Dynamic Arrest in Polymer Melts: Competition between Packing and Intramolecular Barriers

We present molecular dynamics simulations of a simple model for polymer melts with intramolecular barriers. We investigate structural relaxation as a function of the barrier strength. Dynamic correlators can be consistently analyzed within the framework of the Mode Coupling Theory (MCT) of the glass transition. Control parameters are tuned in order to induce a competition between general packing effects and polymer-specific intramolecular barriers as mechanisms for dynamic arrest. This competition yields unusually large values of the so-called MCT exponent parameter and rationalize qualitatively different observations for simple bead-spring and realistic polymers. The systematic study of the effect of intramolecular barriers presented here also establishes a fundamental difference between the nature of the glass transition in polymers and in simple glass-formers.

Since they do not easily crystallize, polymers are probably the most extensively studied systems in relation with the glass transition phenomenon. Having said this, their macromolecular character, and in particular chain connectivity, must not be forgotten. Its most evident effect is the sublinear increase of the mean squared displacement (Rouse-like) arising after the decaging process, in contrast to the linear regime found in non-polymeric glass-formers. Another particular ingredient of polymers is that, apart from fast librations or methyl group rotations, every motion, as local as it be, involves jumps over carbon-carbon rotational barriers and/or chain conformational changes.
In this work we investigate, by means of molecular dynamics simulations, the decisive role of intramolecular barriers on the glass transition of polymer melts, by systematically tuning barrier strength in a simple bead-spring model. We discuss the obtained results within the framework of the Mode Coupling Theory (MCT) of the glass transition [1]. Initially derived for monoatomic hard-sphere systems, the theory has been further developed for more complex systems, including fully-flexible bead-spring chains as simple models for polymer melts [2]. MCT asymptotic laws have been tested in different polymeric systems. The values of the associated dynamic exponents exhibit significant differences between the limits of fully-flexible bead-spring chains [3] and fully-atomistic polymers [4]. In particular, the so-called exponent parameter takes standard values λ ∼ 0.7 for the former case and values approaching the upper limit λ = 1 for chemically realistic polymers [4]. While the former λ-values are characteristic of systems dominated by packing effects, as the archetype hard-sphere fluid, the limit λ = 1 arises at higher-order MCT transitions [5]. The latter, or more generally tran-sitions with λ < ∼ 1, arise in systems with different competing mechanisms for dynamic arrest. These systems include short-ranged attractive colloids [6,7] (competition between short-range attraction and hard-sphere repulsion) or binary mixtures with strong dynamic asymmetry [8,9] (bulk-like caging and confinement).
Motivated by these analogies, we argue that values λ < ∼ 1 for real polymers also arise from the competition between two distinct mechanisms for dynamic arrest: usual packing effects and polymer-specific intramolecular barriers. Such barriers are not present in fully-flexible bead-spring chains, which exhibit standard λ-values [3]. In order to shed light on this question, we perform a systematic investigation of the interplay between packing and intramolecular barriers. Starting from fully-flexible bead-spring chains, stiffness is introduced by implementing intramolecular bending and torsion terms. The barrier strength is systematically tuned in order to induce competition between the former two mechanisms. We restrict to stiffness for which no orientational order is present, and provide a complete dynamic picture of the isotropic phase as a function of the barrier strength. An extensive test of MCT asymptotic laws is performed. Simulation results are described with consistent sets of MCT exponents. A progressive increase of λ is induced by strengthening the competition between packing and intramolecular barriers, confirming the proposed scenario.
Temperature T , time t, wave vector q, and monomer density ρ are given respectively in units of ǫ/k B (with k B the Boltzmann constant), σ(m/ǫ) 1/2 , σ −1 , and σ −3 . We investigate, at fixed ρ = 1, the T -dependence of the dynamics for different values of the bending and torsion strength, (K B ,K T ) = (0,0), (15,0.5), (25,1), (25,4), and (35,4). The case (K B ,K T ) = (35, 4) is also studied for ρ = 0.93. The total number of chains is N c = 300. Periodic boundary conditions are implemented. Equations of motion are integrated in the velocity Verlet scheme [11]. The system is prepared by placing the chains randomly in the simulation box, with a constraint avoiding monomer core overlap. The initial monomer density is ρ = 0.375. Equilibration consists of a first run where the box is rescaled periodically by a factor 0.99 < f < 1 until the target density ρ is reached, and a second isochoric run at that ρ. Thermalization at the target T is achieved by periodic velocity rescaling. Once the system is equilibrated, a microcanonical run is performed for production of configurations, from which observables are computed. For each state point, the latter are averaged over typically 40 independent samples.
Orientational ordering (induced by chain stiffness) is discarded for all the analyzed cases by measuring the quantity P 2 (Θ) = (3 cos 2 Θ − 1)/2, where Θ is the angle between the end-to-end vectors of two chains, and average is performed over all pairs of distinct chains. In all cases we obtain negligible values |P 2 (Θ)| < 10 −2 .
We compute density-density correlators, defined , the sum extending over the positions r j of all the monomers in the system. Density self-correlators are defined as F s (q, t) = (N N c ) −1 Σ j exp{iq · [r j (t) − r j (0)]}. Results for the former quantities are shown in Fig. 1, at several q-values, for two state points with non-zero barriers, at T close to the critical MCT temperature (see below). As usual, a plateau is observed in the interval corresponding to the caging regime, i.e., the temporary trapping of a particle by its neighbors. This interval is known as the β-regime within the framework of MCT. The second decay, corresponding to full relaxation of density fluctuations of wave vector q, is known as the α-regime, and is often described by an empirical Kohlrausch-Williams-Watts (KWW) function, A q exp[−(t/τ K q ) βq ], where A q , the KWW time τ K q and the exponent β q are q-dependent. Next we summarize the basic predictions of MCT and test them in the present system. In its ideal version, MCT predicts a sharp transition [1] from an ergodic liquid to an arrested state (glass) at a given value of the relevant control parameters -here x = (T, ρ, K B , K T ). When crossing the transition point x = x c the longtime limit of F (q, t) and F s (q, t) jumps from zero to a non-zero value, denoted as the critical non-ergodicity parameter (f c q and f cs q , respectively). MCT predics asymptotic laws for dynamic observables. Such laws are characterized by dynamic exponents that are qand state-point independent. They are univoquely determined by the static correlations at x = x c [1]. Moreover, all the dynamic exponents are univoquely related to a single one, the exponent parameter λ (see below), which is the basic one controlling all MCT asymptotic laws. Now we summarize the main ones.
For ergodic states close to x c , the initial part of the α-process (i.e., the von Schweidler regime) is given by a power law expansion [1]: (and analogously for self-correlators) with 0 < b ≤ 1. The non-ergodicity parameters and the prefactors h q and h (2) q only depend on q and are different for each correlator. The α-relaxation time τ α only depends on the separation parameter |x − x c |. MCT predicts a divergence [12] according to the power law τ α ∝ |x − x c | −γ . In practice τ α can be defined as the time τ z where F (q max , t) decays to some small value z far below the plateau, with q max the q-value at the maximum of the static structure factor S(q) = (N N c ) −1 ρ(q, 0)ρ(−q, 0) . Here we will use τ 0.2 . The exponent γ is given by [1]: As mentioned above, the full α-decay can be described by a KWW function. In the limit of large q MCT predicts [13] for the KWW times a power law τ K q ∝ q −1/b . The exponents a, b, and γ are univoquely related to the exponent parameter λ ≤ 1 through [1]: with Γ the Euler's Gamma function. When numerical solutions of the MCT equations are not available, the former non-ergodicity parameters, prefactors and exponents are obtained as fit parameters from simulation or experimental data. Consistency of the analysis requires that dynamic correlators and relaxation times are described by a common set of exponents, univoquely related through Eqs. (2,3). This consistent test has been done for all the systems here investigated, with different strength of the intramolec-  Fig. 1 shows at fixed ρ = 1 and for a broad q-range, fits to Eq. (1) of density correlators for the state points K B = 15, K T = 0.5, T = 0.81 (S1) and K B = 35, K T = 4, T = 1.33 (S2). A good description is achieved, for all the q-values and over several time decades, with a fixed b-exponent (b = 0.50 and 0.37 for respectively S1 and S2). Fig. 2 displays, for the former barrier strength, the q-dependence of the critical non-ergodicity parameters. The fully flexible case K B = K T = 0 is also included. As deduced from the stronger decay of f c q and f cs q for stronger barriers, chain stiffness induces a weaker localization at fixed density. By making an approximate fit of f cs q to Gaussian behavior, exp(−q 2 l 2 c /6), we estimate, at fixed ρ = 1, a localization length l c = 0.19, 0.21, and 0.23 for respectively (K B , K T ) = (0,0), (15,0.5), and (34,4).
The increase of the barrier strength at fixed ρ also induces a higher critical temperature T c , and a longer relaxation time for fixed ρ and T . This is demonstrated in Fig. 3, which also shows a test of the predictions τ α ∝ (T − T c ) −γ and τ K q ∝ q −1/b (for large q) for the former values of the barrier strength. A good description is obtained with the same b-exponents used for the von Schweidler fits of Fig. 1, and   derived from them through Eqs. (2,3). This demonstrates the consistency of the data analysis. For comparison, Fig. 3 also includes results for the fully flexible case K B = K T = 0. Table I displays the results of the MCT analysis (dynamic exponents and T c ) for all the investigated cases. It also includes the mean chain end-to-end radius at T c , R c ee , as computed from the simulations. R c ee provides a qualitative characterization of chain stiffness. From numerical values in Table I a clear correlation between the exponent parameter λ and chain stiffness is unambiguously demonstrated. The competition between packing effects and intramolecular barriers induces a progressive increase of λ from the value λ = 0.76 for fully-flexible chains to λ = 0.89 for the stiffest investigated chains.
This observation rationalizes the large difference observed between MCT exponents for fully-flexible beadspring chains and chemically realistic polymers. Table  II shows a representative compilation of exponents for glass-formers of very different nature. Exponents for fully-flexible bead-spring chains are similar to those of non-polymeric glass-formers, including the hard-sphere fluid, i.e., the archetype glass-former dominated by packing effects. Chemically realistic polymers of increasing complexity exhibit instead values approaching the limit λ = 1 characteristic of higher-order MCT transitions. The systematic study presented in this work strongly suggests a competition between general packing effects and polymer-specific intramolecular barriers as the origin of this difference. It also suggests a fundamental difference in the nature of the glass transition in real polymers -driven by the former competing mechanisms-as compared to simple glass-formers [14]. Real polymers are thus classified in the family of complex systems as short-ranged attractive colloids [6,7] or binary mixtures with strong dynamic asymmetry [8,9], which are characterized by an underlying higher-order MCT transition -or at least by unusually large values of λarising from a competition between distinct mechanisms for dynamic arrest. Finally, results reported here provide fundamental information for microscopic theories (and in particular for MCT) of the glass transition in polymers, which need to account for the decisive role of intramolecular barriers.