Characterization of basic dynamical properties of the polymer melt. (a) The coherent density-density correlation function F(q 0, t) for all T. The α-relaxation time τα is defined by F(q 0, τα) = 0.2. Symbols are shown for the lowest T to indicate typical intervals at which data are collected. (b) The non-Gaussian parameter α2(t) at each T shows a peak due to the correlated motion that occurs roughly on the time scale of t*, defined by the maximum of α2(t). The inset compares the behavior of τα and t*.
The dynamical average cluster size for (a) high-mobility particles ⟨n M (t)⟩ and (b) low-mobility particles ⟨n I (t)⟩ at all T studied. The data are normalized by the value at t = 0, equivalent the cluster size of the same fraction of particles chosen at random. The definitions of the mobility groups are discussed in the main text. The inset of part (a) shows the percolation probability p of mobile particle clusters as a function of T; the dotted vertical line indicates the temperature where p ≈ 0.5, a standard identifier of the percolation transition in finite-sized systems. 62 For immobile particle clusters, p < 0.2 for all T, so that percolation is not prevalent.
(a) Characteristic time where the cluster size peaks for mobile (black circles) and immobile (red squares) clusters. Note the similarity to the behaviors of τα and t*, shown in Fig. 1(b) . (b) and (c) Parametric plots that show that t M ∼ t* and t I ∼ τα; the dashed lines indicate an equality between these quantities.
Typical examples of (a) the most mobile and (b) least mobile clusters. Different clusters are shown in different colors, and the segments of all chains are shown translucent. (c) The same mobile clusters (all colored red) and immobile clusters (all colored blue) to facilitate comparing their relative spatial distribution.
The distribution of particle cluster sizes P(n) for (a) mobile and (b) immobile particle clusters. The distribution can be described by a power-law with an exponential cut-off, like clusters nearing the percolation transition. The dashed lines indicate a characteristic power-law. Different colors represent different T, as in previous figures.
Scaling of cluster mass n with the radius of gyration R g to define the fractal dimension d f of (a) mobile and (b) immobile particle clusters. The solid lines represent the simulation results, and different colors indicate different T, as in previous figures. Small circular symbols are shown for the lowest T, to indicate the density of data. The bold dashed lines are a guide to the eye, and provide approximate bounds on d f . This is more clearly seen in the insets, which show the T dependence of d f . The dashed lines of the insets indicate the limiting behaviors discussed in the text.
The dynamical string length L(t) for all T studied. The inset shows the T dependence of the characteristic peak value, simply denoted as L.
The characteristic time t L of the peak string length. The inset shows that, like the characteristic time of mobile particle clusters, t L scales nearly linearly with t*.
The distribution of string lengths P(L) follows an exponential law that can be anticipated by analogy with equilibrium polymerization. 73
The scaling to determine the fractal dimension d f of strings. The solid lines represent the simulation results, and different colors indicate different T, as in previous figures. Small circular symbols are shown for the lowest T, to indicate the density of data. The bold dashed lines are a guide to the eye, and provide approximate bounds on d f . This is more clearly seen in the insets, which show the T dependence of d f . The dashed lines of the inset indicate expected limiting behaviors. Specifically, there appears to be a modest change in d f from ≈5/3 for short strings (the approximate value for a self-avoiding walk in 3D) to ≈2 (a simple random walk, also characteristic of many branched polymers). Accordingly, a typical string at low T is less extended, just as in polymer chains in dilute versus concentrated solutions.
The temporal evolution of fractal dimension d f of mobile clusters and strings at the lowest T = 0.30. The mobile-particle cluster structure is relatively insensitive to t, but the string geometry differs dramatically between t* and τα.
Temperature dependence of the activation free energy ΔF(T), evaluated from the relaxation time according to Eq. (6) . The dashed line indicates the high T asymptotic value ΔF A = 2.2. The inset shows values of ΔF, string mass L, and mobile cluster mass n M normalized by their values at T A to facilitate comparison of the growth of these quantities on cooling.
Testing the mobile particles cluster and string sizes for consistency with the AG prediction that the activation energy is linear in the size z of CRR. The cluster size n M grows too rapidly on cooling to fit with the AG picture, the string size L appears consistent. See also Fig. 12 for a direct comparison of L and n M to the relative change in ΔF with no free parameters in the comparison.
Testing S conf ∝ 1/L. The inset checks for the best power-law relation, which yields an exponent slightly larger than one.
The mean radius of gyration ⟨R g ⟩ for the strings and mobile particle clusters. This defines a characteristic length that we can test within the RFOT framework.
Testing Eq. (11) to determine the exponent ψ of RFOT. Both the mobile particle clusters and strings yield a consistent fit with ψ ≈ 1.3. The data for the mobile particle clusters are shifted by two units along the abscissa for clarity of the figure.
Evaluation of the surface scaling exponent θ from the scaling law (Eq. (12) ). The value d f − θ ≈ 1.3 is consistent with ψ, and with the expectations from AG.
Linear scaling between t* and the (inverse) characteristic diffusion time D/T in the Kob-Anderson binary LJ mixture, 49 demonstrating that the time scale t* corresponds to the time scale for mass diffusion.
(a) Identification of the cage size from the mean-squared displacement. The inset shows the logarithmic derivative, which shows a minimum that we use to define . (b) The resulting T dependence of the cage size . Particles with a squared displacement less than are defined as “caged particles.”
Dynamical fraction of caged particles for T < T A .
The normalized cluster size for caged particles for T < T A , where the cage is well-defined.
The characteristic fraction of low mobility particles for several different approaches. All approaches yield similar fractions for all T.
(a) The total entropy S and vibrational entropy as a function of temperature. (b) The configurational component S conf = S − S vib. The line is a guide for the eye.
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