Letter pubs.acs.org/macroletters
Stringlike Cooperative Motion Explains the Influence of Pressure on Relaxation in a Model Glass-Forming Polymer Melt Wen-Sheng Xu,*,†,‡ Jack F. Douglas,*,§ and Karl F. Freed*,†,∥,⊥ †
James Franck Institute, ∥Department of Chemistry, and ⊥Computation Institute, The University of Chicago, Chicago, Illinois 60637, United States § Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, United States ABSTRACT: Numerous experiments reveal that the dynamics of glassforming polymer melts are profoundly influenced by the application of pressure, but a fundamental microscopic understanding of these observations remains incomplete. We explore the structural relaxation of a model glassforming polymer melt over a wide range of pressures (P) by molecular dynamics simulation. In accord with experiments for nonassociating polymer melts and the generalized entropy theory, we find that the P dependence of the structural relaxation time (τα) can be described by a pressure analog of the Vogel−Fulcher−Tammann equation and that the characteristic temperatures of glass formation increase with P, while the fragility decreases with P. Further, we demonstrate that τα for various P can quantitatively be described by the string model of glass formation, where the enthalpy and entropy of activation are found to be proportional, an effect that is expected to apply to polymeric materials under various applied fields.
T
elastic (FENE) potential with the commonly used parameters kb = 30ε/σ2 and R0 = 1.5σ, where ε denotes the energy scale of the LJ potential. All beads have the same mass m. Length, time,
he dynamics of glass-forming liquids are well-known to be greatly altered upon the application of pressure (P), a phenomenon whose deep understanding is of significance in numerous manufacturing applications.1,2 Experiments indicate that nonassociating glass-forming liquids at very large P can vitrify at fixed temperatures (T), at which these liquids are simple fluids at atmospheric pressure and that the P dependence of the structural relaxation time (τα) generally displays a pressure analog of the Vogel−Fulcher−Tammann (PVFT) equation,1,2 where P and the critical pressure P0 replace T and the critical temperature T0 in the conventional Vogel−Fulcher−Tammann (VFT) equation, respectively. The generalized entropy theory (GET) predicts the PVFT relation and further provides a rationale based on the variation of the configurational entropy of the fluid with P.3 Correspondingly, the Adam−Gibbs (AG)4 and string5 models of glass formation imply that the average length of the cooperatively rearranging regions (CRRs) or stringlike collective motion, characteristic of glass-forming liquids, should grow with P in such a way as to explain the PVFT relation. Despite substantial experimental effort, computational studies of the influence of pressure on the dynamics of glass-forming liquids have been surprisingly quite limited. In this Letter, we explore the influence of P on the glass formation of a coarse-grained “bead−spring” model of polymer melts.6,7 The system is composed of 200 linear chains, each with 16 beads. Nonbonded beads interact via a truncated-andshifted Lennard-Jones (LJ) potential with the cutoff distance of rcut = 2.5σ, where σ is the effective diameter of the beads. Bond connectivity is maintained by the finitely extensible nonlinear © XXXX American Chemical Society
and pressure are reported in units of σ, mσ 2/ε , and ε/σ3, respectively. Molecular dynamics (MD) simulations are performed in three dimensions under periodic boundary conditions using the HOOMD-blue simulation package.8−10 We employ the same procedure in our previous work11,12 to explore glass formation along isobars. The simulations are first performed at a constant P in the NPT ensemble with a Martyna−Tobias−Klein barostat-thermostat,13 which enables the determination of the desired density as a function of T for the given P. The simulations are then performed with the computed densities in the NVT ensemble with a Nosé−Hoover thermostat14,15 to obtain the properties of interest. A time step of Δt = 0.002 or 0.005 is used in the high or low T simulations, respectively. We ensure that the quantities reported in the present paper are computed for properly equilibrated polymer fluids, and hence, we do not study nonequilibrium aspects of glass formation. Four independent runs are performed for each state point to improve the statistics. Our analysis begins with the T and P dependence of τα, determined by the common convention as the time at which the self-intermediate scattering function Fs(q,t) decays to 0.2. The self-intermediate scattering function is defined as Received: October 18, 2016 Accepted: November 22, 2016
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Figure 1. (a) Arrhenius plot of the structural relaxation time τα for various pressures P. Pluses and crosses denote the positions of the localization temperatures Tl and the Arrhenius temperatures TA, respectively. Dashed, dotted, and dash-dotted lines indicate the Arrhenius, Vogel−Fulcher− Tammann (VFT), and power-law fits to τα, respectively. (b) VFT collapse of the temperature dependence of τα for various P. The dotted line indicates the VFT relation, ln(τα/τ0) = DT0/(T − T0). N
Fs(q , t ) =
1 ⟨∑ exp{−iq·[rj(t ) − rj(0)]}⟩ N j=1
(1)
where i = −1 , N is the total number of beads, rj(t) is the position of bead j at the time t, and the wavenumber q is chosen to be 7.1, which is close to the first peak of the static structure factor. Figure 1a displays τα as a function of 1/T for all P studied. While the magnitude of τα is strongly influenced by varying P, Figure 1a shows that the T dependence of τα follows the VFT equation in the low T regime for all P studied, namely, τα = τ0 exp[DT0/(T − T0)], where τ0, D, and T0 denote the high T limit of τα, the fragility parameter quantifying the strength of the T dependence of τα, and the VFT temperature at which the extrapolated τα diverges. This universal behavior is further demonstrated in Figure 1b, where a master curve is evident by plotting ln(τα/τ0) as a function of DT0/(T − T0). The variations of D and T0 with P are discussed later. The P dependence of τα for fixed T is also commonly discussed in experiments for glass-forming liquids at elevated pressures.1 Figure 2 illustrates the P dependence of τα for fixed T in our simulations. To analyze our simulation data, we employ the same method demonstrated by Paluch and coworkers.16 Specifically, the PVFT equation1 is used to describe the P dependence of τα for fixed T; that is, τα = τα,0 exp[DP P/ (P0 − P)], where τα,0 is the value of τα at P = 0, DP describes the strength of the P dependence of τα, and P0 is the VFT pressure at which the extrapolated τα diverges. Treating both DP and P0 as fitting parameters (i.e., the two-parameter fit), we find that DP is nearly independent of T for T ≤ 1.2 (see the inset to Figure 2b). The fitted results for T > 1.2 are less faithful since the ranges of τα are quite limited for these T (see the inset to Figure 2a). The analysis of ref 16 also indicates that DP remains approximately a constant for a range of T in a given material so that DP can be fixed in the PVFT equation. This one-parameter fit indeed works very well for all T considered in our simulations, as shown in Figure 2a, where DP is fixed to be 5.59, a value that is determined as the average of DP for T ≤ 1.2 using the two-parameter fits. Figure 2b also displays the fitted results for P0 from both two- and one-parameter fits as increasing with T, a trend that is consistent with experiment.16 In particular, P0 increases nearly linearly with T for the range of T considered in the one-parameter fits. We now discuss the characteristic temperatures of glass formation that serve to specify the temperature range of glass formation in our model polymer melt. These temperatures prescribe the onset at TA, the middle (or crossover) at Tc, and
Figure 2. (a) Relationship between the natural logarithm of indeed works very well for all T considered in othe reduced structural relaxation time ln(τα/τα,0) and DPP/(P0 − P) for various T. The dashed line indicates ln(τα/τα,0) = DPP/(P0 − P), with DP = 5.59. The inset to (a) depicts τα as a function of P for various T. Dashed lines indicate the one-parameter fits of τα to the pressure analog of the VFT equation, τα = τα,0 exp[DPP/(P0 − P)], where DP = 5.59. (b) T dependence of P0 determined from two- and one-parameter fits. Both DP and P0 are treated as fitting parameters in the two-parameter fits, whereas P0 is the only fitting parameter in the one-parameter fits using DP = 5.59. Dashed and solid lines indicate cubic and linear fits to P0 determined from two- and one-parameter fits, respectively. Both lines are a guide to the eye. The inset to (b) depicts the dependence of DP on T in the two-parameter fits. The dotted line indicates the average of DP for T ≤ 1.2, which is then used in the one-parameter fits.
the termination at T0 of glass formation as well as the common glass transition temperature Tg, below which the material falls out of equilibrium. Moreover, prior simulations5,17−22 identify a “localization” or “caging” temperature Tl at which particle caging starts to emerge. The determinations of these characteristic temperatures are briefly described below. Previous simulations5,21,22 demonstrate that “localization” or “caging” is evidenced in the mean-squared displacement, ⟨r2(t)⟩ = 1/N ⟨∑Nj=1|rj(t) − rj(0)|2⟩. Specifically, the localization temperature Tl is identified by the appearance of a minimum 1376
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ACS Macro Letters around 1 time unit in ∂ ln⟨r2(t)⟩/∂ ln(t). The resultant Tl is shown as a plus sign (+) for each P in Figure 1a. The estimation of Tl by this method is evidently difficult by experiments, but our recent work11 suggests another method of estimating Tl based on the Hansen-Verlet freezing rule,23,24 which requires only the static structure factor and, thus, appears to offer a simpler approach to determining Tl experimentally. TA is obtained first by fitting τα at high T for each P to the Arrhenius equation, τα = τ∞exp(ΔE/T), where τ∞ is the usual vibrational prefactor and ΔE is the “activation energy”. The fitted results are represented as dashed lines in Figure 1a. TA is then identified as the highest temperature available in our simulations at which τα − τα(Arrhenius) > 10−3, where τα(Arrhenius) is the expected structural relaxation time from the Arrhenius fits. The resultant TA appears as a cross sign (×) for each P in Figure 1a. For convenience, TA is just called the Arrhenius temperature.12 The crossover temperature Tc of glass formation is estimated by fitting τα to a power-law equation, τα = τc(T − Tc)−γ, where τc and γ are additional fitting parameters. This power-law scaling was first suggested by the mode-coupling theory (MCT) of glass formation,25 and, accordingly, Tc is often termed the “mode-coupling temperature”. The same power-law form is predicted to occur in the observed T range by the GET26 along with the values of the scaling exponent γ. Within the GET, Tc is precisely determined by an inflection point in the product of the configurational entropy density and T. To determine Tc, the power-law fits are performed for a range of τα, with 10 ≤ τα ≤ 103 for each P. The fitted results are shown as dash-dotted lines in Figure 1a. The determined γ lies between 2.68 and 2.84 and is comparable with those for other model glass-forming liquids25 and with the GET.26 Our fitted γ and Tc for P = 0 are also consistent with the MCT analysis of Baschnagel and co-workers,27 who investigate the influence of chain length on glass formation in a simulated glass-forming polymer melt at P = 0. T0 is estimated by fitting τα to the VFT equation, as noted earlier. Specifically, we fit τα(T) using a similar range in τα for each P. The lower bounds for τα are chosen to be τα(TA), and the upper bounds for τα are just the largest τα available for each P. The fitted results are shown as dotted lines in Figure 1a. The glass transition temperature Tg is then estimated from the VFT fits using the common empirical definition of τα(Tg) = 100 s. The reduced variables are mapped to physical units relevant to real polymer materials by approximating 1 time unit as 1 ps. Figure 3a summarizes the characteristic temperatures of glass formation as a function of P. While both Tl and TA increase with P nearly linearly, the other characteristic temperatures evidently grow with P in a nonlinear fashion. These general trends are consistent with experiment1 and the GET.3 In particular, the simple linear equation, Tx(P) = Tx(0) + f xP (x = l or A) with the determined constants f l and fA given in the caption of Figure 3, holds well for the range of P investigated. Inspired by the empirical equations that are used to describe the P dependence of Tg in experiment,1 the P dependence of Tc, Tg, and T0 can be well represented by the following equations: Tx(P) = Tx(0)[1 + (νx /μx )P]1/ νx , x = c, g, or 0
Figure 3. (a) Characteristic temperatures (Tc, Tg, and T0) as a function of P. Dotted and solid lines indicate the fits of Tx to the equations Tx(P) = Tx(0)[1 + (νx/μx)P]1/νx and Tx(P) = Tx(0) + AxP + BxP2 (x = c, g, or 0), where the fitting parameters are determined to be (νc, μc, Ac, Bc) = (1.39, 10.55, 3.70 × 10−2, −2.50 × 10−4), (νg, μg, Ag, Bg) = (1.49, 10.33, 3.30 × 10−2, −2.71 × 10−4), and (ν0, μ0, A0, B0) = (1.52, 10.33, 3.07 × 10−2, −2.65 × 10−4). The inset depicts the P dependence of Tl and TA. Dashed lines indicate fits of Tx to equation Tx = Tx(0) + f xP (x = l or A), where the fitting parameters are determined to be f l = 7.21 × 10−2 and fA = 7.10 × 10−2. (b) VFT fragility parameter KVFT as a function of P. The dashed line indicates the fit of KVFT to the equation KVFT(P) = KVFT(0)/(1 + k1P), where the fitting parameter is determined to be k1 = 6.92 × 10−3. The inset depicts the P dependence of Tr = Tl/T0. The dotted line indicates the fit of Tr to the equation Tr(P) = Tr(0) + t1P, where the fitting parameter is determined to be t1 = 2.53 × 10−2.
where the fitted parameters νx, μx, Ax, and Bx are provided in the caption of Figure 3. Figure 3b displays the VFT fragility parameter of glass formation as a function of P using KVFT ≡ 1/D, a quantity that conveniently increases as the fragility grows. KVFT is shown to decrease as P is elevated. We find that the P dependence of KVFT can be described by the simple equation, KVFT(P) = KVFT(0)/(1 + k1P), with k1 = 6.92 × 10−3. Figure 3b further examines how P influences the breadth of glass formation, a property that has long been advocated as a complementary measure of the fragility of glass formation.26,28−32 In particular, the GET calculations even indicate a universal relationship between the breadth of glass formation and KVFT in polymer melts.33,34 We particularly consider the ratio Tr ≡ Tl/T0 as a measure of the breadth of glass formation since Tl and T0 describe the onset and end of glass formation, respectively. The inset to Figure 3b indicates that Tr increases with P, implying that glass formation becomes broader for higher P. The P dependence of Tr can be described by the simple equation, Tr = Tr(0) + t1P, with t1 = 2.53 × 10−2. Therefore, both measures of fragility, KVFT and Tr, indicate that the fragility of glass formation decreases with increasing P in our model nonassociating polymer melt. This trend agrees with many
(2)
and Tx(P) = Tx(0) + Ax P + Bx P 2 , x = c, g, or 0
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ACS Macro Letters experiments for nonassociating polymers1 and with the general rules for fragilities proposed by Paluch and co-workers.35 It is also noted in ref 35 that the isochoric fragility remains invariant for a given material, a point that remains to be investigated in simulations over a range of fixed densities. We now study how the dependence of τα on P and T can be understood from the influence of P on the molecular dynamics of our model polymer melt. In particular, AG4 heuristically proposed a rationale for relaxation in glass-forming liquids in terms of a temperature dependent activation free-energy barrier whose height was assumed to be proportional to the number of particles in the CRRs. While this conceptual picture is quite appealing, no precise molecular definition of the CRRs was provided by AG. The string model5 provides a quantitative realization of the AG theory, in which the stringlike clusters provide a concrete realization of the CRRs. The string model is further justified by a recent theoretical analysis by Freed,36 where transition state theory is extended to account for stringlike cooperative barrier crossing events in glass-forming liquids. Predictions from the string model have been confirmed in a range of glass-forming polymer systems5,20−22 and metallic glass-forming liquids.37 The central prediction of the string model of glass formation is that the activation free energy ΔG for structural relaxation is proportional to the average string length L, where the proportionality factor becomes unity at an onset temperature. The determination of L follows the previously established procedure for polymer systems.19,38 First, mobile particles are defined for simplicity as the f 0 = 6.5% of particles with the greatest displacement over any chosen interval. Two mobile particles j and k are then considered to be in the same string if min[|rj(t ) − rk(0)| , |rk(t ) − rj(0)|] < δ
Figure 4. Relationship between the natural logarithm of the reduced structural relaxation time ln(τα/τ0) and ΔGo(L/Ll)/kBT for various P. Dashed lines indicate ln(τα/τ0) = ΔGo(L/Ll)/kBT. The inset depicts the relation between ΔHo and ΔSo. The dotted line indicates a fit to the equation ΔHo = H0 − TcompΔSo, where the fitting parameters are determined to be H0 = 7.27 and Tcomp = 2.28.
(4)
where δ = 0.55 is used in this work. Then, the number-averaged string length ⟨s(t)⟩ is calculated as ⟨s(t)⟩ ≡ ⟨∑∞ s=1sC(s)⟩/ ∑∞ s=1C(s), where C(s) is the probability of finding a string of length s, and this quantity displays a maximum L ≡ ⟨s(tL)⟩ at a characteristic time tL, where L defines the characteristic string length.5,17−22 If the onset temperature is taken to be Tl, the string model implies the relation ΔG(T) = ΔGo(L/Ll), where ΔGo = ΔHo − TΔSo is the activation free energy at high T with ΔHo and ΔSo designating the enthalpy and entropy of activation. τα then takes the form ⎛ ΔG L ⎞ τα = τ0 exp⎜ o ⎟ ⎝ kBT L l ⎠
Figure 5. Enthalpy ΔHo and entropy ΔSo of activation as a function of P. Lines indicate the fits of data for ΔHo and ΔSo to the equations ΔHo(P) = ΔHo(0)[1 + (νH/μH)P]1/νH and ΔSo(P) = ΔSo(0)/[1 + (νS/μS)P]1/νS, where the fitting parameters are determined to be (νH, μH) = (2.21, 13.50) and (νS, μS) = (5.17 × 10−2, 11.26).
simulations20−22 indicate that ΔHo and ΔSo both vary proportionally in polymer nanocomposites and thin polymer films. Such a relation is also found in our simulations when P is varied (see the inset to Figure 4). We thus expect the activation enthalpy−entropy relationship to arise in polymers under various applied fields (e.g., electric fields, magnetic fields, shear stress, etc.), a result that has significant implications for the properties of condensed materials subjected to numerous kinds of perturbations during processing or end-use applications. Our work also indicates that the entropic contribution to the activation free energy can be appreciable in glass-forming liquids, in particular, at low P. Therefore, it appears to be inappropriate to neglect the entropy of activation in modeling the dynamics of glass-forming liquids, as assumed in the AG theory4 and the GET.26 In summary, we have investigated the influence of pressure on the dynamics of a model glass-forming polymer melt. The application of pressure to polymeric and other glass-forming liquids can be expected to increase the packing efficiency of the molecules. Based on the general principle of the GET of glass formation that the variation of the fragility of glass-forming liquids derives from packing frustration (materials composed of molecules whose packing is more frustrated due to molecular
(5)
where τ0 can be determined from a knowledge of τα at the onset temperature21,22 so that ΔHo and ΔSo are the only adjustable parameters. Here, we examine whether or not eq 5 holds in our simulations. A detailed analysis for the T and P dependence of L will be presented elsewhere. Figure 4 shows that the string model quantitatively describes our simulation data for various P, thereby implying that the striking change of τα with P can be understood from the influence of P on L and the activation free energy parameters governing relaxation at high T where the fluid is dynamically homogeneous. Figure 5 displays the nonlinear variations of both ΔHo and ΔSo with P, as well described by the empirical equations, ΔHo(P) = ΔHo(0)[1 + (νH/μH)P]1/νH and ΔSo(P) = ΔSo(0)/ [1 + (νS/μS)P]1/νS, where the fitting parameters νH, μH, νS, and μS are provided in the caption of Figure 5. Previous 1378
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shape complexity, chain rigidity, etc. are more fragile),26 we expect the fragility of glass formation to decrease with increasing pressure. This trend is confirmed in our simulations. Likewise, elevating pressure is also expected to increase the cohesive interactions, and we thus observe that all the characteristic temperatures of glass formation, which increase with the cohesive interaction strength,11 grow with P. Beyond these expected qualitative trends, we demonstrate that the string model of glass formation, which is broadly consistent with the AG theory4 and the GET,26 quantitatively describes our relaxation data for all P studied. Our work then provides a remarkable validation of this model and motivates the application of the string model to other important problems with applied fields (e.g., polymers in a gap with applied normal forces, polymers subjected to electric and magnetic fields, etc.). The enthalpy and entropy of activation free energy governing the high temperature relaxation dynamics emerge as proportional to each other as P is varied, an effect as large in scale as changing the activation free energy due to collective motion and thus an effect of central importance in understanding the dynamics of fluids generally. A proportional change between the enthalpy and entropy of activation has recently been observed in dislocation nucleation of metallurgical materials under applied stress.39 The entropic contribution to the activation free energy has also been suggested to be important in modeling plastic deformation processes in polymeric and other glass-forming materials.40,41 These results, along with the present work, thus, suggest that the assumption in the classical Eyring plasticity theory of amorphous materials42 that the entropy of activation is unchanged with deformation is unwarranted. Finally, we stress that some fluids exhibiting hydrogen bonding, such as water and silica,43 can actually become more structured under applied pressures so the results of the present paper cannot be anticipated to apply to polymers with strong hydrogen-bonded interactions. However, polymer models with this type of associative interactions would be an interesting topic for future study to determine if the present theoretical framework describes relaxation in these complex fluids.
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REFERENCES
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. *E-mail:
[email protected]. ORCID
Wen-Sheng Xu: 0000-0002-5442-8569 Present Address ‡
Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831, United States. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank an anonymous reviewer for bringing ref 16 to our attention, which motivates our more thorough analysis for the pressure dependence of the structural relaxation time at fixed temperatures, as shown in Figure 2. We are grateful for the support of the University of Chicago Research Computing Center for assistance with the simulations carried out in this work. This work is supported, in part, by the National Science Foundation (NSF) Grant No. CHE-1363012. 1379
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