Electronic Physics Auxiliary Publication Service (EPAPS) Materials
Veering the Motion of a Magnetic Chemical Locomotive in a Liquid
Krishna Kanti Dey,1 Deepika Sharma,2 Saurabh Basu,2 and Arun Chattopadhyay1,3
1
Centre for Nanotechnology, Indian Institute of Technology Guwahati, Guwahati 781 039, India.
2
3
Department of Physics, Indian Institute of Technology Guwahati, Guwahati 781 039, India.
Department of Chemistry, Indian Institute of Technology Guwahati, Guwahati 781 039, India.
Email: arun@iitg.ernet.in
Experimental Section:
The experimental procedures are briefly described here. 5 g of commercially available
cation exchange resin microbeads (polystyrene divinylbenzene co-polymer, Amberlite- IR
120, MERCK) were at first kept in 20 mL of 3.46 M HCl (MERCK) solution for 5 h. The
beads were then washed thoroughly with water to remove excess HCl. The activated beads,
used for further experiments were of nearly the same sizes (average radius = 0.4 mm) and
weights (average weight = 0.5 mg). These were then divided into five different groups, each
containing 105 activated beads. This was followed by keeping each of these five groups in 10
mL
aqueous
NiCl2.6H2O
(MERCK)
solutions
of
concentrations 0.02 M, 0.01 M, 5  103 M , 1  103 M and 5  104 M for 3 h. The solutions were
then decanted followed by washing the beads with water to remove excess NiCl 2. 20 beads
from each of the above samples were then independently treated with 10 mL of
0.05 M aqueous NaBH4 (MERCK) solution for 1 h. This was followed by washing the beads
with water and air-drying them. Similarly, coated beads were prepared with varying NaBH4
concentrations ( 0.15 M , 0.1 M , 0.05 M and 0.01 M ), keeping the concentration of NiCl2
constant at 5  104 M . Milli Q grade water was used throughout the experiments. Ni coated
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beads were investigated by a LEO 1430 VP scanning electron microscope (SEM). Magnetic
properties of the beads were studied using a vibrating sample magnetometer (VSM; Model
7410, Lake Shore Cryotronics Inc., USA) A DC electromagnet (Model EMU-50V, Scientific
Equipment & Services, Roorkee, India) was used for controlling the velocity of beads in
aqueous H2O2 solution. Videos were captured by a Creative Webcam manufactured by
Creative Technology Limited, Singapore. The beads were found to be coated with thin Ni
film when the concentration of Ni-salt was maintained at 1  103 M and above, keeping the
concentration of NaBH4 fixed at 0.05 M . On the other hand, uniform depositions of Ni NPs
(of diameters 79 nm-130 nm) on the beads occurred at 5  104 M NiCl2. Lowering the
concentration of NaBH4 below 0.05 M resulted in poor quality film or NP depositions.
Results:
FIG. 1: SEM images of two selected samples with formation of Ni film (left) and Ni NPs (right) on the
polymer beads.
FIG. 2:
Energy dispersive X-ray profiles of samples coated with thin film (left) and NPs (right),
confirming the presence of Ni.
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FIG. 3: Magnetization curves for samples coated with Ni thin film and Ni NPs. Here B is the external
magnetic field in Gauss and m is the corresponding magnetization in emu. The wavering behavior in the
saturation region in case of NP-coated bead may be attributed to the inhomogeneity in particle distribution.
Table 1: Observed ferromagnetism in samples coated with Ni thin film (Sample a) and Ni NPs (Sample b).
Physical Quantity
Sample a
Sample b
Sp. Magnetization (emu/gm)
0.73
0.42
Retentivity (emu/gm)
0.34
0.15
Coercivity (kiloOe /gm)
174.06
408.14
FIG. 4: Variation of Specific Magnetization Msp in emu/gm of polymer beads with molar
concentrations of NiCl2 (C) and NaBH4 (C*) solutions.
3
t0s
t4s
t 8 s
FIG. 5: Photographs of typical deviation of a Ni-nanoparticle coated polymer bead in aqueous H2O2
solution, in the presence of an inhomogeneous external magnetic field.
Table 2: Increase in deviation of a magnetic bead in aqueous H2O2 with the increase in magnetic field
gradient.
Applied magnetic
Field gradient
Deviation of thin film
Average deviation of
field (kG)
(Gauss/mm)
coated sample (rad)
NP coated sample
(rad)
0.75
16.80
0.52
0.61
1.09
27.20
0.67
0.89
1.45
39.34
0.68
1.06
1.81
43.54
0.69
1.22
Calculations involved in the modeling:
The force acting on a Ni NP coated polymer bead is given by Fm    m  B 
 m   B ,
where m  4Ai1ˆ is the magnetic moment of the bead and B   B0  x  iˆ is the magnetic field
at a distance x from the origin along X-axis.
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 

 ˆ  ˆ
Fm   iˆ 
j  k   4A1iˆ   B0  x  iˆ  4A1iˆ
y
z 
 x

(1)
The total viscous force acting on the accelerating polymer bead is given by the empirical
relation
1 t
a  t
14

Fv  6 R v    R 3   a  6 R 2   2 
dt   F1  F2  F3
1
23

0  t  t  2
(2)
Here, F1 is equal to the steady-state viscous drag on the sphere acting in the direction
opposite to its motion, F2 has the same magnitude as the resistance of an accelerating sphere
in irrotational motion and F3 is the term signifying the history of the acceleration.
Table 3: Typical instantaneous velocity of the bead and viscous drag on (F1 only) it at various values
of external magnetic field.
B0 (Tesla)
Instantaneous transverse
F1 (N)
velocity v (m/s)
0.08
0.0003
2.26  10 9
0.11
0.0005
3.77 109
0.15
0.0020
1.5110 8
0.18
0.0025
1.89 108
Typical acceleration of the bead is found to be in the range 1.5  104 ms 2 to1.9 104 ms 2 .
Assuming the transverse acceleration to be a constant and of value 1.8  104 ms 2 in all the
cases, we get F2  2.411011 N and F3  1.37 109 N ,where a typical time interval of t  5 s
is considered.
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Table 4: Total transverse viscous drag on a polymer bead at different external magnetic fields.
B0 (Tesla)
F1 (N)
F2 (N)
F3 (N)
Fv (N)
0.08
2.26  10 9
2.411011
1.37 109
3.66 109
0.11
3.77 109
2.411011
1.37 109
5.16 109
0.15
1.5110 8
2.411011
1.37 109
1.65 108
0.18
1.89 108
2.411011
1.37 109
2.03 108
The net transverse acceleration of a polymer bead is thus given by am 
4A1  Fv ˆ
i
mb
(3)
Taking v T be the terminal velocity of the bead, its total vertical displacement in time t is
given by z  vTt
(4)
The total transverse displacement of the bead in time t is given by x 
1 4A1  Fv  z 
From Eqs. (4) and (5), we get x 
 
2
mb
 vT 
(5)
2
2mb vT2
x
Or, z 
4A1  Fv
2
This is a parabola of the form z 2  4ax , where 4a 
1 4A1  Fv 2
t
2
mb
(6)
2mb vT2
4A1  Fv
(7)
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1 4A1  Fv 2
t
mb
4A1  Fv
x 2
tan   

t,
z
vTt
2mb vT
Again
(8)
This shows that at a definite field gradient, the angular deviation increases continuously with
time, till the bead reaches the boundary of the vessel.
Now, the magnetic vector potential of the polymer bead may be approximated as






2  1   1


  b3  a 3  B0
A1 
3

0
2
a
  2  1   2   2      1 
b


(9)
We consider the value of relative permeability of deposited Ni to be   50 [1]
Permeability of free space = 0  1.26 106 NA2 , [2]
Also we assume the thickness of the magnetic coating over the polymer bead surface to be of
the order of 1 nm . Then,
b  0.4mm  0.4 103 m  4 104 m , and a   4 104 - 110-9  m  3.99999 104 m
3
3
 a   3.99999 
  
  0.9999975 and
4
b 

b
3
 a 3    6.4 1011  6.399952  1011  m3  4.8  1016 m3
Putting these values in Eq. (9), we get,
A1  4.19 109 B0
m3
N / A2
(10)
Therefore, from Eq. (6) the trajectory of the polymer bead inside the liquid is given by,
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2mb vT2
2mb vT2
z 
x
x
4 4.19 109  B0   Fv
5.27 108  B0   Fv
2
Where the parabola parameter is given by 4a 
(11)
2mb vT2
5.27 108  B0   Fv
(12)
Now, Average mass of the polymer beads is mb  0.5mg  0.5 106 Kg .
Typical
terminal
velocity
of
the
beads
was
found
to
be
within
the
range vT  0.002 ms -1  0.007 ms -1 . Taking v T  0.004 ms -1 we get 2mb vT2  1.6 1011 kgm 2 s 2
Table 5: Theoretically calculated parameters of the parabolae at different values of external magnetic
field.
B0

(Tesla)
4πA1 γ
Fv
4πA1 γ - Fv
2mb v T2
4a
(Tesla/m)
(N)
(N)
(N)
(Kgm2s-2)
(Theoretical)
0.08
1.68
7.08 109
3.66 109
3.42 109
1.6 1011
0.0047
0.11
2.72
1.58 108
5.16 109
1.06 108
1.6 1011
0.0015
0.15
3.93
3.11108
1.65 108
1.46 108
1.6 1011
0.0011
0.18
4.35
4.13 108
2.03 108
2.10 108
1.6 1011
0.0008
To calculate the values of the parabola parameters experimentally, the 4a values are evaluated
at all the data points of a trajectory. The average of these values is considered as the 4a value
of that particular trajectory.
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Table 6: Theoretical and experimental parabolae parameters at different values of external magnetic
fields.
B0

4a (m)
4a (m)
(Tesla)
(Tesla/m)
(Theoretical)
(Experimental)
0.08
1.68
0.0047
0.038
0.11
2.72
0.0015
0.017
0.15
3.93
0.0011
0.010
0.18
4.35
0.0008
0.0053
Reference
(1) S. Lucyszyn, PIERS Online 4, 686, 2008 and the reference therein.
(2) J. D. Jackson, Classical Electrodynamics, 3rd ed. John Wiley & Sons, Inc., page 782.
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