Dynamics of Polymers: a Mean-Field Theory

We derive a general mean-field theory of inhomogeneous polymer dynamics; a theory whose form has been speculated and widely applied, but not heretofore derived. Our approach involves a functional integral representation of a Martin-Siggia-Rose type description of the exact many-chain dynamics. A saddle point approximation to the generating functional, involving conditions where the MSR action is stationary with respect to a collective density field $\rho$ and a conjugate MSR response field $\phi$, produces the desired dynamical mean-field theory. Besides clarifying the proper structure of mean-field theory out of equilibrium, our results have implications for numerical studies of polymer dynamics involving hybrid particle-field simulation techniques such as the single-chain in mean-field method (SCMF)

Hydrodynamic Self-Consistent Field Theory for Inhomogeneous Polymer Melts

We introduce a mesoscale technique for simulating the structure and rheology of block copolymer melts and blends in hydrodynamic flows. The technique couples dynamic self consistent field theory (DSCFT) with continuum hydrodynamics and flow penalization to simulate polymeric fluid flows in channels of arbitrary geometry. We demonstrate the method by studying phase separation of an ABC triblock copolymer melt in a sub-micron channel with neutral wall wetting conditions. We find that surface wetting effects and shear effects compete, producing wall-perpendicular lamellae in the absence of flow, and wall-parallel lamellae in cases where the shear rate exceeds some critical Weissenberg number.Comment: Revised as per peer revie

Coherent States Formulation of Polymer Field Theory

We introduce a stable and efficient complex Langevin (CL) scheme to enable the first numerical simulations of the coherent-states (CS) formulation of polymer field theory. In contrast with Edwards' well known auxiliary-field (AF) framework, the CS formulation does not contain an embedded non-linear, non-local functional of the auxiliary fields, and the action of the field theory has a fully explicit, finite-order and semi-local polynomial character. In the context of a polymer solution model, we demonstrate that the new CS-CL dynamical scheme for sampling fluctuations in the space of coherent states yields results in good agreement with now-standard AF simulations. The formalism is potentially applicable to a broad range of polymer architectures and may facilitate systematic generation of trial actions for use in coarse-graining and numerical renormalization-group studies.Comment: 14pages 8 figure

Partitioning of a polymer chain between a confining cavity and a gel

A lattice field theory approach to the statistical mechanics of charged polymers in electrolyte solutions [S. Tsonchev, R. D. Coalson, and A. Duncan, Phys. Rev. E 60, 4257, (1999)] is applied to the study of a polymer chain contained in a spherical cavity but able to diffuse into a surrounding gel. The distribution of the polymer chain between the cavity and the gel is described by its partition coefficient, which is computed as a function of the number of monomers in the chain, the monomer charge, and the ion concentrations in the solution.Comment: 17 pages, 6 figure

Tilt grain boundary instabilities in three dimensional lamellar patterns

We identify a finite wavenumber instability of a 90$^{\circ}$ tilt grain boundary in three dimensional lamellar phases which is absent in two dimensional configurations. Both a stability analysis of the slowly varying amplitude or envelope equation for the boundary, and a direct numerical solution of an order parameter model equation are presented. The instability mode involves two dimensional perturbations of the planar base boundary, and is suppressed for purely one dimensional perturbations. We find that both the most unstable wavenumbers and their growth rate increase with $\epsilon$, the dimensionless distance away from threshold of the lamellar phase.Comment: 11 pages, 7 figures, to be published in Phys. Rev.

Microphase separation in polyelectrolytic diblock copolymer melt : weak segregation limit

We present a generalized theory of microphase separation for charged-neutral diblock copolymer melt. Stability limit of the disordered phase for salt-free melt has been calculated using Random Phase Approximation (RPA) and self-consistent field theory (SCFT). Explicit analytical free energy expressions for different classical ordered microstructures (lamellar, cylinder and sphere) are presented. We demonstrate that chemical mismatch required for the onset of microphase separation ($\chi^{\star} N$) in charged-neutral diblock melt is higher and the period of ordered microstructures is lower than those for the corresponding neutral-neutral diblock system. Theoretical predictions on the period of ordered structures in terms of Coulomb electrostatic interaction strength, chain length, block length, and the chemical mismatch between blocks are presented. SCFT has been used to go beyond the stability limit, where electrostatic potential and charge distribution are calculated self-consistently. Stability limits calculated using RPA are in perfect agreement with the corresponding SCFT calculations. Limiting laws for stability limit and the period of ordered structures are presented and comparisons are made with an earlier theory. Also, transition boundaries between different morphologies have been investigated

The effect of shear on persistence in coarsening systems

We analytically study the effect of a uniform shear flow on the persistence properties of coarsening systems. The study is carried out within the anisotropic Ohta-Jasnow-Kawasaki (OJK) approximation for a system with nonconserved scalar order parameter. We find that the persistence exponent theta has a non-trivial value: theta = 0.5034... in space dimension d=3, and theta = 0.2406... for d=2, the latter being exactly twice the value found for the unsheared system in d=1. We also find that the autocorrelation exponent lambda is affected by shear in d=3 but not in d=2.Comment: 6 page

Evidence of a Critical time in Constrained Kinetic Ising models

We study the relaxational dynamics of the one-spin facilitated Ising model introduced by Fredrickson and Andersen. We show the existence of a critical time which separates an initial regime in which the relaxation is exponentially fast and aging is absent from a regime in which relaxation becomes slow and aging effects are present. The presence of this fast exponential process and its associated critical time is in agreement with some recent experimental results on fragile glasses.Comment: 20 Pages + 7 Figures, Revte

Facilitated spin models: recent and new results

Facilitated or kinetically constrained spin models (KCSM) are a class of interacting particle systems reversible w.r.t. to a simple product measure. Each dynamical variable (spin) is re-sampled from its equilibrium distribution only if the surrounding configuration fulfills a simple local constraint which \emph{does not involve} the chosen variable itself. Such simple models are quite popular in the glass community since they display some of the peculiar features of glassy dynamics, in particular they can undergo a dynamical arrest reminiscent of the liquid/glass transitiom. Due to the fact that the jumps rates of the Markov process can be zero, the whole analysis of the long time behavior becomes quite delicate and, until recently, KCSM have escaped a rigorous analysis with the notable exception of the East model. In these notes we will mainly review several recent mathematical results which, besides being applicable to a wide class of KCSM, have contributed to settle some debated questions arising in numerical simulations made by physicists. We will also provide some interesting new extensions. In particular we will show how to deal with interacting models reversible w.r.t. to a high temperature Gibbs measure and we will provide a detailed analysis of the so called one spin facilitated model on a general connected graph.Comment: 30 pages, 3 figure

Orientational phase transitions in the hexagonal phase of a diblock copolymer melt under shear flow

We generalize the earlier theory by Fredrickson [J. Rheol. v.38, 1045 (1994)] to study the orientational behaviour of the hexagonal phase of diblock copolymer melt subjected to steady shear flow. We use symmetry arguments to show that the orientational ordering in the hexagonal phase is a much weaker effect than in the lamellae. We predict the parallel orientation to be stable at low and the perpendicular orientation at high shear rates. Our analysis reproduces the experimental results by Tepe et al. [Macromolecules v.28, 3008 (1995)] and explains the difficulties in experimental observation of the different orientations in the hexagonal phase.Comment: 21 pages, 6 eps figures, submitted to Physical Review