A transport model and constitutive equation for oppositely charged polyelectrolyte mixtures with application to layer-by-layer assembly

Abstract

We develop a general framework for transport of polyions, solvent and salt, with intended application to Layer-by-Layer (LbL) assembly of polyelectrolyte monolayers (PEMs). The formulation for the first time includes electrostatics, chemical potential gradients, and mechanical stress gradients as driving forces for mass transport. The general model allows all species to be mobile throughout the process and avoids the assumptions of stepwise instantaneous equilibrium and/or immobilized structures typical of previous approaches, while reducing to these models in appropriate limits. A simple constitutive equation is derived for a mixture of oppositely charged polyelectrolytes that accounts for network strand dilution and cross-chain ion pairing by appending reactive terms to the Smoluchowski probability diffusion equation for network strand end-to-end vectors. The resulting general framework encompasses the Poisson equation describing the electrostatic potential distribution, an osmotic pressure balance, a stress constitutive equation, and a generalized flux law of polymer transport. The computational domain is split into a PEM phase and an external solution phase with an appropriate boundary condition derived for the interface between the two. The mobile species (water and small salt ions) are taken to be in a state of dynamic equilibrium with their distributions enslaved to the perturbations in the two polyion compositions. The proposed model captures the swelling response of PEM films to external solutions. For the first time, we studied the effects of the temporal evolution of electrostatic and stress distribution on the rate of chain loss and absorption during rinsing and dipping of an idealized and arbitrarily selected and rigid brush layer into external solutions. The temporal evolution provides a kinetic basis for the ability of LbL films to grow under conditions that thermodynamics alone suggests would cause them to be washed away and to account for partial desorption during washing. The proposed transport framework constitutes a solid basis for eventual quantitative modeling of LbL assembly and transport in polyion networks more generally.

Ion Pairing and the Structure of Gel Coacervates

“…[75][76][77][78] Similarly, analytical extensions of the Voorn-Overbeek model to account for chain connectivity, electrostatic fluctuations, and ion binding rely on an underlying mean-field approximation and an assumption of homogeneous structure, which limit predictions for highly-correlated or inhomogeneous structures. [79][80][81][82][83][84] Scaling models, however, offer a versatile approach to predicting phase behavior and structures of coacervates depending on many molecular features, without the numerical complications of other models. A combination of such analytical theories and molecular simulations will enable progress in understanding coacervation phenomena.…”

“…Integral equation formalisms, in particular, liquid-state theories or PRISM, should in principle cover both weak and strong association regimes, but they are restricted by obtuse closure approximations necessary for their analytical and numerical implementation, which do not allow their systematic improvement. − Similarly, analytical extensions of the Voorn–Overbeek model to account for chain connectivity, electrostatic fluctuations, and ion binding rely on an underlying mean-field approximation and an assumption of homogeneous structure, which limit predictions for highly correlated or inhomogeneous structures. − …”