Viscometric characterization of cobalt nanoparticle-based magnetorheological fluids using genetic algorithms

Article abstract of DOI:10.1016/j.jmmm.2005.01.061

The rheological flow curves (shear stress vs. shear rate) of a nanoparticle cobalt-based magnetorheological fluid can be modeled using Bingham-plastic and Herschel-Bulkley constitutive models. Steady-state rheological flow curves were measured using a parallel disk rheometer for constant shear rates as a function of applied magnetic field. Genetic algorithms were used to identify constitutive model parameters from the flow curve data.

Full text of DOI:10.1016/j.jmmm.2005.01.061

ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 293 (2005) 206–214
www.elsevier.com/locate/jmmm
Viscometric characterization of cobalt nanoparticle-based
magnetorheological fluids using genetic algorithms
Ã
Anirban Chaudhuria, Norman M. Wereleya, , Sanjay Kothab,
Ramachandran Radhakrishnanb, Tirumalai S. Sudarshanb
aDepartment of Aerospace Engineering, University of Maryland, 3180 Martin Hall, College Park, MD 20742, USA
bMaterials Modification Incorporated, 2721-D Merrilee Drive, Fairfax, VA 22031, USA
Available online 2 March 2005
Abstract
The rheological flow curves (shear stress vs. shear rate) of a nanoparticle cobalt-based magnetorheological fluid can
be modeled using Bingham-plastic and Herschel–Bulkley constitutive models. Steady-state rheological flow curves were
measured using a parallel disk rheometer for constant shear rates as a function of applied magnetic field. Genetic
algorithms were used to identify constitutive model parameters from the flow curve data.
r 2005 Elsevier B.V. All rights reserved.
Keywords: Magnetorheological fluids; Nanoparticles; Cobalt; Bingham-plastic model; Herschel–Bulkley model; Genetic algorithms;
Rheology
1. Introduction
chains. A finite stress must develop in the fluid to
yield these structures. The field-dependent yield
stress of these fluids is continuously controllable
and this controllability has been the primary
reason for their use in numerous smart actuation
systems [2,3].
MR fluids have been produced with different
types and sizes of magnetic carrier particles. The
majority of existing MR fluids are composed of
micron-scale Fe particles suspended in a nonmag-
netic carrier fluid [4–10]. These MR fluids have
high yield stress (20–100 kPa) due to the strength
of the dipoles created by the particles. The
comparatively higher yield stress over electro-
rheological (ER) fluids (2–5 kPa) is one major
Magnetorheological (MR) fluids are a class of
smart materials whose rheological properties may
be varied by application of a magnetic field. These
fluids are suspensions of soft magnetic particles
(such as iron or cobalt) in a carrier fluid. Each
particle has a dipole moment, the strength of
which is roughly proportional to its diameter [1].
Upon application of a magnetic field, these dipoles
align parallel to the magnetic field and form
Ã
Corresponding author. Tel.: +1 301 405 1927;
fax: +1 301 314 9001.
E-mail address: wereley@aero.umd.edu (N.M. Wereley).
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmmm.2005.01.061

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207
advantage of MR fluids. However, the density of
the particles makes them susceptible to settling in
the absence of frequent mixing due to predomi-
nant gravity forces. Once sedimented, the residual
magnetic attraction between particles makes redis-
persion difficult. The large particles also lead to
unwanted abrasion of the components in contact
with the fluid. Ferrofluids, which are suspensions
of iron particles of less than 10 nm in diameter,
have also been reported [11,12]. However, these
ferrofluids do not present formation of elongated
chain-like microstructures under the application of
a magnetic field and they are unable to provide
significant magnetoviscous effect, especially sig-
nificant yield stress. Nanometer-sized particles
(10–100 nm) have been introduced [13,14] with an
attempt to reduce settling while maintaining useful
yield stress levels. The mixture is seen to overcome
the settling problem due to the predominance of
thermodynamic forces [15,16] but the yield stresses
obtained for the same shear rates and magnetic
field levels are drastically reduced in comparison to
fluids containing micron-sized particles. The yield
shear stresses in these fluids are comparable with
those achieved in electrorheological fluids.
We use the Bingham-plastic (BP) and Herschel–
Bulkley (HB) constitutive models to characterize
the rheological behavior of MR fluid. In both fluid
models, it is assumed that the preyield behavior is
rigid, and that the fluid flows if and only if the
local shear stress is greater than the yield stress.
For a BP model, once the local shear stress exceeds
the yield stress, the postyield behavior is linear in
that the shear stress increases linearly with shear
rate (linear postyield model). The BP model
employs two parameters: yield stress, ty; and
viscosity, m: For the HB model, once the local
shear stress exceeds the yield stress, the postyield
behavior is nonlinear in that the shear stress
increases as a power law of shear rate (nonlinear
postyield model). The HB model employs three
parameters: yield stress, ty; consistency, K, and
flow index, n. The flow index n can be used to
classify the fluid; n41 indicates a shear-thickening
fluid and no1 indicates a shear-thinning flow. The
HB model has been used to characterize the flow
of MR fluids, especially where shear thickening or
thinning is seen [17–19]. A key issue is to identify
model parameters from flow curve measurements,
that is, the shear stress as a function of shear rate
for varying levels of applied magnetic field.
Genetic algorithms (GA) have been widely used
in applications where a globally optimal solution is
required. In conventional estimation methods, a
model structure is chosen and the parameters of
that model are calculated by minimizing an
objective function. Gradient descent techniques
are usually used for the minimization, but these
are very susceptible to initial guesses and the
obtained parameters may only be locally optimal.
On the other hand, GA uses a probabilistically
guided search procedure that simulates genetic
evolution [20,21]. Populations with stronger fitness
are identified and retained, while weaker ones are
discarded. The process ensures that successive
generations are fitter. The algorithm cannot be
trapped in local minima since it employs random
mutation procedures. The overall search proce-
dure is stable and robust and can identify globally
optimal parameters of a system.
The rheological parameters of MR fluid con-
stitutive models can be identified from flow curves
measured using a parallel disk rheometer. Flow
curves were measured as a function of applied field
for MR fluid with a solids loading of 40 weight
percent (wt%) nanometer scale cobalt particles.
The particles were aspherical in shape with an
equivalent diameter of 100 nm (Fig. 1). Both the
BP and HB models were fitted using a simple GA
to estimate model parameters. Constraints were
applied within the algorithm to ensure a mono-
tonically increasing trend of the yield stress with
increase in applied magnetic field. The identified
rheological parameters provided a good model fit
to measured flow curves. The parameter variations
with change in applied magnetic field were smooth.
Comparison of estimation errors suggest that the
HB model is more accurate over the entire range of
measured shear rates, but the BP model is equally
good for the high shear rate regime.
2. Synthesis and testing of MR fluids
A chemical precipitation technique was em-
ployed to produce cobalt nanopowders. In this

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rheological behavior of these materials can be
separated into distinct preyield and postyield
regimes. MR fluids are known to exhibit a strong
field-dependent shear modulus and a yield stress
that resists flow until the shear stress reaches a
critical value. Both the BP and HB models present
such a behavior, and both models present rigid
preyield behavior. The BP model has been used as
a constitutive model for MR fluids [10]. The
simplicity of this two-parameter model, with yield
stress, ty; and postyield viscosity, m; has led to its
wide use for representation of field-controllable
fluids. This model assumes that the fluid exhibits
shear stress proportional to shear rate in the
postyield region and is described by the following
equation:
Fig. 1. SEM picture of cobalt nanoparticles.
_
t ¼ ty þ mg.
(1)
process, ammonium hydroxide was used to pre-
cipitate an aqueous suspension of cobalt nitrate at
a pH of 11. The resulting precipitate (cobalt
hydroxide) was then centrifuged and washed
several times with distilled water and iso-propanol
to remove excess hydroxide. This was followed by
overnight drying at 150 1C to produce cobalt oxide
nanopowders. Cobalt oxide nanopowders were
further reduced under hydrogen at 450 1C for 3
hours to obtain pure elemental cobalt nanoparti-
cles.
However, in cases where the fluid experiences
postyield shear thickening or shear thinning, the
behavior becomes nonlinear and the assumption
of constant plastic viscosity is invalid. The HB
model is more suitable as a constitutive model for
MR fluids [14] and has been applied to analysis of
MR-based devices [18]. The HB model can be
represented as
n
_
t ¼ ty þ Kg ,
(2)
where yield stress, ty; consistency, K and flow
index, n, are the model parameters. The model
assumes that below the yield stress, ty; the
suspension behaves as a rigid solid, similar to the
BP model. The flow index characterizes the
postyield behavior; n41 indicates shear thickening
and no1 indicates shear thinning. The BP model is
a special case ðn ¼ 1Þ of the HB model.
To prepare stable MR fluids, hydraulic oil was
chosen as a carrier fluid. Mobil DTE20 series is
used extensively in high-pressure systems including
industrial, marine and mobile services because of
its excellent anti-wear properties, multi-metal
compatibility and corrosion resistance. Air Pro-
ducts AD01 EnviroGem was used as a surfactant
in the magnetorheological suspensions. The fluid
was mixed in hydraulic oil using a high-speed
emulsifier at nearly 11,000 rpm. Nanopowders
obtained from the microwave plasma synthesis
were added to the oil and the mixing continued.
Concentration of cobalt was 40% (by weight) in
the fluid.
4. Genetic algorithms for model parameter
estimation
The cumulative squared error between measured
and fitted shear stress values as a function of shear
rate was used as the objective function, which
leads to a quadratic minimization problem. Most
algorithms perform a minimization of the cost
function starting at a user-provided initial guess
and use additional information, usually function
3. Rheological models
MR fluids demonstrate nonlinear behavior
when subjected to external magnetic fields. The

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209
gradients, to approach the minima. However, such
techniques only yield a local minimum in the
proximity of the initial guess. In the case of
nonlinear estimation, there may exist multiple
optimal solutions that are equally significant, but
identification of all solution sets is not possible
unless the optimizer is run with different initial
guesses. Function gradients at all points are also
not always available or are computationally costly.
GAs are a tool for finding global solutions in
large parameter spaces by simultaneously evaluat-
ing and refining different solution sets to identify
near-optimal solutions rather than employing
methods to converge to a single solution. GAs
are different from usual optimization and search
procedures [20,21] in that they search from a
population of points and use probabilistic transi-
tion rules, not deterministic rules. An important
consideration is that a GA uses only objective/
fitness function information, and not derivative or
any other auxiliary knowledge. The evolution
scheme selects ‘‘fitter’’ individuals to be members
of the new generation and adds to this generation
the individuals that result from crossover and
mutation of the selected individuals. Convergence
is guaranteed by the selection that makes the best
solution of the new generation better or equal to
the best solution of the previous generation. The
overall search procedure is stable and robust and
can identify globally optimal parameters of a
system. GAs have the ability to solve highly
nonlinear functions where other techniques fail.
We used a simple genetic algorithm (SGA)
[22,23] for identifying the parameters of the
rheological model. A population of P individuals
was generated, each member composed of a binary
string of length N. Each individual was mapped to
a rheological parameter using maximum and
minimum bounds as follows:
used to determine the error and corresponding
fitness value of each population member. For each
iteration, we applied the three genetic operators
(reproduction, crossover and mutation) to create a
new generation. Reproduction was simulated by a
simple roulette wheel-based selection scheme.
Crossover was carried out at a single point for
each parameter of the model. To prevent the
algorithm from getting stuck in local minima, new
genetic information was periodically injected via
mutation. The objective function used was the sum
of squared errors and the corresponding fitness
was evaluated as the reciprocal of the objective
function. For each individual,
Ne
X
2
squared error Ej ¼
wi ꢀ ðti ꢁ t^iÞ ,
(4)
i¼1
fitness value Fj ¼ 1=Ej,
(5)
where Ne is the number of experimentally obtained
points for each value of magnetic field, t^i is the
measured shear stress for a particular strain rate
and ti is the shear stress at the same strain rate for
a particular set of rheological parameters. wi was
used as a weighing factor. To compare the fitted
model with the experimental data, we defined
errors as functions of magnetic field, f; as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
Ne
X
u
t
1
2
Abs error eðfÞ ¼
ðti ꢁ t^iÞ ,
(6)
Ne
i¼1
eðfÞ
Percentage error PðfÞ ¼
ꢀ 100,
(7)
tf max
where tf max is the maximum measured shear
stress at a particular value of magnetic field, f:
The search procedure was carried out for each
value of magnetic field. The flow index, n, was
allowed to vary from 0 to 1, as the measured flow
curves strongly indicated that the cobalt nanopar-
ticle MR fluid exhibits shear thinning. In case of
the yield stress, ty; for each level of applied
magnetic field, we selected the highest measured
shear stress as the upper bound and the estimated
yield stress at the next lowest magnetic field as the
lower bound. This procedure resulted in a mono-
tonically increasing yield stress as a function of
magnetic field. This constraint was included
Parameter value
¼ ðdecoded value of binary stringÞ
ðupper bound ꢁ lower boundÞ
ꢀ
.
ð3Þ
ð2N ꢁ 1Þ
The initial population was generated randomly
and spread over the entire space of possible
parameter values. Model equation (2) was then

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because a yield stress that increases monotonically
with applied magnetic field was physically mean-
ingful.
using a simple genetic algorithm. The algorithm
was run with a population size P ¼ 2000 and using
N ¼ 50 bits for encoding. The probability of
crossover was chosen as Pc ¼ 0:85; and the
Several termination criteria for a genetic search
have been proposed. One simple criterion is to
stop the search when almost all individuals in the
population are nearly identical; another criterion is
to test the improvement in the best fitness score
over successive generations. However, the first
criterion can lead to extensive search times while
the second one is not good for functions char-
acterized by ‘‘plateau-type’’ regions [24]. In our
work, we stopped the GA after a fixed number of
generations, which was chosen as a compromise
between constraints of convergence, computing
time and accuracy.
probability of mutation was chosen as Pm
¼
0:025: Constraints were applied to the search
scheme to ensure that a monotonically increasing
characteristic was obtained for the yield stress with
increasing magnetization. The cost function was
evaluated Eq. (4). The weight, wi; is adjusted so
that the algorithm would provide a better fit to the
flow curve data at the higher shear rates.
The parameters estimated for the BP model are
shown in Fig. 2. The yield stress increases
monotonically from 50 Pa at 0.03 T to 1750 Pa at
0.30 T. The variation in yield stress is smooth and
almost linear, while the viscosity changes smoothly
in a nonlinear manner as magnetic field increases.
The identified model parameters for the HB
model are shown in Fig. 3. The parameters change
smoothly over the range of applied magnetic fields.
Yield stress, ty; presents a monotonically increas-
ing trend with values ranging from 10 to 1450 Pa,
and are lower than those obtained from the BP
model. The estimated consistency parameter (K)
and flow index (n) show considerable scatter and
HB model GA parameter identification results
were also compared to a least mean squares (LMS)
error minimization technique employing the same
constraints as discussed above for the GA technique.
5. Results
The parameters of the constitutive models were
obtained after carrying out global optimization
2
1.8
1.6
1.4
1.2
1
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.8
0.6
0.4
0.2
0
0.25
0.2
0.15
0.1
0.06 0.09 0.12 0.15
0.24 0.27 0.3
0.03
0.21 0.24 0.27
0.3
0.18 0.21
0
0.06 0.09 0.12 0.15
0
0.03
0.18
Magnetic flux density [Tesla]
Magnetic flux density [Tesla]
Fig. 2. Identified parameters for Bingham-plastic model.

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211
50
45
40
35
30
25
1.6
1.4
1.2
1
0.8
0.6
20
15
10
5
0.4
0.2
0
0
0.21
0.24
0.27
0.3
0
0.03 0.06
0.09 0.12 0.15 0.18
0.06 0.09 0.12 0.15
0.03
0.24 0.27 0.3
0.18 0.21
0
Magnetic flux density [Tesla]
Magnetic flux density [Tesla]
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.21
0.27
0.24
0.3
0
0.03 0.06
0.09 0.12 0.15 0.18
Magnetic flux density [Tesla]
Fig. 3. Identified parameters for Herschel–Bulkley model.
their variations do not carry physical significance.
To improve the estimation results, we ran the
optimizer with a fixed flow index value of n ¼ 0:45:
These results are shown in Fig. 4. Both yield stress
and consistency are seen to be monotonically
increasing functions of magnetic field in this case.
Using the identified parameters, we fitted
models to the measured flow curve data (Fig. 5).

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50
45
40
1.5
1.25
1
35
30
25
20
15
10
5
0.75
0.5
0.25
0
0
0.060.09 0.12 0.15
0.24 0.27 0.3
0.18 0.21
0.21
0.24 0.27
0
0.03
0.03 0.06
0.09 0.12 0.15 0.18
0.3
0
Magnetic flux density [Tesla]
Magnetic flux density [Tesla]
Fig. 4. Identified parameters for Herschel–Bulkley model with n ¼ 0:45:
Both the BP and HB models fit the high shear
rate data equally well, but the low shear rate
regions are better represented by the HB model.
Finally, we compare the fitting errors, obtained
using Eqs. (6) and (7), for different models (BP
and HB) and schemes (LMS and GA). The plots
are shown in Fig. 6. The HB model, with errors
limited to 2% of the maximum stress, was more
accurate than the BP model. In addition, the HB
model parameters identified via the GA technique
were improved over those identified via the LMS
technique in that the modeling error was lower.
The HB model with fixed n also had the lowest
modeling errors and is more suitable as a
constitutive equation for the fluid flow since it
results in smoother variation of all the identified
parameters.
curves were measured for a suspension of cobalt
particles, having equivalent diameter of 100 nm, at
different levels of magnetic field. The flow curves
were measured for discrete values of applied
magnetic field ranging from 0.03 to 0.30 T. A
simple genetic algorithm was used to estimate the
parameters of the constitutive model. Constraints
were imposed on the minimization process to
ensure that the yield stress was a monotonically
increasing function of applied magnetic field. The
following points are noted:
(i) The maximum yield stress measured in the
cobalt nanoparticle-based MR fluid at 0.3 T
was 1.75 kPa (BP) or 1.45 kPa (HB), which is
much lower than yield stresses reported for
MR fluids with micron-sized particles at
comparable solids loading and applied mag-
netic field.
6. Conclusion
(ii) The dynamic yield stress is seen to vary from
about 10 Pa at 0.03 T to almost 1450 Pa at
0.30 T for the HB model, and from 50 to
1750 Pa for the BP model.
(iii) For the BP model, the yield stress and
postyield viscosity were both monotonically
increasing functions of magnetic field.
In this study, we identified the rheological
parameters of two constitutive models, Bingham-
plastic (BP) and Herschel–Bulkley (HB), that we
used to characterize the rheological properties of
nanocobalt based magnetorheological fluid. Flow

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213
3
2.5
2
3
Fitted model
0.304T
0.243T
0.304T
0.243T
Fitted model
2.5
2
0.182T
0.152T
0.122T
0.182T
0.152T
0.122T
1.5
1
1.5
1
0.091T
0.091T
0.5
0
0.5
0
0.061T
0.061T
0.030T
1400
0.030T
1400
1200
1600
400 600 800
1000
1200
1600
0
200 400 600 800 1000
Shear rate [s-1]
0
200
Shear rate [s-1]
Fig. 5. Reconstruction of rheological flow curves using (a) Bingham-plastic model, and (b) Herschel–Bulkley model.
7
80
70
60
50
40
30
20
10
0
Bingham Plastic
Bingham Plastic
Herschel-Bulkley (LMS)
Herschel-Bulkley (GA)
Herschel-Bulkley (LMS)
Herschel-Bulkley (GA)
Herschel-Bulkley (GA, n = 0.45)
6
5
4
3
2
1
0
Herschel-Bulkley (GA, n = 0.45)
0.21
0.03 0.06
0.09 0.12 0.15 0.18
0.27
0.3
0.21
0.27
0.24
0.03 0.06
0.24
0.09 0.12 0.15 0.18
0.3
Magnetic flux density [T]
Magnetic flux density [T]
Fig. 6. Errors between reconstructed and experimentally obtained rheological flow curves: (a) absolute error, (b) percentage error.
(iv) For the HB model, choosing a constant value
of n ¼ 0:45 (indicating shear thinning) pro-
duced yield stress and postyield consistency
that were both monotonically increasing
functions of magnetic field. Also, a constant
flow index, that is, a flow index that is not
dependent on field strength, resulted in the
lowest modeling error.

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(v) The percentage errors in model fitting were
[6] G. Bossis, S. Lacis, A. Meunir, O. Volkova, J. Magn.
Magn. Mater. 252 (2002) 224.
less than 2% of the peak shear stress when
GA was used to identify the HB model
parameters. The LMS technique had higher
errors than the GA technique.
[7] M.C. Yang, L.E. Scriven, C.W. Macosko, J. Rheol. 30
(1986) 1015.
[8] G. Bossis, E. Lemaire, J. Rheol. 35 (1991) 1345.
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(1995) 1011.
(vi) Both the BP and HB models can model the
fluid effectively. The HB model had errors
averaging 2%, which is about 1% lower than
the errors observed for the BP model.
[10] S. Genc, P. Phule, Smart Mat. Struct. 11 (2002) 140.
[11] K. Butter, A.P. Philipse, G.J. Vroege, J. Magn. Magn.
Mater. 252 (2002) 1.
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We can conclude that both the Bingham-plastic
and Herschel–Bulkley model can be used as
constitutive equations to describe the behavior of
nanocobalt-based MR fluids, especially in the high
shear rate region. Genetic algorithms can be
efficiently applied to such rheological parameter
estimation problems and the results are compar-
able to those obtained from gradient-based
schemes.
[14] N. Rosenfeld, N.M. Wereley, R. Radhakrishnan, et al.,
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