A phase field model for dynamic simulations of reactive blending of polymers.

A facile way to generate compatibilized blends of immiscible polymers is through reactive blending of end-functionalized homopolymers. The reaction may be reversible or irreversible depending on the end-groups and is affected by the immiscibility and transport of the reactant homopolymers and the compatibilizing copolymer product. Here we describe a phase-field framework to model the combined dynamics of reaction kinetics, diffusion, and multi-component thermodynamics on the evolution of the microstructure and reaction rate in reactive blending. A density functional with no fitting parameters, which is obtained by adapting a framework of Uneyama and Doi and qualitatively agrees with self-consistent field theory, is used in a diffusive dynamics model. For a symmetric mixture of equal-length reactive polymers mixed in equal proportions, we find that depending on the Flory χ parameter, the microstructure of an irreversibly reacting blend progresses through a rich evolution of morphologies, including from two-phase coexistence to a homogeneous mixture, or a two-phase to three-phase coexistence transitioning to a homogeneous blend or a lamellar copolymer. The emergence of a three-phase region at high χ leads to a previously unreported reaction rate scaling. For a reversible reaction, we find that the equilibrium composition is a function of both the equilibrium constant for the reaction and the χ parameter. We demonstrate that phase-field models are an effective way to understand the complex interplay of thermodynamic and kinetic effects in a reacting polymer blend.

Interfacial reaction-induced roughening in polymer thin films.

Reactive blending of immiscible polymers is an important process for synthesizing polymer blends with superior properties. We use a phase-field model to understand reaction dynamics and morphology evolution by diffusive transport in layered films of incompatible, end-reactive polymers. We thoroughly investigate this phenomenon over a large parameter space of interface shapes, layer thicknesses, reaction rates specified by a Damkohler number (Daf), and Flory-Huggins interaction parameter (χ), under static conditions with no external fields. For films of the same thickness, the dynamics of the system is not significantly influenced by the length of the film or the initial shape of the interface. The interface between the polymers is observed to roughen, leading to the formation of a spontaneous emulsion. The reaction progresses slower and the interface roughens later for thicker films, and systems with higher χ. Increasing Daf increases the reaction rate and hastens the onset of roughening. The quasi-static interfacial tension decreases with the extent of reaction, but does not become vanishingly small or negative at the onset of roughening. Simulations with reversible reactions and systems where only a fraction of the homopolymers have reactive end groups show that a critical diblock (reaction product) concentration exists, below which interfacial roughening and spontaneous emulsification is not observed. We also demonstrate that thermal fluctuations accelerate the onset of interfacial roughening, and help sustain the system in an emulsified state.

Efficient field-theoretic simulation of polymer solutions.

We present several developments that facilitate the efficient field-theoretic simulation of polymers by complex Langevin sampling. A regularization scheme using finite Gaussian excluded volume interactions is used to derive a polymer solution model that appears free of ultraviolet divergences and hence is well-suited for lattice-discretized field theoretic simulation. We show that such models can exhibit ultraviolet sensitivity, a numerical pathology that dramatically increases sampling error in the continuum lattice limit, and further show that this pathology can be eliminated by appropriate model reformulation by variable transformation. We present an exponential time differencing algorithm for integrating complex Langevin equations for field theoretic simulation, and show that the algorithm exhibits excellent accuracy and stability properties for our regularized polymer model. These developments collectively enable substantially more efficient field-theoretic simulation of polymers, and illustrate the importance of simultaneously addressing analytical and numerical pathologies when implementing such computations.

Coherent states formulation of polymer field theory.

We introduce a stable and efficient complex Langevin (CL) scheme to enable the first direct 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 nonlinear, non-local, implicit functional of the auxiliary fields, and the action of the field theory has a fully explicit, semi-local, and finite-order 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-CL 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.

Numerical coarse-graining of fluid field theories.

We present a formalism for the systematic numerical coarse-graining of field-theoretic models of fluids that draws upon techniques from both the Monte Carlo renormalization group and particle-based coarse-graining literature. A force-matching technique initially developed for coarse-graining particle-based interaction potentials is adapted to calculate renormalized field-theoretic coupling coefficients in a complex-valued field theory, and a related method is introduced for coarse-graining field-theoretic operators. The viability of this methodology is demonstrated by coarse-graining a field-theoretic model of a Gaussian-core fluid and thereby reducing lattice discretization errors.