Molecular Tracer Diffusion in Nondilute Polymer Solutions: Universal

Article pubs.acs.org/Macromolecules

Molecular Tracer Diffusion in Nondilute Polymer Solutions: Universal Master Curve and Glass Transition Effects Apostolos Vagias,*,† Jennifer Schultze,‡ Mikheil Doroshenko,‡ Kaloian Koynov,‡ Hans-Jürgen Butt,‡ Mario Gauthier,§ George Fytas,†,‡,∥ and Dimitris Vlassopoulos†,∥ †

FORTH, Institute of Electronic Structure and Laser, Heraklion 71110, Crete, Greece Max-Planck-Institute for Polymer Research, 55128 Mainz, Germany § Department of Chemistry, Institute for Polymer Research, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada ∥ Department of Materials Science & Technology, University of Crete, Heraklion 70013, Crete, Greece ‡

ABSTRACT: Using fluorescence correlation spectroscopy, we investigated the diffusion dynamics of a molecular probe in solutions of polystyrene and polybutadiene with different molar masses (Mw) and molecular architectures (stars, combs) over a broad concentration range (c). Whereas the tracer diffusion coefficient D was found to be insensitive to variations in Mw and architecture, the obtained master plot D(c) was polymer-specific, reflecting additional friction effects related to the concentration-dependent glass transition temperature Tg(c). This effect is fully suppressed for the same probe diffusion in poly(dimethylsiloxane) solutions with very low and virtually concentration-independent Tg. Hence, a universal master curve of D vs c can be obtained when Tg(c) is accounted for.

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INTRODUCTION The elucidation of diffusion dynamics in physical polymer networks has been a fundamental problem of long-lasting interest.1−15 Outstanding challenges include understanding the role of polymer and tracer properties such as the polymer concentration (c), the polymer matrix molar mass (Mw), the solution glass transition, and the tracer size toward reliable predictions and possibly scaling relations.3,16−20 Moreover, understanding probe diffusion has a direct impact on several applications such as chromatography,21 drug delivery,22 nanocomposites,23,24 hydrogel-based sensors,25 and microrheology.26 So far, experimental results suggested a dependency of the tracer diffusion coefficient (D) on c, the tracer size, specific matrix−tracer interactions, and macromolecular architecture.27 Although coupling of tracer and polymer matrix mobilities is univocally expected, conclusive experimental evidence about polymer-specific glass transition temperature (Tg) effects on molecular probe dynamics in polymer liquids remains elusive. Some investigators28,19 suggested superposition of the tracer diffusion data vs T − Tg, while others described its failure3 perhaps due to the different Williams−Landel−Ferry (WLF)29 parameters utilized for each examined polymer. In the absence of enthalpic interactions,30 the diffusion of molecular tracers in polymer networks relates to the matrix Tg. However, finding a direct link11,31 to polymer segmental dynamics and elucidating the independent contributions to solute friction remain formidable tasks.32 For nearly athermal solutions of linear polymers with different Mw,33 the diffusion slowdown with concentration was shown to superimpose on a single curve depending solely on c, for a given polymer type and © XXXX American Chemical Society

temperature. However, it remains intriguing to inspect polymer specificities on a parametric investigation in D(c) at temperatures not sufficiently far from the matrix Tg, which could induce variations in the local friction coefficient with concentration and temperature. In order to address this challenge, we employed fluorescence correlation spectroscopy (FCS)34,35 and studied the diffusion of a molecular tracer (terrylene diimide derivative (TDI), see Figure 1) in nondilute solutions of polystyrene (PS) and 1,4polybutadiene (PBD), using diethyl phthalate (DEP) and squalene as good solvents, respectively. We also addressed the effect of macromolecular architecture by using comb-shaped PS and star-shaped PBD in the aforementioned solvents. In the

Figure 1. Chemical structure of the used terrylene diimide derivative (TDI). Received: July 3, 2015 Revised: November 23, 2015

A

DOI: 10.1021/acs.macromol.5b01464 Macromolecules XXXX, XXX, XXX−XXX

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measurements, an ∼10 μL aliquot of sample solution was transferred to a round microscope cover slide. For the measurement of Tg,PS(c), the PS/DEP solutions were prepared following the same protocol as for the FCS experiment. A reusable Attofluor steel chamber served as sample holder for the round microscope cover glass slides holding the polymer solutions. Techniques. Differential Scanning Calorimetry (DSC). We estimated the composition dependence of Tg(ϕPS) of six PS/DEP solutions using a Mettler Toledo DSC-822 (Schwerzenbach, Switzerland) calorimeter; ϕPS is the volume fraction of PS in the mixture. The expected plasticizing effect of DEP, i.e., an increase of the solution’s Tg, with PS composition, is displayed in the main plot of Figure 2. The

FCS experiment, the tracer concentration is low enough (in the nanomolar range) to exclude tracer aggregation or modification of the polymer matrix. The results obtained are discussed on the basis of the WLF29,36 equation, where Tg(c) is associated with the average friction effect on probe diffusion. The notion of an additional Tg(c)-based slowdown effect was corroborated by its suppression in the diffusion of the same molecular probe in polydimethylsiloxane (PDMS) solutions using toluene as solvent; these possess very low and virtually concentrationindependent Tg.

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EXPERIMENTAL SECTION

Materials. The star polymer (Table 1) incorporated a dendritic hard core and a soft shell of PBD arms of different sizes and

Table 1. Molecular Characteristics of the Polymers Investigated linear polymers code

Mw (g/mol)

polydispersity index (Mw/Mn)

c* (g/mL)

PS 1M PS 500k PS 200k PBD 170k PBD 88k

1.2M 611K 217K 170K 88K

1.16 1.13 1.05 1.05 1.05

0.016 0.026 0.054 0.022 0.036

PBD 10k 10K PBD 1k 1K PDMS 60k 67K star polymers code RS64-81 RS64-PBD5 RS64-30 RS64-05 AM01402 RS32-PBD5 comb polymer Mw,branch code (g/mol) PS722

860

0.17 0.87

complementary structure (s) PS 722 (comb) PS722 (comb) PS 722 (comb) RS64−81 (star) AM01402/RS32PBD5/RS64-30 (star) RS64-PBD5 (star) RS64-05 (star)

1.05 1.4 1.16 functionality ( f)

Mw,arm (g/mol)

c* (g/mL)

200 925 163 929 300 300

80K 4.4K 30K 0.5K 30K 30K

0.07 2.2 2 4.76 2.7 0.45

Mw,backbone (g/mol)

q (branches/ backbone)

Mw,total (g/mol)

c* (g/mL)

11.7K

28

1190K

0.013

Figure 2. Composition dependence of the glass transition temperature (Tg) of polystyrene (PS) (MW = 231K)/DEP mixtures (solid squares) measured by DSC. Tg(ϕ) values calculated by the Fox−Flory equation using either the bulk ϕPS (dashed blue line) or an effective ϕeff = ϕs + (1 − ϕs)·ϕPS (black solid line, ϕs = 0.6) are also shown. Inset: typical DSC trace for a PS/DEP solution with 45% PS. The vertical solid line indicates the Tg value.

inset of Figure 2 shows a typical DSC trace for a PS/DEP solution with ϕPS = 0.45. The prediction of the Fox−Flory equation (dashed blue line), 1/Tg(ϕPS) = ϕPS/Tg,PS + (1 − ϕPS)/Tg,DEP, with Tg,DEP being the glass transition temperature of neat DEP, overestimates37 the experimental Tg(ϕPS) as indicated by the blue dashed line. Assuming full solvent removal, a quantitative agreement is obtained by substituting ϕPS by an effective ϕeff = ϕs + (1 − ϕs)ϕPS where ϕs > ϕPS denotes the PS concentration within a (small) volume with dimension of the order of the Kuhn segment length. The black solid line in Figure 2 corresponds to ϕs = 0.6, in agreement with the observed dependence Tg(ϕPS) in PS/dibutyl phthalate mixtures.37 Fluorescence Correlation Spectroscopy (FCS). The measurements were conducted at 22 °C on an commercial instrument (Carl Zeiss, Jena, Germany) using the module ConfoCor2 coupled to an inverted microscope, Axiovert 200. An alpha-Plan-Apochromat 100×/1.46 Oil (Carl Zeiss) objective was used, and a HeNe laser at λ = 633 nm served to excite TDI. The emitted fluorescence was collected by the same objective, and after passing through a long pass 650 nm emission filter and a confocal pinhole, it was delivered to an avalanche photodiode detector capable of single-photon counting. The temporal fluctuations of the detected fluorescence intensity, δF(t), caused by the diffusion of the fluorescent tracers through the limited confocal observation volume were recorded and evaluated in terms of an autocorrelation function given by eq 1:

functionalities but always based on the same coupling chemistry. Welldefined linear PBDs were obtained from Polymer Standard Service (Mainz, Germany) with low dispersity (see Table 1). The linear PS samples were obtained from Polymer Source (Montreal, Canada). The comb-shaped PS were synthesized and kindly provided by J. Roovers (National Research Council, Canada), the linear PDMS by T. Wagner (MPIP-Mainz), and the terrylene diimide derivative (TDI) tracer by Klaus Müllen (MPIP-Mainz). Sample Preparation. The linear and star-shaped PBD samples were stored at low temperature (−65 °C). The other polymers were stored under ambient conditions. Predetermined quantities of the star polymer matrixa few milligramswere transferred to glass bottles, followed by the amount of solvent (toluene for PDMS; squalene for PBD; DEP for PS) required to reach the desired concentration in the solution. Except for the PDMS samples (prepared directly in toluene), at least 10 volumes of THF containing BHT as stabilizer (in the ppm range) and TDI tracer was added to the vial, and argon was blown for 1 min to generate an inert atmosphere. All samples were stirred with a magnetic stirring bar for a period of 4 days at 25 rpm (under ambient conditions) with sealed caps and then for 8 h at 40 °C with the caps removed. After evaporation of the THF, they were placed in vacuum oven for at least another week at 20 °C before performing the FCS measurements for TDI; prolonged evaporation at higher temperatures near Tg,PS led to composition changes due to DEP losses. For the FCS

G(t ) = B

⟨δF(t )δF(t + τ )⟩ ⟨F(t )⟩2

(1) DOI: 10.1021/acs.macromol.5b01464 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules The experimentally measured autocorrelation curves for the TDI fluorophore employed in this paper, G′(t) = G(t) + 1, were represented by the multicomponent diffusion function (eq 2)34 n

G′(t ) = 1 +

1 ⎛⎜ T ′ −t / τT ⎞⎟ 1+ e ∑ ⎝ ⎠ N 1 − T′ i=1

Fi

(1 + ) (1 + ) t τi

t

S 2τi

(2) using for the vast majority of polymer solutions a single component (n = 1). Two components (n = 2) were only necessary for highly concentrated squalene solutions of some star polymers (AM01402, RS64-81, and RS64-PBD5), but the amplitude of the slow component was very small (Fslow ∼ 0.1); hence, whereas it does not really affect the fast mode and the conclusions drawn here, a thorough discussion of its role (Fslow) is beyond the scope of the present work. The parameters T′ and τT are the fraction and the decay time of the triplet state, N represents the average number of diffusing fluorescent species through the FCS observation volume, and Fi and τi are the amplitude and the lateral diffusion time of the ith species through the FCS observation volume, with Di = w02/(4τi) being the corresponding diffusion coefficient. The factor S = z0/w0 (∼6) is the so-called structural parameter, i.e., the ratio between the axial (2z0) and the lateral (2w0) dimensions of the Gaussian confocal observation volume. The values of z0 and w0 were determined by measurements in dilute (10 nM) toluene solutions of the molecular tracer terrylene diimide (TDI) using the known32 value of its hydrodynamic radius RH = 0.8 nm. Furthermore, in all fits the triplet decay time and fraction (amplitude) were used as adjustable fitting parameters. The triplet times obtained for TDI were in the range of 3−5 μs. Adequate statistics were ensured by the accumulation of G(t) for at least 10 min per (x,y) positiona time interval 2 orders of magnitude larger than the characteristic diffusion time of G(t) for TDI, and by probing different (x, y) positions within the solution for each sample. Concerning the representation in terms of D, a single D0 value was used for the normalization of the obtained D everywhere, in each respective solvent. The error bars, when shown (for example, in Figure 4), denote the standard deviation with respect to an average D among three different positions (x, y)at the same z, for a given c. The different (x, y) positions were selected such that they were separated by several multiples of the focal spot waist (∼300 nm) to assess the possible impact of composition heterogeneities.

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RESULTS AND DISCUSSION Figure 3 depicts typical examples of normalized fluorescence intensity autocorrelation functions, G(t) = G′(t) − 1 eqs 1 and 2, from which the translational diffusion coefficient of TDI in the various polymer solutions can be extracted. TDI has RH ≈ 0.8 nm, as previously measured32 by FCS in pure solvent (toluene). For all linear polymer solutions as well as the vast majority of star-shaped polymer solutions, TDI diffusion was represented by a single 3-dimensional Fickian diffusion fit34 including a triplet contribution to account for the well-known fast photophysical relaxation mechanism of TDI (eq 2). The inflection point of each curve denotes the characteristic average time needed by a tracer to diffuse laterally through the confocal volume of the microscope. The influence of increasing polymer concentration on the TDI diffusion (Figure 3a) is demonstrated by the increased time lag in the corresponding G(t) curve. However, the effect of polymer type on the TDI diffusion coefficient is even more striking as seen in the inset to Figure 3a, which compares the TDI diffusion in PS/DEP and PBD/squalene solutions taking into account the different diffusion time τs of TDI in the two solvents; about 40 times stronger slowdown is observed in PS as compared to PBD solutions at c = 0.46 g/mL. Moreover, we find almost no

Figure 3. Normalized fluorescence intensity autocorrelation functions, G(t), for the diffusion of TDI tracer in different polymer solutions at 22 °C. (a) Star-shaped PBD (squares; RS64-81, at c = 0.05 g/mL (red) and c = 0.36 g/mL (blue)) and linear PBD (PBD 170k L, triangles; c = 0.66 g/mL). The arrow points in the direction of increasing concentration and the dashed black line denotes G(t) for TDI in squalene in the absence of polymer. Inset: G(t) vs t/τs for solutions of linear PBD 170k/squalene (blue triangles) and linear PS 1M/DEP (red circles) solutions, both at c = 0.46 g/mL, where τs is the TDI diffusion time in neat squalene and DEP. (b) Different polymer chain architectures: star-shaped PBD (RS64-81, blue squares; c = 0.69 g/ mL) and linear PBD 170k (PBD 170k, black triangles; c = 0.66 g/mL) solutions in squalene as well as reference measurements in squalene (dashed black G(t)). Inset: G(t) for linear PS 1M (red circles, c = 0.46 g/mL) and comb-shaped PS722 (blue triangles, c = 0.43 g/mL) polymer solutions in DEP, as well as reference measurements in DEP (dashed black G(t)). (c) G(t) for linear PDMS at different concentrations: 0.01 g/mL (red triangles); 0.50 g/mL (black squares); 0.70 g/mL (blue circles). Inset: G(t/τs) at c = 0.70 g/mL for solutions of linear PDMS 60k/toluene (black squares) and linear PBD 170k/ squalene (red inverted triangles), where τs is the TDI diffusion time in neat toluene and squalene, respectively. The dashed vertical lines in (a−c) denote the relaxation time obtained from the single Fickian representation (eq 2) of G(t). The molecular characteristics of the polymers are listed in Table 1. C

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Macromolecules quantitative difference on the G(t) of TDI when the polymer architecture is varied for the same polymer chemistry (Figure 3b) and concentration. It should be mentioned, however, that topology effects were reported elsewhere for the diffusion dynamics of the same tracer in poly(isoprene) melts.27 The TDI diffusion slowdown, expressed as D(c)/D0, is plotted against concentration in Figure 4 for linear PS/DEP

superimpose on a single master curve as well. However, it is progressively distinctly slower as compared to the master curve for diffusion of TDI in PBD solutions when the concentration increases. The maximum slowdown observed for PS solutions is roughly 1000-fold. It is much stronger than that for the PBD solutions in the same c range (Figure 4). Moreover, the TDI data for the solutions of comb-shaped PS722 polymers in DEP fall onto the aforementioned master data set for linear PS, suggesting the absence of specific molecular architecture effects as also inferred from Figure 3b. This dependence of the tracer diffusion master curve on chemistry-specific polymer composition may relate to a strong c-dependent local friction. For PS solutions, this friction coefficient is described by a VFT expression41 as ζPS(c) = ζ0(c) exp(BPS/(T − T0,PS(c)) ∼ D(c)−1, denoting that friction increases with increasing T0,PS(c), since Tg increases with PS concentration; Tg,PS = 373 K and Tg,DEP = 180 K. The parameter T0,PS(c) = Tg,PS(c) − c2 represents the ideal VFT temperature, whereas the WLF constant c2 assumes the same value (50 K) for all polymer data sets examined in this work. Finally, B is an activation parameter,36 and the term ζ0(c) denotes a friction coefficient related to pure crowding effects. The single glass transition temperature of the nondilute PS solutions at the examined concentrations was estimated from the interpolation between the experimental Tg values in the range 10−45% (Figure 2). The Fox−Flory expression (using Tg,PS = 373 K and Tg,DEP = 180 K), which is used to describe the composition dependence of Tg,PS(c), overestimates37 the experimental values in the range 0 < ϕ < 0.5. To the contrary, PBD solutions at ambient temperature are far above Tg (by more than 100 K) and Tg,PBD reasonably close to Tg,sq; hence, the diffusion slowdown reflects strong predominance of crowding effects over minute Tg effects. For the PBD solutions with Tg,PBD ∼ 180 K, Tg,sq < 250 K,41 and BPBD = 900 K,42 we estimated ζPBD(c) = ζ0(c) exp(BPBD/(T − T0,PBD(c)) = 540ζ0(c) at T = 293 K. This means that in contrast to the PS/DEP system with very different Tg(c), ζPBD(c) is virtually Tindependent. Hence, the approach of solute friction at length scales of the order of the monomer size, presented by Brochard and de Gennes for tracer diffusion in melts,9 should also apply to the solutions under study.43 We underline the decreasing Tg(c) trend to be the main cause in the progressive speed-up of TDI dynamics as the friction coefficients decrease: ζPS(c) ≫ ζPBD(c). To demonstrate that a collapse of the two master curves (PS onto PBD) is indeed feasible, the diffusion data of the PS solutions were shifted upward by a factor K(c) ≡ ζPS(c)/ ζPBD(c). From the K(c) values, the extracted value of the activation parameter BPS = 1180 ± 150 K (or activation energy of 9.8 ± 1.2 kJ/mol) is similar to that for a PS melt obtained by dielectric spectroscopy (which probes segmental dynamics).44 The length scale of segmental dynamics associated with Tg is on the order of the Kuhn segment length and Mw-independent, rendering the conformity to the present analysis qualitative (master curves in Figure 4). Despite some experimental uncertainties at high concentrations, the c dependence of parameter K unambiguously suggests deviations from the crowding models9,38,39 of tracer diffusion, and we believe that such deviations reflect the Tg(c) effects discussed above. We claim that the distinct value of the prefactor A (in the abovementioned semiempirical D(c) equation) for PS solutions (Figure 4, main plot) as compared to PBD solutions, and the larger than unity slope (≈2) of the solid line in the inset of Figure 4, stem from the additional contribution of Tg(c) effects;

Figure 4. Normalized diffusion slowdown, D(c)/D0, vs c for TDI under good solvency conditions: DEP for PS (black symbols; pentagons for P2722 combs), and squalene for linear PBD at two different Mw (PBD 170k, blue stars and PBD 52k, blue pentagons) and star PBD (Table 1): RS64-30 (Mw,arm = 30K, f = 169; double-crossed blue symbols), RS64-05 (Mw,arm = 0.5K, f = 929; blue triangles), RS32PBD5 (Mw,arm = 30K, f = 300; left-tilted red triangles), RS 64−81 (Mw,arm = 80K, f = 200; red circles), RS64-PBD5 (Mw,arm = 4.4K, f = 925; blue squares), and AM01402 (Mw,arm = 30K, f = 300; blue rhombi). The diffusion slowdown data for TDI in linear PDMS (Mw = 60K)/toluene solutions are depicted by the green squares. The solid black (A ≈ 9.5, β ≈ 1.1), blue (A ≈ 4.2, β ≈ 1.1), and dashed green (A ≈ 2.8, β ≈ 1.1) lines through the data represent the stretched exponential fit D = D0 exp(−Aϕβ); see text. Inset: the friction coefficient ratio log(ln(K(c))) vs log(c), where K(c) = ζPS(c)/ζPBD(c), for vertical shift of the PS data in the main panel. The solid line has a slope of 2.

(black symbols) and of PBD/squalene (colored symbols) solutions; D0 represents the TDI diffusion coefficient in each solvent (amounting to 2.2 × 10−11 m2 s−1 in squalene, 8.8 × 10−12 m2 s−1 in DEP, and 4.9 × 10−10 m2 s−1 in toluene). For TDI in PBD/squalene solutions, the diffusion coefficients vs ϕ (= c/ρ, ρ being the polymer density) superimpose on a single Mw-independent master curve (solid blue line in Figure 4). The slowdown can be described by the semiempirical form D = D0 exp(−Aϕβ). The coefficients A = 4.2 ± 0.2 and β = 1.1 ± 0.1 are independent of polymer architecture. They represent38,39 polymer-Mw-related parameters depending also on tracer size and solvency. However, this appears at odds with the observed Mw independence of TDI diffusion in the present polymer solutions. Alternatively, the physical meaning of A and β might be inferred through D = D0 exp(−RH/ξ), a diffusion expression used to support findings from sedimentation experiments.40 The equation is based on a theory of solute diffusion dynamics in semidilute polymer solutions, with tracer radius RH, in a physical network with mesh size ξ ≈ bϕ−ν/(3ν−1) with ν ≈ 0.59 for good solvents and b being the polymer segment (Kuhn) length. This expression predicts that the tracer diffusion retardation is independent of matrix polymer Mw at polymer concentrations above the overlap concentration (c*). The c-dependent data of TDI diffusion in nondilute DEP solutions of entangled linear PS with various Mw (Table 1) D

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might suggest the presence of other effects, such as favorable tracer−polymer interactions, in the reported biomacromolecular work.45 We note that the rest of the reported data shown in Figure 5 exhibit a c-dependent crossover between the strongly plasticized master curves (PBD/squalene; PDMS/toluene) and the one including stronger Tg effects (PS/DEP). As such, we are confident that the scaling curves are universal, representing two different cases impacting molecular tracer slowdown in polymer solutions in the absence of interactions: two of pure crowding effects (PDMS as genuine subcase; PBD with slight impact from Tg(c) effects) as well as one (PS) where crowding is strongly coupled with Tg(c) effects. We emphasize that the present FCS study is a facile method to probe c-dependent friction effects in entangled polymer liquids requiring only ultralow (nM) probe concentrations as opposed to other spectroscopic methods,3,6,50,51 ruling out potential artifacts such as chemical modification of the polymer matrix or probe aggregation.

this plot is inspired by the corresponding stretched exponential diffusion slowdown of TDI in PS solutions. The effect of polymer-specific Tg(c) is corroborated by the weak c-dependence of the TDI dynamics in polymer solutions with very low and nearly concentration independent Tg. This was exemplified by the FCS measurements of TDI diffusion in toluene solutions of linear PDMS (Mw = 60K) with Tg = 143.5 K (Figure 3c). At the same concentration, TDI diffusion is slightly faster in PDMS/toluene than in PBD/squalene both at c = 0.7 g/mL, as seen in the inset of Figure 3c. Note that TDI diffuses 22 times faster in toluene than in squalene, and thus a representation of G(t/τs) is used in the inset to Figure 3c to account for this effect. The c-dependence of TDI dynamics in PDMS solutions is somewhat weaker than in PBD solutions but drastically weaker than in the PS solutions, as shown in Figure 4. The TDI diffusion slowdown in PDMS/toluene system can also be described by the semiempirical stretched exponential with the coefficients A = 2.8 ± 0.2 and β = 1.1 ± 0.1 (dasheddotted green line in Figure 4). To further support the validity of our conclusions, we compare the diffusion slowdown of literature data restricted to various molecular-sized tracers using different characterization methods45−49 with respect to the three master curves of Figure 4. Three distinct groups of data sets can be discerned: The anthracene data in CCl4 solutions of PS (black triangles, Figure 5) exhibit relatively stronger slowdown following the PS curve

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CONCLUSIONS Friction affects the mobility of noninteracting molecular tracers in polymer liquid matrices in two different ways. Far above Tg, exemplified by the PBD and PDMS solutions, tracer diffusion slowdown is controlled by the local (Mw-independent) viscosity (∼ζ0(c)), which in turn depends on the tracer size,9,18 reflecting only pure topological crowding effects. When Tg(c) effects become non-negligible (for PS solutions), the excess slowdown is attributed to the larger friction ζPS(c). In both cases, a universal, molecular structure (and Mw)-independent master curve is obtained. The relation between D(c) and Tg(c) does not imply direct association of the probe motion with the local structural α-relaxation of the polymer matrix.32

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (A.V.). Notes

The authors declare no competing financial interest.

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Figure 5. D(c)/D0 for different molecular tracers in polymer solutions, reported elsewhere: coumarin in aqueous methylcellulose solutions (solid red squares),45 methyl red in PVAc−toluene (black crosses),46 methyl red/naphthalene in dilute or nondilute PS−toluene solutions (inverted black triangles),46 coumarin in PS−toluene (black hexagons),47 bromonaphthalene in PS−THF (blue rhombi),48 methyl red in PS−THF (blue triangles),48 lysozyme in PEO−water49 (doublecrossed blue symbols), and anthracene in PS−CCl4 (black triangles).48 The solid black (A ≈ 9.5, β ≈ 1.1), blue (A ≈ 4.2, β ≈ 1.1), red45 (A ≈ 40, β ≈ 0.9), and dashed-green (A = 2.78; β = 1.1) lines denote the stretched exponential representation of the tracer diffusion (see text and Figure 4).

ACKNOWLEDGMENTS A.V., G.F. and D.V. were supported by the Greek General Secretariat for Research and Technology (ARISTEIA-RINGS and THALIS (Metaassembly)). M.G. acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC).

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REFERENCES

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(solid black lines in Figures 4 and 5). At the other extreme, the lysozyme diffusion data in aqueous PEO 600 solutions (doublecrossed blue symbols) superimpose on the PBD master curve (denoted by blue in Figures 4 and 5) over the whole concentration range examined, corroborating the notion of almost no Tg(c) effects. Another extreme case is represented by the coumarin diffusion data (red squares) in semidilute aqueous methylcellulose solutions, where the slowdown appears at rather low concentrations in the examined physical networks (Figure 5). The latter is reflected in the large associated parameter (A) in the stretched exponential representation. This E

DOI: 10.1021/acs.macromol.5b01464 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.5b01464 Macromolecules XXXX, XXX, XXX−XXX