Download MODELING OF NEMATIC ELECTROLYTES AND NONLINEAR ELECTROOSMOSIS

MODELING OF NEMATIC ELECTROLYTES AND NONLINEAR ELECTROOSMOSIS
M. CARME CALDERER˚, DMITRY GOLOVATY
:,
OLEG LAVRENTOVICH
; , AND
NOEL J. WALKINGTON
§
arXiv:1601.02318v1 [cond-mat.soft] 11 Jan 2016
Abstract. We derive a mathematical model of a nematic electrolyte based on the Leslie-Ericksen theory of liquid crystal
flow. Our goal is to investigate the nonlinear electrokinetic effects that occur because the nematic matrix is anisotropic, in
particular, transport of ions in a direction perpendicular to the electric field as well as quadratic dependence of the induced flow
velocity on the electric field. The latter effect makes it possible to generate sustained flows in the nematic electrolyte that do
not reverse their direction when the polarity of the applied electric field is reversed. From a practical perspective, this enables
the design of AC-driven electrophoretic and electroosmotic devices. Our study of a special flow in a thin nematic film shows a
very good agreement with laboratory experiments.
Key words. Ericksen-Leslie, liquid crystals, electroosmosis, variational principles, asymptotics
AMS subject classifications. 35Q35, 74E10, 49J40, 74A20, 80M30
1. Introduction. In this article, we derive equations governing the electrokinetics of a nematic electrolyte that consists of ions that diffuse and advect in the nematic liquid crystalline matrix. The nematic
electrolytes are characterized by the unique nonlinear phenomena that occur in these material systems due
to anisotropy of conductivity and permittivity of the matrix.
Electrokinetic phenomena are usually explored for binary systems in which an isotropic fluid with ions,
called electrolyte, is in contact with a solid substrate or contains dispersed solid particles. One distinguishes
two sides of electrokinetics: electrophoresis, defined as motion of particles dispersed in the electrolyte and
electroosmosis, motion of an electrolyte with respect to the walls of a chamber. A necessary condition of
electrokinetics is spatial separation of electric charges of opposite polarities ([1], [14] and [16]). In classic linear
electrokinetics, charges are separated at the solid-electrolyte interface through chemical-physical processes
such as dissociation and selective adsorption and formation of permanent electric double layers [16]. An
externally applied electric field imposes a torque on the electric double layer and drives electrokinetic flows.
The driving force, proportional to the product of charge and field, is balanced by the viscous drag; the
resulting velocities grow linearly with the electric field. As a result, only a direct current (DC) field can be
used to power linear electrokinetics, as an AC field would produce no net displacement.
The studies of the liquid crystal-enabled electrokinetics are a part of a much larger field of liquid crystal
colloids that is currently experiencing a great deal of interest partially as a result of the progress in the field of
nanotechnology. Recent experiments, [5], [6], [9], [10], [11], [15] and [17], demonstrate that when the isotropic
electrolyte is replaced with an anisotropic electrolyte, a liquid crystal containing ions, the electrokinetic flows
become strongly nonlinear, with the velocities growing as a square of the electric field. For such a flow, if the
polarity of the applied field is reversed, the direction of the flow remains unchanged, enabling alternating
current (or AC) -driven electroosmosis and electrophoresis. The nonlinearity disappears as soon as the liquid
crystal is melted into an isotropic phase.
Of a particular interest—from the point of view of this paper—are the experiments in [15] where surface
patterning of the plates bounding the nematic film is used to impose anchoring conditions on the nematic
director. Surface patterning of liquid crystal cells thus allows one to impose a well-defined spatial variation of
the director in the bulk [15] and, consequently, the characteristics of the induced flow. This setup is especially
amenable to theoretical analysis since the surface-induced director patterns in the available experiments are
usually periodic, either one- or two-dimensional.
Our approach to modeling of nematic electrolytes follows the ideas that were originally used to obtain
the Ericksen-Leslie equations of liquid crystalline flow. The work of Leslie in this area spans a lifetime of
research effort both to capture the appropriate physical phenomena and to provide a sound mathematical
theory to describe the flow, with the main focus on understanding the balance of angular momentum. Leslie’s
thermodynamically self-consistent derivation appears to be the most straightforward method to formulate
a consistent model for the nematic electrolytes. Although the exact form of the governing equations may
˚ School
of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA.
of Theoretical and Applied Mathematics, University of Akron, Akron, OH 44325, USA.
; Liquid Crystal Institute, Kent State University, Kent, OH 44242
§ Department of Mathematical Sciences, Carnegie-Mellon University, Pittsburgh, PA 15213
: Department
1
be different if one was to follow a more rigorous procedure, we believe that we are able to capture the
correct form of all principal contributions by imposing the proper thermodynamical structure, along with
appropriate invariances and symmetries.
The variables of the model consist of the velocity field v of the nematic, the pressure p due to the
incompressibility constraint, the unit director field n representing the average molecular orientation at a
given point, the electrostatic potential Φ, and concentrations tck uN
k“1 , N ě 2, associated with the N species
of ions with valences tzk uN
,
respectively.
k“1
Our development of the model follows the two main works by Leslie [12] and [13]. The former simplifies
the previous approach by emphasizing the role of the rate of energy dissipation of the system and its direct
connection with the viscous contributions to both the stress and the molecular force. These, combined with
the variational insights on Leslie’s works by Walkington [19], allow for a more direct approach to the LeslieEricksen model. As it was done by Leslie, we assume that the laws of balance of linear and angular momentum
hold in local form. We supplement these by the local mass balances for the ions and the Maxwell’s equations
of electrostatics and postulate the equation of balance of energy at a time t ě 0 for every subdomain V of
the domain Ω occupied by the nematic. The total energy of the system is now the sum of the Oseen-Frank
free energy density, the entropic contribution due the presence of ions, and the electrostatic energy. The
balance of energy involves the dissipation function that is required to be positive for all processes, according
to the Second Law of Thermodynamics. This condition yields the constitutive equations for the generalized
stress tensor and the molecular force.
In what follows, we assume that the diffusion and dielectric permittivity matrices of the system are
uniaxial. The anisotropy of the diffusion matrix is fundamental in explaining the experimentally observed
AC electroosmosis in a nematic film confined between the patterned plates. However, it is the combination of
this anisotropy with the anisotropy of dielectric permittivity and the anisotropy of the viscosity that provides
a rich variety of possible flow patterns [10]. Note that, in the proposed model, we neglect dependence of the
viscosity coefficients on concentration fields thus omitting electro-rheologic effects that may not be relevant.
The theory of polyelectrolyte gels previously studied [2], [4] provides a rigorous setting to model electromechanic interaction in liquid crystals. Given the dissipative character of these systems, our development
also suggest that an alternate approach to generate governing equations and formulate boundary conditions
may be based on the Onsager’s principle for the Rayleighian functional.
In the second part of the article, we compare the predictions of our model to the experimental results for
electroosmosis in a nematic liquid crystalline film constrained between two parallel plates. In [15] an approach
was developed to generate electrokinetic effects in a nematic electrolyte with surface-imposed distortions of
molecular orientations induced by patterning of the plates. In the presence of the uniform electric field,
these variations produce space charge separation that triggers electroosmotic flows in the liquid crystal. In
particular, for the setup depicted in Fig. 1, the director orientation is periodically varying in the vertical
direction and an AC field is applied in the horizontal direction. It was observed that spatially periodic
horizontal flow proceeds along the ”guiding rails” induced by molecular orientation with the direction of
the flow independent of the sign of the field. Analysis and simulations of our model confirm this behavior
and demonstrate the quadratic dependance of the driving force on the electric field. Here, for simplicity we
assume that the liquid crystal domain is very thin and neglect the anisotropies of viscosity and dielectric
permittivity, with the diffusion being the only source of anisotropy.
2. Leslie-Ericksen Model for a Nematic Electrolyte.
2.1. Standard Nematic Model. We begin by reviewing Leslie’s derivation of the classical LeslieEricksen model. Let Ω, with the piecewise smooth boundary BΩ, denote the domain occupied by the liquid
crystal. Suppose that v “ vpx, tq and n “ npx, tq denote the velocity and director fields, respectively. The
vector fields tpx, tq and lpx, tq represent contact force and contact couple per unit area of a surface element
S, respectively. We assume that there exist the Cauchy stress tensor T and the generalized stress tensor L
(contact torque) such that
(2.1)
tpx, tq “ T px, tqνpx, tq,
lpx, tq “ Lpx, tqνpx, tq.
We set
(2.2)
T “ T e ` T v,
L “ Le ` Lv
2
where ν represents the unit outer normal to a surface S at x, the tensor field T e is the elastic stress, and
T v is the anisotropic part of the viscous stress tensor. Following Leslie, we assume that there is no viscous
torque tensor associated with L, that is, we set Lv “ 0.
We postulate the equations of balance of linear and angular momentum, together with the incompressibility assumption and the unit director field constraint:
(2.3)
ρv9 ´ ∇ ¨ T “ ρf ,
(2.4)
∇ ¨ v “ 0,
(2.5)
χ:
n ` g ´ ∇ ¨ L “ ρg,
(2.6)
n ¨ n “ 1.
Here` ρ is the mass density and˘χ is the density of the moment of inertia of nematic rods. The superimposed
d
represents the material time derivative. The symbols f and g denote the
dot as well as the symbol dt
density per unit mass of an applied external force and torque, respectively. The body torque g can be written
as
g “ gv ` ge ,
(2.7)
where ge and gv denote the elastic and viscous contributions associated with director field rotations.
Below we assume that the nematic energy density is in the Oseen-Frank form
(2.8)
1
1
1
WOF pn, ∇nq “ K1 p∇ ¨ nq2 ` K2 pn ¨ p∇ ˆ nqq2 ` K3 |n ˆ p∇ ˆ nq|2
2
2
2
1
2
2
` pK2 ` K4 qptrp∇nq ´ tr p∇nqq,
2
where the Frank elastic constants Ki , i “ 1, . . . , 4 are assumed to satisfy the Ericksen’s inequalities
(2.9)
K1 ą 0, K2 ą 0, K3 ą 0, K2 ě |K4 |, 2K1 ě K2 ` K4 ,
to guarantee existence of a global minimizer of the total energy
ż
(2.10)
U“
WOF pn, ∇nq
Ω
under appropriate boundary data [18].
2.1.1. Dynamics. Among the different approaches that can be used to derive the constitutive equations
for the fields T e , T v , Le , ge , gv , we choose the line of reasoning proposed by Leslie that starts with postulating
the balance laws (2.3) and (2.5) along with the equation of the energy balance. Following [13], we let RLC
be the rate of viscous dissipation per unit volume and assume that
˙ ż
ż
ż
ż ˆ
d
1
1
2
2
9 `
9 “
9 ` WOF `
(2.11)
ρpf ¨ v ` g ¨ nq
pt ¨ v ` l ¨ nq
ρ|v| ` χ|n|
RLC ,
dt V 2
2
V
BV
V
for every subdomain V Ď Ω with the smooth boundary BV . The local form of (2.11)
(2.12)
9 OF ` RLC ,
T ¨ ∇v ` L ¨ ∇n9 ` g ¨ n9 “ W
follows via the divergence theorem from (2.1), (2.3), and (2.5). Since
(2.13)
∇n9 “ p∇nq¨ ` ∇n∇v,
we have that
(2.14)
9 OF “ BWOF ¨ n9 `
W
Bn
BWOF
“
¨ n9 `
Bn
BWOF
¨ p∇nq¨
B∇n
BWOF
¨ p∇n9 ´ ∇n∇vq ,
B∇n
3
`
˘
where A ¨ B “ tr B T A for any A, B P M 3ˆ3 . Substituting (2.14) into (2.12) yields
T ¨ ∇v ` L ¨ ∇n9 ` g ¨ n9 “
so that (2.2) and (2.7) give
ˆ
T e ` T v ` p∇nqT
(2.15)
BWOF
BWOF
¨ n9 `
¨ p∇n9 ´ ∇n∇vq ` RLC ,
Bn
B∇n
˙
ˆ
˙
BWOF
BWOF
¨ ∇v ` Le ´
¨ ∇n9
B∇n
B∇n
ˆ
˙
BWOF
` ge ` gv ´
¨ n9 “ RLC .
Bn
The Second Law of Thermodynamics in the form of the Clausius-Duhem inequality, together with the
appropriate smoothness assumptions, implies the positivity of the rate of viscous dissipation function
(2.16)
RLC px, tq ě 0,
@x P Ω, t ą 0,
9 Specifically, given npx, tq and ∇npx, tq, the inequality (2.16) must hold
for all dynamical processes tv, nu.
9 This yields the constitutive relations
for arbitrary choices at px, tq of v, ∇v, n9 and ∇n.
T e “ ´pI ´ p∇nqT
(2.17)
BWOF
,
B∇n
BWOF
` λn,
Bn
BWOF
Le “
,
B∇n
ge “
(2.18)
(2.19)
where p and λ are the Lagrange multiplier corresponding to the constraints (2.4) and (2.6), respectively.
It also follows that
RLC “ gv ¨ n9 ` T v ¨ ∇v.
(2.20)
The arguments in [12] then yield the total viscous stress
(2.21)
8 b n ` α3 n b n
8 ` α4 Dpvq ` α5 Dpvqn b n ` α6 n b Dpvqn,
T v “ α1 pDpvqn ¨ nq ` α2 n
and the viscous molecular force
8 ` γ2 Dpvqn,
g v “ γ1 n
(2.22)
where
(2.23)
γ1 “ α3 ´ α2 , γ2 “ α6 ´ α5 ,
8 “ n9 ´ W pvqn is the Lie derivative of n. Further,
and n
(2.24)
Dpvq “
1
1
p∇v ` ∇v T q and W pvq “ p∇v ´ ∇v T q
2
2
represent the symmetric and skew parts of the velocity gradient ∇v, respectively. Note that an even more
8 can be established for (2.21)-(2.22), although we chose
general expression [7] that involves the gradient of n
not to include terms of this type here.
In what follows, we assume that the Parodi’s relation [12] given by
(2.25)
α6 ´ α5 “ α2 ` α3 .
holds. This relation is necessary to ensure the variational structure of the system of equations and thus the
equivalency of the equation of balance of linear momentum (2.3) to that derived via the Onsager’s principle.
Then (2.20) and (2.21)-(2.22) give
(2.26)
8 ¨ Dpvqnq ` α4 |Dpvq|2 ` pα5 ` α6 q|Dpvqn|2 ` γ1 |n|
8 2.
2RLC “ α1 pn ¨ Dpvqnq2 ` 2γ2 pn
4
Ericksen [3] gave sufficient conditions for the positivity of RLC in the following
Proposition 2.1. Suppose that (2.25) and the inequalities
(2.27)
α4 ą 0,
3
α1 ` α4 ` α5 ` α6 ą 0,
2
γ1 p2α4 ` α5 ` α6 q ě γ22 .
γ1 ą 0,
hold. Then
(2.28)
RLC ě 0
and
Tv “
(2.29)
BRLC
BRLC
and gv “
.
B∇v
B n9
8 “ 0.
Moreover RLC ” 0 if and only if ∇v “ 0 and n
2.1.2. Boundary Conditions. Since the energy law (2.11) for the Leslie-Ericksen system must hold
in the entire domain Ω, it follows from the equations (2.3)-(2.6) and (2.17)-(2.20) that
ż
ż
9 .
9 “
tt ¨ v ` l ¨ nu
tT ν ¨ v ` Lν ¨ nu
(2.30)
BΩ
BΩ
9 therefore the boundary conditions on BΩ
This equation should be valid for all dynamical processes tv, nu
should be of the form
(2.31)
T ν “ t̂ or v “ 0,
(2.32)
Lν “ l̂
and
or n “ n̂,
where t̂, l̂ and n̂ are prescribed vector fields on BΩ with |n̂| “ 1 and T and L given by (2.2), (2.17), (2.21)
and (2.19). Observe that the fields t̂ and l̂ can be time-dependent in this formulation.
2.2. Nematic Electrolyte. Suppose now that the domain Ω P R3 is occupied by a nematic electrolyte
that contains ions. Later on, we will assume that some parts of BΩ correspond to conducting electrodes
on which we will prescribe values of the electrostatic potential Φ, while the other parts of the boundary
will be assumed to be electrically insulated. In this section, however, we will impose time-independent,
Dirichlet boundary data on the potential Φ everywhere on BΩ, corresponding to nematic being surrounded
by conductors held at fixed potentials. This problem setup is chosen for simplicity because we do not expect
the boundary data on the electric field to affect the constitutive expressions on electrostatic forces in the
bulk of the nematic electrolyte.
Suppose that there are N ą 1 families of charged ions present in the liquid crystal at concentrations ck ,
with valences zk , where 1 ď k ď N . Let the velocity fields of the ions be denoted by tuk u1ďkďN . In what
follows, we assume that the system is in the dilute regime so that the particles are not subject to mutual
interaction. The continuity equations for the ions are given by
(2.33)
Bck
` ∇ ¨ pck uk q “ 0
Bt
in Ω, k “ 1, ...N.
Motivated by standard results of the theory of isotropic diffusion, we assume that the rate of dissipation
associated with the mobility of ions in the nematic is a quadratic function of the relative velocity of the ions
with respect to the liquid crystalline medium. We set
(2.34)
R “ RLC `
N
ÿ
kB θck Dk´1 puk ´ vq ¨ puk ´ vq,
k“0
where RLC is given by (2.26) the diffusion matrix Dk is anisotropic, reflecting the fact that the mobilities of
the k-th species in the directions parallel and perpendicular to the nematic director are generally different.
The ions also contribute to the free energy of the system via an entropic energy density term
(2.35)
N
ÿ
Wion “ kB θ
k“1
5
ck ln ck ,
where kB is the Boltzmann constant and θ is the absolute temperature [7].
The electric displacement vector D of the nematic liquid crystal is given by
(2.36)
D “ ε0 E ` P,
where ε0 is the dielectric permittivity of the vacuum, E the electric field and P is the induced polarization
of the liquid crystal
(2.37)
P “ ε0 χE.
Here χ denotes the electric susceptibility matrix of the liquid crystal. Substituting (2.37) into (2.36) we
obtain
(2.38)
D “ ε0 εE,
where ε “ I ` χ is the dielectric permittivity matrix. Letting ε} and εK to represent the dielectric permittivities when E is parallel and perpendicular to n, respectively, and denoting εa “ ε} ´ εK , we have
that
(2.39)
εpnq “ εK I ` εa n b n.
The fields E and D satisfy the Maxwell’s equations of electrostatics
(2.40)
E “ ´∇Φ,
∇¨D“
N
ÿ
qzk ck .
k“1
that hold in Ω, subject to time-independent Dirichlet boundary data for Φ on BΩ. The electrostatic energy
N
N
ÿ
ÿ
ε0
1
qzk ck Φ “ ´ εpnq∇Φ ¨ ∇Φ `
qzk ck Φ,
Welec “ ´ D ¨ E `
2
2
k“1
k“1
(2.41)
is clearly nonlocal because Φ is determined by solving the second equation in (2.40) for the given n, ck ,
and the appropriate boundary data on Φ. The arguments of Leslie in the purely mechanical case [12] rely
on formulating a local energy balance (2.11) for a material control volume V Ă Ω with the balance being
9 with a support in V . The following simple proposition
assumed to hold for any dynamical process tv, nu
allows for localization of the time derivative of the total electrostatic energy.
Proposition 2.2. Suppose that Φ satisfies the second equation in (2.40), subject to time-independent
9 is contained in V , that is the rates
Dirichlet boundary data on BΩ. If the support of tv, u1 , . . . , uN , nu
v, u1 , . . . , uN , n9 all vanish in ΩzV̄ , then
d
dt
ż #
ż
Welec “
Ω
V
ż
(2.42)
N
ÿ
`
˘
ε0 dεpnq
´
∇Φ ¨ ∇Φ ´
qzk ck ∇Φ ¨ pv ´ uk q ` ε0 εpnq∇Φ ¨ ∇v T ∇Φ
2 dt
k“1
N
ÿ
`
+
qzk ck Φ pv ´ uk q .ν.
BV k“1
Proof. Using (2.33), (2.41), integration by parts, incompressibility of the flow, our assumptions on the
6
řN
dynamical process, and writing ρe “ k“1 qzk ck , we have
*
ż
ż "
d
d
1
Welec “
´ pε0 εpnq∇Φ ¨ ∇Φq ` ρe Φ
dt Ω
dt Ω
2
"
"
* * ż "
*
ż
dΦ
ε0 dεpnq
Bρe
d
p∇Φq ` ρe
“
´
∇Φ ¨ ∇Φ `
` ∇ ¨ pρe vq Φ `
´ε0 εpnq∇Φ ¨
2 dt
Bt
dt
dt
Ω
Ω
#
+
ż
N
ÿ
ε0 dεpnq
“
´
zk ∇ ¨ pck pv ´ uk qq
∇Φ ¨ ∇Φ ` qΦ
2 dt
V
k“1
*
" ˆ ˙
*
ż "
dΦ
dΦ
`
´ ∇v T ∇Φ ` ρe
´ε0 εpnq∇Φ ¨ ∇
dt
dt
Ω
+
ż #
N
ÿ
` T
˘
ε0 dεpnq
“
∇Φ ¨ ∇Φ ` qΦ
zk ∇ ¨ pck pv ´ uk qq ` ε0 εpnq∇Φ ¨ ∇v ∇Φ
´
2 dt
V
k“1
ż
ż
dΦ
BΦ
`
t∇ ¨ pε0 εpnq∇Φq ` ρe u ´
tε0 εpnq∇Φ ¨ νu ,
dt
BΩ Bt
Ω
and (2.42) follows from (2.40) and the fact that the potential does not depend on time on BΩ.
Remark 1. Note that by using (2.39) and the first equation in (2.40), the first integrand on the right
hand side of (2.42) can be written as
ε0 dεpnq
ε0 εa
¨
9
pn b nq ∇Φ ¨ ∇Φ “ ε0 εa pn ¨ ∇Φq pn9 ¨ ∇Φq “ ε0 εa pE b Eq n ¨ n.
∇Φ ¨ ∇Φ “
2 dt
2
Further, recalling the definition (2.38) of D and using the first equation in (2.40), the third integrand on the
right hand side of (2.42) can be written as
`
˘
ε0 εpnq∇Φ ¨ ∇v T ∇Φ “ D ¨ ∇v T E “ pE b Dq ¨ ∇v,
so that (2.42) takes a form
d
dt
ż #
ż
Welec “
´ε0 εa pE b Eq n ¨ n9 ` pE b Dq ¨ ∇v ´
Ω
V
+
qzk ck ∇Φ ¨ pv ´ uk q
k“1
N
ÿ
ż
(2.43)
N
ÿ
`
qzk ck Φ pv ´ uk q ¨ ν
BV k“1
ş
The material time derivative of V Wion will also enter the energy balance.
Proposition 2.3. Suppose that (2.33) holds. Then
(2.44)
d
dt
ż
ż
Wion “ ´
V
N
ÿ
ż
kB θ∇ck ¨ pv ´ uk q `
V k“1
N
ÿ
kB θ pln ck ` 1q ck pv ´ uk q .ν.
BV k“1
Proof. Using (2.35) for Wion and the mass balances (2.33) gives
#
+ ż
˙*
ˆ
ż
ż
N
N "
ÿ
ÿ
d
d
Bck
Wion “
kB θ
` ∇ ¨ pck vq
ck ln ck “
kB θ pln ck ` 1q
dt V
Bt
V dt
V k“1
k“1
ż ÿ
N
“
kB θ pln ck ` 1q ∇ ¨ pck pv ´ uk qq
V k“1
ż
N
ÿ
“´
ż
kB θ∇ck ¨ pv ´ uk q `
V k“1
N
ÿ
BV k“1
7
kB θ pln ck ` 1q ck pv ´ uk q ¨ ν.
Combining (2.43) and (2.44) we obtain
+
"ż
* ż #ÿ
ż
N
d
Wion `
Welec “
ck ∇µk ¨ puk ´ vq ´ ε0 εa pE b Eq n ¨ n9 ` pE b Dq ¨ ∇v
dt V
Ω
V
k“1
ż ÿ
N
(2.45)
´
ck µk puk ´ vq ¨ ν,
BV k“1
where the quantities
(2.46)
µk “ kB θplnpck q ` 1q ` qzk Φ “
¯
B ´
Wion ` qzk ck Φ ,
Bck
k “ 1, . . . , N,
are the electrochemical potentials of the ions.
In order to establish the set of governing equations, we now extend the procedure carried out above
in the purely mechanical case. We postulate the same local forms of the balance of both linear (2.3) and
angular (2.5) momenta, coupled with the constraint relations (2.4) and (2.6) and assume that the mass
balances (2.33) hold in Ω along with the Maxwell’s equations of electrostatics (2.40) that hold in R3 . The
equation of balance of energy in an arbitrary subdomain V Ă Ω for any isothermal dynamical process
9
tv, u1 , . . . , uN , nu,
with a support in V is then given by
ż
(2.47)
ż
9 `
ρpf ¨ v ` g ¨ nq
V
˜
t ¨ v ` l ¨ n9 ´
BV
N
ÿ
¸
ck µk puk ´ vq ¨ ν
k“1
d
“
dt
"ż ˆ
V
1
1
9 2 ` Wion ` WOF
ρ|v|2 ` χ|n|
2
2
˙
*
ż
ż
Welec `
`
Ω
R.
V
řN
The additional boundary term k“1 ck µk puk ´ vq ¨ ν, which does not appear in (2.11), represents the energy
transported across the boundary by the ions. Equations (2.15), (2.34), and (2.45) allow to express (2.47) in
the local form
ˆ
˙
ˆ
˙
BWOF
e
v
T BWOF
e
T ` T ` p∇nq
´ E b D ¨ ∇v ` L ´
¨ ∇n9
B∇n
B∇n
ˆ
˙
N
ÿ
BWOF
e
v
` g `g ´
` ε0 εa pE b Eq n ¨ n9 ´
(2.48)
ck ∇µk ¨ puk ´ vq
Bn
k“1
“ RLC `
N
ÿ
(
kB θck Dk´1 puk ´ vq ¨ puk ´ vq .
k“0
The necessary conditions for positivity of the dissipation functional R required by the Clausius-Duhem
inequality for an arbitrary admissible dynamical process then gives the following analogs of the constitutive
relations (2.17)-(2.19) which account for the presence of ions and the electric field
(2.49)
(2.50)
(2.51)
(2.52)
T e “ ´pI ´ p∇nqT
BWOF
` E b D,
B∇n
BWOF
´ ε0 εa pE b Eqn ` λn,
Bn
BWOF
Le “
,
B∇n
1
uk “ v ´
Dk ∇µk ,
k “ 1, . . . , N,
kB θ
ge “
along with the relations (2.21) for the viscous stress T v and (2.22) for the viscous molecular force gv .
As in the clasical Ericksen-Leslie system, the fields p and λ in (2.49)-(2.50), are the Lagrange multipliers
8
corresponding to the constraints (2.4) and (2.6), respectively. We are now ready to formulate the full set of
equations governing the evolution of a nematic electrolyte.
Proposition 2.4. Suppose that the continuity equations (2.33), the linear momentum balance (2.3), the
angular momentum balance (2.5), and the Maxwell’s equations of electrostatics (2.40) hold in Ω. Further,
suppose that the energy balance (2.47) holds in every subdomain V Ă Ω for any dynamical process
9
tv, u1 , . . . , uN , nu,
with a support in V . Then the necessary conditions for positivity of the dissipation functional R in (2.34)
are (2.21)-(2.22) and (2.49)-(2.52).
The system of equations for the nematic electrolyte is as follows
ˆ „
˙
Bck
1
` ∇ ¨ ck v ´
Dk ∇µk
(2.53)
“ 0,
Bt
kB θ
N
ÿ
(2.54)
qzk ck ,
´∇ ¨ pε0 εpnq∇Φq “
k“1
ˆ
BWOF
` ε0 p∇Φ b ∇Φq εpnq ` T v
ρv9 ´ ∇ ¨ ´pI ´ p∇nqT
B∇n
(2.55)
“ ρf ,
∇ ¨ v “ 0,
(2.56)
(2.57)
˙
BWOF
χ:
n`
´ ε0 εa p∇Φ b ∇Φq n ` λn ` ∇ ¨
Bn
(2.58)
ˆ
BWOF
B∇n
˙
` gv ` λn “ ρg,
n ¨ n “ 1,
where T v is given by (2.21) and gv is given by (2.22).
Proof. The equation (2.53) immediately follows from (2.52) and (2.33). The remaining equations follow
by substituting the expressions for the appropriate stresses, using the definitions of D and E, and by recalling
that a b Ab “ pa b bqAT for any a, b P R3 and A P M 3ˆ3 .
Remark 2. From the equation (2.53), we identify the tensor
Mk “
(2.59)
qzk
Dk ,
kB θ
as the mobility tensor of the kth species. Note that, when Dk is a multiple of the identity, M is the analog of
Einstein’s mobility relation of electrons in a gas (zk “ ´1). The conductivity matrix (mobility times charge
density) of the kth species is now given by
σk “
(2.60)
1
ck zk2 q 2 Dk .
kB θ
2.2.1. Boundary Conditions. Since the energy law (2.47) for the nematic electrolyte has to hold in
the entire domain Ω, it follows from the equations (2.53)-(2.58) that
+ ż #
+
ż #
N
N
ÿ
ÿ
(2.61)
T ν ¨ v ` Lν ¨ n9 ´
ck µk puk ´ vq ¨ ν “
t ¨ v ` l ¨ n9 ´
jk ,
BΩ
BΩ
k“1
k“1
where jk , k “ 1, . . . , N represents the normal energy flux associated with the transport of the k-th species
9
of ions across the boundary. This equation should be valid for all dynamical processes tv, u1 , . . . , uk , nu
therefore the boundary conditions on BΩ should be of the form
(2.62)
T ν “ t̂ or v “ 0,
and
(2.63)
Lν “ l̂
and
(2.64)
ck µk puk ´ vq ¨ ν “ ĵk
or n “ n̂,
or uk ¨ ν “ v ¨ ν,
9
k “ 1, . . . , N,
where t̂, l̂, n̂, and ĵk , k “ 1, . . . , N are prescribed fields on BΩ with |n̂| “ 1 and T and L given by (2.2),
(2.49), (2.21) and (2.51). Observe that the fields t̂, l̂, ĵk , k “ 1, . . . , N can be time-dependent in this
formulation.
The set of the boundary conditions should be supplemented by the boundary data on the electric field.
Here, we will impose the Dirichlet conditions on the poential on the boundary between the nematic electrolyte
and a conductor
(2.65)
Φ|BΩ “ Φ0 ,
for some prescribed Φ0 . On the boundary between the nematic and an insulating medium [8], we will impose
the condition of the zero jump of the normal component of the displacement D, that is
(2.66)
rD ¨ νsBΩ “ 0,
where r¨sBΩ represent the jump of a quantity in the brackets across BΩ. In this case, the equations of
electrostatics have to be solved in R3 .
2.2.2. Variational Structure. Setting
Wpn, ∇n, Φ, ∇Φ, c1 , . . . , cN q “ WOF pn, ∇nq ` Wion pc1 , . . . , cN q ` Welec pφ, ∇φ, c1 , . . . , cN q
N
ÿ
`
˘ ε0
“ WOF pn, ∇nq `
kB θck lnpck q ` qzk ck Φ ´ εpnq∇Φ.∇Φ,
2
k“1
Maxwell’s equation (2.53) and the balance laws (2.54), (2.55) and (2.57) may be written as
`
˘
Bck
` ∇ ¨ ck pv ´ p1{kB θq∇µk q “ 0,
Bt
˙
ˆ
BW
BW
“
,
∇.
B∇Φ
BΦ
ˆ
˙
BR
BW
BW
ρv9 ´ ∇ ¨ ´pI `
´ p∇nqT
´ ∇Φ b
“ ρf ,
B∇v
B∇n
B∇Φ
ˆ
˙
BR
BW
BW
χ:
n`
´∇¨
`
` λn “ ρg.
B n9
B∇n
Bn
Introducing the following transform of W,
W ˚ pn, ∇n, Φ, ∇Φ, µ1 , . . . , µN q “ ´WOF pn, ∇nq ` kB θ
the dual relations take the form
`
˘
BW ˚
ck “
“ exp pµk ´ qzk Φq{kB θ ´ 1
Bµk
N
ÿ
`
˘ ε0
exp pµk ´ qzk Φq{kB θ ´ 1 ` εpnq∇Φ.∇Φ,
2
k“1
and
µk “
BW
“ kB θplnpck q ` 1q ` qzk Φ.
Bck
Computing ∇W ˚ and rearranging the terms shows
ˆ
˙
ˆ
˙
BW
BW
BW ˚
BW ˚
∇ ¨ ´W ˚ I ´ p∇nqT
´ ∇Φ b
“ ∇ ¨ ´W ˚ I ` p∇nqT
` ∇Φ b
B∇n
B∇Φ
B∇n
B∇Φ
ˆ
˙
ˆ
˙
N
ÿ BW ˚
BW ˚
BW ˚
BW ˚
BW ˚
“ p∇nqT ∇ ¨
´
` ∇¨
´
∇Φ ´
∇µk
B∇n
Bn
B∇Φ
BΦ
Bµk
k“1
ˆ
˙ ˆ
˙
N
ÿ
BW
BW
BW
BW
T
“ ´p∇nq
∇¨
´
´ ∇¨
´
∇Φ ´
ck ∇µk ,
B∇n
Bn
B∇Φ
BΦ
k“1
so the linear momentum equation can be written as
ˆ
˙
ˆ
˙ ÿ
N
BR
BW
BW
˚
T
(2.67)
ρv9 ´ ∇ ¨ ´pp ` W qI `
` p∇nq
∇¨
´
`
ck ∇µk “ ρf .
B∇v
B∇n
Bn
k“1
10
The energy estimate for the coupled system now follows upon multiplying the equations for the concentrations
by µk , Maxwells equation by Φt , and the linear and angular momentum equations by v and n9 respectivly.
Granted appropriate boundary data, this gives
+ ż
ż
ż #
N
ÿ
˘
(
`
`
˘
ck
BR
BR
d
2
2
2
9
`W `
“
p1{2q ρ|v| ` χ|n|
¨ n9 `
¨ ∇v `
|∇µk |
ρ f ¨ v ` g ¨ n9 .
dt Ω
B∇n9
B∇v
k θ
Ω
Ω
k“1 B
The identities required obtain the statement of the linear momentum equation in (2.67) and to pose the
balances of mass for the concentrations in terms of the chemcial potentials are used in an essential fashion
for the development of stable numerical schemes with non–negative concentrations [19].
3. LC-Enabled Electroosmosis. Next, we apply the model developed in the previous section to
replicate predictions of an experiment on electroosmosis in a nematic liquid crystalline film constrained
between two parallel plates. In [15], the authors present an approach to generate electrokinetic effects,
by using as an electrolyte a liquid crystal with surface-imposed distortions of molecular orientations. In
the presence of the uniform electric field, these variations produce space charge separation that triggers
electroosmotic flow in the liquid crystal. In particular, for the setup depicted in Fig. 1, the director
orientation is periodically varying in the vertical direction and an AC field is applied in the horizontal
direction. It was observed that spatially periodic horizontal flow proceeds along the ”guiding rails” induced
by molecular orientation with the direction of the flow independent of the sign of the field. Indeed, for
weak fields, it is known that the driving force of the flow is proportional to the square of the field. We are
interested in establishing a simple model of this process.
50 µm
50 µm
E
(a)
3.5
µm / s
(b)
Fig. 1. Liquid crystal-enabled electroosmotic flows in a flat nematic cell with patterned one-dimensionally periodic director
field [15]. (a) Experimentally determined director pattern (short black dashes) and a schematics of space charge separation
due to director distortions when the electric field is directed from left to right. Electric conductivity of the nematic is higher
along the director than in a direction perpendicular to it. Clouds of separated positive and negative ions are marked by the
”+” and ”-” symbols, respectively. Thick arrows show the direction of the electrostatic forces acting on the charge clouds and
the direction of local electroosmotic flows. Note that reversal of field polarity reverses the polarity of charges but preserves the
directions of the driving electrostatic force and the induced nematic flows. (b) Experimentally determined map of electroosmotic
velocities corresponding to the director pattern in (a).
3.1. A Simplified Model. In what follows, we will represent the film by a domain Ω ˆ r´h, hs Ă R3 ,
where Ω Ă R2 and 0 ă h ! 1. The plates are patterned in a way that enforces strong anchoring of the
director so that the director orientation varies in a prescribed way along the plates while it remains parallel
to the plates. We will assume that the patterns on the top and the bottom plate are identical to each other
and impose slip boundary conditions on both film/plate boundaries Ω ˆ t´h, hu. Choosing a coordinate
system with the z-axis perpendicular to the plates, we impose the Dirichlet condition n|BΩˆtzu “ n|BΩˆthu
for any z P r´h, hs on the lateral boundary of the film BΩ ˆ r´h, hs.
If we let α1 “ α2 “ α3 “ α5 “ α6 “ 0 and α4 ą 0, then by (2.23) the parameters γ1 “ γ2 “ 0 and, hence
by (2.22) the viscous molecular force gv ” 0. By assuming that the dielectric permittivity anisotropy of the
liquid crystal is small and setting εa “ 0, we eliminate direct interaction between the director and the electric
11
filed. To simplify the model further, we adopt the equal elastic constants approximation K1 “ K2 “ K3 “ K,
so that
K
2
WOF pn, ∇nq “ |∇n| .
2
Note that we have also eliminated the term in (2.8) that corresponds to the elastic constant K4 since this
term is a null Lagrangian under the Dirichlet boundary data on the director.
Given these assumptions, (2.57) reduces to the harmonic map equation
(3.1)
∆n “ γn
in Ω ˆ r´h, hs, where n satisfies the Dirichlet conditions on the boundary of the film. Since the director is
assumed to be parallel to the plates on the film boundary, we seek a solution of (3.1) of the form
npθpx, yqq “ pcos θpx, yq, sin θpx, yq, 0q
. By substituting this ansatz into (3.1) we find that
(3.2)
∆θ “ 0
in Ω. In the remainder of this section we will require the director pattern on the plates to satisfy the equation
(3.2) so that npx, yq “ pcos θpx, yq, sin θpx, yq, 0q is a solution of (3.1). In fact, the third component of n can
be neglected and n can be written as
npθpx, yqq “ pcos θpx, yq, sin θpx, yqq.
It is now reasonable to look for a solution of the system of governing equations (2.53)-(2.57) that
is independent of the z-variable and such that the third component of velocity is identically zero, i.e.,
vpx, y, tq “ pupx, y, tq, vpx, y, tq, 0q or
vpx, y, tq “ pupx, y, tq, vpx, y, tqq,
if we drop the trivial component. As an additional simplifying assumption, we consider a case of two ionic
species given by the fields
cp px, y, tq and cm px, y, tq
with zp “ 1 and zm “ ´1, respectively. We further select the anisotropic diffusion matrix for the both
species to be in the form
(3.3)
Dpθq “ D̄ pI ` pλ ´ 1qnpθq b npθqq ,
where D̄ ą 0 and the parameter λ ě 0 determines the strength of anisotropy.
The system (2.53)-(2.57) now takes the form
´
´
¯¯
$
Bcp
qcp
’
`
∇
¨
pvc
q
“
∇
¨
Dpθq
∇c
`
∇Φ
,
’
p
p
Bt
k
θ
’
B
´
´
¯¯
’
’
’
qc
Bc
m
m
’
’
& Bt ` ∇ ¨ pvcm q “ ∇ ¨ Dpθq ∇cm ´ kB θ ∇Φ ,
´∆Φ “ εεq0 pcp ´ cm q ,
(3.4)
’
’
∆θ
“ 0,
’
’
` Bv
˘
’
’
’
ρ
` pv ¨ ∇q v “ ´∇π ` µ∆v ´ q pcp ´ cm q ∇Φ,
’
Bt
%
∇ ¨ v “ 0,
2
in Ω, where π “ p ` kB θ pcp ` cm q ` K
2 |∇θ| and µ :“ α4 .
Ω “ r´L, Ls ˆ r´W, W s, we impose the boundary conditions
$
on
’
’ v“0
’
Bu
’
v
“
0
and
“
0
on
’
By
’ ´
¯
’
’
& ∇cp ` qcp ∇Φ ¨ Dpθqν “ 0 on
kB θ
´
¯
(3.5)
qcm
’
’
∇c
´
∇Φ
¨ Dpθqν “ 0 on
m
’
kB θ
’
’
’
’
Φ px, ˘W, tq “ 0,
’
% y
Φp˘L, y, tq “ ˘Φ0 ptq.
12
Specializing further to a rectangular domain
t´L, Lu ˆ r´W, W s,
r´L, Ls ˆ t´W, W u,
BΩ,
BΩ,
Note that, in writing the impenetrability conditions on cp and cm in (3.5), we took advantage of the symmetry
of the diffusion matrix. Furthermore, the second equation in (3.5) corresponds to a perfect slip condition on
the lateral components of the boundary BΩ.
Remark 3. Since npx, yq “ pcos θpx, yq, sin θpx, yqq, then
2
∇nT ∇n “ ∇θ b ∇θ “
(3.6)
|∇θ|
I ` τ,
2
where, by (3.2), the deviatoric stress tensor τ is divergence-free: ∇ ¨ τ “ 0.
3.2. Nondimensionalization. Next, we nondimensionalize the system (3.4)-(3.5) as follows. Let
(3.7)
x̃ “
x
,
W̄
ṽ “
v
,
ū
t̃ “
ūt
,
W̄
c̃p “
cp
cm
, c̃m “
,
c̄
c̄
Φ̃ “
Φ
,
Φ̄
p̃ “
p
,
p̄
where f¯ denotes a characteristic value of the quantity f . We let Φ̄ “ W̄ L´1 }Φ0 }8 “ W̄ }E0 }8 , where E0
represents the strength of the electric field between the electrodes. Following [15], assume that
ū “
εε0 Φ̄2
.
µW̄
By denoting D̃pθq “ D̄´1 pDij pθqq and dropping all tildes for notational convenience, we obtain the system
of nondimensional equations
$
’
’ Pe pcpt ` ucpx ` vcpy q “ pD11 pθq pcpx ` Fcp Φx qqx
’
’
’
` pD12 pθq pcpx ` Fcp Φx qqy ` pD12 pθq pcpy ` Fcp Φy qqx
’
’
’
’
’
’
` pD22 pθq pcpy ` Fcp Φy qqy ,
’
’
’
’
’
Pe pcmt ` ucmx ` vcmy q “ pD11 pθq pcmx ´ Fcm Φx qqx
’
’
’
’
’
` pD12 pθq pcmx ´ Fcm Φx qqy ` pD12 pθq pcmy ´ Fcm Φy qqx
’
&
(3.8)
` pD22 pθq pcmy ´ Fcm Φy qqy ,
’
’
’
’
’ ´Φxx ´ Φyy “ B pcp ´ cm q ,
’
’
’
’
’
θxx ` θyy “ 0,
’
’
’
’
’
Re put ` uux ` vuy q “ ´px ` uxx ` uyy ´ B pcp ´ cm q Φx ,
’
’
’
’
’
Re pvt ` uvx ` vvy q “ ´py ` vxx ` vyy ´ B pcp ´ cm q Φy ,
’
’
%
ux ` vy “ 0,
“ ´1 ´1 ‰
in Ω “ ´ε , ε
ˆ r´a, as. Here we set
Re “
ρūW̄
,
µ
Pe “
ūW̄
,
D̄
ε“
W̄
,
L
a“
W
,
W̄
F“
q Φ̄
,
kB θ
B“
qc̄W̄ 2
εε0 Φ̄
and assume that p̄ “ µūW ´1 . The boundary conditions can now be written as
(
$
u“v“0
on “ ´ε´1 , ε´1‰ ˆ r´a, as,
’
’
’
’
on ´ε´1 , ε´1 ˆ t´a, au,
v “ 0 and Bu
’
By “ 0
’
(
’
’
’
p∇cp ` F∇Φq ¨ Dpθqi “ 0
on ´ε´1 , ε´1 ( ˆ r´a, as,
’
’
’
& p∇cm ´ F∇Φq ¨ Dpθqi “ 0
on ´ε´1 , ε´1 ˆ r´a, as,
“
‰
(3.9)
D12 pθq pcpx ` Fcp Φx q ` D22 pθq pcpy ` Fcp Φy q “ 0
on ε´1 , ε´1 ˆ t´a, au ,
’
’
“
‰
’
’
’
D12 pθq pcmx ´ Fcm Φx q ` D22 pθq pcmy ´ Fcm Φy q “ 0 on ´ε´1 , ε´1 ˆ t´a, au ,
’
’
’
’
’
0,
’ Φy`px, ˘a, tq “
’
˘
%
´1
Φ ˘ε , y, t “ ˘Φ0 ptq,
where we set Φ̃0 ptq “ Φ0Φ̄ptq and drop the tilde. From now on—unless specified otherwise—we will work with
the nondimensional problem (3.8)-(3.9).
13
3.3. Periodic Flow Pattern. Suppose now that the director field follows a periodic stripe pattern
with the stripes being parallel to the x-axis. This can be modeled by assuming that a “ πn for some n P N
and setting
(3.10)
θ“
y ` y0
,
2
where y0 P p0, πq. One can immediately observe that this function satisfies the fourth equation in (3.8). By
a direct computation we also have
ˆ
ˆ
˙˙
1
sin py ` y0 q
cos py ` y0 q
(3.11)
D“
pλ ` 1qI ` pλ ´ 1q
.
sin py ` y0 q ´ cos py ` y0 q
2
(
Our setup can be(associated with an electrochemical experiment, in which the right— ε´1 ˆr´πn, πns—
and the left— ´ε´1 ˆ r´πn, πns—components of the boundary are identified as a positive and a negative
electrode, respectively. Here the rest of the boundary is assumed to be electrically insulated. As the positive
ions will be attracted to the negative electrode and vice versa, the boundary layers would form next to the
electrodes that would subsequently suppress both the potential difference and flow in the nematic electrolyte
as long as the electrodes potentials remain fixed. In an experiment, this is circumvented by applying the
AC instead of the DC field making the corresponding problem inherently transient. Here we will assume
that the parameter ε is small and that the flux of ions on the timescale of the flow is not large enough to
significantly affect the boundary layers. We will thus solve the ”outer problem” away from the electrodes
and set Φ0 to be constant in time and equal to the half of the potential difference between the ”matching
regions” corresponding the the edges of the right and left boundary layers. Since we are not solving the
equations inside the boundary layer, we can then also ignore the second and the third boundary conditions
in (3.9).
We now seek a solution of (3.8)-(3.9) in the form
cp “ cp pyq,
cm “ cm pyq, v “ pupyq, 0q, Φ “ x ` φpyq.
Then the incompressibility condition trivially holds and the system (3.8)-(3.9) reduces to
$
pF D12 pθqcp ` D22 pθq pcpy ` Fcp φy qqy “ 0,
’
’
’
’
’
’
’ ´ pF D12 pθqcm ` D22 pθq pcmy ´ Fcm φy qqy “ 0,
’
’
’
’ φyy “ ´B pcp ´ cm q ,
’
’
’
’
’
uyy “ B pcp ´ cm q ,
’
’
&
py “ ´B pcp ´ cm q φy ,
(3.12)
’
’
’ px “ 0,
’
’
ż πn
ż πn
’
’
’
’
cp dy “
cm dy “ 2πn,
’
’
’
´πn
’
’ ż´πn
’
πn
’
’
’
%
u dy “ 0,
´πn
subject to the boundary conditions
$
uy p˘πnq “ 0,
’
’
&
F D12 pθp˘πnqqcp p˘πnq ` D22 pθp˘πnqqcpy p˘πnq “ 0,
(3.13)
F D12 pθp˘πnqqcm p˘πnq ´ D22 pθp˘πnqqcmy p˘πnq “ 0,
’
’
%
φy p˘πnq “ 0.
Note that we have added three integral conditions to the problem in order to take into account conservation
of mass of both species as well as to ensure that there is a zero net flow across each cross-section of the
domain. Indeed, since the time-dependent problem has now been replaced by the stationary problem, we
have impose the condition that the total masses of both ionic species are the same as what they were at the
initial time for the time-dependent problem.
14
The fourth and fifth equations in (3.12) are used to determine pressure. By combining the third and the
fourth equations in (3.12), we find that
(3.14)
uyy “ ´φyy ,
so that the nematic liquid crystal velocity is found once the electric field has been computed. To this end,
observe that by integrating the first two equations in (3.12) and using (3.13), we have
F D12 pθqcp ` D22 pθq pcpy ` Fcp φy q “ 0,
´F D12 pθqcm ` D22 pθq pcmy ´ Fcm φy q “ 0
on r´πn, πns. Dividing these equations by D22 pθqcp and D22 pθqcm , respectively, and using (3.11) gives
(3.15)
plog cp ` Fφqy “ ´
F D12 pθq
“ ´ plog Apyqqy ,
D22 pθq
plog cm ´ Fφqy “
F D12 pθq
“ plog Apyqqy
D22 pθq
on r´πn, πns where
(3.16)
F
Apyq “ pλ ` 1 ´ pλ ´ 1q cospy ` y0 qq .
Integrating (3.15), we find
cp “ c0p A´1 pyqe´F φ and cm “ c0m ApyqeF φ ,
where c0p and c0m are arbitrary positive constants. Since φ is determined up to an arbitrary constant, we can
replace φ by φ ´ φ0 and choose φ0 so that
c0p eFφ0 “ c0m e´Fφ0 “ c0 .
It follows that
(3.17)
cp “ c0 A´1 pyqe´F φ and cm “ c0 ApyqeF φ .
Substituting these expressions into the third equation in (3.12), we obtain the equations
`
˘
" 2
φ “ c0 B ApyqeFφ ´ A´1 pyqe´Fφ , y P p´πn, πnq,
(3.18)
φ1 p˘πnq “ 0,
satisfied by φ.
Alternatively, the system of governing equations can be reduced to a single equation for r “ log cp . By
taking the derivative of the first equation in (3.15) and setting
(3.19)
Bpyq “ log pλ ` 1 ´ pλ ´ 1q cospy ` y0 qq,
we obtain
r2 ` Fφ2 “ ´FB 2 pyq.
(3.20)
Since from (3.17) it follows that cm “ c20 {cp “ c20 e´r , using the third equation in (3.12) gives
`
˘
(3.21)
r2 ´ F B er ´ c20 e´r “ ´FB 2 pyq.
This equation should be supplemented by the integral conditions from (3.12) that in terms of r take the form
ż πn
ż πn
(3.22)
er dy “ c20
e´r dy “ 2πn.
´πn
´πn
Even though they are significantly simpler than the original system of PDEs, both the problem (3.18) and
the problem (3.21)-(3.22) are still highly nonlinear and generally need to be solved numerically.
15
3.3.1. Asymptotic Solutions. The behavior of solutions of (3.21)-(3.22) is determined by the sizes
of nondimensional groups B and F. In the experimental setup considered in [10], the physical parameters
had the following values
(3.23)
L “ 1 ¨ 10´2 m,
D̄ “ 4.89 ¨ 10´11 m2 {s,
W “ 5 ¨ 10´4 m,
ĉ “ 1 ¨ 1019 m´3 ,
H “ 5 ¨ 10´5 m,
λ “ 1.417,
q “ 1.6 ¨ 10´19 C, µ “ 0.832 Pa ¨ s,
εε0 “ 5.32 ¨ 10´11 F{m,
ρ “ 1 ¨ 103 kg{m3 , Φ0 “ 400 V,
W̄ “ 5 ¨ 10´5 m.
These yield
ū “ 5 ¨ 10´6 m{s and Φ̄ “ 2 V,
(3.24)
and
(3.25)
Re “ 3 ¨ 10´7 , Pe “ 5,
ε “ 5 ¨ 10´3 , F “ 79, B “ 37.
According to these observations, here we will assume that the nondimensional groups F, B " 1, so that we
can take advantage of a natural small parameter B1 in order to solve the problem (3.21)-(3.22) asymptotically.
We thus suppose that δ :“ F1 , where δ ! 1 and use (3.25) to set B “ b{δ where b “ Op1q. In what follows
we will further assume that y0 “ 0.Then (3.21)-(3.22) takes the form
$
˘
b` r
’
’ δr2 ´
e ´ c20 e´r “ ´B 2 pyq,
’
&
δ
(3.26)
ż πn
ż πn
’
’
r
2
’
%
e dy “ c0
e´r dy “ 2πn.
´πn
´πn
Assuming that
` ˘
r “ δr1 ` O δ 2 ,
c0 “ 1 ´
` ˘
c1
δ ` O δ2 ,
2
we can rewrite (3.26) as
"
r1 “ ´B 2 pyq ` b cş1 ,
ş´2b
πn
πn
r dy “ 2πnc1 ` ´πn r1 dy “ 0,
´πn 1
x P p´πn, πnq,
to the leading order in δ. Then, it immediately follows that c1 “ 0 and that r1 is given by
(3.27)
r1 pyq “
1 2
B pyq.
2b
From (3.15) and (3.27), setting φpyq “ φ0 pyq ` O pδq , we obtain
" 1
φ0 “ ´B 1 pyq, x P p´πn, πnq,
(3.28)
φ10 p˘πnq “ 0.
We conclude that, up to an arbitrary constant of integration,
(3.29)
φ0 pyq “ ´Bpyq.
Finally, from (3.14), the expression for the leading term in velocity upyq “ u0 pyq ` O pδq takes the form
u0 pyq “ Bpyq ` Cu1 y ` Cu2 ,
where Cu1 and Cu2 are arbitrary constants. Since by (3.13) we have that u0y p˘πnq “ 0, then using the integral
condition on velocity in (3.12) it follows that
ż πn
1
Cu1 “ 0,
Cu2 “ ´
Bpyq dy,
2πn ´πn
16
hence
(3.30)
u0 pyq “ Bpyq ´
1
2πn
ż πn
Bpyq dy.
´πn
` ˘
To summarize the asymptotic results above, up to terms O δ 2 , we have
1 2
B pyq,
2B
1 2
(3.32)
B pyq,
cm pyq “ 1 ´
2B
(3.33)
φpyq “ ´Bpyq,
ż πn
1
(3.34)
upyq “ Bpyq ´
Bpyq dy.
2πn ´πn
` ˘
` ˘
Here we have used δ “ b{B and the facts that cp “ er “ 1 ` δr1 ` O δ 2 and cm “ c20 e´r “ 1 ´ δr1 ` O δ 2 .
Note that, since the scaling for u was chosen to be quadratic in the applied field and FB is independent of
the field, the flow u1 is quadratic in the field in terms of the dimensional units. Thus reversing the direction
of the field would not result in flow reversal. We point out that any solution obtained by solving (3.26), also
solves the original system (3.12) (but it does not satisfy all of the boundary conditions in (3.13)).
(3.31)
cp pyq “ 1 `
3.3.2. Applicability of Asymptotic Solutions. Discussion. In Figs 2-6, we have plotted, both,
the asymptotic solutions (3.31)-(3.34) as well as full numerical solutions of (3.12)-(3.13). The numerical
solution was obtained by solving (3.21)-(3.22) in MATLAB and there is an excellent match between the
corresponding numerical and asymptotic solution fields. In Fig. 12 we have plotted on a logarithmic scale
the maximum magnitude of the flow velocity as a function of the applied potential. The dependence on the
field is clearly quadratic,
In general, the behavior of solutions critically depends on the size of nondimensional groups F and B. For
example, observe that, if the applied filed is large enough to cause separation of all charges in the nematic,
the field along the y-direction can be interpreted as a field in a capacitor with an area charge density of
c̄W̄ and the distance between the capacitor plates equal to W̄ . The potential difference between the plates
of such capacitor is equal to c̄W̄ 2 {εε0 . So, we conclude that the nondimensional group B is equal to the
ratio between the characteristic potential Φ̄ and the maximum potential difference that can be supported
by the system via separation of charges. Then, if B " 1 as in the experiment in [10], there is always enough
charges in the system to support quadratic growth of the flow velocity, as the applied field increases. If this
parameter is small, however, then all of the available charges would have been separated already for a given
field and the flow would become proportional to the magnitude of the field. Indeed, in Figs. 7-11 we have
plotted the numerical and asymptotic solutions when c̄ “ 7 ¨ 1015 . The pictures clearly show a full charge
separation. Further, the dependence of the flow velocity on the field in Fig. 13 is closer to being linear,
indicating that the system is close to saturation. The saturated solutions depicted in Figs. 7-11 can also be
obtained via an asymptotic singular perturbation procedure for (3.26) using an appropriate small parameter.
This analysis is beyond the scope of this paper and will be presented elsewhere.
4. Acknowledgements. The authors acknowledge support from National Science Foundation, grant
number dms-dmref 1435372.
17
19
x 10
Concentration of the positive ions
Numerics
Asypmtotics
1.0025
1.002
1.0015
cp, 1/m3
1.001
1.0005
1
0.9995
0.999
0.9985
−4
−3
−2
−1
0
1
2
3
y, m
4
−4
x 10
Fig. 2. Concentration of the positive ions for the parameters in (3.23)
18
x 10
Concentration of the negative ions
Numerics
Asypmtotics
10.015
10.01
cm, 1/m3
10.005
10
9.995
9.99
9.985
9.98
9.975
−4
−3
−2
−1
0
1
y, m
2
3
4
−4
x 10
Fig. 3. Concentration of the negative ions for the parameters in (3.23)
16
6
x 10
Charge concentration
Numerics
Asypmtotics
5
4
cp−cm, 1/m3
3
2
1
0
−1
−2
−3
−4
−5
0
y, m
5
−4
x 10
Fig. 4. Space charge distribution for the parameters in (3.23)
18
Potential
Numerics
Asypmtotics
0.3
0.2
Φ, V
0.1
0
−0.1
−0.2
−0.3
−4
−3
−2
−1
0
1
2
3
y, m
4
−4
x 10
Fig. 5. Electric potential for the parameters in (3.23)
Velocity
−6
1
x 10
Numerics
Asypmtotics
0.8
0.6
0.4
u, m/s
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−5
0
5
y, m
−4
x 10
Fig. 6. Flow velocity for the parameters in (3.23)
16
x 10
Concentration of the positive ions
Numerics
Asypmtotics
3.5
3
2.5
cp, 1/m
3
2
1.5
1
0.5
0
−0.5
−1
−4
−3
−2
−1
0
y, m
1
2
3
4
−4
x 10
Fig. 7. Concentration of the positive ions when c̄ “ 7 ¨ 1015 m´3
19
16
x 10
Concentration of the negative ions
3
Numerics
Asypmtotics
2.5
2
1.5
cm, 1/m
3
1
0.5
0
−0.5
−1
−1.5
−2
−4
−3
−2
−1
0
1
2
3
y, m
4
−4
x 10
Fig. 8. Concentration of the negative ions when c̄ “ 7 ¨ 1015 m´3
Charge concentration
16
6
x 10
Numerics
Asypmtotics
5
4
cp−cm, 1/m3
3
2
1
0
−1
−2
−3
−4
−5
0
5
y, m
−4
x 10
Fig. 9. Space charge distribution when c̄ “ 7 ¨ 1015 m´3
Potential
Numerics
Asypmtotics
0.3
0.2
Φ, V
0.1
0
−0.1
−0.2
−0.3
−4
−3
−2
−1
0
y, m
1
2
3
4
−4
x 10
Fig. 10. Electric potential when c̄ “ 7 ¨ 1015 m´3
20
Velocity
−6
1
x 10
Numerics
Asypmtotics
0.8
0.6
0.4
u, m/s
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−5
0
5
y, m
−4
x 10
Fig. 11. Flow velocity when c̄ “ 7 ¨ 1015 m´3
Maximum velocity vs. potential
−13.95
−14
−14.05
log vmax
−14.1
−14.15
−14.2
−14.25
−14.3
−14.35
−14.4
−14.45
5.75
5.8
5.85
5.9
5.95
log φ
Fig. 12. Maximum flow velocity as a function of Φ0 on a logarithmic scale when c̄ “ 1 ¨ 1019 m´3 . Φ0 P r300, 400s V
Maximum velocity vs. potential
−14.475
log vmax
−14.48
−14.485
−14.49
5.988
5.99
5.992
5.994
5.996
5.998
log φ
Fig. 13. Maximum flow velocity as a function of Φ0 on a logarithmic scale when c̄ “ 7 ¨ 1015 m´3 . Φ0 P r398, 403s V
21
REFERENCES
[1] M. Z. Bazant and T. M. Squires, Induced-charge electrokinetic phenomena, Current Opinion in Colloid and Interface
Science, 15 (2010), pp. 203 – 213.
[2] H. Chen, M.-C. Calderer, and Y. Mori, Analysis and simulation of a model of polyelectrolyte gel in one spatial
dimension, Nonlinearity, 27 (2014), p. 1241.
[3] J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), pp. 381–392.
[4] H.Chen, Y.Mori, C.Micek, and M.C.Calderer, A dynamic model of polyelectrolyte gels, SIAM J.Appl.Math, 73 (2013),
pp. 104–123.
[5] S. Hernandez-Navarro, P. Tierno, J. Ignes-Mullol, and F. Sagues, Ac electrophoresis of microdroplets in anisotropic
liquids: transport, assembling and reaction, Soft Matter, 9 (2013), pp. 7999–8004.
[6] S. ”Hernandez-Navarro, P. Tierno, J. Igns-Mullol, and F. Sagus, Liquid-crystal enabled electrophoresis: Scenarios
for driving and reconfigurable assembling of colloids, The European Physical Journal Special Topics, 224 (2015),
pp. 1263–1273.
[7] M. Kléman and O. D. Laverntovich, Soft matter physics: an introduction, Springer, 2002.
[8] L. Landau and E. Lifschitz, Electrodynamics of Continuous Media (Second Edition Revised and Enlarged), vol. 8 of
Course of Theoretical Physics, Pergamon, Amsterdam, 1984.
[9] O. D. Lavrentovich, I. Lazo, and O. P. Pishnyak, Nonlinear electrophoresis of dielectric and metal spheres in a
nematic liquid crystal, Nature, 467 (2010), pp. 947–950.
[10] I. Lazo and O. D. Lavrentovich, Liquid crystal-enabled electrophoresis of spheres in a nematic medium with negative
dieletric anisotropy, Phi.Trans.R.Soc. A, 371 (2013), p. 20120255.
[11] I. Lazo, C. Peng, J. Xiang, S. V. Shiyanovskii, and O. D. Lavrentovich, Liquid crystal-enabled electro-osmosis
through spatial charge separation in distorted regions as a novel mechanism of electrokinetics, Nature Communications, 5 (2014), p. 047801.
[12] F. Leslie, Continuum theory for nematic liquid crystals, Continuum Mechanics and Thermodynamics, 4 (1992), pp. 167–
175.
[13] F. M. Leslie, Theory of flow phenomena in liquid crystals, Advances in Liquid Crystals, 4 (1979), pp. 1–81.
[14] H. Morgan and N. Green, Ac electrokinetics: colloids and nanoparticles. Research Studies Press, 2003.
[15] C. Peng, Y. Guo, C. Conklin, J. Viñals, S. V. Shiyanovskii, Q.-H. Wei, and O. D. Lavrentovich, Liquid crystals
with patterned molecular orientation as an electrolytic active medium, Phys. Rev. E, 92 (2015), p. 052502.
[16] W. Russel, D. Saville, and W. Schowalter, Colloidal Dispersions, Cambridge University Press, New York, 1989.
[17] Y. Sasaki, Y. Takikawa, V. S. R. Jampani, H. Hoshikawa, T. Seto, C. Bahr, S. Herminghaus, Y. Hidaka, and
H. Orihara, Colloidal caterpillars for cargo transportation, Soft Matter, 10 (2014), pp. 8813–8820.
[18] E. G. Virga, Variational theories for liquid crystals, vol. 8 of Applied Mathematics and Mathematical Computation,
Chapman & Hall, London, 1994.
[19] N. J. Walkington, Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations,
ESAIM: Mathematical Modelling and Numerical Analysis, 45 (2011), pp. 523–540.
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