 1, 1997

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It is useful to contrast optical probe methods that employ probe chains with methods that use probe spheres. The motions of a single probe chain very closely resemble the motions of the matrix chains, even though the probe chains carry correlation holes with themselves. However, the motions of a probe chain are relatively complex; internal modes of probe chains make it more difficult to isolate probe chain center-of-mass motion at short times. On the other hand, while a probe sphere and its entrained depletion zone is physically unlike a probe chain, a probe sphere has the advantage for data interpretation that the scattering object is mechanically rigid and has only translational and rotational modes. Here we have elected to study spheres. For studies of probe chains, note, e.g., results of Bu and Russo.22 Probe diffusion in HPC:water has been studied previously by several authors.18-20 Before making a more detailed analysis of our new data above, we first show that our findings do not disagree with earlier work at the points where comparison is possible. HPC:water was first studied by Brown and Rymden,30 who used quasi-elastic light-scattering to observe 72 nm polystyrene spheres in solutions of Mw 800 kDa HPC at concentrations 0-2 g/g (η e 8 cP). Brown and Rymden report that the apparent hydrodynamic radius

Rh ) kBT/6πηDs

(11)

of their probes fell roughly 2-fold with increasing c. Here kB is Boltzmann’s constant, T is the absolute temperature, and η is the solution viscosity. Brown and Rymden also studied the effect of surfactant addition, finding that Triton X-100 tended to displace adsorbed polymer, thereby greatly reducing probe aggregation via polymer bridging. Effects of TX-100 on polymer adsorption were studied in more detail in our previous paper8 and are exploited experimentally here. Yang and Jamieson21 used optical probe methods to examine diffusion of large (R ) 61, 108, 179 nm) probes in HPC samples having Mw of 1.1 × 105, 1.4 × 105, 4.5 × 105, and 8.5 × 105 Da, in solutions with relative viscosities ηr up to 45 and concentrations c[η] e 4. Spectra were analyzed with a multiexponential fitting program. At elevated concentrations, Ds depended on concentration as a stretched exponential in c; in some systems there was a narrow dilute-solution regime in which Ds was virtually independent of c. In solutions of the two lower-M polymers, the Stokes-Einstein equation worked accurately. In solutions of higher-molecular-weight polymers, the microviscosity ηµ inferred from light scattering and the known probe radius R

ηµ ) kBT/6πDsR

(12)

was markedly less than the measured bulk viscosity. In these large-M systems, ηµ/η decreased with increasing c and increasing M and increased with increasing R, but even for 179 nm radius spheres ηµ/η was appreciably less than 1. Yang and Jamieson21 found distinct small and large-M regimes. Our polymer M is intermediate to theirs. Our large probes show qualitative behavior (θη approximately constant) similar to the behavior (Dη ∼ constant) found by Yang and Jamieson for probes in their lower-M systems. Our smallest (21 nm) probes were substantially smaller than any of their probes and showed a qualitative behavior (θη increasing with increasing c) similar to the behavior that Yang and Jamieson found for larger probes in their large-M (larger than our M) regime. All of our data could be said to be consistent with Yang and Jamieson’s if one took the key variable in the comparison to be R/Rg.

Russo and collaborators18-20 report a series of studies of probe diffusion in HPC:water, emphasizing an interest in HPC as a semirigid (persistence length ≈ 100 Å) polymer that forms liquid crystals in some concentration regimes.18 Their probe species included 79 and 181 nm polystyrene spheres18,20 (for which Ds was measured with quasielastic light scattering), and free dye molecules19 and dye-labeled dextrans and polystyrene sphere22 (for which Ds was measured by fluorescence recovery after photobleaching). In their initial study, Russo et al.19 studied polystyrene spheres in Mw ) 292 kDa HPC, η(c) being obtained with a cone and plate viscometer. η and Ds each varied over 3 orders of magnitude as c was increased. For concentrations below 0.001 g/g of HPC Russo et al. interpret a decrease of Dsη with increasing c as arising from polymer-driver cluster formation of the probes. For the concentration range 0.001-0.04 g/g of HPC, Dsη was independent of c. Reference 19’s results agree with our new data here. The two studies involved polymers of nearly the same Mw. For probes with R ≈ 100 nm (our 109 nm spheres, Russo et al.’s 79 nm spheres forming (their Figure 7) 100 nm clusters), the concentration dependence of the relaxation time scale (our θ, Russo et al.’s Ds) follows closely the concentration dependence of η. Mustafa and Russo20 made a separate study of 91 nm diameter spheres in aqueous solutions of 1 MDa HPC, emphasizing comparative line-shape analysis using cumulants, multiexponential fits, and Laplace inversion methods. They interpreted their spectra as having a bimodal distribution of relaxation times (“fast” and “slow” modes), with multiple analysis methods supporting the same interpretation. The decay rates of the fast and slow modes both scale with scattering angle as q2. Reference 20 noted that an additional, much faster but very weak decay was sometimes apparent. Mustafa and Russo20 report two groups of relaxations, while we here find a single decay mode characterized by θ and β. Nonetheless, Mustafa and Russo’s datasso far as it goessis fundamentally consistent with ours. The seeming disagreement arises entirely from the use of different data analysis methods. They key issue is that a semilog plot of a stretched exponential appears to have an initial, rapidly decaying regime and a longtime near-exponential decay, the rollover from fast to slow decay occurring near t ≈ τ. A Laplace transform of a stretched exponential captures this appearance of fast and slow decays by identifying, in the one stretched exponential, two sets of modessfast and slow. Our C(t) is described to high precision by a stretched exponential, together with a very fast, weak exponential decay. A Laplace transform of our C(t) contains two groups of modes, “fast” and “slow”, corresponding to the early and late parts of the stretched exponential, plus the weak, fast mode discussed above. Our one stretched exponential in t thus leads to both the “fast” and the “slow” modes of Mustafa and Russo.20 Consistent with our interpretation that our decay and their two groups of modes refer to the same spectral entity, our earlier work8 found θ ∼ q2, while Mustafa and Russo found that their “fast” and “slow” modes each scale as q2. Mustafa et al.18 and Bu and Russo22 used FPR to study the diffusion of small probes through HPC solutions. Mustafa et al. found that the retardation of diffusion of fluorescein by HPC depends on c but is independent of HPC molecular weight and that in concentrated solution (weight fraction >0.03) the microviscosity inferred from fluorescein diffusion is a small fraction of the macroscopic solution viscosity. Bu and Russo22 used fluorescence photobleaching recovery to measure Ds of fluorescein dye, dye-labeled dextrans, and dye-labeled 55 nm polystyrene spheres in 300 kDa HPC:water. The dextrans had

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J. Phys. Chem. B, Vol. 101, No. 1, 1997 45

1.3 e Rh e 18 nm. The degree of retardation of the probes, relative to their diffusion rate in pure water, depends on c and R. Comparing the microviscosity with the macroscopic solution viscosity η and solvent viscosity η0, smaller probes (R e 5 nm) had ηµ e 3η0, even at c ) 39 g/L, where η/η0 > 5 × 103. For larger (5 nm e R e 55 nm) probes, Bu and Russo22 found at fixed c that ηµ tends toward η with increasing R. However, even with their largest (R ) 55 nm) probes, ηµ/η e 0.5. Bu and Russo’s22 FPR spectra are fit well by a simple exponential. In contrast, a single-exponential fit to our data would be completely inadequate, perhaps because we operate on shorter distance scales and much shorter time scales than those used by Bu and Russo. The other qualitative physical descriptions of Bu and Russo’s data and our new results here are concordant. Each study used a range of probe sizes; the two size ranges are partially overlapping. Our two largest probes, which are larger than the probes used by Bu and Russo, show a pseudo-Stokes-Einsteinian behavior θη/θ0η0 ≈ 1. Bu and Russo’s largest probes were too small to enter the analogous regime Dη/D0η0. Our 109 nm probes, which are close in size of Russo’s largest probes, reach θη/θ0η0 ≈ 2 at 15 g/L; at this concentration, Bu and Russo’s results imply ηµ/η g 2. Finally, at our largest c our small spheres attain θη/θ0η0 ≈ 10, which is in the range of ηµ/η for the nearest of Bu and Russo’s probe sizes and matrix concentrations. Bu and Russo22 note that our previous study8 of HPC:water found factor-of-100 deviations from Stokes-Einstein behavior, this deviation being much larger than any deviation observed by Bu and Russo. However, we8 encountered θη/θ0η0 ∼ 102 only for c or M much larger than those examined here or in ref 22. In the range of c and M studied here, this work and ref 22 agree as to the observed level of non-Stokes-Einsteinian behavior. It should be emphasized that our microviscosity will never match precisely the ηµ of Bu and Russo, because we parametrize our spectra differently. Bu and Russo used singleexponential fits that cannot be rationally applied to our spectra, while our fitting function was dominated by a single stretched exponential. This laboratory’s earlier work8 with 67 nm probes reports few measurements for the c and M studied here. The 67 nm probes had a diffusion rate intermediate between the 21 and 109 nm spheres, and showed non-pseudo-Stokes-Einsteinian behavior to a 2-fold lesser extent than did the 21 nm spheres. The diminution in pseudo-non-Stokes-Einsteinian behavior between the 21 and 67 nm probes is consistent with Bu and Russo’s observation22 that the extent of this behavior decreases with increasing R. In summary, at each point where a comparison can be made, the new data in this paper are consistent with older results of Brown and Rymden,30 Yang and Jamieson,21 or Russo and collaborators.18-20,22 We now consider what our new results imply about probe dynamics in HPC:water. First, use of spectra having a range of sample times confirmed that the major decay of each spectrum is accurately given by a stretched exponential:

g(1)(t) ) exp(-θtβ)

(13)

perhaps preceded by a weak, rapidly decaying pure exponential. Our measurements sample a finite number of points, to finite accuracy; we cannot prove that this function is unqiue. Second, θ has a stretched-exponential dependence on matrix concentration c. The stretching parameter β decreases with increasing c, being very near to 1.00 for probes in pure water, but declining to 0.9 or 0.8 with increasing c. Qualitatively, this behavior of β is consistent with the coupling model of Ngai

Figure 6. Correlation of θ and solution viscosity η: plot of θη against HPC concentration. Parameters for stretched exponentials (solid lines) are in Table 4; other details are as in Figure 2.

and collaborators,23 in which increasing polymer concentration leads to increasing coupling, manifested as a decline in β. Third, θ and τ can be compared with the solution viscosity. Viscosities here were taken from Phillies and Quinlan,17 who report a systematic study of HPC:water viscosity at 25 °C using capillary viscometers. Phillies and Quinlan studied three highmolecular-weight (Mw ∈ [3.0 × 105, 1.15 × 106] Da) HPC samples for solutions ranging from extreme dilution to η ≈ 3 × 105 cP. Use of multiple viscometer sizes confirmed that shear rates remained low enough to avoid shear thinning. Phillies and Quinlan’s major objective was to study the functional dependence of η on c. They confirmed to good accuracy that η follows the stretched exponential of eq 3 at lower concentration and a power law

η)η j cx

(14)

at higher concentration. Here η0 is the solvent viscosity, R and η j are scaling prefactors, and ν and x are scaling exponents. Reference 17 denoted the regimes where eqs 3 and 14 apply as “solutionlike” and “meltlike”, respectively. The terminology was chosen to avoid ascribing to this purely phenomenological behavior an interpretation in terms of any specific, potentially erroneous, model of polymer dynamics. Phillies and Quinlan found for HPC:water that the transition between these regimes is continuous and analytic; i.e., both η(c) and its first derivative dη/dc are continuous at the transition concentration c+. For the 300 kDa HPC studied here, Phillies and Quinlan report c+ ≈ 18 g/L so c+[η] ≈ 7. In the terminology of ref 17, experiments are here confined to the solutionlike regime c < c+. In HPC:water, the transition between the solutionlike and meltlike regimes is sharp; there is no crossover regime within which neither eq 3 nor eq 14 applies. Solutionlike-meltlike transitions are apparent for polymer viscosity and viscoelasticity in some, but not all, systems in which literature data have been properly reexamined.4,31 However, for some systems31 dη/dc has a step discontinuity at c+. Figures 6 and 7 and Tables 4 and 5 consider the relationship between θ, τ, and η by examining θη and τ/η as functions of c and probe radius. Figure 6 plots θη against c. For our three largest sphere sizes, θη is very nearly a constant. Excluding three outlier points, the interpretation that θη is independent of

46 J. Phys. Chem. B, Vol. 101, No. 1, 1997

Phillies and Lacroix

Figure 7. Correlation of τ and solution viscosity η: plot of τ/η against HPC concentration. Parameters for stretched exponentials (solid lines) are in Table 5; other details are as in Figure 2.

TABLE 4: Fit of θη for Probes of Nominal Radius R to a Stretched Exponential θη ) A0 exp(-rcν) or Exponential (ν ≡ 1) in Concentration R (nm)

A0

R

ν

760 189 102 21

207 604 1.4 × 103 5.6 × 103

1.9 × 10-3 -1.09 × 10-2 -1.81 × 10-2 -6.9 × 10-2

1 1 1 1.29

TABLE 5: Fit of β/η for Probes of Nominal Radius R to a Stretched Exponential β/η ) B0 exp(-rcν) or Exponential (ν ≡ 1) in Concentration R (nm)

B0

R

ν

760 189 102 21

3.6 × 10-3 1.2 × 10-3 5.9 × 10-4 1.6 × 10-4

1.96 × 10-2 3.36 × 10-2 0.27 0.28

1 1 0.56 0.86

c for large spheres is supported by our data to within experimental error. In contrast to the large-sphere behavior, θη for our smallest (21 nm) spheres increases 10-fold with increasing c. η increases more rapidly than θ falls. The solid lines in Figure 6 are stretched-exponential fits corresponding to parameters of Table 4. For all but the smallest spheres, R is so small that the lines are very nearly horizontal. Figure 7 treats the ratio τ/η and its dependence on polymer concentration and probe radius. For each size of probe, τ/η exhibits a stretched-exponential decline with increasing c. Fitting parameters for the solid curves are given in Table 5. For each probe radius, from our measurements the relaxation time τ increases with c more slowly than the viscosity η increases. Even for the largest spheres studied here (nominal diameter 760 nm), τ/η falls by ca. 1/3 with increasing c. For the smallest spheres, τ/η falls roughly 20-fold with increasing polymer concentration. We conclude that η does not determine τ in a simple manner. Fourth, for simple Stokes-Einstein diffusion, one has D ∼ η-1 and g(1)(t) ∼ exp(-Dq2t1). Our spectra are not simple exponentials. In the parametrization of g(1)(t) used here, the stretching exponent β differs considerably from unity at elevated c. While θ and Dq2 both characterize the decay of g(1)(t), θ and Dq2 do not haVe the same dimensions, because they enter their functions in different ways. It is therefore improper to make a simple identification of our θ with a Dq2 for some value of D, because the numerical value of D would depend on the

choice of time units. θ and D characterize the same physical processes but not in the same way. Nonetheless, θ defines a spectral relaxation pseudorate (“pseudo” because θ has units (time)-β). For large spheres θ ∼ 1/η, which we denote pseudoStokes-Einsteinian behavior. For the small probes θ falls less rapidly than η increases, which we denote pseudo-non-StokesEinsteinian behavior. For the systems studied here, a cumulant analysis yielding the light- scattering-intensity-average D h is rapidly convergent.8 Correlations between that well-defined, experimentally reproducible D h and other system properties, e.g., c, η, and M, were studied in the previous paper.8 That study was generally consistent with previous work32 on probe diffusion in polymer solutions, in which we almost always33 found spectral line shapes readily described by low-order cumulant expansions with relatively concentration-independent second cumulants. Here, however, we have a polymer with unusual viscosity properties (notably a solutionlike-meltlike transition at an η/η0 of 102 rather than 104), spectra that become quite nonexponential at large c and M, and a simple functional form that describes our spectra. We found it interesting to explore the parametric behavior of that simple function. Fifth, we have confirmed the conjecture of Bu and Russo22 thatsfor sufficiently large probe particlessthere is a sense in which the relaxation of probe concentration fluctuations is governed by the by macroscopic viscosity η. Bu and Russo showed that their data tended toward this limit, but none of their probe particles were large enough to make StokesEinsteinian behavior manifest. We find θ ∼ η1 for probes of radius 50 nm or larger. Strictly speaking, our new finding is not Stokes-Einsteinian behavior, because θ only has units Dq2 if β ) 1, which is only the case here. Under what conditions do we find pseudo-Stokes-Einsteinian behavior? Interpolating from Yang and Jamieson,21 our HPC sample has Rg ≈ 50-60 nm and an Rh approaching 30 nm. 55 nm radius probes show pseudo-Stokes-Einsteinian behavior, but 33 nm radius probes do not. The apparent criterion for Stokes-Einsteinian behavior is that the probe R must be ≈Rg or > Rh. Finally, we propose that θ and β are the parameters most useful for describing g(1)(t); i.e., they are the parameters most likely to result directly from an ab initio model of a polymer solution. Rationales supporting this description of θ and β and not τ and β, include the following: (i) The probe concentration c(r,t) is a conserved variable. The time constant for the diffusive relaxation of a conserved variable arises from a term like ∇‚D‚∇c(r,t), D here being an elseunspecified diffusion tensor. From the repeated nabla, the time constant for diffusive relaxation should depend on scattering vector as q2. θ has the requisite q dependence; τ does not. (ii) The diffusion of large spheres might plausibly be proposed to be governed by the macroscopic viscosity η, because motion on distance and time scales larger and longer than any physical scale of the system could be argued to involve only low-shear flow. From Figures 4 and 5, for large spheres θη is nearly constant; on the other hand, τ/η has a strong concentration dependence. The plausibly expected viscosity dependence is manifested by θ, but not τ. Acknowledgment. The partial support of this work by the National Science Foundation under Grant DMR94-23702 is gratefully acknowledged. Supporting Information Available: Tables of numerical results (4 pages). Ordering information is given on any current masthead page.

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