Download magnetic permeability and electric conductivity of magnetic emulsions

MAGNETIC PERMEABILITY AND ELECTRIC CONDUCTIVITY OF
MAGNETIC EMULSIONS
Alexander N. Tyatyushkin
Institute of Mechanics, Moscow State University, Michurinskiy Pr., 1, Moscow 117192, Russia
Simultaneous effect of electric and magnetic fields on magnetic emulsions has
peculiarities which are not characteristic for both magnetic and electric fields acting
separately.
In order to obtain a magnetic emulsion, a kerosene-based magnetic liquid is heated
up, then some amount of an oil is added to it, and the obtained mixture is then cooled
down back to the room temperature [1]. The dispersed phase of such an emulsion
consists of small drops of more concentrated magnetic liquid, and its dispersive phase
is a less concentrated magnetic liquid, in which the drops are suspended.
In the absence of electric and magnetic fields the drops of the dispersed phase of a
magnetic emulsion are spherical. An electric or magnetic field distorts the shape of
the drops and thus makes the emulsion anisotropic. The goal of the present work is to
theoretically investigate the influence of the anisotropy induced by simultaneously
acting electric and magnetic fields on the electric conductivity and magnetic
permeability.
Kerosene-based magnetic emulsions are weakly electrically conducting, and the
effect of an electric field on them can be described within the electrohydrodynamic
approximation [2]. Due to weak conductivity, the magnetic field caused by the
electric current in an electric field can be neglected.
Single drop of the dispersed phase. Consider a single drop of the dispersed
phase, whose radius in the absence of fields is R, in simultaneously acting uniform
constant electric and magnetic fields. Let the viscosity, electric conductivity, dielecric
and magnetic permeability of the dispersed and dispersive phases be, respectively, i,
i, i,  i, e, e, e, e, the surface tension of the interface between the phases be 
and the magnetic and electric intensity vectors be, respectively, H and E. If i and e
are sufficiently small, then, to the first order approximation with respect to the
dimensionless small parameters H2R/ and E2R/, the surface of the deformed
drop is an ellipsoid whose semi-axes, a1, a2 and a3, are determined by the following
expressions [3]
a
2
1
a
2
2
 R2  R2  E 2  H 2  3D R 12  ,
(1)
 R2  R2  E 2  H 2  3D R 12  ,
(2)
a
(3)
2
3
 R 2  R 2  E 2  H 2 R 6  .
The orientation of the ellipsoid is given by the unit vectors, l1, l2 and l3, directed
along its main axes
l1  sin HE  sin EH  E  H ,
(4)
l1   cosHE  cosEH  E  H ,
(5)
l1  E  H  E  H ,
(6)
Here, , , D,  and  are determined as follows
2
9e 16e  19i  i e   e i 9 e  i  e 


,
2
8 5e  i  2e  i 
8  2e  i 
9e  i  e 

8  2e  i 
D
E
2
(7)
2
,
(8)
 H 2   4 E  H  ,
2
2
(9)
sin 2   2 E  H E  H H 2 D,
(10)
cos2    2 E  H E  H H 2 D .
(11)
Here,  and  denote the scalar and vector products, respectively, and |b| denotes the
module of the vector b. The conditions for the parameters of the problem, R, i, i, i,
i, e, e, e, e, , Ha and Ea, which should be satisfied in order (1)–(11) to be valid,
are written down in Ref. [3].
Electric conductivity and magnetic permeability. Consider a sufficiently dilute
monodispersed magnetic emulsion. Applying the method used in [4] for obtaining the
electric permeability of dilute suspension of spherical particles, one obtains the
following expressions for the tensors of electric conductivity and magnetic
permeability,  and , of such an emulsion
  eI  I i  e   N e  c ,
1
(12)
μ  e I  I i  e   N e  c ,
1
(13)
where c is the volume concentration of the dispersed phase, I is the unit tensor, S-1
denotes the inverse tensor with respect to the tensor S, and N is the demagnetization
tensor (see [4]) of a single drop. To a first order approximation with respect to the
small parameters (ai2-R2)/R2 (i=1,2,3),
  I 3   T  3R 2I   5R 2  ,

 

(14)
where T = a12l1l1 + a22l2l2 + a32l3l3 (bd denotes the dyadic product of the vectors b
and d). Substituting (14) into (12) and (13) and using the smallness of the parameters
(ai2-R2)/R2, one obtains
  0 I 
2
52e  i 
2
R
2
c,
9e i  e  T  3R 2 I
(15)
2
μ  0 I 
where
9e i  e  T  3R 2 I
52e  i 
2
R
2
c,
(16)
0  e  3e i  e c 2e  i  ,
(17)
0  e  3e i  e c 2e  i  ,
(18)
Transverse current and magnetization. Due to the anisotropy, the electric
current density, j=Ea, and the magnetization, M=(-I)Ha/(4), in a magnetic
emulsion have, in general, both longitudinal and transverse components relative to Ea
and Ha, respectively (Sb denotes the contraction of the tensor S with the vector b.
Using (1)–(11) and (14)–(16), one obtains the following expression for the
longitudinal and transverse components
jl  0 E 


9e i  e  2E 4  3 E  H   E 2 H 2 R
2
52e  i 
2
2
6E
2
cE
, (19)
9e i  e  8RE  H E  H   E
2
jt 
Ml 

52e  i 
E
2
0  1
2
c
,
(20)
H
4
2
2
9  e  i   e  2 H 4  3 E  H   E 2 H 2 R
52  e   i 


24H 2
2
9e i  e  2RE  H H  E   H
cH
,
(21)
2
Mt 
52e  i 
2
H
2
c,
(22)
It follows from (20) and (22) that the transverse components are equal to zero when
Ea and Ha are parallel or perpendicular and are maximal for given values of Ea and
Ha when the angle between Ea and Ha is equal to 45 or 135.
Acknowledgements. This work is supported by Russian Foundation for Basic
Research grants No. 01-01-00010 and No. 01-01-00423.
References
1. Yu. I. Dikanskii, N. G. Polikhronidi and K. A. Balabanov,
Magnetohydrodynamics 24 (1988) 211.
2. J. R. Melcher and G. I. Taylor, Ann. Rev. Fluid Mech. 1 (1969), 111.
3. A. N. Tyatyushkin and M. G. Velarde, J. Colloid Interface Sci. 235 (2001) 46.
4. L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media, (Pergamon
Press, 1984).