Focusing on isotropic elastic networks we propose a novel simple-average expression $G(t) = mu_A - h(t)$ for the computational determination of the shear-stress relaxation modulus $G(t)$ of a classical elastic solid or fluid and its equilibrium modul
us $G_{eq} = lim_{t to infty} G(t)$. Here, $mu_A = G(0)$ characterizes the shear transformation of the system at $t=0$ and $h(t)$ the (rescaled) mean-square displacement of the instantaneous shear stress $hat{tau}(t)$ as a function of time $t$. While investigating sampling time effects we also discuss the related expressions in terms of shear-stress autocorrelation functions. We argue finally that our key relation may be readily adapted for more general linear response functions.
We revisit the relation between the shear stress relaxation modulus $G(t)$, computed at finite shear strain $0 < gamma ll 1$, and the shear stress autocorrelation functions $C(t)|_{gamma}$ and $C(t)|_{tau}$ computed, respectively, at imposed strain $
gamma$ and mean stress $tau$. Focusing on permanent isotropic spring networks it is shown theoretically and computationally that in general $G(t) = C(t)|_{tau} = C(t)|_{gamma} + G_{eq}$ for $t > 0$ with $G_{eq}$ being the static equilibrium shear modulus. $G(t)$ and $C(t)|_{gamma}$ thus must become different for solids and it is impossible to obtain $G_{eq}$ alone from $C(t)|_{gamma}$ as often assumed. We comment briefly on self-assembled transient networks where $G_{eq}(f)$ must vanish for a finite scission-recombination frequency $f$. We argue that $G(t) = C(t)|_{tau} = C(t)|_{gamma}$ should reveal an intermediate plateau set by the shear modulus $G_{eq}(f=0)$ of the quenched network.
A coarse-grained simulation model eliminates microscopic degrees of freedom and represents a polymer by a simplified structure. A priori, two classes of coarse-grained models may be distinguished: those which are designed for a specific polymer and r
eflect the underlying atomistic details to some extent, and those which retain only the most basic features of a polymer chain (chain connectivity, short-range excluded-volume interactions, etc.). In this review we mainly focus on the second class of generic polymer models, while the first class of specific coarse-grained models is only touched upon briefly.
The scaling of the bond-bond correlation function $C(s)$ along linear polymer chains is investigated with respect to the curvilinear distance, $s$, along the flexible chain and the monomer density, $rho$, via Monte Carlo and molecular dynamics simula
tions. % Surprisingly, the correlations in dense three dimensional solutions are found to decay with a power law $C(s) sim s^{-omega}$ with $omega=3/2$ and the exponential behavior commonly assumed is clearly ruled out for long chains. % In semidilute solutions, the density dependent scaling of $C(s) approx g^{-omega_0} (s/g)^{-omega}$ with $omega_0=2-2 u=0.824$ ($ u=0.588$ being Florys exponent) is set by the number of monomers $g(rho)$ contained in an excluded volume blob of size $xi$. % Our computational findings compare well with simple scaling arguments and perturbation calculation. The power-law behavior is due to self-interactions of chains on distances $s gg g$ caused by the connectivity of chains and the incompressibility of the melt. %
Whereas the first part of this paper dealt with the relaxation in the beta-regime, this part investigates the final (alpha) relaxation of a simulated polymer melt consisting of short non-entangled chains above the critical temperature Tc of mode-coup
ling theory (MCT). We monitor the intermediate incoherent as well as the coherent chain and coherent melt scattering functions over a wide range of wave numbers q. Upon approaching Tc the coherent alpha-relaxation time of the melt increases strongly close to the maximum of the static structure factor of the melt. At q corresponding to the radius of gyration of the chain the melt relaxation time exhibits another maximum. The temperature dependence of the relaxation times is well described by a power-law with a q-dependent exponent in an intermediate temperature range. The time-temperature superposition principle of MCT is clearly bourne out in the whole range of wave numbers. An analysis of the alpha-decay using Kohlrausch-Williams-Watts (KWW) functions reveals that the collective melt KWW-stretching exponent and KWW-relaxation times are modulated with the structure factor. Furthermore, both incoherent and coherent KWW-times approach the large-q prediction of MCT at q comparable to the maximum of the structure factor. At small q a power law with exponent -3 is found for the coherent chain KWW-times similar to that of recent experiments.
We report results of molecular-dynamics simulations of a model polymer melt consisting of short non-entangled chains in the supercooled state above the critical temperature of mode-coupling theory (MCT). To analyse the dynamics of the system we compu
ted the incoherent, collective chain and melt intermediate scattering functions as well as the Van Hove correlation functions. We find good evidence for the space-time factorization theorem of MCT. From the critical amplitudes we could derive typical length scales of the beta-dyamics. In an extensive quantitative analysis the leading order description of MCT was found to be accurate in the central beta-regime. Higher order corrections extend the validity of the MCT approximation to a greater time window. Indications of polymer specific effects on the length scale of the chains radius of gyration are visible in the beta-coefficients.
