Finite Element-Based Brownian Dynamics Simulation of Nanofiber Suspensions Using Monte Carlo Method1

Abstract

This paper presents a computational approach for simulating the motion of nanofibers during fiber-filled composites processing. A finite element-based Brownian dynamics simulation (BDS) is proposed to solve for the motion of nanofibers suspended within a viscous fluid. We employ a Langevin approach to account for both hydrodynamic and Brownian effects. The finite element method (FEM) is used to compute the hydrodynamic force and torque exerted from the surrounding fluid. The Brownian force and torque are regarded as the random thermal disturbing effects which are modeled as a Gaussian process. Our approach seeks solutions using an iterative Newton–Raphson method for a fiber's linear and angular velocities such that the net forces and torques, including both 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 fiber position and orientation as a function of time. Instead of remeshing the fluid domain as a fiber migrates, the essential boundary condition is transformed on the boundary of the fluid domain, so the tedious process of updating the stiffness matrix of finite element model is avoided. Since the Brownian disturbance from the surrounding fluid molecules is a stochastic process, Monte Carlo simulation is used to evaluate a large quantity of motions of a single fiber associated with different random Brownian forces and torques. The final fiber motion is obtained by averaging numerous fiber motion paths. Examples of fiber motions with various Péclet numbers are presented in this paper. The proposed computational methodology may be used to gain insight on how to control fiber orientation in micro- and nanopolymer composite suspensions in order to obtain the best engineered products.