Mathematics of soft matter, with applications in energy devices and biology

Soft matter refers to materials that are easily deformed by thermal fluctuations and external forces. They include liquid, colloids, polymers, gels, liquid crystals, etc. Soft matter plays an important role in a variety of applications, such as energy devices (battery, capacitor, and fuel cell) and biology systems (neuron, protein, and DNA). The mathematical description of soft matter is challenging because of the multiscale structure, and the non-local interaction between the particles.

In this talk, I will present how mathematics contributes to the understanding of soft matter, with a focus on electrolyte solutions and the bacteriophage DNA. For equilibrium configuration, our modeling follows the density functional theory; and for dynamics, we follow the framework of energetic variational approach, which describes both the energy and dissipation of the system. Bacteriophages densely pack their long dsDNA genome inside a protein capsid. The ordered conformation of the viral genome inside the capsid is consistent with a hexagonal liquid crystalline structure, surrounded by mobile ions. The Coulomb interaction between charges causes the correlation between their density distributions and various correlation effects. We develop a model and efficient numerical method to study the role of ions in the spatial configuration of the viral genome. We also consider the DNA-like liquid crystal filament in a cylindrical capsid, where multiple solutions exist and bifurcation analysis is carried out.