Surprising Temperature Scaling of Viscoelastic Properties in Polymers

Jun 28, 2018 - The data (Figure 1B) were analyzed in terms of the creep compliance: (3)where ... (35−37) Indeed, this scaling shows rather universal...
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Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Surprising Temperature Scaling of Viscoelastic Properties in Polymers Alexander L. Agapov,† Vladimir N. Novikov,†,‡ Tao Hong,† Fei Fan,† and Alexei P. Sokolov*,†,§ †

Department of Chemistry, University of Tennessee, 1420 Circle Drive, Knoxville, Tennessee 37996, United States Institute of Automation and Electrometry, Russian Academy of Sciences, 1 Koptyug ave., Novosibirsk 630090, Russia § Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States

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‡

ABSTRACT: We present detailed studies of segmental dynamics and viscosity in polystyrene (PS) and poly(2-vinylpyridine) (P2VP) with different molecular weights (MW). Analysis reveals that the molecular weight dependence exhibits very different temperature scaling for segmental and chain dynamics: while segmental relaxation in samples with different MW forms a master curve when presented vs T − Tg, the viscosity of the same samples falls on a master curve when presented vs Tg/T (here Tg is the glass transition temperature). This result indicates significant difference in the friction mechanisms for chain and segmental dynamics. Even more puzzling is that the absolute values of viscosity appear to be essentially MW independent when presented vs Tg/T up to surprisingly high molecular weight. In particular, viscosity at Tg does not show any appreciable molecular weight dependence up to MW ≈ 30 000 g/mol. We speculate that molecular scale relaxation (chain dynamics) in polymers behaves similar to the structural relaxation in small molecular liquids, while additional slowing down mechanism contributes to the segmental relaxation in the same polymers. decoupling usually increases with MW.14−18,28 Apparently, the increase of molecular weight affects the segmental and chain dynamics in polymers in a different manner. In this paper we present a detailed analysis of MW dependence of viscosity and segmental relaxation in two polymers: poly(2-vinylpyridine) (P2VP) and polystyrene (PS). We find that MW dependences of viscosity and segmental dynamics exhibit qualitatively different scaling with respect to temperature. While the temperature dependence of segmental relaxation time for all studied MW falls on a master curve when plotted as a function of T − Tg(M), viscosity falls on a master curve when plotted vs Tg(M)/T; here M is molecular mass. Even more surprising, for a given polymer, η(M) scales well vs Tg/T in an absolute value up to a certain molecular weight. Moreover, the viscosity at Tg does not show any appreciable molecular weight dependence up to M ≈ 30 000 g/mol for both PS and P2VP. Our results suggest that at temperatures close to Tg the segmental dynamics and viscosity of short chains are governed by the mechanism similar to that observed in small molecular liquids. The same mechanism seems to control behavior of viscosity at all MW, while an increase in MW leads to an additional slowing down in segmental dynamics. The latter depends on the chain rigidity and leads to an increase in steepness of the temperature dependence of segmental relaxation time τα(T) (fragility).

I. INTRODUCTION Polymers present a class of materials with unique viscoelastic properties not observed in other materials. Despite their widespread applications, the fundamental understanding of the relationship between polymer dynamics on various length and time scales remains far from clear.1−13 Classical textbook knowledge suggests that both local segmental dynamics and macroscopic viscous flow are controlled by the same monomeric friction mechanism.1−3 However, it was demonstrated several decades ago that segmental and chain dynamics in many polymers have different temperature dependence,14−22 emphasizing some difference in the underlying friction mechanisms. It was shown that segmental dynamics and its temperature dependence are very sensitive to a number of parameters: monomer chemistry, microstructure, chain rigidity, and degree of polymerization. On the other hand, the macroscopic viscosity controlled by global chain relaxation in polymers has temperature dependence (scaled by the glass transition temperature Tg) that is less sensitive to these microscopic parameters.14−26 As a result, a strong decoupling of segmental and chain dynamics is observed in polymers with the degree of decoupling being a material specific parameter.17,18,25 Several models have been proposed to describe this decoupling,9,10,14,20−22,26−30 but the microscopic picture of the difference in the friction mechanisms at different length and time scale remains elusive. One of the possible approaches to unravel this puzzle is to study the role of molecular weight (MW) in polymer dynamics. Oligomers are expected to have properties similar to that of small molecules where structural relaxation and viscosity are coupled, and the © XXXX American Chemical Society

Received: March 2, 2018 Revised: May 16, 2018

A

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Table 1. Molecular Weight and PDI of the Studied Polymers, Their Glass Transition Temperature Tg Estimated Using DSC and BDS, Stretching Parameter β, Fragility of Segmental Dynamics ms and of Viscosity mη, and Viscosity at Tg MW (g/mol)

PDI

Tg(DSC) (K)

Tg(BDS) (K)

β

ms

mη

η(Tg) (Pa·s)

700 780 1640 2200 3860 4880 9120 12610 31800 1020 2140 3030 4050 9100 35900

1.3 1.24 1.24 1.23 1.23 1.15 1.11 1.09 1.06 1.05 1.03 1.02 1.01 1.05 1.07

264 279 321.5 336 350 353.5 364.5 368.5 371 287.5 333 341 348.5 357.5 366

262.5 278.5 320 333.5 349.5 355 365 369.5 371.5 289.5 326 336.5 343 355 364

0.6 ± 0.1 0.6 ± 0.1 0.55 ± 0.1 0.55 ± 0.1 0.6 ± 0.1 0.65 ± 0.1 0.5 ± 0.1 0.5 ± 0.1 0.4 ± 0.1 0.55 ± 0.1 0.5 ± 0.1 0.45 ± 0.15 0.5 ± 0.2 0.5 ± 0.1 0.5 ± 0.1

76 83 104 109 127 145 142 160 135 68 78 82 87 90 97

84 86 91 98 93 102 98 91 86 85 89 93 96 89 84

0.8 × 1010 0.5 × 1010 0.4 × 1010 0.6 × 1010 0.25 × 1010 0.5 × 1010 0.4 × 1010 0.3 × 1010 1.66 × 1010 0.9 × 1011 0.7 × 1011 1 × 1011 2.3 × 1011 0.2 × 1011 1 × 1011

PS

P2VP

Figure 1. (A) Normalized dielectric spectra of P2VP; numbers in the legends correspond to the molecular weight of the samples. (B) Examples of timedependent compliance for PS sample with MW ∼ 30K measured at different temperatures to estimate viscosity from the long time behavior (line). parameter α was in the range ∼0.7−1 for all the studied samples, while β varies between 0.65 and 0.4 (Table 1). The characteristic Havriliak−Negami relaxation times were converted to the relaxation time corresponding to the maximum of the loss peak using the relation31 ÄÅ ÉÑ1/ α ÅÅ παβ πα ÑÑÑ Å τα = τHNÅÅÅsin sin ÑÑ ÅÅÇ 2(β + 1) 2(β + 1) ÑÑÑÖ (2)

II. EXPERIMENTAL SECTION P2VP was purchased from Polymer Source and PS was purchased from Scientific Polymer Products. The MW and polydispersity (PDI) for PS samples have been measured using GPC (Tosoh EcoSEC GPC, 2 × TSKgel SuperMultiporeHZ-M columns) while those of P2VP samples were provided by the supply company (Table 1). Tg was estimated using differential scanning calorimetry (DSC, Q100 DSC from TA Instruments). Temperature scans were performed on cooling cycle at a rate of 10 K/min with the midpoint in heat capacity change taken as Tg(DSC). Detailed studies of segmental dynamics were performed using the Novocontrol Alpha Concept 80 system equipped with broadband impedance analyzer and active ZGS sample interface. All broadband dielectric spectroscopy (BDS) measurements were done on heating and cooling cycles in parallel plate geometry using gold plated round electrodes with a Teflon ring acting as a spacer. The temperature was controlled by the Novocontrol Quattro setup with accuracy better than ±0.1 K. Representative dielectric loss spectra of P2VP are shown in Figure 1, where three processes, conductivity, α-relaxation (segmental), and β-relaxation, are identified. For quantitative analysis, the dielectric spectra have been fit by two Havriliak−Negami (HN) relaxation distribution functions and a conductivity contribution:

ε″(ω) = ε0 +

∑

σ Δε + 0 iωε0 (1 + (iωτHN)α )β

The temperature at which τα = 100 s was taken as Tg (BDS). Viscosity of the same samples was measured using an AR-2000ex rheometer (TA Instruments) in parallel plate geometry with plates of 8 or 25 mm in diameter and a sample thickness of ≈1 mm. The samples were subjected to a small stress, σ, and the deformation and flow of the sample, γ(t), was monitored as a function of time. The data (Figure 1B) were analyzed in terms of the creep compliance: J(t ) =

γ(t ) = Jg + Jd ψ (t ) + t /η σ

(3)

where Jg is the glassy compliance, Jd is the steady-state compliance, ψ(t) is a normalized function for retarded elasticity of the sample which describes the form of the time-dependent recoverable deformation, η is the viscosity of the system, and t is time. Long time behavior of the compliance was used to estimate the viscosity of the samples (Figure 1B). Additional details of the viscosity measurements can be found in our earlier paper.32

(1)

where Δε is the dielectric strength of the process, τHN is the characteristic HN relaxation time, and the exponents α and β describe the asymmetry and width of the spectra, respectively, and σ0 is the dc conductivity. The obtained from the fit HN shape parameters did not show any systematic temperature dependence. The asymmetry

III. RESULTS The temperature dependence of segmental relaxation time and viscosity of P2VP with different MW are shown in Figure 2 B

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Figure 2. Temperature dependence of segmental relaxation time τα(T) (A) and viscosity η(T) (D) of P2VP with different molecular weights. The same data scaled vs Tg/T (B, E) and vs T − Tg (C, F). Respective molecular weights are shown in parts B and E. For comparison, some literature data are shown by black solid lines: segmental relaxation time for P2VP with Mw = 18 kg/mol38 (A) and viscosity of P2VP with Mw = 33K and 8.4K (D, left and right lines, respectively).39

appear to be essentially independent of molecular weight up to MW ∼ 5K (Figure 3E). Thus, the presented analysis reveals a surprising result: while segmental dynamics in molecules with different MW scales well vs T − Tg (the fact known earlier34−37), viscosity (i.e., the global molecular scale relaxation) of the same samples scales better vs Tg/T. This clearly indicates a significant difference in friction mechanisms controlling segmental and chain dynamics in the same polymers.

which also includes several literature data for comparison. Both τα and η increase with MW, as usually observed for polymers. The temperature dependence of structural relaxation or viscosity is usually characterized by the so-called steepness (fragility) index:33 ms =

∂ log τα(T ) ∂(Tg /T )

or T = Tg

mη =

∂ log η(T ) ∂(Tg /T )

T = Tg

IV. DISCUSSION AND THEORY The scaling of segmental dynamics vs T − Tg can be explained using rather simple picture. Usually τα(T) can be well described by the Vogel−Fulcher−Tammann (VFT) behavior:

(4)

Thus, to compare the steepness index of polymers with different molecular weights, the relaxation time is usually analyzed vs Tg/T. This scaling shows well-known increase in fragility with molecular weight for segmental dynamics (Figure 2B). However, this scaling reveals surprising behavior of viscosity (Figure 2E): not only does it show the same temperature dependence, it has almost the same value for all MW between 1K and 9K. A significant difference appears only for the sample with MW = 36K. Even in this sample viscosity at Tg remains comparable to the viscosity of lower MW samples, but it varies slightly weaker with T (Figure 2E). This slight decrease in fragility of the viscosity mη with increase in MW is in stark contrast to the increase in fragility ms observed for the segmental dynamics (Figure 2B). The scaling of segmental dynamics vs T − Tg was suggested in ref 34 and was confirmed recently for several polymers.35−37 Indeed, this scaling shows rather universal behavior of segmental dynamics in P2VP with different MW (Figure 2C). At the same time, viscosity shows significant difference for samples with different MW in this scaling (Figure 2F). The same results, good scaling of viscosity vs Tg/T and good scaling of segmental dynamics vs T − Tg, are observed also for PS samples with different MW (Figure 3). Also in that case the absolute values of viscosity

ji B zyz τα(T ) = τ0 expjjj z j T − T0 zz k {

(5)

where B is a parameter characterizing the bare (very high T) activation energy, while T0 is the VFT temperature at which relaxation time should diverge. We note that in many glassforming liquids a single VFT function cannot describe the temperature dependence of τα(T) in the entire temperature range41,42 two VFT functions are required, one at low temperatures and another one at high temperatures. However, in some of the glassforming liquids a single VFT function describes τα(T) reasonably well, in particular in hydrogen-bonding liquids, such as glycerol, in ionic liquids and in many polymers.42 For simplicity, we assume the validity of the VFT equation in polymers. We assume also that the parameter B is MW independent because at very high T it is a bare activation energy of segmental dynamics. This assumption is in a reasonable agreement with the direct fit of the experimental data in PS, from both the literature40 and our data from Figure 3A, and in P2VP with the data of Figure 2A. C

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Figure 3. Temperature dependence of segmental relaxation time τα(T) (A) and viscosity η(T) (D) of PS with different molecular weights. The same data scaled vs Tg/T (B, E) and vs T − Tg (C, F). Respective molecular weights are shown in parts B and E. For comparison, some literature data are shown by black solid lines: segmental relaxation time for PS with Mw = 760, 2300, 3700, and 6400 (A, from right to left)40 and viscosity of PS with Mw = 1100, 3400, and 15 700 (D, from right to left, respectively).34

Taking into account that by definition τα(Tg) = constant, we can rewrite the VFT equation as ÅÄÅ ÑÉ Å τ (T ) ÑÑÑ B B ÑÑ = lnÅÅÅÅ α = B ÅÅÇ τ0 ÑÑÑÖ T − T0(M ) T − Tg(M ) + ÅÅÄ τα(Tg) ÑÑÉ lnÅÅÅÅ ÅÅÇ

=

B T − Tg(M ) + A

τ0

ÑÑ ÑÑ ÑÑÖ

(6)

In this case, indeed segmental dynamics should scale well vs T − Tg. Thus, the observed scaling of segmental dynamics indicates that the increase in MW leads to a significant shift of T0 without affecting other parameters. In the case of the Adam−Gibbs approach, this result suggests that molecular weight affects strongly the temperature at which configurational entropy controlling segmental relaxation should vanish. So, MW affects the temperature dependence of the configurational entropy accessible on the time scale of segmental dynamics. This leads to the shift of T0 and to the increase in fragility with MW. This scaling also explains why polymers with strong MW dependence of Tg also exhibit strong increase in fragility, while polymers with weak MW dependence of Tg have much weaker variation of fragility.28,40,43 The most intriguing result of our analysis is the scaling of viscosity vs Tg/T. In general, it means that fragility of the global molecular scale relaxation mη does not change significantly with MW (Figures 4A and 4B), the effect already recognized in some earlier publications.17,18,28 Moreover, it has been emphasized that mη does not change significantly between different polymers, although their segmental fragility does.17,26 This point was partially challenged in the comment.18 Most puzzling is that even absolute value of viscosity in this scaling does not change much with molecular weight up to rather high MW ∼ 5K.

Figure 4. Molecular weight dependence of fragility for segmental relaxation (red symbols) and viscosity (blue symbols) in PS (A) and in P2VP (B). The open symbols in (A) are the literature data from ref 18. Lines in (A) and (B) present the fit to the eq 16 as described in the text. (C) Molecular weight dependence of viscosity at Tg for P2VP (stars) and for PS (squares).

Moreover, the viscosity at Tg for both polymers is almost MW independent up to even higher MW ∼ 30K (Figure 4C). The classical Rouse model suggests that viscosity should scale linear with molecular weight at MW below entanglement (our case). D

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in low molecular weight PS and poly(methylphenylsiloxane).14,22,47 The importance of the temperature variations of Je was discussed in refs 14, 19, 20, 22, 34, 44, and 47−50. These changes make it even more surprising that viscosity of short PS and P2VP chains presented vs Tg/T appears essentially MW independent (Figures 2E and 3E). Several models have been proposed to describe the difference in temperature dependence of chain and segmental modes. They include a coupling model that relates the difference to the variation in coupling parameter of different modes.14,20−22,44,48−50 The latter affects the stretching of the relaxation processes. Indeed, predictions of the coupling model have been confirmed for some polymers.14,20−22,44,48−50 However, contradictions with the coupling model predictions have been reported in the cases of PIP.24 Ideas of dynamic heterogeneity and decoupling of translational and rotational motions have been also involved26 and were partially challenged in the following comment.27 Thus, the microscopic details of the decoupling of chain and segmental modes remain the subject of discussion. However, a new idea placing the emphasis on the unusually high segmental fragility has been suggested recently.28 It has been proposed that the extremely high segmental fragility in relatively rigid polymers can be related to a nonergodic behavior of macromolecules on time scale of segmental relaxation. In simple terms, a segment in rigid polymers cannot relax completely on the segmental relaxation time because it extends to a distance larger than the characteristic size of the cooperatively rearranging regions (for details, see ref 28). As a result, the accessible configurational entropy controlling segmental relaxation vanishes with temperature much faster than it does on the molecular (chain relaxation) time scale. This leads to a sharp increase in segmental fragility with MW. In contrast, the system remains ergodic and configurational entropy remains fully accessible on the time scale of global molecular (chain) relaxation. This leads to the steepness index of the molecular scale dynamics mη being rather independent of MW (Figures 4A and 4B). To make this idea more quantitative, let us consider a simple model based on the configuration entropy (Adam−Gibbs) approach. Following the earlier idea,28 we assume that the configurational entropy accessible for segmental relaxation in a polymer, SS(T,M), is smaller than the total configurational entropy of the system SC(T). It can be written as

This clearly contradicts to the observed here scaling. We note that failure of the Rouse model at temperatures close to Tg has been noticed by Plazek and co-workers many years ago.14,44 We want to emphasize that the difference in temperature dependence of segmental and global molecular scale (chain) dynamics, the so-called thermorheological complexity, has been discussed in many earlier papers.14−22 The best illustration of the difference in temperature dependence of characteristic relaxation times comes from BDS of some polymers, e.g., polyisoprene (PIP) or poly(propylene glycol) (PPG), where chain relaxation appears as a normal mode.17,21,24 In that case one could clearly see the difference in temperature variations of chain and segmental modes without use of any time−temperature superposition. It has been shown that the temperature dependence of the chain modes has essentially the same fragility, independent of MW.17,24 This is similar to the essentially MW independent fragility for viscosity (Figure 4). However, the characteristic relaxation times of the chain modes in BDS show clear changes with MW even when presented vs Tg/T,17,24 while viscosity appears to be scaling even in absolute values at low MW (Figures 2E and 3E). It is difficult to rationalize why the increase in molecular weight does not lead to an increase in viscosity when it is presented vs Tg/T. Recently it was suggested28 that behavior of molecular scale relaxation (chain relaxation for polymers) is similar in small molecular liquids and in polymers. Indeed, fragility of chain dynamics usually does not exceed m ∼ 80−100,26 the same as the fragility in most of the van der Waals liquids. However, fragility of segmental relaxation in the same polymers can be much higher, ms ∼ 150 and higher.45 Detailed analysis of our results reveals that fragility of segmental dynamics increases significantly with MW for both P2VP and PS, while steepness index of viscosity shows only weak changes with MW (Figures 4A and 4B). So, the apparently viscosity of PS and P2VP with MW < 5K behaves as viscosity of small van der Waals liquids, and the increase in molecular weight affects only Tg (as in small van der Waals liquids46) without significant changes in steepness index and even absolute value of viscosity at the same T/Tg values. In other words, the increase in molecular mass affects only characteristic energy scale (Tg). However, independence of viscosity at Tg on molecular weight up to MW ∼ 30K (Figure 4C) still remains a puzzle. It is equal to the viscosity of small molecules and oligomers at Tg. This value is usually controlled by structural relaxation. This value of viscosity for small molecules is often used as a definition of Tg. However, this is not correct for polymers in general because in that case viscosity will be strongly dominated by chain dynamics.1,3,19 Yet we find the viscosity at Tg to be essentially molecular weight independent up to MW ∼ 30K (Figure 4C). Following the idea presented in ref 19, we can write η(T) = ηG(T) + ηR(T), where ηG is glassy (structural) and ηR is rubbery (chain modes) contributions to viscosity. Obviously, ηG dominates in short chains, and no chain modes were actually found in rheological spectra of PS with MW ∼ 1K.19 But it might also dominate at Tg of intermediate MW chains (up to MW ∼ 5K−10K) due to stronger temperature variations of segmental dynamics than that of the chain dynamics. Thus, the same value of η(Tg) probably means that viscosity even at intermediate molecular weight is strongly stiffened by the segmental dynamics at temperatures approaching Tg. This idea of encroachment of segmental modes into the spectrum of chain modes upon approaching Tg in short polymer chains has been proposed in refs 14 and 16. It appears in strong temperature dependence of the steady state recoverable compliance Je upon approaching Tg

SS(T , M ) = SC(T , M ) − ΔS(M )

(7)

Here ΔS(M) represents the configurational states inaccessible on segmental relaxation time scale. It should be larger for the rigid chains and smaller or even negligible for flexible chains. The total configurational entropy is maximal for a system of oligomers. Connecting monomers to a chain should lead, in general, to a decrease of the available configurational states even for fully flexible chains. For simplicity, we neglect here this relatively small difference between the full configurational entropy of oligomeric system and SC(T,M) that corresponds to the configurational states of chains accessible on the molecular time scale, i.e., time scale of viscous flow. In other words, we assume that the total configurational entropy does not depend on MW. For simplicity, we assume also that ΔS is essentially temperature independent but varies with MW. Using VFT behavior of viscosity, we assign T0η to the temperature where the full configuration entropy becomes zero, SC(T0η) = 0, while the VFT temperature of segmental relaxation, T0s, corresponds to vanishing of configurational entropy accessible on segmental time scale, SS(T0s,M) = 0. Expanding the function TSC(T) E

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Macromolecules around T0η and keeping only linear term in T − T0η, one has TSC(T) = T0ηSC′(T0η)(T − T0η), where SC′(T) = dSC(T)/ dT. As a result, we obtain the following VFT equation for viscosity: log

Bη η D = ≈ η0 TSC(T ) T − T0η

where Bη =

One can see that always ms > mη, and the difference between ms and mη increases with k, i.e., with an increase of the fraction of the configurational entropy not accessible on the segmental time scale. Figure 5 shows the expected dependence of ms/mη on the

(8)

D . T0ηSC′ (T0η)

For segmental relaxation one has TSS(T,M) = T[SC(T,M) − ΔS(M)] ≅ T0ηSC′(T0η)(T − T0η) − TΔS(M). This linearized with respect to T expression leads to the following VFT equation for segmental relaxation time: log

Bs τα D = ≈ TSS(T ) T − T0s τ0

(9)

where Bs =

Bη 1−k

,

T0s =

T0η 1−k

Figure 5. Model prediction (eq 16) for the ratio of segmental and chain fragility as a function of the fraction of configurational entropy k not accessible on segmental time scale for different values of chain fragility. Even at a fraction of a few percent the difference between chain and segmental dynamics can be significant.

(10)

and k=

ΔS T0ηSC′ (T0η)

(11)

parameter k for a few values of mη. It is obvious that even at relatively small k the difference between viscosity and segmental fragilities can be significant, and it increases with increasing mη. To find the dependence of ms on MW, we note that ΔS and k should first increase with MW and then saturate at sufficiently large MW, like, e.g., Tg saturates when the chain becomes Gaussian.11 Assuming some simple form of saturation, e.g., similar to the Ueberreiter−Kanig expression for Tg51

The dimensionless parameter k reflects how large is the fraction of the configurational entropy not accessible on time scale of segmental relaxation. We note that T0s > T0η and Bs > Bη. Fragilities associated with viscosity and segmental relaxation can be expressed in the following way: mi =

Bi Tg (Tg − T0i)2

1/k = 1/k∞ + C /M

(12)

where i = η or s, respectively. Using expressions for Bη, Bs, and T0s given above, it is easy to show that 2 ms 1 (Tg − T0η) = mη 1 − k (Tg − T0s)2

with some constant C, we can describe molecular weight dependence of fragility for PS assuming molecular weight independent mη ∼ 80 (Figure 4). The fit (Figure 4A) gives k∞ ∼ 0.042 ± 0.002. Thus, the proposed model qualitatively captures the molecular weight dependence of segmental fragility and ascribes it to an increase in the fraction of conformational entropy not accessible on time scale of segmental dynamics with increase in molecular weight. We note that in the case of P2VP the segmental fragility ms depends on MW much weaker than for PS. Actually, at small M, ms is even a bit smaller than the chain fragility mη; thus, the relations between ms and mη eqs 16 and 17 formally correspond to a negative k. We have no clear explanation to this observation and whether hydrogen bonding in P2VP can play any role. We assume that this inconsistency can be ascribed to the difference in the configurational entropy of oligomeric system and the total configurational entropy of flexible chains that we neglect as described above. However, it might be relevant in the case of P2VP, and in this case we should consider eqs 16 and 17 as relations between the segmental fragility ms(M) and the oligomeric fragility, i.e., segmental fragility at small MW, ms(M0) instead of mη, where M0 is the mass of the monomer. Taking for P2VP ms(M0) ∼ 65, the best fit (Figure 4B) gives k∞ = 0.044 ± 0.003.

(13)

The fragility of segmental relaxation can be expressed as usual via Tg and T0s: m0 ms = 1 − T0s/Tg (14) where m0 = log τα(Tg)/τ0 ∼ 17 is the minimal fragility. We emphasize that similar expression is not practical for fragility of polymer viscosity because the latter does not have a universal value at Tg. Using the relations T0η/T0s = 1 − k and Tg/T0s = ms/ (ms − m0), one can rewrite eq 13 as ms (m0 + k(ms − m0))2 = mη m0 2(1 − k)

(15)

Now with the help of eq 15, we can write a relation between the segmental and viscosity fragilities: ÄÅ ÉÑ 2kmη 4kmη ÑÑÑ m0 2(1 − k) ÅÅÅÅ ÑÑ ÅÅ1 − − 1− ms = Ñ m0 m0 ÑÑÑ 2mηk 2 ÅÅÅÅ (16) ÑÖ Ç

V. SUMMARY To summarize, the presented analysis reveals strongly different scaling of the temperature dependence for segmental relaxation and viscosity in polymers with different molecular weights. In agreement with earlier reports,34−37 our analysis shows that

At small k, in linear approximation in k it can be represented as ms ≈ mη(1 + k(2mη /m0 − 1))

(18)

(17) F

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Macromolecules segmental relaxation time as a function of T − Tg falls on a master curve in a more or less molecular weight independent manner. This leads to an increase in steepness of the temperature dependence of τα (fragility) with molecular weight. Viscosity, on the other hand, falls on a master curve when plotted vs Tg/T. We speculate that the observed difference in temperature scaling maybe related to a nonergodic behavior of the polymers on segmental relaxation time scale, while ergodicity is restored on the global molecular (chain) relaxation time scale. As the result, the configurational entropy controlling segmental relaxation decreases upon cooling much faster than the total configurational entropy that controls molecular scale relaxation. Moreover, the fraction of configurational entropy not accessible on time scale of segmental relaxation increases with chain length, leading to the observed increase of segmental fragility with MW. This explains the very high segmental fragility in many polymers that is not observed in small molecules, as was already emphasized in ref 28. Surprisingly, for a given polymer η(T) scales vs Tg/T even in absolute values up to a certain molecular weight. Most puzzling, the viscosity at Tg does not show any appreciable molecular weight dependence up to MW ≈ 30 000 g/mol for both PS and P2VP. Such behavior is not predicted by any theory of polymer viscoelasticity. A possible explanation may be that segmental dynamics dominate the behavior of viscosity at temperatures close to Tg up to rather high MW.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (A.P.S.). ORCID

Vladimir N. Novikov: 0000-0002-1658-6638 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. Y. Wang (ORNL) for the help with viscosity measurements and useful discussions. This work was supported by the NSF Polymer program (Grant DMR-1408811).

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DOI: 10.1021/acs.macromol.8b00454 Macromolecules XXXX, XXX, XXX−XXX