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Research Papers

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

[+] Author and Article Information
Dongdong Zhang

Department of Mechanical Engineering,
Prairie View A&M University,
Prairie View, TX 77446
e-mail: dozhang@pvamu.edu

Douglas E. Smith

Department of Mechanical Engineering,
Baylor University,
Waco, TX 76798
e-mail: douglas_e_smith@baylor.edu

2Corresponding author.

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MICRO- AND NANO-MANUFACTURING. Manuscript received October 3, 2014; final manuscript received August 26, 2015; published online September 25, 2015. Assoc. Editor: John P. Coulter.

J. Micro Nano-Manuf 3(4), 041007 (Sep 25, 2015) (12 pages) Paper No: JMNM-14-1067; doi: 10.1115/1.4031492 History: Received October 03, 2014; Revised August 26, 2015

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.

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Figures

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Fig. 3

(a) Three-dimensional finite element model with the fiber's centroid at the origin of the x′y′z′ system and the long axis along the x′ axis, (b) mesh model in the x′z′ plane with three applied essential boundary conditions (BC1, BC2, and BC3), and (c) velocity distribution of fluid domain in the x′z′ plane with the applied simple shear flow

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Fig. 2

Definition of fiber orientation and position in the finite element-based method

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Fig. 1

(a) Injection molding process of center-gaited disk mold, (b) fiber suspensions within the flow in the mold cavity, and (c) definition of a single fiber orientation in Jeffery's theory

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Fig. 4

Results of the 3D motion of a single ellipsoidal fiber in a simple shear flow (one period): (a) evolution of fiber orientation (ϕ,θ,ψ) and (b) evolution of fiber position (xc,yc,zc)

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Fig. 5

Comparison of 3D FEM in planar motion and 2D FEM: (a) fiber orientation ϕ and (b) fiber position zc and yc

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Fig. 8

Disturbance of fiber motion on fluid velocity, pressure, and stress around fiber surface at ti=0.8 with Brownian motions: (upper 1) Uz (−4.5 to 6.5); (upper 2) Uy (−2.88 to −0.196); (upper 3) p (−7.37 to 3.05); (upper 4) γ˙ (0.07–9.4); (lower 1) σzz (−6.5 to 10); (lower 2) σyz (−0.74 to 6.7); (lower 3) σyy (−3.5 to 7.1); and (lower 4) σzz−σyy (−9.78 to 6.1)

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Fig. 6

Disturbance of fiber motion on fluid velocity, pressure, and stress around fiber surface at ti=0.8: (upper 1) Uz ( − 4.5 to 6.5); (upper 2) Uy (−0.6 to 0.6); (upper 3) p (−3.8 to 1.3); (upper 4) γ˙ (0.63–4.22); (lower 1) σzz (−2.2 to 4.9); (lower 2) σyz (−0.008 to 2.5); (lower 3) σyy (−2.02 to 4.67); and (lower 4) σzz−σyy (−3 to 2.3)

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Fig. 7

Distribution of fiber angles at ti = 5 for small and large Péclet numbers: (a) Per=0.003 and (b) Per = 3000

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