^{1}, Jack F. Douglas

^{2}and Srikanth Sastry

^{3}

### Abstract

We carefully examine common measures of dynamical heterogeneity for a model polymer melt and test how these scales compare with those hypothesized by the Adam and Gibbs (AG) and random first-order transition (RFOT) theories of relaxation in glass-forming liquids. To this end, we first analyze clusters of highly mobile particles, the string-like collective motion of these mobile particles, and clusters of relative low mobility. We show that the time scale of the high-mobility clusters and strings is associated with a diffusive time scale, while the low-mobility particles' time scale relates to a structural relaxation time. The difference of the characteristic times for the high- and low-mobility particles naturally explains the well-known decoupling of diffusion and structural relaxation time scales. Despite the inherent difference of dynamics between high- and low-mobility particles, we find a high degree of similarity in the geometrical structure of these particle clusters. In particular, we show that the fractal dimensions of these clusters are consistent with those of swollen branched polymers or branched polymers with screened excluded-volume interactions, corresponding to lattice animals and percolation clusters, respectively. In contrast, the fractal dimension of the strings crosses over from that of self-avoiding walks for small strings, to simple random walks for longer, more strongly interacting, strings, corresponding to flexible polymers with screened excluded-volume interactions. We examine the appropriateness of identifying the size scales of either mobile particle clusters or strings with the size of cooperatively rearranging regions (CRR) in the AG and RFOT theories. We find that the string size appears to be the most consistent measure of CRR for both the AG and RFOT models. Identifying strings or clusters with the “mosaic” length of the RFOT model relaxes the conventional assumption that the “entropic droplets” are compact. We also confirm the validity of the entropy formulation of the AG theory, constraining the exponent values of the RFOT theory. This constraint, together with the analysis of size scales, enables us to estimate the characteristic exponents of RFOT.

J.F.D. acknowledges support from National Institutes of Health (NIH) Grant No. 1 R01 EB006398-01A1. F.W.S. acknowledges support from National Science Foundation (NSF) Grant No. CNS-0959856 and ACS-PRF Grant No. 51983-ND7.

I. INTRODUCTION

II. MODEL AND SIMULATION DETAILS

III. DYNAMICAL CLUSTERS APPROACHING THE GLASS TRANSITION

A. Mobile and immobile clusters

B. Cluster size distribution and fractal dimension

C. String-like cooperative motion

IV. DYNAMICAL SCALES AND RELAXATION

A. Summary of the AG and RFOT predictions

B. Testing the Adam and Gibbs approach

C. Testing the RFOT approach

V. DISCUSSION AND CONCLUSION

## Figures

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 τ_{α} is defined by *F*(*q* _{0}, τ_{α}) = 0.2. Symbols are shown for the lowest *T* to indicate typical intervals at which data are collected. (b) The non-Gaussian parameter α_{2}(*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 α_{2}(*t*). The inset compares the behavior of τ_{α} and *t**.

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 τ_{α} is defined by *F*(*q* _{0}, τ_{α}) = 0.2. Symbols are shown for the lowest *T* to indicate typical intervals at which data are collected. (b) The non-Gaussian parameter α_{2}(*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 α_{2}(*t*). The inset compares the behavior of τ_{α} 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.

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 τ_{α} and *t**, shown in Fig. 1(b) . (b) and (c) Parametric plots that show that *t* _{ M } ∼ *t** and *t* _{ I } ∼ τ_{α}; the dashed lines indicate an equality between these quantities.

(a) Characteristic time where the cluster size peaks for mobile (black circles) and immobile (red squares) clusters. Note the similarity to the behaviors of τ_{α} and *t**, shown in Fig. 1(b) . (b) and (c) Parametric plots that show that *t* _{ M } ∼ *t** and *t* _{ I } ∼ τ_{α}; 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.

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.

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.

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 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 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 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 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 τ_{α}.

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 τ_{α}.

Temperature dependence of the activation free energy Δ*F*(*T*), evaluated from the relaxation time according to Eq. (6) . The dashed line indicates the high *T* asymptotic value Δ*F* _{ A } = 2.2. The inset shows values of Δ*F*, 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.

Temperature dependence of the activation free energy Δ*F*(*T*), evaluated from the relaxation time according to Eq. (6) . The dashed line indicates the high *T* asymptotic value Δ*F* _{ A } = 2.2. The inset shows values of Δ*F*, 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.

Confirmation that the entropy representation of the AG theory (Eq. (9) ) is valid for the present system. The inset shows the *T* dependence of *S* _{conf} and the extrapolating vanishing temperature *T* _{ K }, which is consistent with *T* _{0} of the VFT fit (Eq. (1) ) to τ.

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 Δ*F* with no free parameters in the comparison.

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 Δ*F* 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.

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.

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.

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.

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.

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.”

(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 }.

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 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*.

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.

(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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