First-principle approach to rescale the dynamics of simulated coarse-grained macromolecular liquids

Abstract

We present a detailed derivation and testing of our approach to rescale the dynamics of mesoscale simulations of coarse-grained polymer melts (I. Y. Lyubimov, J. McCarty, A. Clark, and M. G. Guenza, J. Chem. Phys. 132, 224903 (2010)). Starting from the first-principle Liouville equation and applying the Mori-Zwanzig projection operator technique, we derive the generalized Langevin equations (GLEs) for the coarse-grained representations of the liquid. The chosen slow variables in the projection operators define the length scale of coarse graining. Each polymer is represented at two levels of coarse graining: monomeric as a bead-and-spring model and molecular as a soft colloid. In the long-time regime where the center-of-mass follows Brownian motion and the internal dynamics is completely relaxed, the two descriptions must be equivalent. By enforcing this formal relation we derive from the GLEs the analytical rescaling factors to be applied to dynamical data in the coarse-grained representation to recover the monomeric description. Change in entropy and change in friction are the two corrections to be accounted for to compensate the effects of coarse graining on the polymer dynamics. The solution of the memory functions in the coarse-grained representations provides the dynamical rescaling of the friction coefficient. The calculation of the internal degrees of freedom provides the correction of the change in entropy due to coarse graining. The resulting rescaling formalism is a function of the coarse-grained model and thermodynamic parameters of the system simulated. The rescaled dynamics obtained from mesoscale simulations of polyethylene, represented as soft-colloidal particles, by applying our rescaling approach shows a good agreement with data of translational diffusion measured experimentally and from simulations. The proposed method is used to predict self-diffusion coefficients of new polyethylene samples.

Challenges in Multiscale Modeling of Polymer Dynamics

“…According to Stokes's law [216], the friction coefficient, ζ, is related to the hydrodynamic radius, r hydr , through the solvent viscosity, η solvent , as ζ = 6πη solvent r hydr . Thus, in the coarse-graining process, the internal friction coefficient between monomers is typically also changed, leading to incorrect dynamic behavior of the coarse-grained system [37,217,218]. It is therefore necessary to perform dynamic mapping (i.e., rescale the dynamics) in order to simulate the correct behavior.…”

“…Very recently, Guenza and her co-workers developed another super coarse-grained model by coarse-graining the polymer chain into a sphere with radius equal to R G , based on the Ornstein-Zernike equations [71,72,87,88,217,218,[268][269][270][271][272]. The newly derived model is different from the previous IBI models or super coarse-grained model in four aspects: (i) the model was derived analytically through the Ornstein-Zernike equation [11]; (ii) it is not state-dependent, in contrast with the effective potential functions derived through the IBI method; (iii) the analytical solution does not need further optimization against the more detailed model; and (iv) the thermodynamic quantities (as well as the self-diffusion coefficient) of the super coarse-grained model can also be analytically determined.…”

“…In deriving this super coarse-grained analytical model, Guenza and her co-workers first mapped an all-atomistic polymer chain onto a freely-rotating chain model [217,218]. They then applied the hard-bead approximation to evaluate the integrals of the Ornstein-Zernike equation [273].…”

“…For N > 30, the approximation of h cc given above is exact [73,274]. The inter-particle potential in this super coarse-grained model is then obtained using a hypernetted-chain closure approximation [175,218]:…”

“…The center of the soft-colloidal particle lies on the center of mass of its corresponding polymer chain. The total intermolecular correlation function between these centers can be formulated for long chains (N → ∞) as [218]:…”