PHYSICAL REVIEW E 71, 056611 共2005兲
Equilibrium orientation of an ellipsoidal particle inside a dielectric medium with a finite electric
conductivity in the external electric field
Yu. Dolinsky* and T. Elperin†
The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering,
Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel
共Received 7 May 2004; revised manuscript received 19 October 2004; published 23 May 2005兲
We study the stability of the orientation of an ellipsoidal dielectric particle immersed into a host dielectric
medium under the action of the external electric field. It is assumed that the particle and the host medium have
a finite electric conductivity. We demonstrate that an equilibrium orientation of the ellipsoidal particle changes
with time in a stationary electric field with a constant direction. It was found that during time interval T1 an
equilibrium orientation of the spheroidal particle with a finite electric conductivity remains the same as the
equilibrium orientation of an ideal dielectric particle. During time interval T2, where T = T1 + T2 is a period of
the external electric field, the equilibrium orientation of the axis of symmetry of the particle is normal to the
initial equilibrium direction.
DOI: 10.1103/PhysRevE.71.056611
PACS number共s兲: 41.20.⫺q, 77.22.⫺d
I. INTRODUCTION
The dynamics of solid or liquid particles in a host medium
under the action of an external electric field is of theoretical
and technological interest. Technological application includes manipulation of microparticles in biotechnology and
genetic engineering 关1兴, nanotechnology 关2,3兴, and noncontact measurements of physical properties of particles. Interaction of an external electric field with an inclusion embedded into a host medium is important for understanding the
mechanisms of the electric breakdown of dielectrics, in atmospheric physics and aerosol dynamics 关4–6兴. The results
obtained in numerous theoretical and experimental studies on
particle dynamics under the action of the external electric
field were summarized in several survey papers and monographs 关6–9兴.
One of the issues that warrant theoretical and experimental studies is the rotation of liquid or solid particles embedded into a weakly conducting host medium. This issue has
been considered in a number of publications for the case
when a particle has a spherical and a spheroidal shape 共see,
e.g., 关10–14兴兲, and the rotation of ellipsoidal particles was
analyzed in 关15,16兴. These studies were concerned mainly
with applications, and some important aspects of the dynamics of particles in the external electric field were not addressed. Rotation of ellipsoidal particles with a shell in the
nonstationary external field was studied in 关15,16兴 using a
simplified approach that did not require a comprehensive
analysis of the dynamics of the particle. In this study we
obtained a general expression for an instantaneous moment
of forces acting at an ellipsoidal particles as a function of the
orientation of its principal axes.
For a case of an ideal dielectric the mathematical formulation of the problem is known 关17,18兴. The situation is different for the case of a particle with a finite electric conduc-
*Email address: [email protected]
†
Email address: [email protected]
1539-3755/2005/71共5兲/056611共10兲/$23.00
tivity. Indeed, here an instantaneous moment of forces acting
at a particle depends not only upon its instantaneous orientation but also on its orientation during the earlier time moments. The reason for this behavior is as follows. The total
ជ acting upon a dielectric ellipsoidal particle with a
torque M
ជ =M
ជ +M
ជ . The
finite conductivity is the sum of two terms, M
␧
␴
ជ
ជ
ជ
ជ
first term is M ␧ = P␧ ⫻ E0, where P␧ is a dipole moment deជ is
termined by the initial polarization of the medium and E
0
ជ =P
ជ ⫻ Eជ ,
an applied electric field. The second term is M
␴
␴
0
ជ is a dipole moment caused by a flow of an electric
where P
␴
charge from the external source to the surface of a particle.
ជ and the torque M
ជ settle during a
The dipole moment P
␧
␧
short time interval of a local relaxation while the dipole moជ and the torque M
ជ settle during time ␶ of a macment P
␴
␴
␴
roscopic relaxation that depends upon the conductivities of
particle and a medium. If an applied electric field is normal
to the axis of symmetry, for the case of an ideal dielectric the
ជ =P
ជ and M
ជ =M
ជ = 0. For the case of a
total dipole moment P
␧
␧
nonideal dielectric, the dipole moment associated with a free
ជ of a rotating particle is not aligned with the apcharge P
␴
plied electric field because of the finite relaxation time so
ជ =P
ជ ⫻ Eជ ⫽ 0ជ . This difference in the directions of the
that M
␴
␴
0
ជ is associated
external electric field and the dipole moment P
␴
with Quincke rotation that was extensively discussed in the
literature 关7,10,13,14兴. In this study we investigate a torque
acting at a stationary particle as a function of its orientation
共direction of its axis of symmetry兲 with respect to the applied
electric field. Thus we assumed that the angular velocity of a
ជ = 0ជ .
particle ⍀
We show that in a medium with a finite electric conductivity, a torque acting at the particle in a stationary electric
field can change the orientation of a particle even when the
direction of the field is fixed. Thus, if initially the particle
was in a state of a stable equilibrium, then after some time
the initial orientation of the particle loses its stability. Our
analysis shows that there exist two time intervals, T1 and T2,
056611-1
©2005 The American Physical Society
PHYSICAL REVIEW E 71, 056611 共2005兲
YU. DOLINSKY AND T. ELPERIN
FIG. 1. Ellipsoid with semiaxes a1 , a2 , a3 共a1 = a2 ⬍ a3: prolate
spheroid; a1 = a2 ⬎ a3: oblate spheroid兲 and electric permittivity ␧2
and conductivity ␴2 inside a host medium with permittivity ␧1 in
ជ .
the external electric field E
0
such that during time T1 the stable orientation of the particle
is the same as for the case of an ideal dielectric. During time
interval T2, where T = T1 + T2 is a period of the external electric field, the direction of stable orientation is normal to that
for the case of an ideal dielectric.
This paper is organized as follows. In Sec. II we present a
mathematical formulation of the problem and discuss the underlying physics. Special attention is given to those features
in the formulation of the problem that arise due to a finite
conductivity of a host medium. In particular, we elucidate the
physical aspects that constitute the difference between the
problem for the case of a weakly conducting dielectric and
an ideal dielectric case. In Sec. III we calculate the basic
parameters required to determine the electric field and the
electric current of a dielectric ellipsoid with permittivity ␧2
and conductivity ␴2 that is embedded into a host medium
with permittivity ␧1 and conductivity ␴1. In Sec. IV we investigate stability of the orientation of particle in the external
electric field.
II. MATHEMATICAL FORMULATION OF THE PROBLEM
Let us consider an ellipsoidal particle with permittivity ␧2
and conductivity ␴2 embedded into a host medium with permittivity ␧1 and conductivity ␴1 in the external electric field
ជ 共see Fig. 1兲.
with a strength E
0
In a conducting medium a potential component of an elecជ = −ⵜ
ជ ␸ is determined by the following system of
tric field E
equations:
ជ =␳ ,
ⵜជ · D
ex
⳵␳ex ជ ជ
+ ⵜ · j = 0,
⳵t
共1兲
ជ and electric current density ជj
where electrostatic induction D
are determined by the following relations:
ជ = ␧ ␧Eជ ,
D
0
ជj = ␴Eជ ,
ជ = − ⵜជ ␸ .
E
the time required for a charge redistribution taking into account a finite conductivity is much larger than the characteristic time of microscopic relaxation of the dipole moments
induced by local polarization.
The host medium with an embedded particle can be considered as a piecewise homogeneous medium. Since a charge
is localized at the inhomogeneities, in the case of a piecewise
homogeneous medium it accumulates at the interface boundaries. Density of a surface free charge ␥ is determined by the
following relations:
冕
␳exdV =
冕
␥dS or ␳ex = ␥␦共u兲兩ⵜជ u兩,
where ␦共u兲 is a Dirac’s delta function, u = F共x , y , z兲 and u
= 0 is an equation of the surface. Equations 共1兲 and 共3兲 yield
boundary conditions at the interface boundary:
ជ ·D
ជ 兴 = ␥,
关N
ជ · ជj兴 = − ⳵␥ .
关N
⳵t
共4兲
Here 关A兴 = A+ − A−, A+ and A− are values of a function A at
ជ is the
the external and internal surfaces, respectively, and N
external unit normal vector.
As mentioned earlier, the main difference between a
weakly conducting medium and an ideal dielectric is charge
transport from the external source to the interface boundary.
Time variation of a charge constitutes the principal difference between a leaky dielectric and an ideal dielectric model
关17,18兴. In the following section we determined time dependencies of this charge and electric potential ␸.
III. ELECTRIC FIELD AND ELECTRIC CURRENT IN A
MEDIUM WITH AN ELLIPSOIDAL INCLUSION
In this section we determine an electric field of an ellipsoid immersed into a medium with a finite electric conductivity. Using the obtained results we investigate variation of
the electric field during electric charge flow from the external
source to the surface of the ellipsoid, variation of the electric
charge ␥ at the surface of the ellipsoid and dependence of the
electric charge relaxation time upon the geometrical parameters of the ellipsoid.
Consider an ellipsoidal inclusion with the half-lengths of
the axes a1 , a2 , a3, permittivity ␧2 and electric conductivity
␴2 that is immersed instantaneously into a host medium
with permittivity ␧1 and electric conductivity ␴1 in the external electric field Eជ 0 共see Fig. 1兲.
The solution of an electrostatic problem is performed in a
system of coordinates associated with an ellipsoid. In this
system of coordinates the equation of a surface of the ellipsoid and the components of the electric field are determined
by the following equations:
3
共2兲
Hereafter we assume that a particle is at rest. Formula 共2兲 for
an electrostatic induction implies that a characteristic time
required to attain an equilibrium polarization is substantially
smaller than other characteristic times in the problem. Thus
共3兲
u=
兺
i=1
x2i /a2i − 1,
ជ=
E
3
Eieជ i,
兺
i=1
eជ i = ⵜជ xi .
共5兲
If before the insertion of an ellipsoidal particle, the electric
field was homogeneous then electric potential ␸ can be written in the following form:
056611-2
PHYSICAL REVIEW E 71, 056611 共2005兲
EQUILIBRIUM ORIENTATION OF AN ELLIPSOIDAL…
3
␸ = 兺 ␸i = − 兺 E0ixi„1 + Fi共␰,t兲…,
i=1
共6兲
Ni =
i=1
where ␰ 共and coordinates ␩ , ␵ used below兲 are the ellipsoidal
coordinates determined through x1 , x2 , x3 by formulas presented in 关17兴 共Chap. 1, Sec. 4兲 and ␰ is chosen such that ␰
= 0 corresponds to a surface of the ellipsoid u = 0. The potential ␸ in each medium is determined by the Laplace equation:
ⵜ2␸ = 0.
共7兲
The expression for Fi共␰ , t兲 can be written as follows:
Fi共␰,t兲 = Fi1共␰,t兲␪共␰兲 + Fi2␪共− ␰兲,
where
␪共x兲 =
再
1, x 艌 0,
0, x ⬍ 0.
共8兲
冎
Using formula 共6兲 and solving Eq. 共7兲 with continuity condition for the potential ␸ we arrive at the following equation:
Fi1共␰,t兲 =
Fi2共t兲Ii共␰兲
,
Ii共0兲
共9兲
I i共 ␰ 兲 =
冕
␰
ds
,
共s + a2i 兲R共s兲
R共s兲 = 冑共s + a21兲共s + a22兲共s + a23兲,
共10兲
3
共11兲
i=1
where ␥i共␩ , ␵ , t兲 is a free electric charge accumulated at the
surface of an ellipsoid due to the ith component of an electric
field. For the case of an ellipsoid in a homogeneous external
electric field E0i共t兲, it is also convenient to represent
␥i共␩ , ␵ , t兲 as
␥i共␩,␵,t兲 =
␧0␧1 E0i共t兲xi共0, ␩,␵兲
˜␥i共t兲,
h1共0, ␩,␵兲
2a2i
共12兲
where h1共0 , ␩ , ␵兲 is a Lamé’s coefficient along the coordinate
␰ at ␰ = 0, xi共0 , ␩ , ␵兲 = 兩xi共␰ , ␩ , ␵兲兩␰=0, h1共0 , ␩ , ␵兲 = 冑␩␵ / 2R共0兲
3
共x2i / a4i 兲1/2.
and in a Cartesian coordinate system h1共0兲 = 21 兺i=1
In order to derive Eq. 共12兲 for ␥i共␩ , ␵ , t兲 one can use Eq.
共6兲 for a potential ␸ and Eq. 共2兲 that determines a relation
between a potential and electric field. At the surface
ជ ·ⵜ
ជ 兲␸ = 兩共1 / h 兲共⳵␸ / ⳵␰兲兩
of the ellipsoidal particle 共N
1
␰=0 and
1
2
兩⳵xi / ⳵␰兩␰=0 = 2 关xi共0 , ␩ , ␵兲 / ai 兴. The latter relations and Eqs.
ជ ·ⵜ
ជ 兲␸ ⬀ E 共t兲x 共0兲 / 2a2h 共0 , ␩ , ␵兲.
共8兲 and 共10兲 imply that 共N
0i
i
i 1
Coefficients ˜␥i共t兲 depend only upon time.
ជ
The unit normal vector at the surface of the ellipsoid N
ជ
can be represented as N = 兺 N eជ 共see, e.g., 关17兴兲 with
共i兲
i i
,
i = 1,2,3.
ជ,
␥ = ␥ជ ⬘ · N
共13兲
共14兲
where the formula for the components of the vector ␥ជ ⬘
= 兺共i兲␥⬘i eជ i reads
␥⬘i = ␧0␧1E0i共t兲˜␥i共t兲.
共15兲
These components have a simple physical meaning. They are
equal to the magnitudes of the electric charge at the apexes
ជ = eជ , N
ជ = eជ , and N
ជ = eជ . Hereafter we
of the ellipsoid: N
1
1
2
2
3
3
ជ = 共1 , 0 , 0兲, N
ជ
will write as it is generally agreed that N
1
2
ជ = 共0 , 0 , 1兲. Then ␥⬘ , ␥⬘ , ␥⬘ are the magni= 共0 , 1 , 0兲, and N
3
1
2
3
tudes of the free charge at these locations.
Coefficients ˜␥i共t兲 are determined from the boundary conditions 共4兲. The first equation in the boundary conditions 共5兲
yields
F2i共t兲 = −
˜␥i共t兲ni + f i␧
,
1 + f i␧
共16兲
where ni = R共0兲Ii共0兲 / 2 is a depolarization factor, 0 ⬍ ni ⬍ 1,
3
兺i=1
ni = 1, f i␧ = ␬␧ni, and ␬␧ = ␧2 / ␧1 − 1. The second equation
in the boundary conditions 共4兲 implies that
冉
and function Fi2共t兲 is determined from the boundary conditions 共4兲. In order to determine the function Fi2共t兲 it is convenient to represent a free electric charge ␥共␩ , ␵ , t兲 as
␥共␩,␵,t兲 = 兺 ␥i共␩,␵,t兲,
2a2i h1共0, ␩,␵兲
Equations 共11兲–共13兲 imply the following expression for a
free electric charge accumulated at the surface of the ellipsoid due to the external electric field, ␥共␩ , ␵ , t兲:
where 共for details of this solution see 关17兴, Chap. 1, Sec. 4兲
⬁
xi共0, ␩,␵兲
3
冊
Ė 共t兲
␧ 0␧ 1n i ˙
˜␥ + ˜␥i 0i
− f i␴
E0i共t兲
␴1
,
F2i共t兲 =
1 + f i␴
共17兲
where f i␴ = ␬␴ni and ␬␴ = ␴2 / ␴1 − 1.
Formulas 共16兲 and 共17兲 yield the following equation for
˜␥i共t兲:
冉
冊
␬␴ − ␬␧ 1
Ė 共t兲 1
˜␥˙ i共t兲 + ˜␥i共t兲 0i +
=
,
E0i共t兲 ␶i
1 + f i␧ ␶0
共18兲
where ␶0 = ␧0␧1 / ␴1 and ␶i = ␶0关共1 + f i␧兲 / 共1 + f i␴兲兴.
Assuming that the initial free electric charge of the particle is zero, the expression for ˜␥i共t兲 can be written as
˜␥i共t兲 = ˜␥i共⬁兲⌸i共t兲,
⌸i共t兲 =
1 1
E0i共t兲 ␶i
冕
t
e−␶/␶iE0i共t − ␶兲d␶ ,
0
共19兲
where ˜␥i共⬁兲 = 共␬␴ − ␬␧兲 / 共1 + f i␴兲. If the initial free electric
charge of the particle is not zero one must account for the
electric field produced by this charge. Hereafter it is assumed
that initially the particle was not charged.
Formula 共19兲 allows us to determine the functions ⌸i共t兲
for given E0i共t兲 and parameters ␶i that characterize a system.
In a case when E0i共t兲 = const the result is presented below. In
the case of a stationary field the result is given by Eq. 共47兲.
In this study we expressed the considered physical characteristics through functions ⌸i共t兲.
056611-3
PHYSICAL REVIEW E 71, 056611 共2005兲
YU. DOLINSKY AND T. ELPERIN
Substituting Eqs. 共18兲 and 共19兲 into Eq. 共16兲 yields
Fi2共t兲 =
1 − ⌸i共t兲 ⌸i共t兲
+
− 1.
1 + f i␧
1 + f i␴
共20兲
The magnitudes of the electric fields and currents can be
determined using formulas 共2兲, 共5兲, and 共20兲:
ជ =
E
2
3
兺
i=1
E0i共t兲eជ i
冉
冊
1 − ⌸i共t兲 ⌸i共t兲
+
,
1 + f i␧
1 + f i␴
ជj2 = ␴2Eជ 2,
ជ = ␧ ␧ Eជ .
D
2
0 2 2
共21兲
The value of the accumulated charge at the surface can be
determined from formulas 共11兲, 共12兲, and 共19兲:
3
␥=
E0i共t兲xi共0兲 ␬␴ − ␬␧
␧ 0␧ 1
⌸i共t兲.
h1共0兲 i=1
1 + f i␴
2a2i
兺
共22兲
ជ can be written as
According to Eq. 共21兲 the electric field E
2
ជ describes
a sum of two fields, Eជ 2 = Eជ ␧ + Eជ ␴, where the field E
␧
ជ
a renormalization of the external electric field E0 due to polarization. At the initial time, t = 0, Eជ ␧共0兲 recovers the known
formula for the electric field of the dielectric ellipsoid with a
permittivity ␧2 imbedded into the host medium with a perជ describes a
mittivity ␧1 共see, e.g., Ref. 关17兴兲. The term E
␴
renormalization of the external field due to an accumulation
of the electric charge at the surface. If the external field is
constant, E0i = const, then ⌸i共t兲 = 1 − e−t/␶i, and at t → ⬁, the
configuration of the electric field is identical to the configuration of the electric field produced by an ellipsoidal inclusion with electric conductivity ␴2 imbedded into the host
medium with electric conductivity ␴1.
Formula 共22兲 describes a free charge at the surface of a
particle. The total charge is
ជ.
␥c = ␧0关Eជ 兴 · N
The latter formula can be rewritten as
3
␥c =
冉
共23兲
冊
␶a − ␶0 n−1
b + ␬␴
= −1
.
␶b − ␶0 na + ␬␴
Equations 共25兲 and 共26兲 and conditions −1 ⬍ ␬␴ ⬍ ⬁, 0 ⬍ ni
⬍ 1 imply that when ␬␧ ⬎ ␬␴ and na ⬍ nb, then ␶a ⬍ ␶b, while
when ␬␧ ⬍ ␬␴ and na ⬍ nb then ␶a ⬎ ␶b. It is known 关17兴 that
polarization factors n1,n2,n3 and half-lengths of the axes of
ellipsoid a1,a2,a3 are related by the following condition:
when a1 ⬎ a2 ⬎ a3 then n1 ⬍ n2 ⬍ n3. Therefore if a relaxation
time of a free electric charge inside an ellipsoid is less than a
characteristic relaxation time in the host medium, ␶i ⬍ ␶0 or
␬␧ ⬍ ␬␴, then relaxation of a free electric charge occurs faster
along the shorter axes. Alternatively, when ␶i ⬎ ␶0, charge
relaxation proceeds faster in the direction of the longer axes.
For a cylinder with the axis directed along the coordinate
axis x3, n1 = n2 = 21 and n3 = 0. The relaxation time along the
coordinate axis x3, ␶3 = ␶0 and relaxation times along axes x1
and x2, ␶1 = ␶2 = ␶0关共␬␧ + 2兲 / 共␬␴ + 2兲兴. In the case of a thin disk
with the axis directed along the coordinate axis x3, n1 = n2
= 0, n3 = 1 and relaxation times are ␶3 = ␶0关共␬␧ + 1兲 / 共␬␴ + 1兲兴,
␶ 1 = ␶ 2 = ␶ 0.
Polarization factors ni can be expressed as functions of
the ratios of the half-lengths of the axes of ellipsoid to a
half-length of one of the axes. Hereafter we expressed ni as
ni = ni共a1⬘ , a⬘2兲, where a⬘1 = a1 / a3 and a2⬘ = a2 / a3. In Fig. 2 we
showed the dependence of n1 as a function of parameters a⬘1
and a⬘2. Since n1共a1⬘ , a⬘2兲 = n2共a2⬘ , a⬘1兲, the same set of the
curves describes the dependence of n2共a2⬘ , a⬘1兲 by a change of
the parameters, a⬘1 → a2⬘ and a2⬘ → a⬘1. In Fig. 3 we showed the
dependence n3共a⬘1 , a2⬘兲 by presenting the set of curves n3
= n3共a1⬘兲 for different values of parameter a2⬘.
Consider now the behavior of a total surface charge ␥c
which is determined by expression 共24兲. As in the case of a
free charge expression for ␥c can be written similarly to Eq.
共14兲:
where
共24兲
Thus at the initial time ⌸i共t兲 = 0, the total charge coincides
with the polarization charge that is formed during microscopic time by local polarization of the material.
Since time variation of electric charges and currents is
essentially determined by the magnitude of the relaxation
times ␶i it is of interest to analyze the dependence of these
relaxation times on geometrical parameters of the ellipsoid.
Expression for ␶i 关see formulas after Eq. 共18兲兴 yields
␶i − ␶0 = ␶0
共␬␧ − ␬␴兲ni
.
1 + ␬ ␴n i
共26兲
ជ,
␥c = ␥ជ c · N
xi共0兲E0i共t兲 „1 − ⌸i共t兲…␬␧ ⌸i共t兲␬␴
␧0
+
.
h1共0兲 i=1
1 + f i␧
1 + f i␴
2a2i
兺
not a subject of this study. Equation 共25兲 allows us to determine a ratio of relaxation times along different axes of the
ellipsoid, a and b:
共25兲
Equation 共25兲 implies that when ␬␧ ⬍ ␬␴, ␶i ⬍ ␶0 for an arbitrary direction i. It is known that conditions ␶i ⬎ ␶0 or ␬␧
⬎ ␬␴ are the necessary conditions for Quincke rotation that
has been extensively discussed in the literature 关7,14兴 and is
␥ci = ␧0Eoi
冉
共27兲
冊
␬␧„1 − ⌸i共t兲… ⌸i共t兲␬␴
+
.
1 + f i␧
1 + f i␴
共28兲
The values ␥ci are the magnitudes of the total surface charge
ជ = 共1 , 0 , 0兲, N
ជ = 共0 , 1 , 0兲 and N
ជ = 共0 , 0 , 1兲
at the locations N
1
2
3
at the surface of the ellipsoid.
In a particular case of a sphere ni = 31 and the coefficients
f i␧ and ⌸i共t兲 are independent of the direction i. In this case
␥ci / ␥ck = E0i / E0k, and the electric field inside a sphere is directed along the external electric field. In the case of an
ellipsoid ␥ci / ␥ck ⫽ E0i / E0k, and the direction of the internal
electric field varies with time even when the direction of the
external electric field Eជ 0 is constant.
Equations 共27兲 and 共28兲 imply that a charge at any location at the ellipsoid’s surface is determined by three components ␥ci. In Fig. 4 we showed the time dependence of the
056611-4
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EQUILIBRIUM ORIENTATION OF AN ELLIPSOIDAL…
FIG. 2. Dependence of polarization factor n1 vs the nondimensional lengths a⬘1 and a⬘2, a⬘3 = 1
关共1兲 a⬘2 = 2; 共2兲 a⬘2 = 5; 共3兲 a⬘2 = 10;
共4兲 a⬘4 = 20兴.
surface charge, ␥c1共t兲, for different values of a1⬘ and a⬘2 in the
ជ when Eជ · eជ = 0 and the
case of the constant external field E
0
0 2
ជ
angle ␪ between E0 and eជ 3, ␪ = ␲ / 4. In Fig. 5 we showed the
time dependence of the surface charge ␥c3共t兲 for the same
values of the parameters.
Inspection of Figs. 4 and 5 shows that when ␬␧ ⬎ ␬␴ the
total surface charge ␥c共t兲 decreases with time. The cause for
this behavior is that the sign of the free charge ␥共t兲 flowing
to the surface is opposite to the sign of the polarization
charge ␥c共0兲. This is exactly the situation which occurs in the
case of Quincke rotation. When ␬␧ ⬍ ␬␴, the sign of the free
charge ␥共t兲 flowing to the surface coincides with the sign of
the polarization charge ␥c共0兲, and the total surface charge
␥c共t兲 grows with time.
FIG. 3. Dependence of polarization factor n3 vs the nondimensional lengths a⬘1 and a⬘2, a⬘3 = 1
关共1兲 a2⬘ = 2; 共2兲 a2⬘ = 5; 共3兲 a2⬘ = 10;
共4兲 a⬘4 = 20兴.
056611-5
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YU. DOLINSKY AND T. ELPERIN
ជ
FIG. 4. Time dependence of a surface charge ␥c1 at location N
ជ
ជ
= 共1 , 0 , 0兲 in a constant external electric field E0. Vector E0 is loជ · eជ = 0兲 and is directed by the angle ␪
cated in the plane eជ 1, eជ 3 共E
0 2
= ␲ / 4 with the axis of symmetry of the spheroid 共䊊: a1⬘ = a2⬘ = 0.05,
␬␧ / ␬␴ = 0.1, prolate spheroid; 䊐: a⬘1 = a⬘2 = 50, ␬␧ / ␬␴ = 0.1, oblate
spheroid; 〫: a⬘1 = a⬘2 = 1, ␬␧ / ␬␴ = 0.1, sphere; 䉭: a⬘1 = 50, a⬘2 = 0.05,
␬␧ / ␬␴ = 0.1, ellipsoid; 쎲: a⬘1 = a2⬘ = 0.05, ␬␧ / ␬␴ = 10, prolate spheroid; 䊏: a1⬘ = a2⬘ = 50, ␬␧ / ␬␴ = 10, oblate spheroid; ⽧: a1⬘ = a2⬘ = 1,
␬␧ / ␬␴ = 10, sphere; 䉱: a⬘1 = 50, a⬘2 = 0.05, ␬␧ / ␬␴ = 10, ellipsoid兲.
ជ
FIG. 6. Time dependence of a free charge ␥⬘1 at location N
ជ
= 共1 , 0 , 0兲 in a constant external electric field. Vector E0 is located in
ជ · eជ = 0兲 and is directed by the angle ␪ = ␲ / 4 with
the plane eជ 1, eជ 3 共E
0 2
the axis of symmetry of the spheroid 共䊊: a1⬘ = a2⬘ = 0.05, ␬␧ / ␬␴
= 0.1, prolate spheroid; 䊐: a⬘1 = a⬘2 = 50, ␬␧ / ␬␴ = 0.1, oblate spheroid;
〫: a⬘1 = a⬘2 = 1, ␬␧ / ␬␴ = 0.1, sphere; 䉭: −a⬘1 = 50, a⬘2 = 0.05, ␬␧ / ␬␴
= 0.1, ellipsoid; 쎲: a1⬘ = a2⬘ = 0.05, ␬␧ / ␬␴ = 10, prolate spheroid; 䊏:
a1⬘ = a2⬘ = 50, ␬␧ / ␬␴ = 10, oblate spheroid; ⽧: a1⬘ = a⬘2 = 1, ␬␧ / ␬␴ = 10,
sphere; 䉱: a⬘1 = 50, a⬘2 = 0.05, ␬␧ / ␬␴ = 10, ellipsoid兲.
ជ
FIG. 5. Time dependence of a surface charge ␥c3 at location N
= 共0 , 0 , 1兲 in a constant external electric field. Vector Eជ 0 is located in
ជ · eជ = 0兲 and is directed by the angle ␪ = ␲ / 4 with
the plane eជ 1, eជ 3 共E
0 2
the axis of symmetry of the spheroid 共䊊: a⬘1 = a⬘2 = 0.05, ␬␧ / ␬␴
= 0.1, prolate spheroid; 䊐: a⬘1 = a⬘2 = 50, ␬␧ / ␬␴ = 0.1, oblate spheroid;
〫: a1⬘ = a2⬘ = 1, ␬␧ / ␬␴ = 0.1, sphere; 䉭: a1⬘ = 50, a⬘2 = 0.05, ␬␧ / ␬␴
= 0.1, ellipsoid; 쎲: a1⬘ = a2⬘ = 0.05, ␬␧ / ␬␴ = 10, prolate spheroid; 䊏:
a⬘1 = a⬘2 = 50, ␬␧ / ␬␴ = 10, oblate spheroid; ⽧: a⬘1 = a⬘2 = 1, ␬␧ / ␬␴ = 10,
sphere; 䉱: a⬘1 = 50, a⬘2 = 0.05, ␬␧ / ␬␴ = 10, ellipsoid兲.
ជ
FIG. 7. Time dependence of a free charge ␥⬘3 at location N
= 共0 , 0 , 1兲 in a constant external electric field. Vector Eជ 0 is located in
ជ · eជ = 0兲 and is directed by the angle ␪ = ␲ / 4 with
the plane eជ 1, eជ 3 共E
0 2
the axis of symmetry of the spheroid 共䊊: a⬘1 = a⬘2 = 0.05, ␬␧ / ␬␴
= 0.1, prolate spheroid; 䊐: a⬘1 = a⬘2 = 50, ␬␧ / ␬␴ = 0.1, oblate spheroid;
〫: a1⬘ = a2⬘ = 1, ␬␧ / ␬␴ = 0.1, sphere; 䉭: a1⬘ = 50, a2⬘ = 0.05, ␬␧ / ␬␴
= 0.1, ellipsoid; 쎲: a1⬘ = a2⬘ = 0.05, ␬␧ / ␬␴ = 10, prolate spheroid; 䊏:
a⬘1 = a⬘2 = 50, ␬␧ / ␬␴ = 10, oblate spheroid; ⽧: a⬘1 = a⬘2 = 1, ␬␧ / ␬␴ = 10,
sphere; 䉱: a⬘1 = 50, a⬘2 = 0.05, ␬␧ / ␬␴ = 10, ellipsoid兲.
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EQUILIBRIUM ORIENTATION OF AN ELLIPSOIDAL…
FIG. 8. Time dependence of a magnitude of the electric field
inside spheroid in a constant external electric field. Vector Eជ 0 is
located in the plane eជ 1, eជ 3 共Eជ 0 · eជ 2 = 0兲 and is directed by the angle
␪ = ␲ / 4 with the axis of symmetry of the spheroid 共䊊: a⬘1 = a⬘2
= 0.05, ␬␧ / ␬␴ = 0.1, prolate spheroid; 䊐; a1⬘ = a2⬘ = 50, ␬␧ / ␬␴ = 0.1,
oblate spheroid; 〫: a⬘1 = a⬘2 = 1, ␬␧ / ␬␴ = 0.1, sphere; 䉭: a⬘1 = 50, a⬘2
= 0.05, ␬␧ / ␬␴ = 0.1, ellipsoid; 쎲: a⬘1 = a⬘2 = 0.05, ␬␧ / ␬␴ = 10, prolate
spheroid; 䊏: a⬘1 = a⬘2 = 50, ␬␧ / ␬␴ = 10, oblate spheroid; ⽧: a⬘1 = a⬘2
= 1, ␬␧ / ␬␴ = 10, sphere; 䉱: a1⬘ = 50, a2⬘ = 0.05, ␬␧ / ␬␴ = 10, ellipsoid兲.
In Figs. 6 and 7 we showed the behavior of the components ␥⬘i that according to Eqs. 共14兲 and 共15兲 is completely
determined by the behavior of a surface free charge ␥共t兲.
Inspection of these figures shows that for the same values of
parameters the sign of ␥⬘i 共t兲 is opposite to the sign of ␥ci共0兲
when ␬␧ ⬎ ␬␴ and the sign of ␥⬘i 共t兲 coincides with the sign of
␥ci共0兲 when ␬␧ ⬍ ␬␴.
ជ 共t兲
Time behavior of the electric field inside an ellipsoid E
2
is shown in Figs. 8 and 9. In Fig. 8 we showed time variation
of the magnitude of the electric field E2共t兲 while in Fig. 9 we
showed the time dependence of the angle ␣共t兲
= tan−1共E21 / E23兲. Inspection of these figures reveals that
when ␬␧ ⬎ ␬␴, the magnitude of the electric field E2共t兲 grows
while for ␬␧ ⬍ ␬␴ it decreases with time. The reason for this
behavior is that when ␬␧ ⬎ ␬␴ the electric field produced by
ជ and
the free charge ␥共t兲 is directed along the external field E
0
it partially compensates the field produced by the polarization charge.
IV. STABILITY OF THE ORIENTATION OF THE
ELLIPSOIDAL PARTICLE IN THE EXTERNAL
ELECTRIC FIELD
FIG. 9. Time dependence of the angle between the internal electric field and the axis of symmetry of spheroid in a constant external
ជ is located in the plane eជ , eជ 共Eជ · eជ = 0兲 and
electric field. Vector E
0
1 3
0 2
is directed by the angle ␪ = ␲ / 4 with the axis of symmetry of the
spheroid 共䊊: a⬘1 = a2⬘ = 0.05, ␬␧ / ␬␴ = 0.1, prolate spheroid; 䊐: a1⬘
= a2⬘ = 50, ␬␧ / ␬␴ = 0.1, oblate spheroid; 〫: a1⬘ = a2⬘ = 1, ␬␧ / ␬␴ = 0.1,
sphere; 䉭: a⬘1 = 50, a⬘2 = 0.05, ␬␧ / ␬␴ = 0.1, ellipsoid; 쎲: a⬘1 = a⬘2
= 0.05, ␬␧ / ␬␴ = 10, prolate spheroid; 䊏: a⬘1 = a⬘2 = 50, ␬␧ / ␬␴ = 10, oblate spheroid; ⽧: a1⬘ = a2⬘ = 1, ␬␧ / ␬␴ = 10, sphere; 䉱: a1⬘ = 50, a2⬘
= 0.05, ␬␧ / ␬␴ = 10, ellipsoid兲.
ជ =␧ P
ជ ជ
M
1 ⫻ E0 ,
共29兲
ជ is a total dipole moment of the ellipsoid. Using
where P
ជ can be written as
formulas 共3兲 the expression for P
ជ=
P
冕
␥crជ dS,
共30兲
where integration is performed over the surface of an ellipsoid. Substituting Eq. 共24兲 into Eq. 共30兲 we find that
ជ =␧ V
P
0
3
E0ieជ i
兺
i=1
冉
冊
„1 − ⌸i共t兲…␬␧ ⌸i共t兲␬␴
+
.
1 + f i␧
1 + f i␴
共31兲
For t Ⰶ ␶i, in the constant external field ⌸i共t兲 ⬃ t / ␶i, and formula 共31兲 recovers the known expression for a dipole moment of a dielectric ellipsoid 共see 关17兴, Chap. 2, Sec. 9兲.
Equations 共29兲 and 共31兲 yield a formula for a torque acting at
the ellipsoid for an arbitrary orientation of the external electric field and axes of the ellipsoid:
3
3
冉
冊
ជ = ␧ ␧ V 兺 兺 E E ␧ eជ „1 − ⌸i共t兲…␬␧ + ⌸i共t兲␬␴ ,
M
0 1
0i 0k ikm m
1 + f i␧
1 + f i␴
i=1 k=1
Let us now analyze the stability of the orientation of a
particle by considering the dependence of the torque acting
at the particle upon the orientation of the particle with respect to the direction of the external electric field. To this end
we use the following formula for a torque acting at the particle:
共32兲
where ␧ikm is a fully nonsymmetric unit tensor.
Let us consider a spheroid with a coefficient of the depolarization n1 = n2 = 21 共1 − n兲, n = n3, f 1␧ = f 2␧, and f 1␴ = f 2␴. In a
case of a prolate in the direction of eជ 3 spheroid, n3 ⬍ n1 , n2,
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PHYSICAL REVIEW E 71, 056611 共2005兲
YU. DOLINSKY AND T. ELPERIN
while for an oblate ellipsoid n3 ⬎ n1 , n2. The limiting cases of
a cylinder 共n3 Ⰶ n1 , n2兲 and of a disk 共n3 Ⰷ n1 , n2兲 were considered earlier.
Let us define angle ␪ in the plane spanned by vectors
ជ can be represented as
Eជ 0 , eជ 3 共see Fig. 1兲. The electric field E
0
follows:
Eជ 0 = E0 cos ␪eជ 3 − E0 sin ␪eជ 1 .
冋冉
␧0␧1VE20 sin共2␪兲
1 − ⌸3共t兲 1 − ⌸1共t兲
␬␧
−
2
1 + f 3␧
1 + f 1␧
+ ␬␴
冉
⌸3
⌸1
−
1 + f 3␴ 1 + f 1␴
冊册
冊
␧0␧1VE20 sin共2␪兲 ␬2␧共3n − 1兲
.
4
共1 + f 1␧兲共1 + f 3␧兲
+ b0␻␶0共␬␴b1 + b2␬␧␻2␶20兲sin共2␻t兲兴,
␧0␧1VE20 sin共2␪兲 ␬␴2 共3n − 1兲
.
4
共1 + f 1␴兲共1 + f 3␴兲
where
a1 = ␬␴ + ␬␧ +
cos共␻t兲 + ␻␶i sin共␻t兲
1+
␻2␶2i
1
.
cos共␻t兲
where
a2 =
3␬␧ − ␬␴ + 3d1␬2␧ + d2␬3␧
,
共1 + f 3␴兲共1 + f 1␴兲
共36兲
共37兲
共38兲
n共1 − n兲
1+n
,
+ ␬␴
2
2
冉
冊
1+n
共1 − n兲2
n共1 − n兲
1 + ␬␴
,
+ n2 +
4
2
2
共35兲
Substituting Eq. 共37兲 into Eq. 共34兲 we arrive at the following formula for the total torque acting at a particle:
M = M␧ + M␴ ,
n+1
␬ ␴␬ ␧,
2
d1 =
d2 =
Comparing Eqs. 共35兲 and 共36兲 shows that equilibrium orientations at t = 0 and t → ⬁ coincide. It can be shown that at the
intermediate times 0 ⬍ t ⬍ ⬁ the sign of M ⬁ is the same as the
sign of M共0兲 even in the cases with a strong anisotropy, ␶1
Ⰷ ␶3 or ␶3 Ⰷ ␶1.
Consider now a stationary external electric field E0共t兲
= Ē0 cos共␻t兲. Substituting this expression into Eq. 共19兲 yields
a formula for ⌸i共t兲 that in the limit t Ⰷ ␶1 , ␶2 reads
⌸i共t兲 =
共40兲
共41兲
Equation 共35兲 recovers the known formula for a torque acting at the dielectric spheroid as a function of the angle between the axis of symmetry of the spheroid and the direction
of the external electric field Eជ 0 共see, e.g. 关17,18兴兲. Two orientations when the torque vanishes, ␪ = 0 and ␪ = ␲ / 2, correspond to stable and unstable equilibrium orientations for n
⬍ 1 / 3 and, inversely, to unstable and stable equilibrium orientations for n ⬎ 1 / 3 共for details see the Appendix兲.
Let us consider now stability of equilibrium orientations
for t → ⬁. For t / ␶1 Ⰷ 1 and t / ␶2 Ⰷ 1, ⌸1共t兲 = ⌸2共t兲 = 1 and
M共t → ⬁兲 = M ⬁ is determined by the following formula:
M⬁ =
␧0␧1VĒ20 sin共2␪兲
.
2
M 0共␬␧ − ␬␴兲共3n − 1兲
关共a1 + a2␻2␶20兲cos2共␻t兲
2L
M␴ = −
共34兲
.
M0 =
The expression for M ␴ can be written as
In a constant electric field at t = 0, ⌸3共0兲 = ⌸1共0兲 and
M共0兲 =
3n − 1 ␬2␧ cos2共␻t兲
,
2 共1 + f 3␧兲共1 + f 1␧兲
共39兲
共33兲
In the adopted coordinate system 共see Fig. 1兲 a total torque
ជ
acting at the particle is directed along the eជ 2 axis, i.e., M
= Meជ 2. Using Eqs. 共32兲 and 共33兲 we arrive at the following
formula for M:
M=−
M␧ = M0
b0 =
b1 = 2 + ␬␴
共42兲
1 共1 + f 3␧兲共1 + f 1␧兲
,
2 共1 + f 3␴兲共1 + f 1␴兲
1+n
,
2
b2 = 2 + ␬␧
1+n
,
2
共43兲
L = 共1 + f 3␴兲共1 + f 1␴兲共1 + f 3␧兲共1 + f 1␧兲共1 + ␻2␶21兲共1 + ␻2␶23兲.
共44兲
The Eqs. 共39兲 and 共40兲 yield the following expression for a
total torque:
M = M0
3n − 1
关M c + 2M s tan共␻t兲兴cos2共␻t兲,
2
共45兲
where
Mc =
␬2␧
␬␴ − ␬␧
+
共a1 + a2␻2␶20兲
共1 + f 1␧兲共1 + f 3␧兲
L
共46兲
b 0␻ ␶ 0
共b1␬␴ + b2␬␧␻2␶20兲共␬␴ − ␬␧兲.
L
共47兲
and
Ms =
When ␻ → 0, M s → 0 and M c = ␬␴2 / 共1 + f 1␴兲共1 + f 3␴兲, i.e., expression 共36兲 is recovered.
Equation 共45兲 yields the following condition for the
change of an orientation of a particle with respect to its orientation at the initial moment t = 0:
M c + 2M s tan共␻t兲 ⬍ 0.
共48兲
Using Eqs. 共39兲 and 共45兲 it can be shown that an inequality
共48兲 is equivalent to the condition that M共t兲 / M共0兲 ⬍ 0. In the
Appendix we demonstrated that this condition implies the
change of the direction of the stable orientation of the ellipsoid.
056611-8
PHYSICAL REVIEW E 71, 056611 共2005兲
EQUILIBRIUM ORIENTATION OF AN ELLIPSOIDAL…
Let us determine now a time interval inside the period of
the external electric field T during which the condition 共48兲
is satisfied. Consider only the case when M s ⬍ 0, e.g., 共␬␴
⬍ ␬␧兲 and M c ⬎ 0. In this case the inequality 共48兲 implies that
there exist two time intervals where this inequality is satisfied:
tan−1
冉 冊
冉 冊
Mc
␲
⬍ ␻t ⬍
2兩M s兩
2
and ␲ + tan−1
3␲
Mc
.
⬍ ␻t ⬍
2兩M s兩
2
共49兲
During two time intervals determined by inequalities 共49兲 the
direction of the stable orientation becomes normal to the
direction of the initial stable orientation, i.e., ␪s → ␪s + ␲ / 2,
where ␪s is an angle between the external electric field and
the principal axis of the ellipsoid in the initial stable equilibrium. For a prolate spheroid ␪s = 0, while for an oblate spheroid ␪s = ␲ / 2. Define two time intervals, T1 and T2 = T − T1.
During time interval T1 the stable orientation of the particle
coincides with the initial stable orientation while during time
interval T2 关sum of two time intervals determined by Eqs.
共49兲兴 it changes. Equations 共49兲 imply that
T1 =
冠
冉 冊冡
1
Mc
␲ + 2 tan−1
␻
2兩M s兩
,
T2 =
冉 冊
2兩M s兩
2
tan−1
.
␻
Mc
共50兲
Assume that an ellipsoid is subjected to random perturbations uniformly distributed in time. Then a probability p1 that
an equilibrium orientation of the ellipsoid is the same as in
the case of an ideal dielectric p1 = T1 / 共T1 + T2兲 is
p1 =
冉 冊
1 1
Mc
+ tan−1
,
2 ␲
2兩M s兩
共51兲
while the probability that its equilibrium orientation is normal to that in the case of an ideal dielectric p2 = T2 / 共T1
+ T2兲 is
p2 =
冉 冊
2兩M s兩
1
tan−1
.
␲
Mc
共52兲
Consider a limiting case when 兩M s兩 Ⰶ M c and ␻␶0 Ⰶ 1. This
situation occurs in the ranges ␬␴ 艌 1, ␻␶0 Ⰶ 1 and ␬␴ Ⰶ 1,
␻␶0 Ⰶ ␬␴. In these limiting cases
Ms =
and
b 0b 1␬ ␴共 ␬ ␧ − ␬ ␴兲 ␻ ␶ 0
,
L
Mc =
冉
␬␴2
,
共1 + f 1␴兲共1 + f 3␴兲
共53兲
冊
n+1
␻␶0 2 + ␬␴
2
1 ␬␧ − ␬␴
.
p2 =
␲ ␬␴ 共1 + f 1␴兲共1 + f 3␴兲
共54兲
Thus the probability to detect an ellipsoid with an orientation
normal to the orientation in the case of an ideal dielectric
depends only upon the parameters of the system and the
frequency of an applied electric field and does not depend
upon its amplitude. It must be noted that independent of the
magnitudes of M c and 兩M s兩, p2 ⬍ p1, p2 ⬍ 1 / 2, and p1 ⬎ 1 / 2.
V. CONCLUSIONS
We studied the moment of forces acting on a stationary
dielectric ellipsoidal particle imbedded in a host dielectric
medium with a finite electric conductivity under the action of
a homogeneous, time independent or varying with time, electric field. Using a dipole moment approach we showed that
in a constant electric field stable orientations of an ellipsoid
for an ideal dielectric and a dielectric with a finite electric
conductivity are the same.
We demonstrated that an equilibrium orientation of the
ellipsoidal particle changes with time in a stationary electric
field with a constant direction. It was found that during time
interval T1 an equilibrium orientation of the spheroidal particle with a finite electric conductivity remains the same as
the equilibrium orientation of an ideal dielectric particle.
During time interval T2, where T = T1 + T2 is a period of the
external electric field, the equilibrium orientation of the axis
of symmetry of the particle is normal to this direction.
The derived expressions for electric fields and currents
关Eqs. 共21兲 and 共22兲兴 and dipole moment 关Eq. 共31兲兴 are valid
also for the case of a rotating ellipsoid. In the latter case xi共0兲
and E0i共t兲 are spatial coordinates and components of electric
field in a frame attached with the ellipsoid. Transformation
into the laboratory frame is performed using the formulas
E0i = Oik共t兲Gk共t兲 and xi = Oimx̄m, where Gk共t兲 and x̄m are components of the electric field and coordinates in a laboratory
frame. The orthogonal matrix Oik is related with angular veជ by an equation Ȯ = ␧ ⍀ O , where ␧ is the
locity ⍀
ik
iml m lk
iml
Levi-Civita tensor. Using these formulas and expressions for
ជ 共t兲 allows to obtain equations
⌸i共t兲 and dipole moment P
governing the dynamics of a dipole moment similar to equations used in the studies of the Quincke rotation 关13,14兴.
The obtained results imply a possibility of the existence
of a mechanism of rotation of liquid particles that is alternative to the known effect of Quincke rotation. Indeed, the
external electric field causes the deformation of a liquid particle along the direction of the field, and we showed that the
direction of the equilibrium orientation of the ellipsoidal particle changes with time in a stationary external field.
In this study we considered an isotropic medium. The
situation is completely different in the case of an anisotropic
material. In the latter case the torque depends not only upon
the orientation of the ellipsoid with respect to the direction of
the external electric field but upon the mutual orientations of
the principal axes of the medium, principal axes of the ellipsoid and the direction of the external electric field. Clearly, in
this case our approach that uses the ellipsoidal coordinates
and expansion of the external electric field into independent
components directed along the principal axes of the ellipsoid
is not valid. In this study we also considered only the case
when M s ⬍ 0, e.g., 共␬␴ ⬍ ␬␧兲 and M c ⬎ 0. Analysis of this
problem in other cases is a subject of ongoing investigation.
APPENDIX
Neglecting inertial effects the equation describing dynamics of an orientation of a spheroid in the vicinity of the equilibrium position reads
056611-9
PHYSICAL REVIEW E 71, 056611 共2005兲
YU. DOLINSKY AND T. ELPERIN
␩␪˙ = M共␪兲,
共A1兲
where a coefficient ␩ ⬎ 0 depends upon the viscosity of a
medium, M is a torque, and ␪˙ is angular velocity.
Equation 共A1兲 implies that the angle of a stable orientation ␪s is determined by two conditions:
M共␪s兲 = 0,
冏 冏
⳵M
⳵␪
␪=␪s
⬍ 0.
共A2兲
In the vicinity of the equilibrium
M共␪兲=
冏 冏
⳵M
⳵␪
␪=␪s
共␪ − ␪s兲.
共A3兲
In all cases considered in this study the dependence of the
torque M共␪ , t兲 can be written as
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M共␪,t兲 = M 0共␪兲M̃共t兲,
M0 =
␧0␧1VE20 sin共2␪兲
, 共A4兲
2
where M̃共t兲 is some normalization function that does not
depend on angle ␪ 关see Eqs. 共34兲, 共36兲, and 共39兲兴.
In the vicinity of the equilibrium position, where Eqs.
共A2兲 and 共A3兲 are valid, the change of the sign of M共␪ , t兲
implies the change of the sign of the derivative ⳵ M / ⳵␪.
Therefore if a particle had initially a stable orientation at t
= 0, then at time t1 such that M共␪ , t1兲 / M共␪ , 0兲 ⬍ 0, the stable
orientation of a particle is normal to its orientation at t = 0.
Therefore, Eq. 共A4兲 implies that the change of the sign of a
total torque M共␪ , t兲 causes the change of the stable orientation to the direction normal to the initial one while the initially unstable orientation of the particle becomes stable.
关9兴
关10兴
关11兴
关12兴
关13兴
关14兴
关15兴
关16兴
关17兴
关18兴
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