Chemically driven soft bioinspired systems and the variational formulation of physics-informed neural networks

Abstract

Living organisms excel in converting intermolecular interaction energy into mechanical work to generate deformations. To design and engineer soft bioinspired materials for applications such as drug delivery and soft robotics, it is essential to understand and integrate such chemomechanical energy convergence. In this thesis, we describe two bioinspired systems which are driven by chemical interactions between their components. First, we examine polyacrylic acid (PAA) hydrogels infused with divalent copper ions. When exposed to a strong acid stimulus, the gel releases copper and swells rapidly at a rate exceeding the characteristic solvent absorption. We explain this behavior by introducing gel diffusiophoresis, where interactions between the polymer and released ions drive a diffusio-osmotic solvent intake countered by diffusiophoretic motion of the polymer network, enabling the gel to swell at superdiffusive rate. We present a linear theory validating our model with experimental observations and then extend the theory to nonlinear deformations induced by the gel diffusiophoresis. Second, we investigate protocells — giant unilamellar lipid vesicles that preceded the first unicellular organisms. We analyze the spontaneous formation of subcompartments within these protocells. Using a continuum elastohydrodynamic theory, we demonstrate how attractive van der Waals interactions between lipid membrane, aqueous solvent, and Aluminum surface lead to the emergence of an elastohydrodynamic instability. This leads to the formation of protocell colonies with enhanced mechanical stability and ability to capture vital ingredients such as DNA from the environment. Our findings provide new insights into the role of surface interactions in the emergence of the first unicellular organisms. Finally, we propose a new computational framework based on neural networks to solve differential equations. Differential equations are essential in developing a mathematical description of the bioinspired systems that we have studied in this work. In the conventional formulation, physics-informed neural networks (PINN) solve differential equations by minimizing a phenomenological loss function constructed based on these equations. However, higher order derivatives present in many differential equations lead to increased computational cost. Additionally, solving coupled differential equations using PINN is complex due to manually or algorithmically determined ad hoc weight factors appearing in the loss function. Hence, we propose obtaining the solution to the differential equation by optimizing corresponding functionals such as Lagrangian, Hamiltonian or Rayleighian. This variational formulation naturally uses lower order derivatives, and the ad hoc weight factors are replaced by rigorous physical scales. This also allows us to examine the stability of the solutions, and we find that the conventional minimization algorithms are not suited for variational problems with unstable solutions. To that end, we propose an optimization algorithm based on Newton's method to find accurate solutions regardless of their stability for linear and nonlinear ordinary differential equations. Further investigations into partial differential equations are currently underway. This thesis provides new insights into the role of chemical interactions in shaping dynamic responses in bioinspired soft materials, and the proposed numerical methods may provide new pathways for advanced material design inspired by nature.