This dissertation is primarily a survey of the zero-temperature string method, a minimum energy path search algorithm, applied to novel barrier-crossing problems in polymer field theory. I apply the method to both self-consistent field theory (SCFT) and a phase-field model (the Landau-Brazovskii model). In the case of SCFT, the focus is on defect annealing problems in homo+copolymer melts; in the case of the Landau-Brazovskii model, the focus is on finding critical nuclei for the disorder-to-lamellar transition, which is known to be a fluctuation-induced first-order phase transition.

In SCFT, applying the string method is computationally demanding in both processing time and memory, especially for fully 3-dimensional simulations at industrially relevant system sizes. I successfully address these challenges on state-of-the-art massively parallel computing architectures (NVIDIA graphics processing units). As a result our group is able to identify free energy barriers and transition mechanisms for a wide variety of defect annealing problems relevant to industrial directed self-assembly (DSA).

Nucleation in the Landau-Brazovskii model presents its own challenges. The string method as originally formulated is inefficient for nucleation problems, since many images are wasted tracing out unphysical configurations once the nucleus grows to the edges of the simulation cell. I devise a new truncation-based energy weighting (TBEW) scheme that resolves this issue, and will prove valuable to future researchers using the string method to find critical nuclei.

Since the bare Landau-Brazovskii model predicts a second-order transition between disorder and lamellae at a mean-field level, naive application of the zero-temperature string method to this model fails to find a barrier. To circumvent this, I instead apply the string method to a renormalized model that incorporates fluctuations at a mean-field level. Using TBEW and the renormalized model, I investigate nucleation pathways for the disorder-to-lamellar transition, finding anisotropic nuclei in agreement with previous predictions and experimental observations. I also conduct a comprehensive search for experimentally observed nuclei containing various exotic defect structures.

Finally, I evaluate the validity of the nucleation pathways obtained from the renormalized model by numerically simulating the bare model with explicit fluctuations. I find that the renormalized model makes good predictions for certain quantities, including the location of the order-disorder transition. However, due to sharp dependence of critical nucleus size on proximity to the order-disorder transition, even slight errors in the predicted ODT lead to large errors in predicted nucleus size. I conclude that the renormalized Landau-Brazovskii model is a poor tool for predicting critical nuclei in the fully fluctuating bare theory at experimentally accessible parameters, and recommend that future studies work with the fluctuating bare theory directly. I recommend several strategies to extract barriers and rates.