Recently, experimentalists have developed nanostructured, responsive hydrogels that make use of electrostatic interactions in order to form physical cross-links. Due to the ability of these hydrogels to respond to a variety of stimuli including pH, temperature and ion concentration, they are particularly suited for a wide range of biomedical applications including drug delivery and tissue scaffolding. Before use in practical applications, one must first learn how to tune the hydrogels for specific applications. To this end, we study these novel materials using both analytic techniques and computer simulations. The well established technique of self-consistent field theory (SCFT), which makes use of the saddle point approximation, completely fails to capture the electrostatic correlations necessary to accurately simulate the hydrogels. An alternative to SCFT is to use field-theoretic simulations with complex Langevin (CL) sampling, which simulate the unapproximated model.

However, as a newer technique, CL simulations present numerous challenges including the lack of methods for simulating stress-free configurations. Although there has been a concerted effort to make CL simulations more amenable, at their current state of development, we found these challenges prohibitive. Thus, we have taken a different approach. Inspired by an analytic approximation to the model, we develop an entirely new model that implicitly incorporates the essential electrostatic correlations such that the SCFT technique can be used. Using our new model, we generate phase diagrams for the hydrogel system. We find that the phase diagram includes various microphases as well as regions of phase coexistence. Using full CL simulations we then validate our model by reproducing selected points on the phase diagram.

Finally, we directly compare our predictions to recent experimental results and discuss how parameters such as charge density and solvent quality can be used to tune the hydrogel systems. In addition to our simulations of hydrogels, we also performed a systematic investigation of the performance of algorithms to solve the modified diffusion equation, the most computationally expensive step for both SCFT and CL, and found that the most computationally efficient algorithm is highly dependent on the simulation technique used. We also studied anisotropic diblock copolymer colloidal particles and developed a model the reproduces the experimentally observed trend of increasing aspect ratio with increasing particle size. The agreement of our calculations with experiment suggest that thermodynamic, rather than kinetic, effects drive the particle anisotropy.