Finite Element-Based Brownian Dynamics Simulation of Nano-Fiber Suspensions in Nano-Composites Processing Using Monte-Carlo Method

This paper presents a computational approach for simulating the motion of nano-fibers during polymer nano-composites processing. A finite element-based Brownian dynamics simulation is proposed to solve the motion of nano-fibers suspended within a viscous fluid. In this paper, a Langevin approach is used to account for both hydrodynamic and Brownian effects. We develop a stand-alone Finite Element Method (FEM) for modeling the hydrodynamic effect exerted from the surrounding fluid. The Brownian effects are regarded as the random thermal disturbing forces/torques, which are modeled as a Gaussian process. Our approach seeks solutions using an iterative Newton-Raphson method for the fiber’s linear and angular velocities such that the net forces and torques, i.e. the combination of hydrodynamic and Brownian effects, acting on the fiber are zero. In the Newton-Raphson method, the analytical Jacobian matrix is derived from our finite element model. Fiber motion is then computed with a Runge-Kutta method to update the fiber positions and orientations as a function of time. Instead of re-meshing the fluid domain as fiber moves, we applied the transformed essential boundary conditions on the boundary of fluid domain, so the tedious process of updating stiffness matrix of finite element model is avoided. Since Brownian disturbance from the fluid molecules is a stochastic process, Monte-Carlo simulation is used to evaluate the motion of a great many fibers associated with different random Brownian forces and torques. The final fiber motion is obtained by averaging a numerous fiber motion paths. Examples of fiber motions with various Péclet numbers are presented in this paper. The proposed computational methodology will be used to gain insight on how to control fiber orientations in micro- and nano-polymer composite suspensions in order to obtain the best engineered products.