Field-theoretic modeling offers a powerful framework for the theoretical investigation of polymeric systems, but computational study of polymer field theories has primarily been limited to mean-field self-consistent field theory calculations that neglect fluctuation physics. An important recent computational advance has been the introduction of field-theoretic simulations that use complex Langevin sampling to capture the full fluctuation physics of polymer field theories. Such simulations offer great promise for exploring the physics of solvated polymer and polyelectrolyte systems in which fluctuations make essential physical contributions that are not captured by mean-field methods. However, many theoretical and computational difficulties encountered when executing field-theoretic simulations have thus far limited the widespread adoption of the technique.

This thesis presents several methodological advances that facilitate efficient, large-scale, non-pathological field-theoretic simulations of polymer solutions. A theoretical regularization framework is introduced that eliminates ultraviolet divergence pathologies from the field-theoretic models used as the basis for field-theoretic simulations of polymers. An exponential differencing algorithm is presented for integrating the complex Langevin equations employed in field-theoretic simulations, and is shown to have favorable accuracy and stability properties compared to previously introduced algorithms. Two completely separate frameworks for the systematic numerical coarse-graining of polymer field theories are presented, intended to facilitate efficient simulations of large multiscale systems. The first framework uses force-matching concepts originally developed for the coarse graining of particle-based molecular models to generate coarse-grained statistical field theories, and is shown to enable field-theoretic simulation of polymer solutions on coarsely-spaced computational lattices with reduced lattice discretization error. The second framework uses relative entropy minimization to map density probability distributions rather than field probability distributions, and is applied to field-theoretic simulations of polyelectrolyte coacervate complexation, in which pathological field-density relationships interfere with the force-matching coarse-graining method.

In addition to these methodological advances for field-theoretic simulations of polymers, this thesis also includes a study of concentration fluctuations in polymer solutions in extensional flow. It is well-known that shear flows under appropriate conditions can exhibit dramatically amplified concentration fluctuations, but the behavior of fluctuations in extensional flows has been less well investigated. A two-fluid continuum model with a realistic extensive constitutive equation is used to calculate predicted structure factors for a variety of flow conditions, which compare well with the limited experimental data available.