Analytical insights into the phase transition of polyelectrolyte brushes induced by an oppositely charged surfactant

We present a mean-field analytic theory with closed-form analytical solutions and explore the phase transition of polyelectrolyte brushes in the presence of an oppositely charged surfactant, a phenomenon crucial for designing stimulus-responsive smart materials but not considered by analytic theory previously.

Extending the concept of electrostatic-binding-induced hydrophobic effect from polyelectrolyte solutions to polyelectrolyte brushes, we show that electrostatic binding between charged monomers and surfactant ions generates hydrophobic aggregation among the surfactant tails, yielding an effective attraction between surfactant-bearing ionic monomers of polyelectrolyte brushes. The equilibrium adsorption of surfactants leads to a concentration-dependent χ-function, describing effective interactions within the brush. This framework predicts a discontinuous collapse transition followed by reentrant swelling at higher surfactant concentrations. Analytical treatment within a minimal free-energy model, neglecting self-volume increase from surfactant ion-adsorption, reveals that collapse and reswelling transitions share the same thermodynamic signature. For brushes with low densities and long chains, we derive an analytical spinodal approximation and identify the parameter space for discontinuous transitions. Our theory explains why phase transitions occur only when the surfactant tail length exceeds a threshold, and how self-micelle formation of surfactants competes with brush-surfactant binding, governing whether a full reentrant transition or only collapse occurs. This work provides the first analytical framework for understanding and predicting phase transitions of polyelectrolyte brushes interacting with oppositely charged surfactants.