Nonuniversal Coupling of Cage Scale Hopping and Collective Elastic

Dec 15, 2016 - We formulate a theoretical mechanism for the physical origin of the massive dynamic fragility range observed in long chain glass-formin...
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Nonuniversal Coupling of Cage Scale Hopping and Collective Elastic Distortion as the Origin of Dynamic Fragility Diversity in GlassForming Polymer Liquids Shi-Jie Xie and Kenneth S. Schweizer* Departments of Materials Science and Chemistry, University of Illinois, 1304 West Green Street, Urbana, Illinois 61801, United States ABSTRACT: We formulate a theoretical mechanism for the physical origin of the massive dynamic fragility range observed in long chain glass-forming polymer melts within the context of the force-level elastically collective nonlinear Langevin equation theory of coupled local−nonlocal activated segmental relaxation. The hypothesis involves how the cage scale barrier hopping process on the three Kuhn segment length scale is quantitatively coupled to the longer range collective elastic distortion required to sterically allow large-amplitude events to occur. The key nonuniversal aspect is proposed to be an effective microscopic jump distance, a dynamical quantity associated with the activation barrier, which is influenced by nanometer-scale conformational transition physics and monomer chemistry. By introducing a single numerical factor that breaks the universality of the jump distance in our mapping of polymers to liquids of disconnected Kuhn-sized hard spheres, one can account rather well, and simultaneously, for the vitrification temperatures and dynamic fragilities of 17 polymer liquids of diverse chemistry. The very low fragilities of polyisobutylene (PIB) and polyethylene are suggested to be a consequence of suppression of the collective elastic distortion effect. The large fragility variations displayed by polymeric materials appears special to long chain melts, consistent with their absence in molecular and oligomer liquids. Connections between cooperativity and fragility are identified. The theory very accurately captures segmental relaxation time experimental data for PIB, polypropylene, and polycarbonate melts over 11−13 decades. The present work sets the stage for attempting to understand the failure of time−temperature superposition in the deeply supercooled regime. universal description is devoid of fit parameters and has no divergences at finite temperature. The approach has been generalized to treat mobility gradients and other phenomena in confined free-standing thin films.25,26 ECNLE theory plus the minimalist mapping allows no adjustable parameter calculations to be performed for the temperature dependent α relaxation time of diverse molecular and polymer liquids. The theory accurately captures the α relaxation time of (often globular) molecular liquids over 14 decades.21−23 It also provides a theoretical basis for the recently experimentally established universalities of molecular liquids27,28 (also found for polymer melts29). Extension to polymer liquids was achieved based on a disconnected Kuhn segment model,24 resulting in good predictions for the glass transition temperature (Tg), dynamic fragility (m), α relaxation time temperature dependence, and shear elasticity for many (not all) polymers, including the chain length dependence of Tg as the consequence of the molecular weight dependence of backbone stiffness as quantified by the characteristic ratio, CN. For low molecular weight polymers,11−13,15,17,20,30 Tg and fragility (m ∼ 70−80) values are comparable to those of most

I. INTRODUCTION Cooling glass-forming liquids leads to a spectacular increase of the structural relaxation time and viscosity by 14 orders of magnitude.1−4 Despite many decades of theoretical attempts,1−10 understanding even relatively simple liquids composed of rigid nonpolar molecules remains a grand challenge. Polymers bring additional complications and richer phenomenology such as chain length dependences and exceptionally large quantitative variations of dynamical quantities.1−3,11−20 Mirigian and Schweizer recently proposed a quantitative, force-level theory of the α relaxation of supercooled molecular21−23 and polymer liquids,24 the “elastically collective nonlinear Langevin equation” (ECNLE) theory. Tractability and predictive power are achieved based on an a priori minimalist mapping to an effective hard sphere liquid. The activated relaxation event is of a mixed local−nonlocal spatial nature involving coupled cage scale uncooperative hopping and a longer range collective elastic distortion of the surrounding liquid, resulting in two inter-related, but distinct, barriers. The two-barrier idea yields rich dynamic behavior, encompassing the apparent Arrhenius, crossover, and deeply supercooled regimes, each separated by characteristic crossover times and temperatures. Connections to thermodynamics are a consequence, not the driver, of slow dynamics. The resulting quasi© XXXX American Chemical Society

Received: October 18, 2016 Revised: November 29, 2016

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Macromolecules

Figure 1. (a) Schematic of the fundamental relaxation event for spheres which involves a local, large-amplitude cage scale hopping motion on the ∼3 particle diameter length scale leading to an irreversible rearrangement and a nonlocal, spatially long-range collective elastic motion to accommodate the local rearrangement. (b) Cage scale rearrangement is described by the dynamic free energy (units of thermal energy) as a function of particle displacement in units of its diameter. Its spatial derivative defines the force on a tagged particle due to the local surroundings. The transient localization length (rloc), barrier location (rb), jump distance (Δr), and local cage barrier height (FB) are indicated. The shown dynamic free energy curves are for PIB at two reduced temperatures. Quantitatively, the dynamic free energy is nonuniversal even at fixed Tg/T. (c) Polymers are treated as disconnected Kuhn segments composed of Ns rigidly moving units. The image is a cartoon for PIB which has two methyl side groups per monomer.

sequences on Tg and fragility are explored for a large number of polymer liquids. Section II first briefly recalls key elements of ECNLE theory and the effective sphere mapping for molecules and polymers. All technical details and extended discussions are given in prior papers21−24 and are not repeated. How cooperativity enters ECNLE theory is contrasted with two other approaches. For context, prior polymer predictions for Tg and fragility, and their confrontation with experiment, are summarized. Section III presents the conceptual extension of ECNLE theory to introduce nonuniversal coupling between the cage and longrange collective elastic aspects of the activated α relaxation process. Numerical results for T g , fragility, degree of cooperativity, and the segmental relaxation time for a large number of polymer melts are presented and discussed in section IV. The paper concludes with a discussion in section V.

nonpolar molecular liquids and are well predicted by quasiuniversal ECNLE theory. However, for long chains the theory predicts fragilities vary only over a narrow range of ∼80−100.24 The unusually low and high fragilities of certain polymers, which are not typical of molecular liquids, are not captured, nor is the uniquely wide fragility range of m ∼ 46−200.30 Possibly relevant physical effects that are coarse-grained over include complex monomer shape and packing frustration,10,17,18,31,32 relative stiffness of side groups versus the backbone,10,17,31 asymmetry of side group substitution,19,32 and tacticity and intrachain torsional isomerism.10,31 More general consequences of connectivity may also be important, including nonergodic effects.33 Unfortunately, using statistical mechanical theory to predict the influence of such chemically complex aspects on the α relaxation time has not been achieved, so various qualitative ideas remain vague and hard to test. Moreover, fragility is a subtle quantity given it is a ratio of an inverse temperature derivative of the segmental relaxation time to Tg, and both factors vary with polymer chemistry. Besides its fundamental interest, understanding polymer fragility is important for other unsolved problems such as the chemically specific breakdown of time−temperature superposition (thermorheological complexity) in deeply supercooled melts.34−38 Correlations exist between higher fragility and stronger violations of time−temperature superposition (TTS) or “decoupling” of the temperature dependence of the segmental and chain relaxation times.20,35,37,38 Theoretically, whether fragility deeply relates to the enduring, but hard to quantify, concept of dynamic cooperativity remains unclear. The goal of this article is to break the quasi-universal nature of ECNLE theory based on a specific physical hypothesis for the leading order origin of how chemical structure impacts activated polymer relaxation. A rigorous treatment is intractable due to the complexity of conformational degrees of freedom and monomer shapes of real polymers. Rather, we formulate an approach that introduces just one material-specific parameter of clear physical meaning. The critical nonuniversality is argued to be of a dynamical origin associated with the degree of the local cage expansion required to realize an uncooperative hop which, in turn, sets the amplitude of the longer range collective elastic distortion contribution to the activation barrier. Its con-

II. QUASI-UNIVERSAL ECNLE THEORY II.A. Theory and Mapping. ECNLE theory describes activated relaxation as a mixed local−nonlocal rare event composed of a cage scale large-amplitude particle jump which is coupled to a relatively long-range, spontaneous, collective elastic fluctuation needed to sterically accommodate a hop.21 Figure 1a shows a cartoon of the physical mechanism. The foundational quantity for a spherical particle (diameter, d) liquid of packing fraction ϕ is an angularly averaged displacement (r) dependent dynamic free energy, Fdyn(r) = Fideal(r) + Fcaging(r), the derivative of which determines the effective force on a moving tagged particle due to its surroundings. The “ideal” term, βFideal(r) = −3 ln(r/d), favors unbounded diffusion. The localizing “caging” contribution, Fcaging(r), is constructed from the static intermolecular pair correlation function, g(r), or its Fourier space analogue, the structure factor S(k), and captures the effect of forces and kinetic constraints on the nearest-neighbor cage scale. A sample dynamic free energy is shown in Figure 1b. To execute a large-amplitude jump over the local barrier (height, FB) requires a small amount of extra space be created which is realized via a spontaneous collective elastic fluctuation of the particles outside the cage region which is roughly 3 sphere diameters in size.21,22 The corresponding radially B

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Macromolecules Table 1. System Parameters and Theoretical and Experimental Resultsa polymer

Ns

ac

TA(th)

1,4-PBD PVME PDMS PIB PE PP PSF PS PVAC PC PVC PMMA PEE PVCH 1,2-PBD 1,4-PI 3,4-PI

5.6 14.2 12.6 13.6 7.3 9 32.7 38.4 22.8 30.2 15 31.9 11.2 30.4 14.8 6.4 19.2

1 1 1 0.05 (1) 0.1 (1) 4 (1) 3.5 (1) 1 1 4 (1) 2 (1) 1 0.7 (1) 1 1 1 0.5 (1)

407 470 455 611 462 474 849 829 646 735 645 754 515 726 606 435 668

Tg(th) 174 212 218 222 177 258 473 432 327 424 362 408 245 397 278 190 292

(292) (218) (230) (427)

(381) (343) (253)

(312)

Tg(expt)

m(th)

m(expt)

R(Tg)

175, 171 267, 248 145, 150 201 237, 153 263 459, 463 371, 375 311 423 354 367 24555 40017 27345 20045 28756

86 86 93 41 (92) 46 (93) 130 (93) 130 (97) 99 96 140 (99) 118 (104) 102 91 (101) 111 93 87 75 (93)

85, 107 75 100, 79, 115, 85 46 46 137, 122 141 116, 143, 97, 121 95 132, 150 191 103, 145, 115

1.4 1.4 1.4 0.2 0.3 2.5 2.3 1.4 1.4 2.5 1.9 1.4 1.2 1.4 1.4 1.4 1.0

12017 13045 55,30 62,45 7617

a

The number of interaction sites in a Kuhn segment Ns, theoretically (th) calculated TA, Tg, and fragility (m) values based on the choice of ac that “best” reproduces both the experimental (expt) Tg and m values; the corresponding values of Tg and m if ac = 1 are shown in parentheses. The cooperativity parameter R at Tg as defined in eq 13 is also given. Wth the exception of PEE, PVCH, 1,2-PBD, 3,4-PI, and 1,4-PI, the original references for the experimental values of fragility and Tg cited above are all given in ref 24. Abbreviations used for polymer identification: 1,4polybutadiene (1,4-PBD), poly(vinyl methyl ether) (PVME), polydimethylsilioxane (PDMS), polyisobutylene (PIB), polyethylene (PE), polypropylene (PP), polysulfone (PSF), polystyrene (PS), poly(vinyl acetate) (PVAC), polycarbonate (PC), poly(vinyl chloride) (PVC), poly(methyl methacrylate) (PMMA), poly(ethylethylene) (PEE), polyvinylcyclohexane (PVCH), 1,2-polybutadiene (1,2-PBD), 1,4-polyisoprene (1,4-PI), 3,4-polyisoprene (3,4-PI).

symmetric strain field, u(r), decays as an inverse square power law of distance:21,39−41 ⎛ rcage ⎞2 ⎟ , u(r ) = Δreff ⎜ ⎝ r ⎠

r > rcage ≈ 3d /2

do not directly enter the determination of the local cage barrier, FB . The local and elastic collective barriers are intimately coupled via the dynamic free energy. The total barrier for the α process is

(1)

Ftotal = FB + Felastic

where rcage is the location of the first minimum of g(r). The strain field amplitude is set by the orientationally averaged mean cage expansion length, Δreff, which is typically less than the small transient “localization” length or vibrational amplitude, rloc (the location of the minimum of Fdyn(r)):21 Δreff ≈ 3Δr 2/32rcage ≤ rloc

The elastic barrier increases more strongly with increasing density or cooling than its cage analogue and dominates the growth of the α relaxation time as the glass transition is approached. A generic measure of the mean structural relaxation time, τα, follows from a standard Kramers calculation of the mean first passage time over the barrier21,42,43 as

(2)

where Δr ≡ rb − rloc ≈ 0.2−0.4d is the single particle jump distance. The numerical prefactor of 3/32 in eq 2 follows from assuming each spherical particle in the cage hops in a random direction by Δr. This picture is consistent with the local dynamic mean field foundation of the NLE theory but is a quantitative simplification for estimating cage expansion for nonspherical molecules or Kuhn segments. The elastic barrier follows by summing over all harmonic particle displacements outside the cage region:21 βFelastic = ρ(K 0/2)

τα =1+ τs

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

(5)

where KB is the absolute magnitude of the barrier curvature in units of kBT/d2, and τs is the time scale for the non-activated short time process. The theory is rendered quantitatively predictive for molecular liquids by a mapping22,23 to an effective hard sphere fluid guided by the requirement that it exactly reproduces the equilibrium dimensionless density fluctuation amplitude (compressibility) of the liquid, S0(T) = ρkBTκT. The latter thermodynamic quantity sets the amplitude of nanometer-scale density fluctuations and follows from the experimental equation of state (EOS). This mapping determines a material-specif ic, temperature dependent effective hard sphere packing fraction, ϕeff(T). In practice, four known chemically specific parameters enter:22,23 A and B (interaction site level entropic and cohesive energy EOS parameters, respectively), the number of elementary sites that define a rigid molecule, Ns (e.g., Ns = 6 for benzene), and hard sphere diameter, d. Only translational, not rotational, motions enter the mapping. Knowledge of

⎛ rcage ⎞3 2 ⎟ Δr dr 4πr 2u 2(r ) ≃ 12ϕ⎜ K ⎝ d ⎠ eff 0 cage

∫r

(4)

∞

(3)

where r is relative to the cage center, V is the liquid volume outside the cage, and K0 = 3kBT/rloc2 is the curvature (harmonic stiffness) of the dynamic free energy at its minimum. The second equality is an accurate analytic expression. It shows that Felastic is determined by the measure of harmonic stiffness of the transient localized state (directly related to the dynamic glassy shear modulus21,22) and the effective jump distance.21 These two key factors are properties of the dynamic free energy but C

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Macromolecules ϕeff(T) allows g(r) and S(k) to be computed using standard integral equation theory,22,44 which determines Fdyn(r), from which all dynamical results follow, including the fragility: m ≡ d(log τα(T ))/d(Tg /T )|Tg

involved in the rearrangement event which grows upon cooling due to loss of configurational entropy. Specifically τα ≈ exp(z(T )EA /kBT ) = [exp(EA /kBT )]z(T ) τ0 (10)

(6)

The two-barrier ECNLE theory can be roughly written in the AG form as22

Independent of the dynamic theory, the above mapping immediately predicts a simple (approximate) relation for the glass transition temperature22

Tg ∝ B Ns

τα ∝ exp((FB(T ) + Felastic(T ))/kBT ) ∝ [exp(FB(T )/kBT )]z(T ) (11a)

(7)

and an analytic relation for the fragility has also been derived: m ∝ c + 4A Ns

z( T ) ≡ 1 +

(8)

(11b)

where the local cage barrier is predicted to be only weakly temperature dependent and roughly corresponds to Arrhenius relaxation. Here, z(T) in eq 11b grows monotonically with cooling and is a measure of “cooperativity” since the barrier ratio quantifies the importance of how local cage expansion induces a stronger (scale-free) collective elastic strain field. Of course, although eqs 10 and 11 “look similar”, the underlying physics is very different. A highly phenomenological scheme that has been widely applied to analyze experimental data for decades is the “coupling model” (CM) of Ngai.20,46,47 It posits

where c is a constant significantly larger than unity.22 The cohesive energy parameter, B, does not enter the fragility, providing a theoretical basis for why experimental attempts to identify a Tg−m “correlation” fail.45 Polymers have all the complexities of molecules plus additional ones such as rotational isomerism. Many workers have employed the Kuhn length as the relevant coarse-graining scale for segmental dynamics. This idea is adopted as a minimalist model whence the polymer liquid is replaced by a fluid of disconnected Kuhn segments modeled as noninterpenetrating hard spheres composed of a number of interaction sites, Ns (Figure 1c). For polymers with simple backbones, one has24 n l n Ns = k s ≈ CN s lbb nbb nbb

Felastic(T ) FB(T )

τα(T ) ≈ [tc−n(T )τ0(T )]1/(1 − n(T ))

(12)

Here, τ0(T) is a “primitive” α (or Johari−Goldstein β) relaxation time of unspecified molecular origin and temperature dependence (perhaps Arrhenius for some systems), tc is a temperature-independent crossover time (of order a picosecond) that signals the onset of “many body coupling” or “intermolecular cooperative effects”, and n is the crucial systemspecific “coupling parameter” (a temperature dependent function in general). The latter is argued to reflect how elementary primitive relaxation events become coupled on larger length and time scales. The value of n is extracted by fitting the KWW formula to experimental relaxation spectra18,20,47 where the stretching exponent βK ≡ 1 − n. At an ad hoc level, one sees some correspondence between eqs 11 and 12 if the “primitive α process” is identified with the local cage uncooperative hopping process of NLE theory and the coupling parameter with the influence of the longer range collective elastic barrier associated with the “coupling” of the local hopping event and cage expansion achieved via a spontaneous elastic fluctuation. Formally, one can write

(9)

where lk is the Kuhn length, lbb is the backbone bond length, ns is the number of sites per monomer, and nbb is the number of bonds per monomer. The Ns values for most polymers studied here are given elsewhere,24 and the complete set is listed in Table 1. This number is determined by both monomer bulkiness (mass) and backbone stiffness. The “Kuhn sphere” diameter follows as πd3/6 ≡ C∞Vmonomer/nbb, where the monomer volume is calculated from the melt mass density. The parameters A and B at 1 atm are given elsewhere.24 With this mapping, ECNLE theory makes dynamical predictions with no adjustable parameters. A center-of-mass level, Kuhn hard sphere model is expected to be most accurate at low molecular weights for multiple interrelated reasons, and their glassy dynamics are very similar to that of molecules, as previously discussed.24 The replacement of molecules or Kuhn segments by effective spheres assumes sufficient “dynamic self-averaging” to render a translational motion model useful. Chemically specific errors must be incurred, which we expect are more severe as molecules become more aspherical or polymer chains are longer with more spatially extended conformational transitions which ultimately set the elementary time scale of the α process. This latter effect will be our hypothesis for the leading origin of dynamic nonuniversality: for interpenetrating long chains, the amount of cage expansion depends on chemical details not present in the effective hard sphere description. II.B. Cooperativity Measure and Phenomenological Models. Adam and Gibbs (AG),5 based on the entropy crisis scenario, posit that the α relaxation time is a magnified version of an underlying Arrhenius (single particle or uncooperative local barrier, EA) activated process. The effective barrier of a “configurationally rearranging region” (CCR) is taken to be Eeff = z(T)EA, where z(T) describes the number of particles

F (T ) 1 = 1 + elastic 1 − n(T ) FB(T )

n(T ) =

R (T ) , 1 + R (T )

R (T ) ≡

(13)

Felastic(T ) FB(T )

where R (and hence the defined parameter n) grows with cooling. Based on the quasi-universal ECNLE theory, at Tg one has R ∼ 1.4, which implies n(Tg) ∼ 0.58. The latter is a typical value deduced from applying the CM to fit relaxation spectra, except for polymers with very low or very high fragilities.18,20,47,48 A caveat to the above discussion is that it seems ambiguous to compare the phenomenological AG model and/or the CM scheme with a microscopic, force-based dynamical approach such as ECNLE theory. Moreover, ECNLE theory at present does not take into account a distribution of relaxation times, D

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Macromolecules which is an implicit crucial element of the CM (but not the AG model). The question of whether there is a deep connection between KWW exponents and the temperature dependence of the α relaxation time remains unclear theoretically and experimentally. For example, in the classic molecular glassforming liquids o-terphenyl (OTP) and tris(naphthyl)benzene (TNB), the apparent stretching exponent is βK ≈ 0.5, independent of temperature in the deeply supercooled regime.49,50 This implies eq 12 cannot capture the strong super-Arrhenius temperature dependences of OTP and TNB if the primitive relaxation process is Arrhenius. Also, although fits of the KWW function over a limited window of data very often result in a temperature dependent βK for polymer liquids (suggesting a breakdown of time−temperature superposition), the actual data can be shifted very well to construct master curves.51,52 The latter behavior would seem to be in qualitative conflict with the idea that the KWW function is of fundamental importance. More generally, accurately extracting KWW exponents from data fitting is often difficult and can depend on the correlation function measured. Despite these caveats, the above comparisons perhaps again highlight how apparently qualitatively different physical ideas about the α process can be viewed in a common framework.21 II.C. Model Calculations and Experimental Confrontations. Despite its quasi-universal and coarse-grained nature, ECNLE theory does contain chemical information via the material EOS, number of rigid moving interactions sites (Ns), and Kuhn length. The first 12 entries in Table 1 summarize prior theoretical results24 in the long chain limit. Values for the temperature where a barrier first emerges (TA) are shown, along with results based on using the universal 3/32 prefactor in eq 2 (indicated by ac = 1 in the table) for the kinetic glass transition temperature (Tg), defined as when the α relaxation time is 100 s, and the fragility (m); the corresponding experimental Tg and m values are also listed. Cross-plots of theory versus experiment are shown in Figure 2, which include calculations for five new systems (described in section IV). Although the Tg predictions are not quite as quantitatively good as found for molecules (∼10−20% deviations22,23), they are encouraging given the simplicity of the coarse-grained description and no adjustable fitting parameters. The situation is very different for the fragility (though note the often high experimental uncertainty). As seen in Table 1, the quasi-universal polymer theory predicts24 a weak dependence on monomer chemistry, with m ∼ 80−100 per molecules, in contrast to the observed remarkably large variation of m ∼ 46−191. The fragilities of long chain PBD, PDMS, PMMA, and PVAC are well captured, but the very low (PIB, PE) and very high (PP, PC, PS, PSF, PVC) values are not. Different accuracy of a theory for Tg and fragility is perhaps unsurprising given the latter is a more subtle ratio quantity and the prior22 analytic analysis of ECNLE theory which suggests no simple Tg−m correlation. For context, recall the minimalist effective hard sphere model has no explicit information about how monomer shape, specific attractions for chemically heterogeneous monomers (e.g., PVC, PDMS), conformational transitions, or connectivity effects beyond the Kuhn scale (∼1−2 nm) affect packing and dynamics.

Figure 2. Comparison of theoretical and experimental values for (a) Tg and (b) dynamic fragility based on the quasi-universal ECNLE formulation that uses ac = 1 for all 17 systems defined in Table 1. The dashed green line represents perfect agreement.

ECNLE theory guided by a clear physical hypothesis within the tractable Kuhn segment mapping scheme. The theory predicts in the deeply supercooled regime the elastic physics typically dominates the determination of fragility. The collective elastic barrier involves two very different quantities: (i) the cage expansion amplitude, Δreff in eq 2, which varies as the square of the jump length, Δr, and sets the amplitude of the strain field, and (ii) the harmonic stiffness, K0 in eq 3, set by the small transient localization length (rloc ≪ Δr). The numerical prefactor of Δreff in eq 2 was computed for hard spheres. Though plausibly accurate for colloids and globular molecules, it seems highly likely quantitative errors will be incurred for some polymers. Moreover, Δreff is a property of the barrier which likely depends on nonuniversal torsional motions required for a conformational transition. For long interpenetrating chains, monomer shape and space-filling “corrugation”, tacticity, symmetrical versus asymmetrical side group substitution, and the geometric nature of trans and gauche states, all could matter in determining the effective “jump distance”. How to a priori compute the dynamical consequences of such effects seems an intractable task. With the above motivation, we propose to phenomenologically account for the complexities in the effective jump distance by introducing one adjustable, temperature-independent, chemistry-specific parameter, λ, that scales the microscopic jump distance as Δr → λΔr. Thus, the elastic barrier in eq 3 is changed by a constant numerical factor:

III. NONUNIVERSAL CAGE-ELASTICITY DYNAMIC COUPLING MECHANISM Our primary goal is to explore the consequences for the polymer fragility problem of breaking the universality of

Felastic → λ 4Felastic ≡ acFelastic

(14)

Physically, this assumes that the key nonuniversal aspect of the activated event lies at the interface between the cage region (of E

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Macromolecules diameter ∼3−5 nm for polymers) and surrounding liquid, which determines the absolute and relative importance of the collective elastic distortion. This is a dynamical idea, not a thermodynamic one which focuses on the minima of the potential energy landscape, nor directly on changes of equilibrium packing. Mathematically, this idea modifies the relative importance of the temperature and density dependent local cage and collective elastic barriers by a constant numerical factor. Given ECNLE theory predicts the elastic barrier scales as the fourth power of the microscopic jump distance, such a “correction” can have a large effect. We expect λ is greater than or less than unity depending on polymer chemistry and thus can either simultaneously increase or decrease both Tg and fragility relative to the quasi-universal (λ = 1) theory. In reality, λ likely depends to some extent on temperature and chain length. Here we pursue a minimalist approach by only considering long chains and focusing on the fragility at Tg where the degree of cooperativity is maximized. It is instructive to discuss the two extreme limits. If ac = 0, then all collective elastic effects vanish and ECNLE theory reduces to the uncooperative NLE hopping theory. For the latter, the α relaxation time is22−24 roughly of an Arrhenius form with a lower bound fragility of m ∼ 37 (weakly chemically specific) and a lower bound Tg. This is not literal Arrhenius behavior, as occurs for ultrastrong network glass-formers such as silica, since molecular and polymer liquids do significantly densify upon cooling at constant pressure. The opposite, “maximally cooperative” limit is ac → ∞. Here, the barrier abruptly changes from zero to infinity at the onset temperature, TA (analogue of the critical temperature of mode coupling theory). Thus, Tg approaches a finite upper bound (TA), and fragility diverges. Increasing ac to high values will thus have a larger effect on fragility compared to Tg, consistent with the notion fragility is a measure of collective physics.

Figure 3. (a) Dependence of Tg on ac for representative polymers over the modest range of ac typically relevant for improving the theoretically calculated dynamic fragility. (b) Same Tg calculations plotted versus (ac)0.1. The dashed lines demonstrate that Tg is proportional to (ac)0.1 over a very wide range of ac.

upper limit (TA). The overall change of Tg with ac is a factor of ∼1.6 for all systems. If one plots the Tg shift from the ac = 1 value and normalizes it by Tg(ac = 1), all calculations almost collapse onto a master curve up to very high values of ac ∼ 20 (not shown). Relative to the ac = 1 case, Tg reductions are more significant than enhancements. Figure 3b shows results which establish there is a wide range of ac where Tg ∼ (ac)0.1. Even in terms of the microscopic jump distance scale factor λ, this is a weak variation, Tg ∼ λ0.4. Ultimately, Tg approaches TA as ac diverges, but only for extremely large (likely unphysical) values of ac. Figure 4a presents the analogous fragility results. Over the modest ac range shown, fragility varies from ∼35 to 120−150, with PVC showing the largest sensitivity and PDMS and PIB the weakest. The fragility reductions for ac ≪ 1 are larger than the fragility enhancements for ac ≫ 1. But in all cases, the factor ∼4 fragility variation is significantly larger than the factor of ∼1.6 variation of Tg. Hence, fragility is more sensitive to ac variations than Tg, as qualitatively argued in section III. Figure 4b plots the fragility shift divided by m(ac = 1) over an exceptionally large change of ac from 0 to more than 700 (corresponding to λ up to ∼5.5). A near collapse of results is found in this representation, with PVC showing the largest variation and PIB the weakest. For this extended range of ac, three regimes appear as a consequence of the two-barrier picture of ECNLE theory: (i) a nearly constant and large fragility reduction for ac < 0.25 corresponding to dominance of the local cage barrier; (ii) a steep, quasi-sigmoidal variation for intermediate ac values spanning ∼0.25−10 where the local and

IV. RESULTS We now numerically implement the new idea of section III by searching for the optimal value of ac that simultaneously “best” accounts for the experimentally observed Tg and fragility values of specific polymer liquids. There is no a priori guarantee this exercise will be successful. We consider the 12 polymers previously analyzed24 and 5 new ones in the saturated hydrocarbon and diene classes: PEE, PVCH, 1,2-PBD (also known as PVE), 3,4-PI, and1,4-PI (also known as cis-PI). Polymer acronyms are defined in Table 1. For the five new systems, the EOS parameters A and B used for the mapping are taken to be those of24 polypropylene and 1,4-polybutadiene, respectively, given their almost identical chemical composition and lack of adequate published EOS data. Our predictions for fragility are not sensitive to these EOS parameters. Table 1 shows the number of interaction sites per Kuhn segment, Ns, computed using the known monomer structure and characteristic ratio.24,53 The barrier onset temperatures, TA, are also shown, which are all very high, as also predicted for molecular liquids.22,23 IV.A. Tg and Fragility: Model Calculations and Trends. To explore the chemistry dependent sensitivity of the predicted Tg values to the parameter ac, Figure 3a presents representative results for eight polymers. The increase of ac from 1 to 4 corresponds to a maximum increase of the microscopic jump length by only a very modest factor of ∼1.4. The functional shapes of Tg(ac) are all the same. The tendency to become more weakly increasing at higher ac is due to the existence of an F

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Macromolecules

the connection between the detailed chemical structure and the parameter ac, it is not possible to definitively explain why the theory does not capture the fragilities of these two polymers. We do note they have distinctive structural features: PVC is chemically very heterogeneous given the chlorine substitution, and 1,2-PBD has a double bond side group. More generally, Figures 3 and 4 suggest that the present model does not appear capable of simultaneously predicting accurate values of Tg and the rare ultrahigh fragilities of polymers such as PVC and poly(ether imide) (m ∼ 214). A fundamental reason for this is presently unclear but likely points to the importance of an additional parameter that must be taken into account if one adopts a coarse-grained spherical Kuhn segment model. For PEE and 3,4-PI, we cannot find experimental values for fragility, so our results are predictions. For Tg, there is nontrivial improvement relative to the prior24 ac = 1 calculations, but it is modest compared to the fragility. Typical deviations are (well) below 12%, with two obvious exceptions; PIB and PE are much better, PP and PC are modestly better. No system is described worse than obtained based on the universal ac = 1 model, although PDMS and PVME are still not that well captured with the theory deviating from experiment by ∼33% and 25%, respectively. Figures 5 and 6 globally document how well the theory does by plotting our results versus the experimental data (with error

Figure 4. (a) Dependence of dynamic fragility on ac for various polymers. (b) Change of dynamic fragility, m(ac) − m(1), normalized by its value for the baseline ac = 1 case, as a function of the parameter ac raised to the one quarter power.

elastic barriers are (somewhat) comparable; (iii) a far weaker, roughly linear growth for large ac where the collective barrier is dominant. Examination of a log−log plot (not shown) of fragility versus ac supports this three regime picture. In the rather tight crossover region around ac = 1 (roughly ac ∼ 0.5− 3), a power law m ∝ ac0.22 is found, while for ac ∼ 10−1000 we find m ∝ ac0.12. IV.B. Optimized Predictions and Comparison to Experiment. To make quantitative contact with experiment, ac is varied to find a value that “best” reproduces both the Tg and fragility compared to measured values. Given the experimental uncertainties of these quantities in the literature (especially fragility), we do not determine the “best” value based on a precise quantitative metric. The resulting ac and predicted Tg and fragility values are listed in Table 1. About one-half of the systems are unchanged relative to the prior study based on quasi-universal ECNLE theory;24 i.e., ac = 1 is the “best” choice. The other half of the systems have ac values significantly smaller or larger than unity. Remarkably, the “strong” polymers PIB and PE are consistently well described, both their low Tg and very low fragilities, by nearly “turning off” the collective elastic barrier (ac ≪ 1). On the other hand, the theory captures the behavior of the high-Tg and high fragility polymers PC, PVC, and PSF (all have stiff backbones) better by modestly increasing the role of collective elasticity (ac ∼ 2−4). To within experimental uncertainty, all fragility calculations agree with measurements to within 15% or better with the exceptions of PVC and 1,2-PBD, the fragilities of which are both strongly underpredicted. Given our coarse graining to the Kuhn scale, and the fact we do not have an a priori theory for

Figure 5. Cross-plot of theoretical and experimental values of the glass transition temperature for all 17 polymers studied (from Table 1, best choice of ac). The dashed green line represents perfect agreement.

Figure 6. Cross-plot of theoretical and experimental dynamic fragility values for all 17 polymers studied (from Table 1, best choice of ac). The dashed green line represents perfect agreement.

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DOI: 10.1021/acs.macromol.6b02272 Macromolecules XXXX, XXX, XXX−XXX

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Finally, given the rather good agreement between theory and experiment for the full temperature dependence of the segmental relaxation time, one would expect the theoretical “data” can be well fit by the free-volume-based WLF equation or other empirical formulas. Indeed, this has already been shown to be true for the WLF formula when applied to underlying hard sphere fluid which is the fundamental underpinning of our approach to describing thermal liquids.21 We have verified this is also true for PC, PP, PE, PIB, PSF, PVC, PS, and 1,4-PBD studied here. The extracted C1 and C2 parameters of the fitted WLF function are found to be nonuniversal, and they are of an absolute magnitude and vary with chemistry in a manner in good accord with experimental studies. This immediately implies the WLF-based relation m = (C1/C2)Tg holds in ECNLE theory. However, given the empirical nature of the WLF formula and this fitting exercise, the unclear fundamental meaning of C1 and C2, and the fact that our force-based theory does not invoke the rather illdefined concept of a “dynamic free volume”, we believe little new is learned from these fitting-based connections beyond providing further support for our claim that ECNLE theory can capture the α relaxation time of polymer liquids over a wide temperature range. IV.C. Degree of Cooperativity. Given the discussion in section II.B, Table 1 shows values of the cooperativity R parameter (ratio of cooperative to uncooperative activation barriers) defined in eq 13 at Tg based on the optimal value of ac. One sees R varies enormously, from ∼0.2 (PIB) to 2.5 (PP, PC), with an almost exact one-to-one correspondence with fragility. Figure 8a presents a model calculation for the underlying hard sphere fluid of how R varies with ac over both a very wide (mainframe) and experimentally relevant more narrow (inset) range. For the latter, R varies from 0 to ∼2.5, corresponding to “coupling parameters” in eq 13 of n ∼ 0−0.7. No simple mathematical form for R(ac) is evident. Figure 8b cross-plots the fragility versus R parameter at Tg. Polystyrene parameters are employed, but chemistry effects are minimal for this question. For modest values of R (inset), or equivalently m ∼ 37−150, the relation is very close to being linear. This linearity is interesting since R is an objective measure of collective effects in ECNLE theory, and it supports the idea that fragility is strongly correlated with cooperativity. For larger values of R and fragility, the form of the correlation becomes more slowly varying. If one takes this result seriously, it suggests that achieving massive fragilities (m > 200) is possible if the elastic barrier dominates its local analogue. Concerning precise polymer chemistry design, how to achieve this is unclear. However, mechanistically it suggests one should search for polymers where conformational transitions and monomer structure considerations are such that the required local cage expansion is very large which will allow maximal coupling of the cage scale barrier hopping process with the longer range elastic strain field.

bars). In both an absolute sense, and relative to Figure 3, there is a massive improvement for fragility and a smaller but significant improvement for Tg. Overall, we conclude that one can enormously improve fragility predictions, and account for their wide range, by introducing the broken universality idea associated with the amplitude of cage expansion, without degrading the rather good prior results for Tg. We view this as nontrivial support for our physical hypothesis of the leading nonuniversal correction in ECNLE theory. Although not the primary topic here (see ref 24 for a detailed study based on ac = 1), Figure 7 shows representative results for

Figure 7. Predictions for the full temperature dependence of the segmental relaxation time in the Angell representation based on the optimized value of ac for the subset of the polymers studied that do not have ac = 1. Here, τα(Tg) ≡ 100 s. Predictions for PP, PS, PC, and PVC are nearly perfectly overlapped due to their close fragilities. Empty symbols are taken from VFT or WLF fits to experimental data for the relaxation time of long chain PIB (red circles),17 PP (blue triangles),57 and PC58 (green squares). The corresponding theoretical curves are not fits.

the full temperature dependence of the segmental relaxation time in the Angell representation based on the optimized value of ac for polymers with ac values not equal to unity (PE, PIB, PP, PSF, PC, PVC, PEE, and 3,4-PI). For all systems with nearly identical fragility, the entire temperature dependence is almost identical; this is why the calculations for PP, PSF, PC, and PVC (m ∼ 120−140) very nearly collapse onto a single curve on the scale of the plots over all temperatures. Such a simplicity is a nontrivial prediction of ECNLE theory and is associated with the predicted deep connection between the local cage and long-range collective elastic barriers,21−23 as first discussed and verified experimentally for molecular liquids.22,23 The theoretical calculations for PIB, PP, and PC are compared to experimental data over a very wide range of time (11−13 decades) and temperature (almost a factor of 2 for PIB). By properly accounting for the fragility at Tg, the theory makes excellent predictions for these three polymers at all times and for all temperatures. The latter rather remarkable result further emphasizes a fundamental idea underlying ECNLE theory that the activated dynamics in all temperature regimes is deeply related. For both PIB and PC, rather small deviations of less than a decade are seen at the highest temperatures; such deviations are smaller for PP. These modest disagreements seem unsurprising since at high temperature one expects fine scale chemical details that have been coarse grained over matter more than in the collective low temperature regime.

V. DISCUSSION We have addressed an open problem in polymer physics within the context of the ECNLE theory of coupled local−nonlocal, two-barrier activated segmental α relaxationthe origin of the massive range of dynamic fragility in long chain polymer melts. We hypothesized the answer lies in how the cage scale hopping process (which is on the three Kuhn segment diameter length scale) is coupled to the longer range collective elastic distortion outside this local region. Hence, our proposal is that it is at the H

DOI: 10.1021/acs.macromol.6b02272 Macromolecules XXXX, XXX, XXX−XXX

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Of course, the fragility or Tg of some polymer liquids (e.g., PVC) remains not well described, likely for variable chemical/ physical reasons. It is an outstanding challenge to understand in terms of angstrom-scale physical chemistry how the precise value of the effective jump distance is determined for specific polymers and how, or if, the correction depends on temperature. Solving these highly nonuniversal problems is difficult, but for sure chain length matters given oligomer fragilities are comparable to those of molecular liquids, systems for which the quasi-universal ECNLE theory is accurate with no adjustments.24 More broadly, the progress reported here suggests one now has an accurate theory (with one adjustable parameter for some systems) for the segmental α relaxation time of polymer melts as a function of temperature. The comparison of theory and experiment for PIB, PP and PC in Figure 7, and the prior comparisons,24 provide evidence for such a conclusion. The present advance sets the stage for addressing the failure of TTS as manifested in chemically specific differences in the temperature dependences of the segmental and chain relaxation processes in the deeply supercooled regime.20,34−38 Progress will be reported in the near future. The present work may also help understand whether a correlation exists between fragility and Tg reduction in free-standing polymer films.54

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (K.S.S.).

Figure 8. (a) Dependence of the cooperativity parameter of eq 13 (barrier ratio R) on ac over a very wide (main frame) and experimentally relevant more narrow (inset) range for the hard sphere fluid. (b) Fragility of PS as a function of barrier ratio R over a wide (main frame) and experimentally relevant more narrow (inset) range. The red dashed line in the inset is a guide to the eye.

ORCID

Shi-Jie Xie: 0000-0003-3317-4720 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the U.S. DOE-BES via Oak Ridge National Lab. We acknowledge helpful discussions with Steve Mirigian and Alexei Sokolov.

interface between the locally compact rearranging cage region and the surrounding liquid that the most important information enters about chemically specific processes associated with conformational transitions. The latter are averaged over in our minimalist mapping to a liquid of effective Kuhn hard spheres. The microscopic jump distance, a dynamical quantity associated with the barrier, is allowed to vary based on introducing one material-specific numerical parameter. It is found that by allowing this parameter to be smaller or larger than the a priori value of unity appropriate for a hard sphere fluid (which works well for about one-half of polymers studied24), one can account rather well, and simultaneously, for Tg and fragility of 17 polymer liquids of diverse chemistry. Moreover, the entire temperature dependence of the segmental relaxation time over 11−13 orders of magnitude is well predicted for three specific polymer melts of widely varying Tg and fragility. The ability to understand the very low fragility values of PIB and PE as a consequence of a very large suppression of the elastic distortion effect is especially notable. More generally, we conclude that massive fragility variation for nonpolar or weakly polar polymeric materials is special to long chain melts. This deduction is based on the fact that no such material-specific adjustment of the jump distance is required to predict very well using the quasi-universal (no fit parameters) version of ECNLE theory the α process of diverse systems of colloidal suspensions,21,23 molecular liquids,22,23 and short chain melts24 based on the a priori effective hard sphere mapping.

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REFERENCES

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