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.

References

References
1.
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
.
2.
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
.
3.
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
.
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
.
5.
Taylor
,
G. I.
,
1923
, “
The Motion of Ellipsoidal Particles in a Viscous Fluid
,”
Proc. R. Soc. London, Ser. A
,
103
(
720
), pp.
58
61
.
6.
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
.
7.
Batchelor
,
G. K.
,
1970
, “
Slender Body Theory for Particles of Arbitrary Cross-Section in Stoke Flow
,”
J. Fluid Mech.
,
44
(
3
), pp.
419
440
.
8.
Folgar
,
F.
, and
Tucker
,
C. L.
, III
,
1984
, “
Orientation Behavior of Fibers in Concentrated Suspensions
,”
J. Reinf. Plast. Compos.
,
3
(
2
), pp.
98
119
.
9.
Petrie
,
C. J. S.
,
1999
, “
The Rheology of Fibre Suspensions
,”
J. Non-Newtonian Fluid Mech.
,
87
(
2–3
), pp.
369
402
.
10.
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
.
11.
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
.
12.
Cintra
,
J. S.
, and
Tucker
,
C. L.
, III
,
1995
, “
Orthotropic Closure Approximations for Flow-Induced Fiber Orientation
,”
J. Rheol.
,
39
(
6
), pp.
1095
1122
.
13.
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
.
14.
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
.
15.
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
.
16.
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
.
17.
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
.
18.
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
.
19.
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
.
20.
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
.
21.
Ermak
,
D. L.
, and
Buckholz
,
H.
,
1980
, “
Numerical Integration of Langevin Equation: Monte Carlo Simulation
,”
J. Comput. Phys.
,
35
(
2
), pp.
169
182
.
22.
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
.
23.
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
.
24.
Junk
,
M.
, and
Illner
,
R.
,
2007
, “
A New Derivation of Jeffery's Equation
,”
J. Math. Fluid Mech.
,
9
(
4
), pp.
455
488
.
25.
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
.
26.
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
.
27.
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
.
28.
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
.
29.
Kim
,
S.
, and
Karrila
,
S. J.
,
1991
,
Microhydrodynamics: Principles and Selected Applications
,
Butterworth-Heinemann
,
Oxford, UK
, pp.
61
67
.
30.
Chapra
,
S. C.
,
2012
,
Applied Numerical Methods With MATLAB for Engineers and Scientists
,
3rd ed.
,
McGraw-Hill
,
New York
, pp.
533
572
.
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