0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Qian, D. , Dickey, E. C. , Andrews, R. , and Rantell, T. , 2000, “ Load Transfer and Deformation Mechanisms in Carbon Nanotube-Polystyrene Composites,” Appl. Phys. Lett., 76(20), pp. 2868–2870. [CrossRef]
Biercuk, M. J. , Llaguno, M. C. , Radosavljevic, M. , Hyun, J. K. , Johnson, A. T. , and Fischer, J. E. , 2002, “ Carbon Nanotube Composites for Thermal Management,” Appl. Phys. Lett., 80(15), pp. 2767–2769. [CrossRef]
Du, F. , Fischer, J. E. , and Winey, K. I. , 2005, “ Effect of Nanotube Alignment on Percolation Conductivity in Carbon Nanotube/Polymer Composites,” Phys. Rev. B, 72, pp. 1–4.
Jeffery, G. B. , 1922, “ The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid,” Proc. R. Soc. London, Ser. A, 102(715), pp. 161–179. [CrossRef]
Taylor, G. I. , 1923, “ The Motion of Ellipsoidal Particles in a Viscous Fluid,” Proc. R. Soc. London, Ser. A, 103(720), pp. 58–61. [CrossRef]
Mason, S. G. , and Manley, R. St J. , 1956, “ Particle Motions in Sheared Suspensions: Orientation and Interactions of Rigid Rods,” Proc. R. Soc. London, Ser. A, 238(1212), pp. 117–131. [CrossRef]
Batchelor, G. K. , 1970, “ Slender Body Theory for Particles of Arbitrary Cross-Section in Stoke Flow,” J. Fluid Mech., 44(3), pp. 419–440. [CrossRef]
Folgar, F. , and Tucker, C. L., III , 1984, “ Orientation Behavior of Fibers in Concentrated Suspensions,” J. Reinf. Plast. Compos., 3(2), pp. 98–119. [CrossRef]
Petrie, C. J. S. , 1999, “ The Rheology of Fibre Suspensions,” J. Non-Newtonian Fluid Mech., 87(2–3), pp. 369–402. [CrossRef]
Wang, J. , O'Gara, J. F. , and Tucker, C. L., III , 2008, “ An Objective Model for Slow Orientation Kinetics in Concentrated Fiber Suspensions: Theory and Rheological Evidence,” J. Rheol., 52(5), pp. 1179–1200. [CrossRef]
Advani, S. G. , and Tucker, C. L., III , 1987, “ The Use of Tensors to Describe and Predict Fiber Orientation in Short Fiber Composites,” J. Rheol., 31(8), pp. 751–784. [CrossRef]
Cintra, J. S. , and Tucker, C. L., III , 1995, “ Orthotropic Closure Approximations for Flow-Induced Fiber Orientation,” J. Rheol., 39(6), pp. 1095–1122. [CrossRef]
Jack, D. A. , and Smith, D. E. , 2005, “ An Invariant Based Fitted Closure of the Sixth-Order Orientation Tensor for Short-Fiber Suspensions,” J. Rheol., 49(5), pp. 1091–1115. [CrossRef]
Hinch, E. J. , and Leal, L. G. , 1972, “ The Effect of Brownian Motion on the Rheological Properties of a Suspension of Non-Spherical Particles,” J. Fluid Mech., 52(4), pp. 683–712. [CrossRef]
Hinch, E. J. , and Leal, L. , 1976, “ Constitutive Equations in Suspension Mechanics—Part 2: Approximate Forms for a Suspension of Rigid Particles Affected by Brownian Rotations,” J. Fluid Mech., 76(1), pp. 187–208. [CrossRef]
Tao, Y. , den Otter, W. K. , Padding, J. T. , Dhont, J. K. G. , and Briels, W. J. , 2005, “ Brownian Dynamics Simulations of the Self- and Collective Rotational Diffusion Coefficients of Rigid Long Thin Rods,” J. Chem. Phys., 122(24), p. 244903. [CrossRef] [PubMed]
Yamamoto, T. , and Kasama, H. , 2009, “ Brownian Dynamics Simulation of Multiphase Suspension of Disc-Like Particles and Polymers,” Rheol. Acta, 49(6), pp. 573–584. [CrossRef]
Meng, Q. , and Higdon, J. J. L. , 2008, “ Large Scale Dynamics Simulation of Plate-Like Particle Suspensions. Part II: Brownian Simulation,” J. Rheol., 52(1), pp. 37–65. [CrossRef]
Somasi, M. , Khomami, B. , Woo, N. J. , Hur, J. S. , and Shaqfeh, E. S. G. , 2002, “ Brownian Dynamics Simulations of Bead-Rod and Bead-Spring Chains: Numerical Algorithms and Coarse-Graining Issues,” J. Non-Newtonian Fluid Mech., 108(1–3), pp. 227–255. [CrossRef]
Tang, W. , and Advani, S. G. , 2008, “ Dynamic Simulation of Carbon Nanotubes in Simple Shear Flow,” Comput. Model. Eng. Sci., 25(3), pp. 149–164.
Ermak, D. L. , and Buckholz, H. , 1980, “ Numerical Integration of Langevin Equation: Monte Carlo Simulation,” J. Comput. Phys., 35(2), pp. 169–182. [CrossRef]
Bretherton, F. P. , 1962, “ The Motion of Rigid Particles in a Shear Flow at Low Reynolds Number,” J. Fluid Mech., 14(2), pp. 284–304. [CrossRef]
Zhang, D. , Smith, D. E. , Jack, D. A. , and Montgomery-Smith, S. , 2011, “ Numerical Evaluation of Single Fiber Motion for Short-Fiber-Reinforced Composite Materials Processing,” ASME J. Manuf. Sci. Eng., 133(5), p. 051002. [CrossRef]
Junk, M. , and Illner, R. , 2007, “ A New Derivation of Jeffery's Equation,” J. Math. Fluid Mech., 9(4), pp. 455–488. [CrossRef]
Koelman, J. M. V. A. , and Hoogerbrugge, P. J. , 1993, “ Dynamic Simulations of Hard-Sphere Suspensions Under Steady Shear,” Europhys. Lett., 21(3), pp. 363–368. [CrossRef]
Sun, S. P. , Wei, M. , and Olson, J. R. , 2011, “ Rheological Behavior of Needle-Like Hydroxyapatite Nano-Particle Suspensions,” Rheol. Acta, 50(1), pp. 65–74. [CrossRef]
Fujara, F. , Geil, B. , Sillescu, H. , and Fleischer, G. , 1992, “ Translational and Rotational Diffusion in Supercooled Orthoterphenyl Close to the Glass Transition,” Z. Phys. B Condens. Matter, 88(2), pp. 195–204. [CrossRef]
Reddy, J. N. , and Gartling, D. K. , 2010, The Finite Element Method in Heat Transfer and Fluid Dynamics, 3rd ed., CRC Press, Boca Raton, FL, pp. 161–275.
Kim, S. , and Karrila, S. J. , 1991, Microhydrodynamics: Principles and Selected Applications, Butterworth-Heinemann, Oxford, UK, pp. 61–67.
Chapra, S. C. , 2012, Applied Numerical Methods With MATLAB for Engineers and Scientists, 3rd ed., McGraw-Hill, New York, pp. 533–572.

Figures

Grahic Jump Location
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

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
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

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
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)

Grahic Jump Location
Fig. 7

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

Grahic Jump Location
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)

Grahic Jump Location
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)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In