Dynamical Theory of Segmental Relaxation and Emergent Elasticity in

Article pubs.acs.org/Macromolecules

Dynamical Theory of Segmental Relaxation and Emergent Elasticity in Supercooled Polymer Melts Stephen Mirigian and Kenneth S. Schweizer* Departments of Materials Science, Chemical and Biomolecular Engineering, Chemistry and Frederick Seitz Materials Research Laboratory, University of Illinois, 1304 West Green Street, Urbana, Illinois 61801, United States ABSTRACT: We generalize the force-level Elastically Collective Nonlinear Langevin Equation theory of supercooled molecular liquid dynamics to polymer melts based on mapping chains to disconnected and noninterpenetrating Kuhnsized spheres. This allows first-principles, no adjustable parameter calculations to be performed for the temperature-dependent mean segmental relaxation time of chemically diverse van der Waals polymers over a wide range of molecular weights. Despite the simplicity of the mapping, the theory does a good job of a priori predicting the glass transition temperature (Tg), the dynamic fragility, and full temperature dependence of the α-relaxation time for some high molecular weight polymers and the chain length dependence of Tg as the consequence of the molecular weight dependence of backbone stiffness. The minimalist model does not capture the unusually low and high fragilities of certain long chain polymers which are not typical of van der Waals molecular liquids. This seems likely due to the simple coarse graining adopted which ignores longer range chain connectivity and nonuniversal factors on the sub-Kuhn length scale. Elasticity, not of an entropic single chain origin, emerges in deeply supercooled polymer liquids due to transient segmental localization and is studied at the microscopic stress-tensor level. Calculations of the frequency-dependent dynamic storage modulus close to Tg appear to be qualitatively consistent with recent measurements. supercooled “regimes”, each separated by characteristic crossover times and temperatures. The theory predicts a growing length scale upon cooling, but no divergences below jamming nor above zero Kelvin. Connections to thermodynamics are a consequence of dynamics and not an a priori assumption. Quantitative comparisons to experiments and simulations on the benchmark hard sphere fluid and colloidal suspension systems reveal good agreement over ∼5 orders of magnitude of relaxation time.16 The theory is rendered both predictive and computationally tractable for thermal liquids by coarse graining a (rigid) molecule into an effective hard sphere. The volume fraction of the corresponding hard sphere fluid is chosen to exactly reproduce the thermal density fluctuation amplitude of the liquid, as encoded in its equation-of-state (EOS).15,17 This mapping results in a “quasi-universal” description which has been quantitatively confronted with experiments on molecular liquids with significant success.17 It also provides a theoretical basis for the remarkable experimentally established universalities of vdW molecular liquids.18,19 The goal of the present paper is to take the first step in extending ECNLE theory to treat segmental relaxation and glassy elasticity in polymer liquids. We adopt a minimalist model by coarse graining the polymer at the Kuhn scale and

I. INTRODUCTION The cooling of glass-forming liquids leads to a spectacular increase of the structural relaxation time and viscosity by 14 or more orders of magnitude.1 Many conceptually distinct models and theories exist for slow activated dynamics,2−6 which are largely phenomenological and crafted for particular temperature regimes.1,7−9 Their confrontation with experiment is almost always carried out using multiple fit parameters, which strongly limits the ability to falsify them as well as their predictive power. This situation even applies to arguably the simplest class of glass-forming liquids composed of nonpolar, rigid, van der Waals (vdW) molecules. While at first glance polymers bring a host of additional complications, their phenomenology at zeroth order is not atypical.1,10 We believe definitive progress requires the creation of a dynamical theory formulated at the level of forces which is devoid of adjustable parameters. We have pursued this goal recently based on extending the microscopic Nonlinear Langevin Equation (NLE) theory11−14 of local uncooperative hopping in spherical particle fluids to include collective motion associated with the long-range elastic distortion required to accommodate the cage-scale irreversible rearrangement. The basic excitation in this “Elastically Collective NLE” (ECNLE) theory15−17 is of a mixed local−nonlocal spatial nature and involves a (coupled) two barrier description. The latter feature is the key to the rich dynamic behavior predicted, which covers over 14 orders of magnitude in relaxation time and encompasses the apparent Arrhenius, crossover, and deeply © XXXX American Chemical Society

Received: October 30, 2014 Revised: January 19, 2015

A

DOI: 10.1021/ma5022083 Macromolecules XXXX, XXX, XXX−XXX

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Table 1. Equilibrium Parameters Used as Input to ECNLE Theory (Symbols Defined in the Text) with Resulting Theoretically Calculated Dynamical Quantities (Th) and Their Experimental Analogues (Expt) polymer PBD PVME PDMS PIB PE PP PSF PS PVAC PC PVC PMMA

Ns 5.6 14.2 12.6 13.6 7.3 9 32.7 38.4 22.8 30.2 15 31.9

d (Å)70 4.5 8.5 8.2 8.9 7.5 8.0 10.7 11.6 9.4 10.4 7.4 10.0

CN70 5.6 7.1 6.3 6.8 7.3 6 n/a34 9.6 7.6 n/a34 7.7 9.1

A29 0.486 0.456 0.675 0.627 0.919 0.858 0.558 0.62 0.762 0.674 1.097 0.898

B (K)29 1205 946 1057 1353 1427 1332 1343 1297 1285 1280 1684 1460

TA (K) (th)

20

407 470 455 611 462 474 849 829 646 735 645 754

175, 267,12 145,20 20127 23727 26327 459,27 371,20 31127 42327 35427 36727

171 24827 15030

46312 37527

m≡

Tg (K) (th)

m (expt)

m (th)

85, 107 7527 100,27 79,32 115,20 8532 4627,28 4627 137,27 12231 14127 139,27 116,32 143,28 97,33 12120 95,27 132,32 15028 19127 103,27 145,28 11532

80 81 84 85 86 86 87 90 90 90 94 96

20

174 212 214 285 218 225 420 425 327 374 336 408

27

d(log τα(T )) d(Tg /T )

Tg

(2)

There is some evidence fragility correlates well with the ratio of the empirically determined high-temperature apparent “Arrhenius” regime activation energy to the glass temperature, EA/kTg. This ratio varies from ∼4 to 20 for polymers28 versus being approximately constant (∼10) for vdW molecules.18 Why this might be so is unclear. In going between the short and long chain limits, the magnitude of dynamical property changes with molecular weight (e.g., Tg, m) and the chain length at which the asymptotic limit is achieved are very polymer-specific. For example, relative to oligomer behavior the fragility can strongly decrease (e.g., PIB), remain roughly the same (e.g., polydimethylsilioxane (PDMS), polyisoprene (PIP)30,41), or strongly increase (e.g., PS, PMMA, polypropylene (PP)20,42) with molecular weight. This suggests it is unlikely that the poorly defined “chain end free volume” effect43 is the fundamental explanation. It does appear that the magnitude of the chain length dependences of Tg and fragility are correlated. For decades it has been suggested that there is a polymerspecific crossover temperature, roughly ∼1.2Tg, below which extra elasticity emerges in the cold melt.44−47 This contribution to the storage modulus occurs at relatively low laboratory frequencies, is larger than the rubbery plateau modulus, and is of a different physical origin; it is of significant practical importance in polymer processing. This problem has been incisively revisited recently by Wang and co-workers.48 Its origin remains largely mysterious, but a connection to the emergent glassy rigidity on the segmental scale in the deeply supercooled regime seems a reasonable hypothesis. In this article we extend ECNLE theory to vdW polymers and apply it to study the issues outlined above. Section II reviews the physical ideas and mapping of the theory for hard spheres and molecules. The extension to polymers is presented in section III. Numerical calculations for the segmental relaxation time of polymers over a wide range of chemistry, time, and temperature are presented in section IV, including the

M0 kBNavcCN 2lbb3

27

sequences of chain connectivity may also be relevant, including coil interpenetration. Attempts to empirically correlate polymer fragility with Tg are not convincing.27 This seems unsurprising given fragility is a ratio of a temperature derivative to the glass temperature which both vary with chemistry

then disconnecting the chain, resulting in a liquid of (noninterpenetrating) spheres. This mapping allows dynamical predictions to be made with no adjustable parameters. The question then becomes, how much of the rich behavior of glassy polymer dynamics can be captured? In order to frame the problem, we first summarize the distinctive experimental behaviors of vdW polymer liquids of diverse chemistry without long side chains. For low molecular weight polymers,20−23 the values of Tg and dynamic fragility (m ∼ 70−80) are comparable to those of most small molecule liquids, even for the notoriously atypical polymer polyisobutylene (PIB).21 Reasons for this could include reduced conformational degrees of freedom, low backbone stiffness, and little interchain interpenetration. The latter is quantified by the “self-concentration”, which on the Kuhn scale (lK = CNlbb, where CN is the characteristic ratio and lbb is a mean backbone bond length) is given by24 ϕs ≡

Tg (K) (expt)

(1)

where M0 is the monomer mass and kB, NA, and c are the Boltzmann constant, Avogadro’s number, and the mass density, respectively. Typical values of ϕs for long chains24 are 0.45 for polybutadiene (PBD), 0.25 for poly(vinyl methyl ether) (PVME), polystyrene (PS), and poly(methyl methacrylate) (PMMA), and ∼0.1 for polycarbonate (PC). However, since the characteristic ratio of short chains of polymers such as PS and PMMA is roughly a factor of 2 smaller than the long chain limit,25,26 self-concentrations become of order unity implying little interpenetration on the Kuhn length scale. For long chains there are large differences between polymers in both vitrification temperature and fragility, as shown in Table 1. Tg can vary from ∼150 to 500 K, and fragilities cover a remarkably wide range27,28 of m ∼ 23−200. This is in contrast to vdW molecules, where, although Tg varies significantly, fragilities typically do not (m ∼ 70−80) although there are exceptions. Interestingly, m ∼ 70−80 also describes oligomers and is consistent with the quasi-universality discussed by Rossler18 and predicted by ECNLE theory.17 Physical effects that have been discussed in the literature which might matter for the dynamic fragility and temperature dependence of the α-relaxation include complex monomer shape and packing frustration,35−38 backbone stiffness,37,39 relative stiffness of side groups versus the backbone,32,37 symmetry of side-group substitution,37,40 tacticity, and intrachain torsional barriers.37,38 More general dynamical conB


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Figure 1. (a) Schematic of the fundamental relaxation event for spheres which involves a local, large amplitude cage-scale motion leading to irreversible rearrangement, and a nonlocal, spatially long-range collective elastic motion to accommodate the local rearrangement. (b) Polymers are treated as disconnected Kuhn segments composed of Ns ∝ CN rigidly moving units mapped to an effective hard sphere.

Fdyn(r), which quantifies the effective force on a moving tagged particle due to the surrounding fluid:

role of molecular weight. Section V studies segmental elasticity in cold liquids, and the paper concludes with a summary and discussion of open issues in section VI.

βFdyn(r ) = −3 ln(r ) +

II. ECNLE THEORY FOR HARD-SPHERE FLUIDS AND THERMAL MOLECULAR LIQUIDS ECNLE theory is based on coupled local cage and long-range collective elastic barriers and was initially developed for hardsphere fluids as described in great technical detail.16 The highly viscous regime is controlled by the collective barrier, which reflects the penalty for a small local volume creation in the surrounding liquid that is necessary to accommodate cage scale rearrangement. The collective barrier depends on the glassy elastic response of the liquid and reflects longer range continuum-like particle displacements. It is determined by both the dynamic (plateau) shear modulus (or inverse square of the transient localization length) and the hop jump length which both grow with densification (or cooling in a thermal liquid). The cage scale barrier computed using NLE theory depends on local packing. For hard spheres (diameter, d, and mass, M), the original NLE theory of local hopping11 is based on an overdamped equation-of-motion for the scalar (angularly averaged) displacement of a tagged particle from its initial position, r(t) ∂Fdyn(r(t )) d r (t ) 0 = −ζs + δf (t ) − dt ∂r(t )

g (d )d 24ϕ

πM ⎡ d3 ⎢1 + 36πϕ kBT ⎣

∫0

∞

dk k 2

2

⎤ ⎡ k 2r 2 (1 + S −1(k))⎥ × exp⎢ − ⎦ ⎣ 6

(5)

where β ≡ (kBT)−1 and ρC(k) = 1 − S−1(k). The emergence of a local minimum and barrier in Fdyn(r) occurs at ϕ ≡ ϕA ≈ 0.43. This does not signal any divergence of the relaxation time or literal arrest, but rather the emergence of a barrier and the beginning of the potential importance of transient localization and activated motion. In practice,11 barriers are essentially negligible (thermal energy or smaller) up to ϕ ≈ 0.5. Above ϕA, the dynamic free energy is characterized by three important length scales: the transient localization length rloc, barrier location rB, and “jump distance” Δr ≡ rB − rloc. The three key energy scales are the well (barrier) curvature K0 (KB) in units of kBT/d2, and barrier height, FB(ϕ), in units of kBT. The localized state determines the glassy shear modulus as16,52 G=

kBT 60π

2

∫0

∞

⎛ k 2r 2 ⎞ ⎡ d ⎤2 dk ⎢k 2 ln(S(k))⎥ exp⎜ − loc ⎟ ⎣ dk ⎦ ⎝ 3S(k) ⎠

(6)

Equation 6 is an appropriate measure of rigidity on a time scale before activated events occur. Analytic analysis of hard-sphere NLE theory allows the derivation of many connections between dimensionless features of the dynamic free energy:52

(3)

where r(t = 0) = 0 and ζs is the short time friction constant associated with “renormalized” binary collision physics which defines the short relaxation time:49,50 τs =

⃗

∫ (2dπk)3 ρ1C+(Sk−)S1((kk))

d 2K 0 =

(S(k) − 1)2 ⎤ ⎥ S(k) + b(k) ⎦

3d 2 Gd3 ∝ ∝ (βFB)2 ∝ ν∞2 kBT rloc 2

(7)

Here, ν∞ = 96π ϕg (d) quantifies the strength of the effective mean-square force caging a particle. Importantly, embedded in eq 7 are connections between the short and long time dynamics, local elasticity, α time, and barrier. The most recent confrontation of NLE theory53,54 for hard spheres against colloidal experiments and simulations suggest it captures rather well roughly the first ∼3 decades of slow relaxation (crossover regime) up to ϕ ≈ 0.57−0.58, where FB ∼ 7−8 kBT. However, at higher ϕ the theory increasingly underpredicts the relaxation time and is not nearly fragile enough, presumably because it misses collective physics. These are approximately included in the ECNLE approach based on the physical picture that the α-relaxation event in real space 2

(4)

Here, the packing fraction is ϕ ≡ ρπd3/6 where ρ is the number density, the contact value of the pair correlation function (computed using Percus−Yevick (PY) theory51) is g(d) = (1 + ϕ/2)/(1 − ϕ)2, S(k) is the static collective structure factor, b(k) = [1 − j0(kd) + j2(kd)]−1, and the thermal noise satisfies ⟨δf(0)δf(t)⟩ = 2kBTζsδ(t). The second term in the brackets of eq 4 is meant to describe, in a non-self-consistent manner, the initial weak effects of caging which are dominant in the normal fluid regime where activated hopping is either not present or is a small correction. The key quantity is the dynamic free energy, C

2


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Macromolecules involves relatively large amplitude (Δr ≈ 0.25d − 0.4d) particle hopping on the cage scale, and accommodation of this motion requires elastic distortion of the increasingly rigid surroundings.15,16 Thus, the activation event and total barrier consists of two contributions: a core or cage region of diameter 2rcage ≈ 3d where particles undergo nonvibrational hopping displacements plus a harmonic strain field that dresses the local excitation (Figure 1a). The collective elastic barrier is explicitly computed based on microscopically generalizing and quantifying the phenomenological “shoving model” idea of Dyre.55,56 Specifically, a selfconsistent dynamic mean field picture of incoherent single particle hopping inside the cage is invoked which requires a local spherically symmetric expansion (dilation) of the surrounding matrix outside the cage16 by an amount

Figure 2. ECNLE theory local barrier (blue circles), elastic barrier (red squares), and total barrier (yellow diamonds) in units of the thermal energy as a function of volume fraction for the hard sphere theory.16 Inset: transient localization length (blue circles) rloc, hop distance Δr (red squares), and effective cage expansion length Δreff (yellow diamonds, right axis) in units of the segment diameter, d.

2 2 ⎡ ⎤ rcageΔr 3 Δr 4 ⎥ 3 ⎢ rcage Δr Δreff = + − 192 3072 ⎥⎦ rcage 3 ⎢⎣ 32

≈

2 3 (rB − rloc) ≪ Δr 32 rcage

(8)

As seen from the calculations in Figure 2, the collective barrier is essentially negligible up to ϕ ≈ 0.55 and begins to increase faster than the local barrier at ϕ′ ≈ 0.57−0.58, thereby defining a crossover volume fraction as when the rate of change (derivatives) of the two barriers are equal. This smooth crossover occurs when Ftotal ∼ 10kBT, corresponding to τα ≈ 104τs ≈ 10−8 s in thermal liquids based on τs ≈ ps. At higher volume fractions the collective barrier grows rapidly and dominates the total barrier growth rate. However, its magnitude does not exceed the local barrier until close to kinetic arrest which occurs at ϕ ∼ 0.61 when Ftot ∼ 30 kBT. Though the two barriers involve physically distinct motions (see Figure 1), we again emphasize that they are not independent since they arise from a single underlying dynamic free energy and physical process. In the deeply supercooled regime, when Ftotal ≥ 10kBT, they are precisely related mathematically as Ftotal ∝ FB(1 + bFB), where b ≈ 0.15.16 A generic mean α time is then calculated as

This cage expansion is smaller than the transient localization length, implying a continuum harmonic description of the elastic distortion field, u(r), is valid16 ⎛ rcage ⎞2 ⎟ , u(R ) = Δreff ⎜ ⎝ R ⎠

R > rcage

(9)

where R is the distance from the cage center. The dynamic free energy cost due to the harmonic motion of a particle located a distance R from the cage center is thus ΔF(R) ≈ K0u2(R)/2. The total elastic dynamic free energy cost follows by summing the contributions of all particles outside the cage region βFelastic = 4π

⎛ rcage ⎞3 ⎟ K dR R2ρg (R )ΔF(R ) ≈ 12ϕΔreff 2⎜ ⎝ d ⎠ 0 cage

∫r

∞

(10) 2

where K0 = 3kBT/rloc . The total activation barrier is the sum of the local cage rearrangement and collective barrier contributions, Ftotal = FB(ϕ) + Felastic(ϕ). The physical reason is that we view the α process as a single event, correlated in space and time, where in order for the local, large amplitude, irreversible, activated hop to be possible, an expansive elastic cage fluctuation is required. The two barriers are thus treated on equal footing with no causal relationship implied. As discussed previously,16,17 the total barrier initially increases sufficiently slowly with volume fraction that it does not become the primary determinant of the α-relaxation time until ϕ ≈ 0.53. One might ask whether there should be an upper cutoff of the integral in eq 10 since on the α time scale the material outside the irreversibly rearranging region cannot be treated as an elastic solid to arbitrarily large distances. This point has been discussed in depth by Dyre,57,58 who argues the relevant “solidity length” is of order a micron near Tg for real liquids. Alternatively, but in the same physical spirit, we estimate the relevant length scale over which the response will be elastic or glassy as l ∼ cτα where c is the speed of sound. For a typical c ∼ 1000 m/s, this length is l ∼ 100 nm when τα = 10−10 s and obviously grows strongly with further cooling. Either of these cutoff length estimates are so large that effectively no error is incurred by extending the upper limit in eq 10 to infinity.

τα =1+ τs

⎛ F + Felastic ⎞ 2π exp⎜ B ⎟ K 0KB ⎝ kBT ⎠

(11)

Thus, all our predictions are controlled by a single barrier which has coupled local and long-range aspects. We remind the reader our approach does not predict any divergent relaxation time below the strictly incompressible random close packing state. Liquids of rigid molecules are mapped17 to an effective hard sphere fluid based on matching the thermodynamic-statedependent amplitude of thermal density fluctuations, i.e., the dimensionless compressibility51 ⎛ ∂βP ⎞−1 HS S0expt = ρkBTκT = ⎜ ⎟ ≡ S0 (ϕeff (T )) ⎝ ∂ρ ⎠ =

(1 − ϕeff (T ))4 (1 + 2ϕeff (T ))2

(12)

where the last equality follows from using PY theory. This mapping defines a system and thermodynamic-state-dependent effective hard sphere volume fraction, ϕeff(T), which grows with cooling and encodes, in an average sense, the thermodynamic consequences of repulsive and attractive forces and molecular shape. The mapping results in a quasi-universal picture that D


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theory are literally determined by long wavelength density fluctuations. Indeed, as discussed in depth previously,52 all the features of the dynamic free energy are dominated by quantities determined by local packing correlations. Independent of the dynamic theory, the above mapping predicts a simple approximate relation for Tg of chemically homologous molecules (same A and B). Since S0(ϕg ≈ 0.615) ≈ 0.0044, and to a good approximation A(NsS0(ϕg))1/2 ≪ 1, one obtains Tg ∝ B√Ns. Thus, Tg scales essentially as the square root of the molecular mass at fixed chemistry, as experimentally established for several homologous series including low molecular weight polymers.59 This relationship breaks down for polymers at high enough molecular weights. As we argue below, this occurs because at high molecular weight the size of the rigidly moving dynamical subunit (identified with the Kuhn length) saturates, so that the relevant Ns eventually becomes chain length independent. ECNLE theory also predicts multiple theoretically welldefined characteristic volume fractions and their corresponding time scales.16,17 The initial crossover from the normal liquid to one where barriers are nonzero occurs corresponds to the ̈ (naive) mode coupling theory transition at11 ϕ = ϕA ≈ 0.43. From eq 14, this defines a temperature, TA, where SHS 0 (ϕA) = 0.031, which is far above Tg and often experimentally inaccessible. Below TA the barrier initially grows in very slowly and the time scale associated with activated relaxation is faster than the most local process time scale determined by effective or renormalized binary collisions. Therefore, a practical onset for the activated regime is when the hopping time scale equals the renormalized binary collision time scale of eq 4, defining the crossover temperature Tx via τs(Tx) ≡ τhop(Tx), which is predicted to occur at a τα ≈ 10−10 s for molecular liquids. Two additional, theoretically well-defined crossover temperatures16,17 which indicate the change from a local to a collective hopping process are when Felastic(T*) ≡ FB (T*) and (d/dT)Felastic(T′) ≡ (d/dT)FB(T′). The temperature T′ indicates where the growth rate of the collective barrier first exceeds that of the local barrier and occurs for molecular liquids at τα ≈ 10−7±1 s; this temperature correlates with diverse empirical estimates of the dynamical crossover temperature.17,60 The temperature T* occurs at a lower temperature where τα ≈ 0.01 s (much closer to the glass transition at τα(Tg) ≡ 100 s) and correlates well with a more recently proposed empirical crossover temperature, TR.18 Numerical calculations of the dynamic fragilities of molecular liquids based on ECNLE theory are well described by the (derived) analytic relation17

connects thermodynamics, structure, and dynamics, where the relaxation of all liquids follow from that of a hard sphere fluid to within the nonuniversal prefactor of eq 11 associated with short time dynamics. The hard sphere mapping is expected (and explicitly shown17) to work best for vdW liquids and to be less accurate as chemical/structural complexity is introduced, e.g., via hydrogen bonding. Coupled with the ECNLE theory of hard sphere fluids, the mapping provides a zero adjustable parameter, quantitative theory for the α-relaxation of real molecular liquids over 14 or more decades in time. The theory has no singularities above absolute zero. In practice, ECNLE theory is implemented and rendered predictive for real materials by using the experimental liquid EOS. It is physically instructive, and computationally convenient, to adopt an accurate analytic expression that can be rationalized from the vdW EOS:17 1 S0expt

≈

a B′ −2≡ − A′ b T

(13)

The parameters B′ and A′ are the molecular level liquid cohesion and an entropic packing contributions, respectively. Solely to reveal the distinct dynamical consequences of molecular size versus “intrinsic” chemical effects, a molecule is parsed into Ns rigidly bonded “interaction sites” (e.g., a site in benzene is a CH group). One then has Ssite = ρskBTκT ≡ 0 NsρkBTκT = NsSexpt 0 , which implies the site level dimensionless compressibility is 1 S0sites(T )

=

⎛ B⎞ Ns ⎜ −A + ⎟ ⎝ T⎠

(14)

where A = A′/Ns and B = B′/Ns. Using eqs 12−14 yields S0expt(T )

ϕeff (T ; A , B , Ns) = 1 + −

S0expt(T ) + 3 S0expt(T )

(15)

The 1-to-1 mapping between temperature and ϕeff (or S0) of the reference hard sphere fluid then implies T (ϕeff ) =

B A+

1 NsS0HS(ϕeff )

(16)

The cohesion parameter, B, sets the energy scale for Tg but cancels out in ratios, a feature that has many implications.17 Values of A and B for real polymers are given in Table 1. Polymer acronyms were defined earlier in the text except for polysulfone (PSF), polyvinylacetate (PVAC), polycarbonate (PC), polyvinylchloride (PVC), and polyethylene (PE). As a parenthetical remark, eq 16 and other general analytic relations derived previously and cited below are not sensitive to the precise equilibrium theory adopted to compute the static structure factor, S(k). Recent studies53,54 have shown even for the α-relaxation time that improving the PY structural input has very modest effects when high barriers control dynamics. We expect that the sensitivity to structural input is further suppressed due to our mapping procedure which encodes the experimental behavior of real liquid density fluctuations via an effective hard sphere volume fraction that reproduces the long wavelength behavior. We emphasize that although the mapping between the temperature in a thermal liquid and an effective hard core volume fraction proceeds via a thermodynamic property, this does not mean the dynamical predictions of the

m≡

d(log τα(T )) d(Tg /T )

∝ 1 + cA Ns Tg

(17)

where c is a numerical factor. The attraction strength parameter, B, does not enter the fragility which is controlled by the combined parameter, A√Ns. Given to leading order Tg ∝ B√Ns, a rough connection between Tg and fragility is A m ∝ 1 + c Tg B

(18)

However, the apparent linear correlation between fragility and glass transition temperature is largely an illusion since the ratio A/B is material-specific. This provides a theoretical basis for why a Tg−m “correlation” is quite poor in practice.27 E


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elementary, lightly coarse-grained dynamical unit with its own “segmental friction constant” of presumed local physical origin. One should view our adopted model as taking this picture seriously in order to do microscopic statistical mechanics. For polymers with simple backbones, the number of interaction sites in a Kuhn segment is given by

We emphasize that in our mapping only translational, not rotational, motions enter and the molecules are presumed rigid. We are in essence aiming to describe an “effective” hop or irreversible event at a lightly coarse grained, center-of-mass level. We are not saying the α-relaxation process literally only involves translational hops on the atomistic scale. Obviously there must be some dynamical cooperation of translational and rotational degrees of freedom in a dense cold liquid. The few published simulations of molecular liquids in the dynamic crossover regime61−65 do find complex activated features, though the role of translation versus rotation in real molecular liquids is no doubt nonuniversal and remains to be clarified. Given our focus on single particle hopping, we have also ignored the phenomenon of decoupling of the temperature dependence of self-diffusion and rotational relaxation or viscosity (or other collective properties).66,67 We view this decoupling effect as of second-order importance in the sense that it corresponds to a ∼2 orders of magnitude difference over a temperature range where the mean α time grows by 10 or more decades. Moreover, as we previously discussed,15−17 many experimental measurements probing diverse single- and many-particle correlation functions, which couple to translational and rotational motion very differently, nevertheless find relaxation times that vary with temperature in an almost identical manner and are often even quantitatively close. It is this experimental fact that underlies the widely presumed notion that a quasi-universal theory can exist, at least in the deeply supercooled regime. Thus, the thesis underlying our replacement of chemically complex molecules by effective hard spheres is that in reality there is sufficient self-averaging of the details of molecular trajectories to render our mapping and effective translational motion model useful. It is for these reasons that we believe that despite its simplifications, ECNLE theory can capture quite well the α-relaxation in real vdW molecular liquids over 14 decades.17 In a practical sense, without some simplifying idea, every system would require a distinct microscopic dynamical theory that depends on the precise repulsive and attractive interactions, molecular shape, packing, and other details. It is only by adopting our present “quasi-universal” picture that the problem of describing activated hopping for real liquids in a predictive, no adjustable parameter manner becomes tractable.

Ns =

n lk ns ≈ CN s lbb nbb nbb

(19)

where lbb is the length of a backbone bond, lk is the length of the Kuhn segment, ns is the number of sites per monomer, and nbb is the number of bonds per monomer. Note that, in contrast to molecules, Ns can take noninteger values since the Kuhn segment is defined statistically. In the second approximate equality we take CN ≈ lk/lbb which is the polymer and chain length dependent characteristic ratio, ignoring the projection along the backbone direction.70 Values of Ns are given for a variety of polymers in Table 1. Importantly, changing Ns directly impacts the temperature mapping via eq 16 and varies significantly with molecular weight for many short chain polymers via the characteristic ratio. It is determined by both the monomer bulkiness (mass) and backbone stiffness. The space-filling diameter of the “Kuhn sphere” follows directly as π 3 CN d ≡ Vmonomer 6 nbb

(20)

where the monomer volume is calculated from the melt mass density. With this mapping, and the known equilibrium chain and melt EOS properties as input, ECNLE theory can make dynamical predictions with no adjustable parameters. Having adopted such a simple mapping, one can ask when it might be most reasonable and why? Of course, answering this question is not an easy task, but we can offer some physically intuitive arguments. Likely, a space-filling Kuhn sphere model is most accurate at low molecular weights for multiple interrelated reasons: (i) the polymer is conformationally more rigid than for long chains and hence more “hard-sphere-like”; (ii) the characteristic ratio is sufficiently small that most low-MW polymers have high self-concentrations which suppresses a defining feature of high polymers, coil interpenetration; (iii) the interchain packing structure (pair correlation function) will be closer to that of the mapped hard sphere fluid due to the absence or small importance of the long-range correlation hole; and (iv) explicit dynamical consequences of chain connectivity must be smaller. Moreover, as discussed in the Introduction, at low molecular weights the glassy dynamics of polymer melts is known experimentally to be very similar to that of small molecules, a fact that is consistent with our perspective. Given the above arguments, we expect that for long chain polymers the Kuhn sphere mapping will be less accurate. However, it could still be useful for a more general physical reason: “dynamic self-averaging”. By this we mean, for example, that simulations of PE71,72 and PBD73 melts find a fast partial conformational relaxation via dihedral rotations that tend to “sphericalize” the segment mass distribution without substantive translational motions, a process likely related to the generic beta or “primitive α”74 relaxation. Under cold enough conditions where the cooperative α process is much slower, the picture of a roughly spherical, space-filling Kuhn segment relaxing via translational motion may be reasonable. Our demonstration below that the theory can a priori predict well the chain length dependence of Tg based on our adopted model

III. EXTENSION TO POLYMERS Polymers have all the complexities of molecules, plus additional ones such as rotational isomerism and long-range chain connectivity. The relevant coarse-graining scale for glassy segmental dynamics in polymeric liquids is a priori unclear.39 Many workers have adopted the statistically defined Kuhn length, which appears to be consistent with the dynamical “segment” deduced from viscoelastic measurements,68 and has also been employed with considerable success in the selfconcentration model for blends.24 We employ this choice and adopt the minimalist model of replacing the polymer by a liquid of noninterpenetrating hard spheres of mass defined by the Kuhn length (Figure 1b). Obviously, the disconnection of a polymer into Kuhn spheres which only undergo translational motion independent of other parts of the connected chain is a strong approximation that cannot be literally true. However, it does allow us to make no adjustable parameter predictions which can be quantitatively confronted with experiment. Moreover, all classic polymer physics dynamical theories69 build on the concept of an F


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contrast to the older NLE polymer work which, though physically motivated, employed an adjustable parameter, ac, meant to mimic how chain stiffness amplified the bare single site local barrier. (ii) The prior polymer NLE theory adopted an analytically simple, but crude, Gaussian thread model for the required structural correlations (e.g., S(k)) which ignored cage scale packing correlations. This is in strong contrast to our present work built on the effective hard sphere model. Crucially, it is this feature that allows us to now construct a unified dynamical theory of colloids,16 molecules,17 and polymers. Without the adoption of this underlying “quasiuniversal” picture, every system would require a distinct dynamical theory, each depending on intractable microscopic details. (iii) The barrier of the prior polymer NLE approach was entirely local, in qualitative contrast to the present ECNLE theory which includes both local and long-range collective elastic effects. This difference has already been shown for hard sphere fluids and molecular liquids to have enormous consequences.16,17

provides support for the usefulness of the disconnection simplification. However, as discussed in the Introduction, the experimental reality of a highly nonuniversal relationship between the short and long chain limiting behaviors of polymer melts clearly implies that chemically specific features must play a complex role. Of course, the minimalist mapping must always incur errors, which are a priori not easy to precisely anticipate. At high temperatures where the α and β processes merge, and outside the deeply supercooled regime, local chemical details likely matter more and our coarse-graining procedure, which ignores (among other things) conformational transitions and rotational isomerism, will presumably be less accurate. Additionally, the short time relaxation process (τs in eq 4) is not purely translational as in hard spheres, a concern likely most important at relatively high temperatures. On the other hand, long chains interpenetrating on the Kuhn scale may be especially relevant at low temperatures when considering collective motion and fragility, since the amount of cage expansion required to accommodate a local hop in such circumstances likely depends on chemical details not present in the effective hard sphere description. Overall, one should be prepared for polymer-specific deviations between theory and experiment at both high and low temperatures. Our goal is to identify these by performing no adjustable parameter calculations with the mapping described above. As far as we know, a first-principles, predictive dynamical calculation of Tg and fragility has never been attempted for the polymer systems we consider here. We note that equilibrium lattice cluster theory can approximately compute configurational entropy for more chemically detailed (but still coarse grained) models, but as the authors state, it involves a plethora of system model parameters that limits full a priori predictability.36−38 The relevant parameters are often chosen by fitting the Tg and other dynamic properties of a specific system, and then predicting variations within a chemically homologous class of polymers (e.g., alkyl-substituted polyolefins) based on the phenomenological Adam−Gibbs model for the α time.9 Considerable successes have been achieved following this approach, including broad qualitative trends for how fragility depends on the absolute and relative stiffness of the backbone and side groups. However, our aim here is quite different from that described above. We hope to produce a zeroth-order path to a forcebased predictive calculation of dynamical properties for real polymeric liquids using no adjustable or fit parameters. Learning what our approach can capture and what it cannot will help establish what approximations are reasonable and where further work is needed. Inherent in this is, necessarily, the failure of the theory to describe some experimental features of some systems. Finally, we note that the present approach follows the prior polymer NLE theory work of Saltzman and Schweizer12,14 in both the adoption of the long wavelength density fluctuations (encoded in S0(T)) as the fundamental variable used to create a mapping from real polymers to a tractable simple model, and also the use of the NLE statistical mechanical idea of a dynamic free energy to compute the local barrier. However, there are three fundamental differences relative to the new ECNLE approach proposed in this paper. (i) Our present mapping is based on the a priori identification of the Kuhn length as the fundamental dynamical unit and its consistent equilibrium and dynamic treatment as an effective hard sphere. This is in

IV. SEGMENTAL RELAXATION We now apply the theory to specific polymers at 1 atm pressure. Our primary focus is the deeply supercooled regime, roughly when τα > 10−7 s. For many polymers, experiments at high temperatures where relaxation is faster are not feasible due to degradation or other experimental limitations. Some workers fit the supercooled regime data to the Vogel−Fulcher− Tammann (VFT) formula and extrapolate to higher temperature. This extrapolation may not be reliable given there can be empirical “high” and “low” temperature VFT regimes.76,77 The polymers we study are listed in Table 1 along with key parameters and theoretical results. We focus on vdW polymers, but within this category the systems are diverse (vinyl chains, dienes, symmetric vs asymmetric substituted, silicones). Three main topics are addressed: (i) Tg and fragility in the long chain limit, (ii) the full temperature dependence of the α time covering 8−17 orders of magnitude depending on the polymer system, and (iii) the molecular weight dependence of Tg and fragility. We also contrast our polymer calculations with that of small molecule liquids. Our overall goal is to establish what aspects of the polymer problem our dynamical theory plus simple mapping can ab initio describe and what aspects it cannot. A. Long Chain Limit: Tg and Fragility. Our results for Tg and fragility in the long chain limit for 12 polymers are given in Table 1. Although not as quantitatively good as our previous results17 for vdW molecules (∼10−20% deviations), overall the predictions are encouraging given the lack of any adjustable parameters or fitting and the simplicity of our coarse-grained description. Figure 3 shows a graphical comparison of theory and experiment. One sees deviations in both directions with no obvious correlations to tacticity, monomer chemistry, backbone stiffness, or self-concentration except, perhaps, that symmetrically substituted polymers (PIB and PDMS) show the largest deviations. The latter trend may indicate the importance of more regular packing on the sub-Kuhn length scale. The situation is quite different for fragility. We do note, as shown in Table 1, the often high experimental uncertainty of fragility. The theory predicts a rather weak dependence on monomer chemistry, with fragilities varying over the vdW molecule-like range of m ∼ 80−96, in contrast to the observed variation of m ∼ 46−191. To within the experimental uncertainties, the theory accurately predicts the fragility of G


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Figure 4. Mean segmental relaxation time (seconds) as a function of scaled inverse temperature for PS of molecular weights 106 (solid blue circles, monomeric), 1350 (open red circles), and 546 000 (hatched yellow circles) as calculated from the VFT fits to experimental data.20 Dashed lines of corresponding color are the theory results. Inset: OTP data78 (blue circles). Dashed red and yellow curves are VFT fits to experimental data for PS20 that have been extrapolated to faster times than where the actual measurements were taken. The dashed red curve is for a MW of 106 (monomeric), and the dashed yellow curve is for a high MW of 546 000. The solid blue curve is ECNLE theory17 for OTP.

Figure 3. Comparison of theoretical and experimental values (from Table 1) for Tg. The dashed black line represents perfect agreement.

PBD, PDMS, PMMA, and PVAC. However, it does not capture the very low (PIB, PE) and very high (PP, PC, PS, PSF) values. This is perhaps unsurprising given the subtle nature of fragility as a Tg-normalized temperature derivative and the “quasiuniversal” nature of our Kuhn segment mapping. The precise origin of the overall (but not always for specific polymers) poorer predictions for dynamic fragility are presently unclear, but they do not seem to correlate with differences in intramolecular conformational barriers, density fluctuation EOS parameters (A, B), or self-concentrations in any obvious way. Empirically, experiments find roughly28 m ∝ kTg/EA, where EA is the empirically extracted “high temperature” Arrhenius activation energy, a quantity that carries some ambiguity in determination. Values of kTg/EA have been reported to range from28 ∼ 0.052 to 0.22 for long chain polymers, specifically, 0.052 (PIB), 0.083 (PDMS), 0.093 (PE), 0.12 (PBD), 0.13 (PP), 0.17 (PVAC), 0.18 (PVC), and 0.22 (PMMA). In contrast, we find (consistent with measurements18 and our results17 for molecules) that the “quasi-universal” ECNLE theory predicts17 a narrow range of EA/kTg ≈ 0.097−0.12 for all polymers. The fragilities of low molecular weight polymers such as PBD, PIB, PS, and PMMA that fall in the range of m ∼ 70−80 are therefore well described by our theory. We emphasize that ECNLE theory predicts that the leading factors that determine Tg and fragility are very different (cohesive energy, B, versus the entropic packing factor, A, in eqs 14−18). In the absence of the collective elastic barrier, the local hopping NLE theory predicts very low fragilities, m ∼ 34, which establishes that collective physics is the leading contributor to fragility within our approach for both molecules and polymers. Presumably, the missing nonuniversal physics lies there. B. Temperature Dependence of the Segmental Relaxation Time. The mainframe of Figure 4 presents experimental data in the deeply supercooled regime along with our no adjustable parameter calculations for PS melts of three molecular weights in the temperature normalized Angell format. It is curious to note that despite the fact that the experimental fragility varies from m ∼ 80−120, there is not much visible difference in the various experimental curves (and even less in the theory) for the three chain lengths over the slowest 8−9 orders of magnitude of relaxation. Although the theory predicts a too low fragility for long chain PS, the overall temperature dependence is reasonable. Excellent agreement is found for the lowest molecular weight system.

The inset of Figure 4 presents PS data over a wider temperature range based on VFT fit extrapolations; data for oterphenyl (OTP) are also shown along with our theoretical prediction which is in excellent agreement with experiment.17 In the deeply supercooled regime, one sees very little difference between OTP and both short and long chain PS melts. Deviations seem largest at higher temperatures, where the longest chain system has a significantly longer α time. OTP and low-MW PS liquids agree over 11 orders of magnitude, down to an α time of ∼0.1 ns, consistent with their nearly identical value of fragility. Overall, it seems nonuniversal long chain effects are perhaps largest at high temperatures outside the deeply supercooled regime where collective effects are subdominant to more local physics. It is perhaps worth noting from the inset to Figure 4 that the prefactor associated with the VFT fit to the low MW PS data is unphysically small, ∼10−19 s. This seems to indicate a crossover to a different, high-temperature VFT law (outside the experimental data range), in analogy with some small molecule behavior. In contrast, the VFT prefactor time scale of long chain PS is ∼10−12 s, a physically plausible value. This may suggest that a single VFT fit may “work” over all temperatures for long chains, a qualitatively distinct behavior from small molecules. A single VFT fit working for all temperatures may imply the dynamical crossover is physically different, obscured, or “smeared out” in long chain melts, a feature not captured by our present theory. Figure 5 shows results for the low Tg and intermediate fragility PDMS melt at three molecular weights in the deeply supercooled regime. Note the weak dependence on chain length and almost complete absence of it in the theory. The calculations are in good agreement with experiment over 9 orders of magnitude down to 100 ns, and fragilities are reasonable (Table 1), though the accuracy decreases with increasing N. The mainframe of Figure 6 presents results for three chemically distinct long chain polymers of intermediate (PVAC, PBD) and high (PP) fragility where α time data H


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Overall, we conclude that disagreement between theory and experiment for the temperature dependence of the segmental relaxation time can occur at low temperature if the fragility is strongly under- or overpredicted, which presumably reflects issues associated with the Kuhn sphere mapping and/or collective elastic effects. On the other hand, at high temperatures the deviations likely have a different physical origin, presumably related to more local, sub-Kuhn scale dynamics and/or packing. C. Chain Length Dependence. The EOS parameters (A and B) in eq 14 have negligible dependence on molecular weight at 1 atm pressure. Thus, upon coarse graining to the Kuhn sphere scale, the sole origin of N-dependence in our model/theory enters via the number of rigidly moving sites, Ns, which for a specific polymer chemistry varies only via the backbone characteristic ratio in eq 19. The question we now explore is whether this aspect alone can account for the experimentally observed variation of polymer Tg and fragility with degree of polymerization. Such an idea for Tg has been suggested by Sokolov and co-workers,81 and indirectly by Novikov and Rossler.59 It conflicts with the phenomenological view that extra free volume at chains ends is the key. Figure 7 shows Tg results for PDMS, PMMA, and PS where the experimental variation of characteristic ratio (inset) is

Figure 5. PDMS segmental relaxation time (seconds) data from VFT fits20 at a MW of 311 (solid blue circles), 2490 (open red circles), and 232 000 (hatched yellow circles). Dashed curves of corresponding color are the theory results.

Figure 6. Experimental segmental relaxation time data versus scaled inverse temperature for PP (blue circles),31 PVAC (red squares),79 and PBD (yellow triangles)80 covering many orders of magnitude in time. Solid curves of corresponding color are the theory results. Inset: points are taken from VFT fits to experimental data for the relaxation time as a function of normalized inverse temperature of long chain PS20 (blue symbols), PDMS20 (red squares), and PIB21 (yellow diamonds). Curves of corresponding color are theory. Figure 7. Tg normalized by its long chain limiting value versus the logarithm of the polymer molecular weight for PS20 (blue circles), PMMA (red squares82), and PDMS (yellow diamonds30). Theory results are shown as curves of corresponding color. Inset: characteristic ratio for PS25 (blue circles), PMMA26 (red squares), and PDMS83 (yellow curve) as a function of the number of backbone bonds. The lines are a fit to the experimental data that facilitate interpolation to all molecular weights. The saturation of Tg correlates with the saturation of the characteristic ratio, CN.

exist over an extraordinary range of 14−17 orders of magnitude. Strong differences in the data are clear at very high temperatures well outside the deeply supercooled regime. The theoretical prediction for PVAC is rather good over the slowest 6−7 orders of magnitude, consistent with the accurately predicted fragility (see Table 1). Upon further cooling, the theory underpredicts the α time, ultimately by 2 orders of magnitude. For PP, the theory both under- and overpredicts the relaxation time depending on temperature. The latter is expected given the significantly smaller fragility calculated for PP. However, in an average sense, the level of agreement over 11 decades from ∼ns to 100 s is encouraging. For PBD, the theory predicts fragility rather well and is quite good over the slowest 8 orders of magnitude of relaxation. At higher temperatures, the theory again underpredicts the relaxation time, though agreement is recovered at ultrahigh temperatures where the α time is ∼ps. The inset of Figure 6 compares long chain data in the deeply supercooled regime of a very low (PIB), intermediate (PDMS), and high (PS) fragility melt. Obviously, the theory is very poor for PIB given its does not capture the observed low fragility. However, for PS and PDMS, rather good agreement is again found.

employed as input to our Kuhn sphere construction. Given there is no fitting in the dynamical theory, the level of agreement between theory and experiment seems remarkable, including the monomer-specific magnitude of the variation and the asymptotic saturation molecular weight (MW). Figure 8 performs the analogous comparison for the fragility of the same three polymers in Figure 7. There is a lowtemperature regime of growing fragility for all three systems. The predicted trends are qualitatively correct, including the location of the crossover to saturation. For PS and PMMA, a crossover occurs around M ≈ 104 g/mol and at a lower 103 g/mol for PDMS due to the milder molecular weight dependence of its characteristic ratio. As with Tg, the magnitude of the change is much smaller for PDMS than for PMMA or PS, again because of the relatively weak MW dependence of the I


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V. SEGMENTAL ELASTICITY For decades there have been reports44−47 of the emergence of “extra elasticity” in cold polymer melts beyond the usual rubber-like entropic contribution starting roughly at a (nonuniversal) temperature of ∼1.2Tg. The most plausible interpretation is that this is due to the onset of transient localization at the segmental scale (a transient “glassy network”) related to the dynamic crossover in supercooled liquids.85 The amount of extra elasticity is usually discussed relative to the entropic modulus, GR = ρskBT for the Rouse model (typically ∼10−20 MPa) or Ge = ρskBT/Ne ∼ 0.1−2 MPa from entanglements, where ρs is the segmental number density and Ne the entanglement crossover degree of polymerization. Very recent stress−strain experiments48 find such extra (linear regime) elasticity emerges even closer to Tg, at ∼(1.05−1.1)Tg, where it first exceeds Ge, ultimately approaching the GPa scale at Tg. While entropic elasticity weakens with cooling, glassy elasticity is associated with intermolecular forces and increases (typically rapidly) with cooling. Schematically, the total shear stress relaxation function has both interchain and entropic intrachain contributions:

Figure 8. Experimental dynamic fragility data for PS20 (blue circles), PDMS20 (red squares), and PMMA84 (yellow diamonds). Dashed horizontal lines represent the long chain limit of the data of corresponding color. Solid curves are the theory.

characteristic ratio. As found for the temperature dependence of the α time, the theory is most accurate for low-MW chains where predicted fragility values are in good agreement with experiment. However, the amplitude of the fragility variation with MW is obviously strongly underpredicted in Figure 8. This is inevitable given the absolute magnitude of the computed fragility for long chains is too small for these polymers. Although the variation of fragility is quantitatively underpredicted in Figure 8, it is of interest to test the validity of the key qualitative prediction of the theory that fragility and Tg are controlled primarily by the common variation of characteristic ratio and Ns with chain length. Thus, we replot in Figure 9 the experimental data of Figures 7 and 8 against the square root of the characteristic ratio, as suggested by eqs 16,17, and 19. To within the considerable noise of the data, one sees a rather good linear correlation for both properties, per the qualitative prediction of the theory. This suggests that whatever the missing physics is that leads to the muted variation in fragility in the long chain limit as predicted by ECNLE theory, it is closely related to the growth of the characteristic ratio.

G(t ) = Gintra(t ) + Ginter(t )

(21)

Often the interchain contribution is of the form Ginter(t ) ≈ Ge−(t / τα)

βK

75

(22)

where the stretching exponent is well below unity, and G is a glassy modulus of order a GPa close to, or at, Tg. This modulus can be computed using the microscopic stress tensor and eq 6. However, a typical viscoelastic experiment in the frequency domain detects both elasticity and relaxation, and in the simplest Maxwell model for glassy segmental relaxation, the dynamic storage modulus is G′(ω) ≈ G

(ωτα)2 1 + (ωτα)2

(23)

In this section we briefly consider this emergent glassy elasticity problem. We make no claim about quantitative accuracy (e.g., non-Maxwell model effects), nor do we aim for a precise comparison with the recent experiments. However, our calculations seem germane with regards to the problem broadly and the new measurements.48 The mainframe of Figure 10 presents dynamic elastic modulus calculations for polystyrene at four frequencies; the entanglement modulus is shown for comparison. We find the glassy modulus first exceeds Ge ≈ 0.2 MPa at temperatures that vary from ∼1.14 to 1.03 times Tg as the frequency decreases from 104 to 0.01 Hz. This calculated crossover temperature range is reasonable compared to recent measurements.48 Below the crossover, the elasticity sharply increases, reaching ∼5 GPa at Tg. This value is also experimentally reasonable at the glass temperature (where typically G ∼ 1−3 GPa), though quantitatively a bit too large for PS. The latter does not seem surprising given the absolute value of the modulus scales inversely as the cube of the Kuhn sphere size, which is estimated from the bulk density. The inset of Figure 10 shows results at 1 Hz for PIB, PMMA, PS, and PVC melts. Their elasticity exceeds Ge at ∼(1.04−1.06) Tg. The predicted modulus at Tg lies in the range ∼3−10 GPa, reasonable though again modestly too high. We caution that in

Figure 9. Experimental fragility plotted versus the square root of the characteristic ratio for PS20 (blue circles), PMMA84 (red squares), and PDMS20 (yellow diamonds). Lines of corresponding color are linear fits, the functional form predicted by ECNLE theory. Inset: experimental Tg versus the square root of the characteristic ratio. Symbols have the same meaning, and dashed lines are linear fits. J


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Specifically, the amplitude of the elastic distortion field is set by the jump distance of the elementary activated event on the cage scale. The latter is presumably sensitive to local chemical effects such as the symmetry of the side-group substitution, tacticity, and perhaps details of the elementary conformational transitions, all of which in turn will modify the elastic perturbation of the matrix required to facilitate cage scale rearrangements. How to theoretically capture such nonuniversal coupling of the local activation event to its surroundings remains a challenge. Deviations between theory and experiment at high temperatures where relaxation is fast and intermolecular barriers are low would seem, rather, to be related to the more atomistic scale chemical structural features lost in our coarse graining to a Kuhn sphere. A hint to the missing physics for the low-temperature dynamic fragility problem, and the breakdown of our quasiuniversal description, may lie in the empirical experimental finding28 that polymer fragility seems to be correlated with the ratio of the “high” temperature apparent Arrhenius barrier to the glass transition temperature, EA/kBTg. Perhaps EA is determined, in part, by conformational dynamics inside the Kuhn scale, which sets the elementary time scale to partially sphericalize segments with little center-of-mass translation. Concerning emergent elasticity in deeply supercooled polymer liquids, our preliminary calculations seem a promising starting point. However, much work remains to be done to create a real theory of viscoelastic response even in the glassy segmental dynamical regime. More broadly, one can ask about the implications of our work for chain relaxation, including the problem of failure of time−temperature superposition or decoupling of segmental and chain relaxation in the supercooled regime.87,89−91 We believe that to address this problem requires first considering dynamic heterogeneity in polymer liquids and that an initial step is to understand diffusionrelaxation decoupling in molecular liquids.66−68 As recently demonstrated for thin films,91 the ECNLE theory can be naturally generalized to predictively treat confined glassy liquids, including the α-relaxation, vitrification, and mobility gradients. Finally, its predictions for the scaling of Tg and fragility with the characteristic ratio (at “fixed chemistry” and chain length) have been recently confirmed using simulations of simple bead-spring models.92

Figure 10. Dynamic storage modulus (in GPa) for PS at frequencies of 0.01 (blue), 1 (red), 100 (yellow) and 10 000 Hz (green) plotted against normalized temperature. The horizontal dashed line is the entanglement modulus.70 Inset: dynamic storage modulus at a frequency of 1 Hz for PIB (blue), PMMA (red), PS (yellow), and PVC (green). Horizontal dashed lines are the corresponding entanglement moduli.70

experiments there may be a connection between at what temperature relative to Tg the emergent extra elasticity is first observed and melt fragility, as true for the problem of the nonuniversal decoupling of the temperature dependence of segmental and chain relaxation in the deeply supercooled regime.87,88 However, these details seem largely beyond the present theory given its inability to capture the very high and very low dynamic fragilities of some polymers.

VI. SUMMARY AND DISCUSSION We have extended the force-level ECNLE theory of glassy dynamics in colloidal suspensions and molecular liquids15−17 to polymer melts based on the a priori, but strong, approximation of mapping polymer chains to a liquid of disconnected and noninterpenetrating Kuhn spheres. No adjustable parameter calculations have been performed for the temperature-dependent segmental relaxation time of a chemically diverse set of van der Waals polymers. Despite the simplicity of the mapping, overall the theory does a good job of predicting the glass transition temperature, dynamic fragility, and the full temperature dependence of the α time for some high molecular weight polymers (those with m ∼ 80−100). The chain length dependence of Tg and (less so) fragility are also reasonably well described as a consequence of the molecular weight dependence of backbone stiffness. The biggest weakness of the present theory is its inability to predict very low and very high fragilities of some long chain polymers, which appear to be qualitatively different from those of molecular liquids. This is likely due to the crude mapping which ignores a litany of effects on the sub-Kuhn scale, such as chain interpenetration, changes in liquid structure due to longer range connectivity, and sidegroup packing, among many other potentially relevant and chemically specific effects. These polymeric features are muted as the chains become shorter and the characteristic ratio decreases, better mimicking a rigid molecule. Given the theory is accurate for low molecular weight polymers which largely do not interpenetrate on the Kuhn scale, we suspect the key missing physics in the deeply supercooled regime is associated with local chemistry and/or larger scale chain interpenetration and how it affects the longrange collective elastic barrier in a nonuniversal manner.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (K.S.S.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Basic Energy Sciences, Materials Science Division via Oak Ridge National Laboratory. Stimulating discussions with Alexei Sokolov are gratefully acknowledged.

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REFERENCES

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