Research

I study how mathematical and computational modeling can be used to understand the behavior of complex materials. My research combines tools from ordinary/partial differential equations, the calculus of variations, mathematical analysis, and numerical simulation to connect physics-based models with predictions about how materials respond to external forces and their surroundings. My doctoral research primarily focuses on continuum models of bent-core smectic-A ferroelectric (SmAP_F) liquid crystals.

Ferroelectric liquid crystals

Liquid crystals are an unusual (and cool) state of matter: they can flow like liquids, while their molecules retain some of the orientational order found in solids. In smectic liquid crystals, the molecules additionally organize into layers. In ferroelectric liquid crystals, the molecules also exhibit a spontaneous electrical polarization, which introduces an additional interaction between molecular orientation and applied electric fields. In bent-core liquid crystals, individual molecules are bent (like a banana or chevron) which also contributes to this polarization through the underlying geometry of how electrons are placed.

This creates a competition between several different physical mechanisms: elasticity may prefer the molecules to be oriented one way, electric forcing may prefer another orientation, and interactions between the molecules and the boundaries of the surfaces containing them may prefer yet another.

Different smectic liquid crystal phases — Examples of several smectic liquid-crystal phases. SmAP_F phase is shown lower left. Note the bent-core structure. Image source: Gill et. al. (2025), *Materials Horizons*, [link].

Why liquid crystals?

From a mathematical perspective: liquid crystals lead to a surprisingly rich collection of analytical problems. Pierre-Gilles de Gennes won the 1991 Nobel Prize in physics for his part in extending classical phase transitions theory to liquid crystals and other complex polymers [link].

In the real world: liquid crystals are useful in engineering applications because their molecular orientation can be controlled by external forces, in particular an applied electric field. This makes it possible to translate a change in electrical input into a change in the material's optical properties.

Every time you use an LCD (liquid-crystal display) screen -- whether on a laptop monitor, television, calculator, or other display -- a voltage is applied across a thin cell of liquid crystal molecules. The resulting electric field changes the orientation of the molecules, which changes how light passes through the display. By controlling this response pixel by pixel, the LCD produces the images you see on your screen.

Diagram of a LCD screen — Diagram of an LCD screen. The liquid crystal layer is located fourth from front, and a polarizing layer surrounds the LC cell on both sides separated by a filtering layer. Image source: PTCLED (2024), [link].

What I Study

How do competing effects from material properties, externally applied electric fields, and interactions with the boundaries of a device influence the structure and evolution of SmAP_F liquid crystals?

I use mathematical and numerical models to determine which configurations are possible, when they are unique, how they change as physical parameters vary, and how the material approaches equilibrium over time.

Equilibrium Structure

The first part of my doctoral research asks what equilibrium configurations of a thin SmAP_F liquid crystal cell are possible under an applied electric field, and how those configurations depend on the competing physical effects within the material.

We describe the molecular orientation using an angle \(\theta(x)\), where \(x\) denotes position across the cell. Starting from a free energy model, we account for elastic deformation of the director, interactions with the cell surfaces, and ferroelectric coupling to an externally applied electric field. Equilibrium configurations correspond to stationary points of this energy and satisfy a nonlinear boundary value problem.

\[ \begin{aligned} \widetilde{\mathcal{I}}[\theta] &= \int_0^1 \underbrace{\frac{1}{2}(\sin^2\theta + \kappa\cos^2\theta)\theta'^2}_\text{elastic} \,dx \\ &\qquad + \int_0^1 \underbrace{\left(\frac{L}{\xi}\right)^2\left(\frac{1}{2}\cos^2\theta-\widetilde{E}_B\cos\theta\right)}_\text{electrostatic} \,dx \\ &\qquad + \underbrace{\widetilde{W}_S\left(\frac{L}{\xi}\right)(\cos\theta(1)-\cos\theta(0))}_\text{surface} \end{aligned} \]

Here, \(x \in [0,1]\) denotes normalized position across the cell, \(\vec{p}(x)= (\cos\theta(x),\sin\theta(x),0)\) is the polar director, \(\kappa\) measures elastic anisotropy, \(L/\xi\) is the ratio of the cell thickness to the elastic-electrostatic intrinsic length scale, \(\widetilde{E}_B\) is the dimensionless applied electric bias, and \(\widetilde{W}_S\) measures surface anchoring strength.

This formulation allows us to ask how competition between bulk forcing from the electric field and anchoring at the cell surfaces determines the equilibrium director profile. Of particular interest is when the electric field becomes strong enough to overcome the surface effects and align the material with the applied field.

Analytically, I establish results on existence and uniqueness, monotonicity and symmetry of solutions, and the admissible molecular configurations associated with different parameter regimes. Numerically, I use second-order finite differences together with gradient-flow and Newton methods to compute equilibrium solutions and explore their dependence on the physical parameters.

The figures below illustrate this competition through the polar director component \(p_x = \cos\theta\). Increasing the applied electric bias favors alignment in the \(+x\) direction throughout the cell, while increasing the surface anchoring strength more strongly enforces the preferred orientations at the two cell boundaries.

Effect of electric bias on liquid crystal alignment — **Electric bias.** An applied electric field in the \(+x\) direction favors alignment of the liquid crystal director with the field. Fixed parameters: \(\widetilde{W}_S=0.5, \kappa = 0.2, L/\xi = 4\)

Effect of surface anchoring on liquid crystal alignment — **Surface anchoring.** Increasing the surface anchoring strength enforces the preferred boundary orientations more strongly, driving the director toward \(p_x = 1\) at \(x=0\) and \(p_x=-1\) at \(x=1\). Fixed parameters: \(\widetilde{E}_B=0, \kappa = 0.2, L/\xi = 4\)

Dynamics and Relaxation

The equilibrium model describes stationary configurations of the free energy, but it does not tell us how the liquid crystal approaches those states. My current work extends this model to study the time-dependent relaxation of the director.

We use a Landau–Khalatnikov evolution in which the same free energy that determines the equilibrium structure also drives the dynamics. The resulting nonlinear partial differential equation describes how the molecular orientation evolves while the system dissipates energy and relaxes toward equilibrium.

Mathematically, I study existence and uniqueness of solutions, preservation of physically admissible director orientations, and the long-time behavior of the evolution. I also use numerical simulations to investigate how physical parameters such as the applied electric bias affect both the transient dynamics and the rate of relaxation.

The heatmaps below show the director component \(p_x = \cos\theta\) as a function of normalized position \(x\) and time \(t\). Color indicates the local molecular orientation, while the horizontal lines mark characteristic times at which the solution has completed 90% of its relaxation according to several measures.

Heatmap of liquid crystal relaxation at zero bias — **Relaxation under zero applied bias.** With \(\widetilde{E}_B=0\), the evolution is governed primarily by elastic and surface effects. The director gradually relaxes toward an equilibrium profile that is antisymmetric about the cell midpoint \(x=\frac{1}{2}\). Fixed parameters: \(\widetilde{W}_S=0.5,\ \kappa=0.2,\ L/\xi=4\).

Heatmap of liquid crystal relaxation at high bias — **Relaxation under strong applied bias.** With \(\widetilde{E}_B=2\), the electric field strongly favors alignment in the \(+x\) direction. The director therefore evolves toward a more uniformly positive \(p_x\) state and reaches its equilibrium configuration more rapidly. Fixed parameters: \(\widetilde{W}_S=0.5,\ \kappa=0.2,\ L/\xi=4\).

My Approach

Across these problems, I combine physics-based modeling, mathematical analysis, and numerical computation. I start from a physics-based energy principle, derive the governing differential equations, analyze the resulting differential equations mathematically, develop numerical methods to compute their solutions, and use those solutions to understand how physical parameters influence the behavior of complex materials (in my doctoral research setting, SmAP_F liquid crystals).

Physical model → Differential equation → Analysis → Computation → Physical interpretation

Research

Research Overview

Ferroelectric liquid crystals

Why liquid crystals?

What I Study

Equilibrium Structure

Dynamics and Relaxation

My Approach

Preprints

In Preparation

Other Papers

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Selected Talks