COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China
©2006 Tsinghua University Press & Springer
Numerical Simulation of the Microstructure of Magnetorheological Fluids
in Magnetic Fields
X. H. Peng *, H. T. Li
Institute of Intelligent Structures, Chongqing University, Chongqing, 400044 China
Email: xhpeng@cqu.edu.cn
Abstract The formation of the chains of ferromagnetic particles in a static magnetorheological fluid subjected to
applied magnetic fields is simulated. Based on the analysis of the dynamics of the particles, a model for the motion of
particles is proposed, and the corresponding numerical approach is developed. Several approaches are used to
accelerate the simulation process, including Velocity-Verlet algorithm, combined link-cell and Verlet list method,
combined fixed and variable time step method, etc. It shows that the developed micro-to-macro approach is of good
convergence and computational efficiency. The chain structure and its variation and the corresponding macroscopic
response of the MRF under a static magnetic field and/or external shear displacement are simulated, and the results are
in satisfactory agreement with experimental results.
Key words: magnetorheological fluid, shear yield stress, micro-to-macro analysis, numerical approach
INTRODUCTION
A magnetorheological fluid (MRF) is a suspension consisting of ferromagnetic particles suspended in some kind of
carrier liquid. When an MRF is set in a parallel static magnetic field, the particles will be polarized and interact to each
other, leading to an anisotropic chain structure. The apparent viscosity of the MRF may increase remarkably, and its
fluidity may decrease and even disappear, which is so called magnetorheological effect.
Great progress has been made in the research on the mechanical properties of MRFs, most of which focused on the
experimental investigation. Skjeltorp [1] studied the microstructures of MRFs confined to thin layers between two
glass plates and observed the chain structures formed under an external magnetic field. Based on Skjeltorp’s work,
Popplewelly [2] analyzed the yield shear stress of MRF composites at different strain rate. Rong [3] provided a
technique of flexible fixturing to study the mechanical properties of MRFs and found that the yield shear stress
increases significantly when the single-chain structure was changed into a thick column structure. Through the
observation on a direct measurement of lateral interactions between particle chains using optical trapping, Furst [4]
found that the lateral interactions determines the long-time microstructure and influences the rheological response of
MRFs.
Besides these experimental investigations, simulation methods were also used to study the properties of MRFs. Ly [5]
provided a computational technique to perform particle dynamics simulations. In his model, a potential theoretic
formulation, where the boundary integral equations were solved with a fast multi-pole approach, was used to compute
the magnetic forces. Based on the interaction potentials, Yang [6] proposed a Monte Carlo numerical simulation to
analyze the influence of magnetic field on aggregate structures of MRFs. Fang [7] reported a non-thermal molecular
dynamics simulation to study the field-induced structures of MRFs under a rotating magnetic field. Similar work was
conducted by Melle [8]. Since a microscopic-to- macroscopic simulation for the magnetic-mechanic behavior of an
MRF involves a large number of ferromagnetic particles, a numerical algorithm of high efficiency is always required.
However, less introduction of accelerating methods for the simulation can be found in literature.
In this paper, a microscopic-to-macroscopic approach is proposed for the investigation of the constitutive behavior of
MRFs. Three kinds of forces on each particle in an MRF subjected to a parallel static magnetic field are considered,
and the process of chain formation is analyzed with the viewpoint of micromechanics and making use of some
concepts of molecular dynamics and discrete element methods. Different from the conventional molecular dynamics
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methods, a new approach is proposed with the following considerations. (a) The size of each ferromagnetic particle is
about several micrometers, the concept of conventional molecular dynamics is, therefore, no longer available, and the
Brownian motion can be neglected due to its slight effect on the motion of the particles. (b) In molecular dynamics, two
neighboring molecules (atoms) can never contact each other but keep a certain distance due to the balance of attracting
and repelling forces, while two neighboring magnetized particles are allowed to contact each other without overlap. (c)
The interactions between the molecules (atoms) depend on the distance between them instead of their relative
orientation. While the interaction between two magnetized particles depend on their relative orientation as well as the
velocities of the particles. These differences lead to additional difficulties in the stability of numerical simulation. In
this work, a combined fixed and flexible time step method is put forward. Different from the conventional discrete
element method [9], long distance interaction exists in MRFs. In order to reduce computing time, the searching region
is reduced to a minimum by defining a cut-off distance, provided the distance between two particles is sufficiently
large so that the long distance interaction is negligible. A combined link-cell and Verlet list method is employed in the
simulation, so that the algorithm has a time complexity of O(N). It proves that the developed algorithm is of
satisfactory convergence and efficiency, and is capable of computing the response of an MRF with sufficiently large
number of ferromagnetic particles suspended in viscous liquid. The microstructures of an MRF simulated with the
developed program are in satisfactory agreement with experimental results.
EQUATIONS OF MOTION
The following assumptions are adopted in the simulation: (a) All the ferromagnetic particles are of identical size and
physical and mechanical properties; (b) Magnetization time is sufficiently short compared with the time for the
formation of chains, and is negligible during simulation.
Applying a constant static magnetic field, the particle in the magnetic field will be magnetized as a dipole with the
moment
m = VM =
4
π R3χ H
3
(1)
where R and V=4πR3/3 are respectively the radius and the volume of the particle, H and M = χH are respectively the
intensity of the applied magnetic field and the intensity of magnetization, and χ is susceptibility.
The point-dipole approximation is used to compute the magnetic force on particles. The magnetic force on particle i
should be the sum of the interaction from all other dipoles exerted by all the other particles [10] and expressed as
⎡ 3μ
⎤
3μ 0
Fim = ∑ ⎢ 05 (m i ⋅ m j − 5mir m jr )rij +
m
m
m
m
+
(
)
ir
j
jr
i ⎥
4π rij4
j ≠i ⎣
⎢ 4π rij
⎦⎥
(2)
where μ 0 is the vacuum permeability, rij is the vector describing the relative position from dipolar particle j to dipolar
particle i, rij＝║rij║, mir and mjr denote the projections of magnetic moment mi and mj on the direction of rij.
Assuming that both the particles and the container walls are stiff, the repelling force occurs on particle i as it contacts
the other particles or the wall of the container. A simple model for such kind of force can be expressed as [11]
3μ 0 mm
2π (2 R) 4
2
Fir =
⎡
⎛ rij
⎞⎤ rij
⎣
⎝
⎠⎦
∑ exp⎢⎢− β ⎜⎜ 2R − 1⎟⎟⎥⎥ r
j ≠i
(3)
ij
where β is the material constant and mm =║m║.
Neglecting the flow instability that may occur at a sufficiently high velocity, and considering a slow movement of
particles in liquid, the viscous resistance can approximately be described with Stokes’ drag as [12]
Fiv = −6πRη (
du i
−vf )
dt
(4)
where η is the coefficient of viscosity of the fluid, dui /dt is the velocity of particle i, and v f is the velocity of liquid at
the position of particle i.
The Brownian motion of a particle in the range of temperature concerned is neglected since the corresponding thermal
energy is much smaller than the magnetic and mechanical energies so that it should have a negligible effect on the
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evolution of the particle structure [2]. The motion of particle i, therefore, can be described with the second Newtonian
law of motion as
d 2u i
m 2 = Fim + Fir + Fiv
dt
(5)
where m is the mass of the particle.
ALGORITHM FOR THE INTEGRATION OF EQUATIONS OF MOTION
Before a magnetic field is applied, the particles distribute randomly with initial velocity vi＝0 (i =1,2,…,N) without
considering Brownian motion. After a magnetic field is applied, the particles are magnetized and become dipoles,
inducing magnetic forces on particles. Driven by the forces, the particles are accelerated, with which the velocity and
displacement can be derived. This process continues until the corresponding steady-state is achieved. In order to get
the steady-state result, the Velocity-Verlet algorithm [13] is used, in which both the position and the velocity of each
particle are derived simultaneously. The Velocity-Verlet integrator is stated as follows:
(1) The initial position of particle i is
{u
i t =0
= uˆ i } ,
( i = 1, 2,..., N )
(6)
in which uˆ i (i =1,2,…,N) is generated with a random function.
The initial velocity of particle i is
{v
i t =0
= 0} ,
( i = 1, 2,..., N )
(7)
(i =1,2,…,N)
(8)
The acceleration of particle i is
a i (t ) =
f i (t )
m
where fi (t ) = Fim (t ) + Fir (t ) + Fiv (t ) is the resultant force on particle i, which can be derived with the position and
velocity of particle i at instant t.
(2) Assuming the computation up to instant t has been finished, given an increment of time Δt, The position and the
velocity of particle i at instant t+Δt can be determined respectively with
1
u i (t + Δt ) = u i (t ) + v i (t )Δt + a i (t )Δt 2
2
v i (t + Δ t ) = v i (t ) +
(9)
1
[ai (t ) + ai (t + Δ t )] Δ t
2
(10)
(3) Reallocate the particles according to their new positions and update the Verlet list of particles (see the explanation
given for the combined link-cell and Verlet list method in section 4).
(4) Repeat the loop from (ii) to (vi) until the corresponding steady-state is reached.
It can be seen that the data input are the position and the velocity of each particle. The most remarkable virtue
compared with other methods (e.g., Verlet algorithm and Leap-frog algorithm) is obtaining the position and velocity
synchronously.
METHODS TO REDUCE COMPUTATIONAL TIME
It should be noted that a complete simulation is time consuming, especially in the computation of magnetic force
between dipolar particles, because the computation of the force on each particle is related to the interaction between the
particle and all other particles. For a system having N particles, N(N-1)/2 interactions should be computed in each
increment without any accelerating approach.
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On the other hand, it can be seen from Eq.(2) that the interaction between two dipolar particles decreases quickly with
the increase of the distance between them. If the distance between two dipolar particles is sufficiently large, the
interaction between the two dipolar particles will tend to vanish.
Link-cell method [14], which is of O(N) time complexity and was developed in molecular dynamics simulation, is used
in the developed approach. In order to further improve the efficiency, Verlet list method is adopted in addition to
link-cell method, as shown in Fig.1. In Fig.1, the computation region is separated into many small square cells of size
rc, which is equal to the cut off distance. It can be seen in Fig.1 that if the distance between two dipolar particles r=6R,
the magnetic force between the two dipolar particles is negligible. rc=6R is, therefore, used in computation. Each
particle (e.g., particle i) should be located in a cell, any other particles making contribution to fi should be located in the
shadow region, which involves at most 3×3 cells for a 2-D problem or 3×3×3 cells for a general 3-D problem. In other
words, in simulation, the search for the related particles only covers 3×3 cells for a 2-D problem or 3×3×3 cells for a
general 3-D problem. Subsequently a judgement is used to determine whether the other particles in these cells are out
of the cut off circle or not, and a Verlet list for particle i is produced at the same time. The main steps of this approach
are stated as follows:
Figure 1: Schematic plot for the combined link-cell and Verlet list method
(1) Set up the relationship between the link cells and the positions of particles at the initial of each time increment Δtk+1,
t=tk. At this step, for each particle, the search of its surrounding particles should cover all of the other particles in its
neighboring cells, noticing that the largest number of neighboring cells is nine. For some particles, which are located
on the boundary, the number of surrounding neighbors is smaller.
(2) Compute the distance between particle i and each of the other particles in the surrounding cells. If the distance is
less than the cut off radius, put the identification serial number of this particle into the list related to particle i. In the
computation of interactions between particle i and the other particles, the search can be constrained in the list related to
particle i.
(3) The displacement of each particle should be superimposed to its coordinates at the end of the time increment until
it reaches a certain value (user defined), then the relationships between the particle and its surrounding particles should
be updated following the steps (1) and (2).
The above mentioned search program helps greatly reducing the computation time, especially in the case involving
large number of particles.
CHOICE OF TIME INCREMENT
A proper choice of time increments is important to the stability and accuracy of computation. In each time increment,
the velocity and the position of each particle should be updated using the equation of motion and the forces on the
particle. A too small time increment may result in an unnecessary increase of the computation time and accumulated
error. On the other hand, a too big time increment may affect the convergence of computation and lead to incorrect
results. Since the magnetic force between dipolar particles involves very strong non-linearity, a proper choice of time
increment is of critical importance. A combined fixed and variable time increment approach is suggested with the
analysis of time increment.
The distance between particles strongly affects the stability of computation. Analysis shows that the magnetic force
increases sharply with the decrease of the distance between two particles when they approach each other. If time
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increment Δt is not suitable, the two particles may overlap and even pass through each other. It was proposed by Martin
[15] that the choice of time increment can be determined by the maximum gradient of the resultant force, which is
roughly given by examining the curvature at the minimum of the potential well when two particles aligned in the
direction of the applied magnetic field. Assuming two nearby still particles start and approach each other due to the
attractive force between the two particles and contact each other inΔt , if the resultant forces on one particle at time t
and t + Δ t are F(t) and F (t + Δ t ) , respectively, the time increment Δ t can be determined with
−1
dF (r )
Δt = 2 m
dr r = r0
(11)
where r0 is the distance between the two particles when they contact each other. It can be seen that at this moment, F’(r)
tends to maximum and Δt is the minimum.
On the other hand, it is difficult to choose a fixed time increment because the positions of particles change
continuously and the relative distances between different particles are various. One method to avoid such problems is
to choose a fixed time increment, for each pair of particles, this fundamental time increment can be subdivided into
microsteps of the sizeΔt . In order to test the stability of this method, two kinds of simulation cases are performed. In
case I the time increment is fixed to be sufficiently small to ensure the stability, while IN CASE II the fundamental time
increment is assigned. For most pairs of particles the fundamental time increment is adopted to calculate the
interaction, but when the two particles approach each other, the given increment was further subdivided into finer
increments related toΔt . Similar chain structures are obtained with the two approaches, however, compared with
Case I, Case II takes much less computation time.
SIMULATION RESULTS
Assuming the computation region is a slot with thickness equal to the diameter of particles, and the ratio of the width W
and the length L equals to 0.3, the problem can be simulated as a two-dimensional problem. The parameters used in
simulation are given in Table 1.
In order to consider the interaction between the slot and the particles, some particles are assumed to be fixed on the
bottom. The interaction between the slot and the particles can, therefore, be approximately considered as the
interaction between particles. Since the domain used in simulation is just a part of the whole region, periodicity
boundary condition is adopted.
Figure 2: Simulation of microstructure in MRF. (a) initial state without external magnetic field. (b) steady state under an
external magnetic field. (c) microstructure in the shear movement when the shear strain is 0.68
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Table 1 Parameters used in simulation
L(μm)
W(μm)
R(μm)
N
μ0(H/m)
rc(μm)
600
90
5
280
4π×10-7
30
ρ (kg /m3)
η (N·s/m2)
H(A/m)
χ
β
Δt(μs)
7.5×103
0.001
1.07×104
1
9
0.1
The chain-formation and the shear process are shown in Fig.2. It can be seen that the particles distribute randomly at
first. When subjected to the applied magnetic field, they begin to move to form chains aligned in the direction of the
field. It can be seen that not all the chains are independent and ideally align in the direction of the applied magnetic
field. when a macroscopic shear strain is applied between the two plates, the chains are inclined and stretched gradually
and tend to rupture. Meanwhile some short chains and ruptured chains will merge to form new chains. In this process,
the macroscopic shear stress of the MR fluid will increase gradually and tend to a steady state corresponding to the
governing magnetic and mechanical parameters. It can be imagined that if the velocity of the rupture and the formation
of chains is approximately equal to each other, i.e., an evolutional balance in the number of the available chains is
achieved, the steady-state macroscopic shear stress will be achieved. The response of shear stress versus shear strain is
shown in fig.3, where the shear rate is 250s-1 and the magnetic induction intensity is 500Gauss.
Figure 3: The response of τ -γ of MRF during a shear deformation
CONCLUSIONS
A model is proposed for the simulation of the formation of dipolar chains at external magnetic field, and the
corresponding efficient algorithm is developed, in which some methods were used to accelerate the simulation. The
formation of chains consisting of dipolar particles in MRFs under applied magnetic field is simulated. The deformation
and the rupture of these chains and the formation of new chains under applied macroscopic shear strain are also
analyzed.
Acknowledgements
This work is financially supported by the NSFC (10472135) and the Commission of Science and Technology of
Chongqing (8414).
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