# A model liquid crystal cell in an RC circuit

Eugene C. Gartland, Jr Department of Mathematical Sciences, Kent State
University, P.O. Box 5190, Kent, OH 44242, USA<EMAIL_ADDRESS>http://www.math.kent.edu/~gartland)

###### Abstract

A nematic liquid-crystal cell subject to an electric field created by
electrodes held at constant potential is modeled as a variable capacitor in an
RC circuit. The state of the system is characterized in terms of the director
field in the cell and the charge on the electrodes. A dynamical system is
developed that couples director dynamics in the cell (with no fluid flow) and
charge dynamics in the circuit. The dynamical equations are derived from
expressions for the total potential energy of the system and a dissipation
involving a single rotational viscosity for the director plus Joule heating
associated with current in the circuit. An effort is made to quantify effects,
in particular the widely differing time scales for the processes involved, and
numerical illustrations are given. The exercise illuminates aspects of the
modeling of equilibrium states of such a system.

Keywords: nematic liquid crystal, electric field, Oseen-Frank model, RC
circuit

##  1 Introduction

We are motivated by a desire to understand better the dynamical stability of
equilibria of a nematic liquid crystal in an electric field created by
electrodes held at constant potential. As modeled in the Oseen-Frank
macroscopic continuum theory, the free-energy functional that governs the
coupled equilibrium states of the liquid crystal and the electric field can,
in the simplest cases, be expressed in the form

$\mathcal{F}[\bm{n},U]=\int_{\Omega}\Bigl{[}W_{\text{e}}(\bm{n},\nabla\bm{n})-\frac{1}{2}{\bm{\epsilon}}(\bm{n})\nabla
U\cdot\nabla U\Bigr{]}\,\text{d}{V}.$ (1)

Here $\bm{n}$ is the director field (unit-length vector field), $U$ the
electric potential (related to the electric field via $\bm{E}=-\nabla U$),
$\Omega$ the domain of the liquid crystal cell, $W_{\text{e}}$ the density of
distortional elastic energy, and ${{\bm{\epsilon}}}$ the dielectric tensor.
These terms are characterized more carefully in the next section. The tensor
${{\bm{\epsilon}}}$ is symmetric positive definite, and as a result, the
problem has an intrinsic “minimax” nature to it, with locally stable
equilibria locally minimizing with respect to $\bm{n}$ but maximizing with
respect to $U$. The assessment of local stability is well understood from a
variational point of view, studied in [8]. However, the negative definiteness
of $\mathcal{F}$ with respect to $U$ causes confusion about how to assess
stability from a dynamical point of view. The source of this confusion, as we
shall see, comes from the modeling assumptions that must be made in order to
put the free energy in the form above. In particular, one must assume either a
static electric field or an electric field that adjusts instantaneously to
changes in the director field in order for (1) to be valid.

The dynamics of such a system can be modeled at different levels of fidelity.
A director field that is evolving in time will have associated with it fluid
flow in the cell, changes in the local electric field (caused by changes in
the dielectric tensor), and changes in the capacitance of the cell (again
caused by the changes in the dielectric tensor), which will in turn cause
changes in the charge distributions on the electrodes. Hydrodynamics is often
of secondary importance (and ignoring it greatly simplifies matters); so we
shall assume no fluid flow. In many cases, the time scale for director
dynamics is orders of magnitude slower than the time scales for the electric
field and circuit dynamics; so a common modeling approximation is to assume
that both the electric field and the charge distributions on the electrodes
adjust instantaneously to any changes in the director field. This leaves a
reduced free energy that is a functional of $\bm{n}$ only, which makes the
“minimax problem” go away. The equilibrium Euler-Lagrange equation for the
reduced model is nonlocal, however—the electric field at a point depends on
the director field everywhere—though the assessment of stability is “normal,”
in that locally stable equilibrium director fields are local minimizers of the
reduced free energy.

There are, however, experiments involving “fast switching” (motivated by
potential applications for light modulators and the like) in which the time
scale for director reorientation is comparable to the time scale for circuit
dynamics—see, for example, [9, 11, 21]. There are, in fact, experiments in
which the circuit dynamics are a limiting factor [1, 14]. The time scale for
the evolution of the electric field, on the other hand, comes from the time-
dependent Maxwell equations and is invariably several orders of magnitude
faster than any other macroscopic time scale present. We examine these issues
more carefully in §4.6.

Here then we assume that the electric field adjusts instantaneously to the
director field, but we model faithfully the coupling between the dynamics of
the director field and those of the charge distribution. The modeling
highlights the role of the voltage source in giving rise to the troubling
minus sign in the free energy in (1) above and the assumptions that must be
made in order to put the free energy in that form. Also, the coupling between
director dynamics and circuit dynamics introduces an additional mechanism for
energy dissipation: Joule heating due to current in the circuit.

We choose a “textbook” model system for illustration (a splay-Fréedericksz
cell) so that all formulas can be worked out explicitly and are not too
cumbersome. Several of our results apply to more general situations, and we
try to indicate this at appropriate points. The model problem, free energy,
and equilibrium equations are presented in §2. There, both the coupled
formulation (in terms of $\bm{n}$ and $U$) and the reduced formulation (in
terms of $\bm{n}$ only) are discussed. The basic equations characterizing RC
circuits are reviewed in §3. In §4, the linkage is made between the state of
the director field in the cell and the state of the electric circuit, with a
combined potential energy and dissipation function and resulting coupled
dynamical system. A simple numerical illustration of the coupled dynamics is
given in §5, and what conclusions can be drawn are discussed in §6.

##  2 Model and equilibrium equations

For a concrete realization of the ideas explored here, we consider a nematic
liquid crystal cell in the splay-Fréedericksz geometry (as depicted in figure
1a) subject to an electric

(a)

(b)

Figure 1: Model problem. Figure 1a: electric-field splay-Fréedericksz-geometry
cell (geometry, coordinate system, ground state) as a capacitor in an RC
circuit, figure 1b. The liquid crystal film is confined to $0<z<d$. Strong
anchoring is assumed on $z=0$ and $z=d$, and the liquid-crystal director is
assumed to remain in $\operatorname{span}\\{\bm{e}_{x},\bm{e}_{z}\\}$. RC
circuit parameters: resistance $R$, capacitance $C$, voltage $V$, charge $Q$
(upper electrode $+$, lower electrode $-$), current $I$ (positive direction
indicated).

field created by electrodes held at constant potential by a voltage source. In
devices, the voltage source is usually just a battery, while in experiments,
the electromotive force could come from a variable power supply. The system is
modeled using the Oseen-Frank macroscopic continuum theory [4, Ch. 3], [19,
Ch. 2], [22, Ch. 3]. The lateral dimensions of the cell are assumed to be much
larger than the cell gap, enabling us to treat all fields as uniform in $x$
and $y$ (the coordinates in the lateral directions), with spatial dependence
only on the $z$ coordinate (the coordinate across the cell gap)—thus we ignore
the influence of fringe fields and effects near the edges of the cell. We
assume strong anchoring conditions and further assume that the director
$\bm{n}$ remains in the $x$-$z$ tilt plane:

$\bm{n}=n_{x}\bm{e}_{x}+n_{z}\bm{e}_{z}=\cos\theta\,\bm{e}_{x}+\sin\theta\,\bm{e}_{z},\quad\theta=\theta(z).$

The free-energy functional will contain contributions from distortional
elasticity plus terms associated with the electric field. The elastic part has
the form

$\mathcal{F}_{\text{e}}[\bm{n}]=\int_{\Omega}W_{\text{e}}(\bm{n},\nabla\bm{n})\,\text{d}{V}=A\int_{0}^{d}W_{\text{e}}(\bm{n},\partial_{z}\bm{n})\,\text{d}{z},$
(2)

where $\Omega$ is the domain of the cell, $A$ the $x$-$y$ cross-section area,
$d$ the cell gap, and

$\displaystyle 2W_{\text{e}}$
$\displaystyle=K_{1}(\operatorname{div}\bm{n})^{2}+K_{2}(\bm{n}\cdot\operatorname{curl}\bm{n})^{2}+K_{3}|\bm{n}\times\operatorname{curl}\bm{n}|^{2}$
(3) $\displaystyle=K_{1}n_{z,z}^{2}+K_{3}n_{x,z}^{2}$
$\displaystyle=\bigl{(}K_{1}\cos^{2}\\!\theta+K_{3}\SS\bigr{)}\theta_{z}^{2}.$

See [4, §3.1.2], [19, §2.2.1], [22, §3.2]. Here we denote
$n_{x,z}=\text{d}n_{x}/\text{d}{z}$, $n_{z,z}=\text{d}n_{z}/\text{d}{z}$, and
$\theta_{z}=\text{d}\theta/\text{d}{z}$. Thus the distortional free energy per
unit area, $\widetilde{\mathcal{F}}_{\text{e}}:=\mathcal{F}_{\text{e}}/A$, as
a function of the tilt angle $\theta$ is given by

$\widetilde{\mathcal{F}}_{\text{e}}[\theta]=\frac{1}{2}\int_{0}^{d}\bigl{(}K_{1}\cos^{2}\\!\theta+K_{3}\SS\bigr{)}\theta_{z}^{2}\,\text{d}{z}.$

The appropriate contribution to the free-energy density associated with a
static electric field arising from electrodes held at constant potential is

$W_{\text{E}}=-\frac{1}{2}\bm{D}\cdot\bm{E},$

where $\bm{D}$ is the displacement field and $\bm{E}$ the electric field. This
is discussed in a general context in [13, §4.7] and [15, §5, §10]; it is
discussed in the particular context of liquid crystals in [2, §3.6.2, §3.6.3],
[3, §10.1], and [23, §7.1]. Another perspective on this expression will be
developed in §4.3. In a system such as ours (a transversely isotropic medium
in the linear regime), it is usually assumed that

$\bm{D}={\bm{\epsilon}}(\bm{n})\bm{E},\quad{\bm{\epsilon}}=\epsilon_{0}\bigl{[}\epsilon_{\scriptscriptstyle\perp}\mathbf{I}+\epsilon_{\text{a}}(\bm{n}\otimes\bm{n})\bigr{]},\quad\epsilon_{\text{a}}:=\epsilon_{\scriptscriptstyle\parallel}-\epsilon_{\scriptscriptstyle\perp},$
(4)

with ${{\bm{\epsilon}}}$ the dielectric tensor and
$\epsilon_{\scriptscriptstyle\perp}$ and
$\epsilon_{\scriptscriptstyle\parallel}$ the relative dielectric
permittivities perpendicular to $\bm{n}$ and parallel to $\bm{n}$, giving

$W_{\text{E}}=-\frac{1}{2}\epsilon_{0}\bigl{[}\epsilon_{\scriptscriptstyle\perp}E^{2}+\epsilon_{\text{a}}(\bm{E}\cdot\bm{n})^{2}\bigr{]},\quad
E=|\bm{E}|.$

The macroscopic modeling of electric fields in liquid crystals is discussed in
[4, §3.3.1], [19, §2.3.1], and [22, §4.1]. We note that the electric field is
in general nonhomogeneous [8] and that the linear relation between static
$\bm{D}$ and $\bm{E}$ fields can be nonlocal in space, though such spatial
dispersion is generally viewed as negligible in macroscopic models of pure
dielectrics—see [13, §I.4] or [15, §103].

The relevant Maxwell equations for electrostatics (assuming no distribution of
free charge in $\Omega$) are

$\operatorname{curl}\bm{E}=\bm{0},\quad\operatorname{div}\bm{D}=0.$

Tangential components of $\bm{E}$ are continuous across a material interface,
while the normal component of $\bm{D}$ suffers a jump equal to the surface
charge density $\sigma_{\text{f}}$ on the interface:

$\llbracket E_{\text{t}}\rrbracket=0,\quad\llbracket
D_{\nu}\rrbracket=\sigma_{\text{f}}.$

Both $\bm{D}$ and $\bm{E}$ vanish inside electrodes. The basic electrostatics
that we require can be found in any of [13, Ch. I, Ch. 1], [15, Ch. I, Ch.
II], [16, Chs. 2–6], or [20, Ch. III]. Using
$\operatorname{curl}\bm{E}=\bm{0}$ and the interface conditions, we conclude
that in our system

$\bm{E}=-U_{z}\bm{e}_{z},\quad U_{z}=\text{d}U/\text{d}{z},\quad
D_{z}(0+)=D_{z}(d-)=-\sigma,$ (5)

where $U=U(z)$ is the electric potential and $\sigma$ is the (uniform) surface
charge density on the upper electrode ($-\sigma$ on the lower electrode).
Given the polarity indicated in our circuit diagram in figure 1b, we have
$\sigma>0$. Thus $W_{\text{E}}$ simplifies to

$W_{\text{E}}=-\frac{1}{2}\epsilon_{0}\bigl{(}\epsilon_{\scriptscriptstyle\perp}n_{x}^{2}+\epsilon_{\scriptscriptstyle\parallel}n_{z}^{2}\bigr{)}U_{z}^{2}.$

For equilibrium states of our model problem, then, the total free energy (per
unit area) expressed as a functional of $\theta$ and $U$ is given by

$\widetilde{\mathcal{F}}[\theta,U]=\frac{1}{2}\int_{0}^{d}\bigl{[}\bigl{(}K_{1}\cos^{2}\\!\theta+K_{3}\SS\bigr{)}\theta_{z}^{2}-\epsilon_{0}\bigl{(}\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS\bigr{)}U_{z}^{2}\bigr{]}\,\text{d}{z}.$
(6)

###  2.1 Coupled system

When viewed as a coupled system in this way, the equilibrium Euler-Lagrange
equations that follow from $\delta_{\theta}\widetilde{\mathcal{F}}=0$ and
$\delta_{U}\widetilde{\mathcal{F}}=0$ are given by

$\displaystyle\frac{\text{d}}{\text{d}{z}}\bigl{[}\bigl{(}K_{1}\cos^{2}\\!\theta+K_{3}\SS\bigr{)}\theta_{z}\bigr{]}=\sin\theta\cos\theta\bigl{[}(K_{3}-K_{1})\theta_{z}^{2}-\epsilon_{0}\epsilon_{\text{a}}U_{z}^{2}\bigr{]},$
(7a)
$\displaystyle\frac{\text{d}}{\text{d}{z}}\bigl{[}\bigl{(}\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS\bigr{)}U_{z}\bigr{]}=0,$
(7b) with boundary conditions $\theta(0)=\theta(d)=0,\quad U(0)=-\Delta
U/2,~{}U(d)=\Delta U/2.$ (7c)

Here $\Delta U$ is the difference between the electric potential on the upper
electrode and that on the lower electrode, and the director field and electric
field in the cell depend only on this difference—an arbitrary constant added
to $U$ has no effect on solutions of (7) (one could just as well impose
boundary conditions $U(0)=0$, $U(d)=\Delta U$). Equation (7b) corresponds to
$\operatorname{div}\bm{D}=0$ for our model cell and emerges in a natural way
as a condition of stationarity of $\widetilde{\mathcal{F}}[\theta,U]$.

The characterization of the stability properties of solutions of the coupled
system (7) is a little nonstandard because of the “minimax” nature of the free
energy $\widetilde{\mathcal{F}}$ in (6). Locally stable solutions are locally
minimizing with respect to $\theta$ but maximizing with respect to $U$; while
globally stable solutions are globally minimizing with respect to $\theta$,
maximizing with respect to $U$. Globally stable solutions can also be
characterized as equilibrium solutions of least free energy. These topics are
taken up in [8]. From the point of view of _dynamical_ stability, the picture
is somewhat confusing, and a goal of this note is to try to understand this
better.

###  2.2 Reduced system

Instead of viewing the system as a coupled system with two state variables,
one can eliminate $U$ and model the system in terms of $\theta$ only. An
integration of (7b) gives

$U_{z}=\Delta
U\biggl{[}\int_{0}^{d}\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS}\biggr{]}^{-1}\\!\\!\frac{1}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS}\,,$
(8)

which when substituted into (7a) produces

$\frac{\text{d}}{\text{d}{z}}\bigl{[}\bigl{(}K_{1}\cos^{2}\\!\theta+K_{3}\SS\bigr{)}\theta_{z}\bigr{]}=\sin\theta\cos\theta\biggl{\\{}(K_{3}-K_{1})\theta_{z}^{2}\\\
{}-\epsilon_{0}\epsilon_{\text{a}}(\Delta
U)^{2}\biggl{[}\int_{0}^{d}\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS}\biggr{]}^{-2}\\!\\!\frac{1}{(\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS)^{2}}\biggr{\\}}.$
(9)

This same equation can be obtained from the first variation of the reduced
free energy $\widetilde{\mathcal{F}}_{\text{r}}$ that results when the
expression for $U_{z}$ above is substituted into (6):

$\widetilde{\mathcal{F}}_{\text{r}}[\theta]=\frac{1}{2}\int_{0}^{d}\bigl{(}K_{1}\cos^{2}\\!\theta+K_{3}\SS\bigr{)}\theta_{z}^{2}\,\text{d}{z}-\frac{1}{2}\epsilon_{0}(\Delta
U)^{2}\biggl{[}\int_{0}^{d}\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS}\biggr{]}^{-1}\\!\\!.$
(10)

If the voltage of the battery is $V$, then in equilibrium $\Delta U=V$, and
the above expression agrees with [19, (3.221)]. When the system is _not_ in
equilibrium, however, $\Delta U$ need not be equal to $V$, and this will be an
issue in what follows.

The formulation in terms of the reduced free energy is natural and has been
widely used. It was used by Deuling in [5], which is recounted in [19, §3.5].
It was used by Hardt, Kinderlehrer, and Lin in their analytical paper [12],
and it was also employed by the author in [8]. The formulation has certain
advantages in terms of stability assessment, in that locally stable states are
local minimizers of $\widetilde{\mathcal{F}}_{\text{r}}$ with respect to
$\theta$, and globally stable states are global minimizers (the second
variation $\delta^{2}\\!\widetilde{\mathcal{F}}_{\text{r}}$ being positive
definite in both cases). Using $\widetilde{\mathcal{F}}_{\text{r}}$ is
equivalent to minimizing the free energy $\widetilde{\mathcal{F}}$ in (6)
subject to the ODE constraint (7b). The approach amounts to viewing the
electric field as slaved to the director field. A disadvantage of the reduced-
free-energy formulation is that the equilibrium Euler-Lagrange equation (9) is
_nonlocal_.

When the system is out of equilibrium and the director field is evolving in
time ($\bm{n}=\bm{n}(z,t)$, $\theta=\theta(z,t)$), then the capacitance of the
liquid-crystal cell will be changing with time as well. In this case, the
battery will need to move charge on or off the electrodes in order to re-
establish the equilibrium potential difference $\Delta U=V$. The associated
currents in the electric circuit will suffer some energy loss, due to Joule
heating, and we need a way to combine these effects with the dynamics and
viscous dissipation in the liquid-crystal cell. We begin by reviewing the
charge dynamics of a standard RC circuit.

##  3 RC circuits

The typical kind of experimental setup that we envision can be viewed as an RC
circuit with the liquid-crystal cell forming a capacitor containing a complex
time-varying dielectric, as depicted in figure 1b. The resistance $R$ could
come from the presence of an actual resistor, or it could just be thought of
as a surrogate to account for the total resistance of all the elements in the
circuit (wires, connectors, conduction layers in the cell, etc.). The equation
governing charge dynamics in such a circuit follows from the Kirchhoff Voltage
Law and the formulas for the potential drops across a resistor and a capacitor
[16, §6.6, §7.8]:

$\Delta U_{\text{res}}=IR,~{}\Delta
U_{\text{cap}}=Q/C~{}\Rightarrow~{}IR+Q/C=V.$

Here $Q(t)$ is the instantaneous total charge on the upper (positive)
electrode, and $I=\text{d}{Q}/\text{d}{t}$ is the current. Appending an
initial condition leads to the IVP

$R\frac{\text{d}{Q}}{\text{d}{t}}+\frac{1}{C}Q=V,~{}~{}Q(0)=Q_{0},$ (11)

the solution of which can be written

$Q(t)=Q_{\infty}+(Q_{0}-Q_{\infty})\exp(-t/\tau),\quad
Q_{\infty}:=CV,~{}~{}\tau:=RC.$ (12)

Thus the steady state of the circuit corresponds to

$Q=CV,~{}~{}\Delta U=V,$

and the characteristic time scale for the dynamics is $\tau=RC$—from here on,
we simply denote $\Delta U_{\text{cap}}=\Delta U$ (as we have used in the
previous section).

The state of the circuit could just as well be characterized in terms of
$\Delta U$ (instead of $Q$), in which case (11) would take the form

$R\frac{\text{d}}{\text{d}{t}}(C\Delta U)+\Delta U=V,~{}~{}\Delta U(0)=\Delta
U_{0}.$

As we shall see in what follows, $C$ depends on $\bm{n}$ (which will be
changing in time, leading to $C=C(t)$); so modeling the circuit in terms of
$Q$ proves to be more convenient (at least for our model problem).

The potential energies of the capacitor and battery, relative to a value of
zero at $Q=0$, are given by

$\mathcal{E}_{\text{cap}}=\frac{1}{2}Q\Delta
U,~{}~{}\mathcal{E}_{\text{bat}}=-QV.$

The relation $Q=C\Delta U$ makes it possible to write these expressions in
various equivalent forms, e.g., $\mathcal{E}_{\text{cap}}=Q^{2}/2C$ or
$\mathcal{E}_{\text{cap}}=C(\Delta U)^{2}/2$. These potential energies
correspond to the work done in a reversible process building up the charge on
the capacitor electrodes in an incremental way. As the charge on the capacitor
is being built up, the electric potential is changing there as well (according
to the relation $Q=C\Delta U$), leading to the factor of $1/2$: the increment
of work done in moving an increment of charge from a location of potential
zero to a location of potential $\Delta U$ is $\delta W=\delta Q\Delta U$,
giving

$W_{\text{cap}}=\int_{0}^{Q_{\text{f}}}\\!\\!\Delta
U\,\text{d}{Q}=\int_{0}^{Q_{\text{f}}}\\!\frac{Q}{C}\,\text{d}{Q}=\frac{1}{C}\int_{0}^{Q_{\text{f}}}\\!\\!Q\,\text{d}{Q}=\frac{1}{2C}Q_{\text{f}}^{2}=\frac{1}{2}Q_{\text{f}}\Delta
U_{\text{f}},$

where $Q_{\text{f}}$ and $\Delta U_{\text{f}}$ are the final (fully charged)
values of $Q$ and $\Delta U$. The battery, on the other hand, always maintains
a constant potential of $V$; so

$W_{\text{bat}}=\int_{Q_{\text{f}}}^{0}\\!V\,\text{d}{Q}=V\\!\int_{Q_{\text{f}}}^{0}\\!\text{d}{Q}=-Q_{\text{f}}V.$

A discussion of these textbook formulas can be found in [16, §6.6]. By
convention, the accounting is done in terms of just the positive electrode,
but it takes into account the contribution of the negative electrode—in
actuality, one has charge $Q$ on the positive electrode at potential $\Delta
U/2$ and charge $-Q$ on the negative electrode at potential $-\Delta U/2$. We
conclude that the total potential energy at any instant, expressed in terms of
$Q$, is given by

$\mathcal{E}:=\mathcal{E}_{\text{cap}}+\mathcal{E}_{\text{bat}}=\frac{1}{2C}Q^{2}-QV.$

The discussion above assumes a _constant_ capacitance $C$. We will revisit
this calculation later with $C=C(Q)$.

We note that whether the system is in equilibrium or not, we always have equal
and opposite total excess charge on the upper and lower electrode surfaces.
This relates to conservation of charge and is a consequence of Gauss’s Law. In
the full time-dependent Maxwell equations (as well as in Maxwell
electrostatics), we always have $\operatorname{div}\bm{D}=0$ in the absence of
any free-charge distribution in $\Omega$ (which we have assumed to be the case
throughout). From this follows

$0=\int_{\Omega}\operatorname{div}\bm{D}\,\text{d}{V}=\int_{\partial\Omega}\\!\bm{D}\cdot\bm{\nu}\,\text{d}{S}=-\int_{\partial\Omega}\\!\sigma_{\text{f}}\,\text{d}{S}.$

In our system, $\sigma_{\text{f}}$ is supported on the top and bottom
electrode surfaces; so the total charge on the top and bottom must sum to zero
($Q$ on the top and $-Q$ on the bottom, in our notation). We also note that we
use interchangeably the terms “potential energy,” “electrostatic energy,” and
“free energy”—all refer to the energy associated with reversible work
processes.

Current flowing through a resistor leads to energy dissipation (by Joule
heating) at a rate $RI^{2}$—see [15, (21.6)] or [20, §1.8, (4)]. The Rayleigh
dissipation function associated with this is

$\mathcal{R}=\frac{1}{2}RI^{2}=\frac{1}{2}R\Bigl{(}\frac{\text{d}{Q}}{\text{d}{t}}\Bigr{)}^{\\!2}.$
(13)

From a variational point of view, then, the circuit dynamics can be obtained
from a dissipation principle:

$\frac{\partial\mathcal{E}}{\partial
Q}+\frac{\partial\mathcal{R}}{\partial\dot{Q}}=0~{}\Rightarrow~{}\frac{1}{C}Q-V+R\frac{\text{d}{Q}}{\text{d}{t}}=0,$

in agreement with (11). Here we follow the formalism of Lagrangian mechanics
with frictional forces and potential energy only (no kinetic energy), denoted
in that setting $L=T-V=-V$ (with $V=\mathcal{E}$ here)—see, for example, [18,
§2.2.1], where a historical perspective and classical references can be found.

One can try to put some of this in context. The total energy dissipated in the
dynamical process is given by

$\mathcal{D}\\!=\int_{0}^{\infty}\\!\\!RI^{2}\,\text{d}{t}=\frac{1}{2C}(Q_{\infty}-Q_{0})^{2},$

using (12), compared to the potential-energy changes of the capacitor and
battery:

$\Delta\mathcal{E}_{\text{cap}}=\frac{1}{2C}\bigl{(}Q_{\infty}^{2}-Q_{0}^{2}\bigr{)},\quad\Delta\mathcal{E}_{\text{bat}}=-(Q_{\infty}-Q_{0})V=\frac{1}{C}(Q_{0}-Q_{\infty})Q_{\infty}.$

Thus we always have

$\Delta\mathcal{E}_{\text{cap}}+\Delta\mathcal{E}_{\text{bat}}+\mathcal{D}=0,$

as we must. In the special case of charging the capacitor from _zero_
($Q_{0}=0$), we have

$\mathcal{D}=\Delta\mathcal{E}_{\text{cap}}=\frac{1}{2}CV^{2},\quad\Delta\mathcal{E}_{\text{bat}}=-CV^{2}.$

In this case, half the work done by the battery goes into the final
electrostatic energy of the capacitor, while the other half is lost to
dissipation. Similar phenomena (in which half of the work is lost to
dissipation) occur in a number of other settings, including simple spring-
mass-damper systems and linear elasticity [7].

At the other extreme, if the initial charge on the capacitor were just
slightly out of equilibrium, $Q_{0}=(1+\varepsilon)Q_{\infty}$, say, then one
would obtain

$\Delta\mathcal{E}_{\text{cap}}=-\varepsilon(1+\varepsilon/2)CV^{2},\quad\Delta\mathcal{E}_{\text{bat}}=\varepsilon
CV^{2},\quad\mathcal{D}=\varepsilon^{2}CV^{2}/2.$

In this case, both $\Delta\mathcal{E}_{\text{cap}}$ and
$\Delta\mathcal{E}_{\text{bat}}$ are $O(\varepsilon)$, but
$\mathcal{D}=O(\varepsilon^{2})$—and the dissipation would essentially be
negligible when $|\varepsilon|\ll 1$. While we always have

$0\leq\mathcal{D}\leq|\Delta\mathcal{E}_{\text{cap}}|,$

we only have $\mathcal{D}=|\Delta\mathcal{E}_{\text{cap}}|$ in the case
$Q_{0}=0$ or in the trivial cases $Q_{0}=Q_{\infty}$ (no dynamics) or
$Q_{\infty}=0$ (no battery).

##  4 Total potential energy and dynamical system

There is a mutual influence between the director field in the cell and the
state of the electric circuit: the director field determines the capacitance
of the cell (which affects the circuit), while the state of the circuit
(whether characterized by $Q$ or by $\Delta U$) affects the electric field in
the cell (and hence the director field). In order to put together a coupled
set of equilibrium equations or dynamical equations, we require expressions
for the total potential energy and for the dissipation of the full system.
These, in turn, require certain “building blocks,” which we now derive.

###  4.1 Capacitance of the cell

Due to the simplicity of our model problem (fields depending on only one space
variable, stratified nature of the dielectric), it is possible to express the
capacitance of the cell in analytical form:

$C[\bm{n}]=\epsilon_{0}A\biggl{[}\int_{0}^{d}\\!\\!\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}n_{x}^{2}+\epsilon_{\scriptscriptstyle\parallel}n_{z}^{2}}\biggr{]}^{-1}\\!\\!.$
(14)

This can be derived in various ways. Using the basic relation for a parallel-
plate capacitor $Q=C\Delta U$ (with $Q$ the total charge on the positive
electrode, $C$ the capacitance, and $\Delta U$ the potential difference
between the electrodes, as before) and the relations in (4) and (5), we have

$D_{z}=\epsilon_{zz}E_{z}=-\sigma~{}\Rightarrow~{}U_{z}=-E_{z}=\frac{\sigma}{\,\epsilon_{zz}}~{}\Rightarrow~{}\Delta
U=\\!\int_{0}^{d}\\!U_{z}\,\text{d}{z}=\sigma\\!\int_{0}^{d}\\!\frac{\text{d}{z}}{\,\epsilon_{zz}}\,,$

which gives

$C=\frac{Q}{\Delta U}=\frac{\sigma
A}{\sigma\\!\int_{0}^{d}(1/\epsilon_{zz})\,\text{d}{z}}=A\biggl{[}\int_{0}^{d}\\!\frac{\text{d}{z}}{\,\epsilon_{zz}}\biggr{]}^{-1}\\!\\!\\!=\epsilon_{0}A\biggl{[}\int_{0}^{d}\\!\\!\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}n_{x}^{2}+\epsilon_{\scriptscriptstyle\parallel}n_{z}^{2}}\biggr{]}^{-1}\\!\\!.$

In this situation, $C$ is simply an integral functional of the director field.
One also sees such expressions derived by approximating the stratified
dielectric as a collection of thin capacitive elements in series [6,
§5.4.3.1]. Note that if the material were isotropic with relative permittivity
$\epsilon_{\text{r}}$ (i.e.,
$\epsilon_{\scriptscriptstyle\perp}=\epsilon_{\scriptscriptstyle\parallel}=\epsilon_{\text{r}}$),
then (14) would simplify to $C=\epsilon A/d$ with
$\epsilon=\epsilon_{0}\epsilon_{\text{r}}$, which is the textbook expression
for the capacitance of a parallel-plate capacitor [16, §6.6].

In more general circumstances (such as a director field that is a function of
more than one space variable), it is not possible to derive a formula such as
the above: there is no explicit analytical expression for the capacitance of a
cell with a two- or three-dimensional inhomogeneity of the dielectric. In such
cases, one can characterize the capacitance in terms of the director field
$\bm{n}$ in $\Omega$ given either $Q$ or $\Delta U$, but to determine $C$, one
must solve an auxiliary problem from an appropriate formulation of Maxwell
electrostatics, then determine $\Delta U$ from this solution (if $Q$ was
given) or $Q$ (if $\Delta U$ was given), and finally compute the ratio
$C=Q/\Delta U$.

###  4.2 Potential energy of the cell

The textbook calculation of the electrostatic energy of a capacitor was
recounted in §3. When combined with the relation $Q=C\Delta U$, it yielded
several equivalent ways of writing this potential energy:

$\mathcal{E}_{\text{cap}}=\frac{1}{2}Q\Delta
U=\frac{1}{2C}Q^{2}=\frac{1}{2}C(\Delta U)^{2}.$ (15)

The calculation of $\mathcal{E}_{\text{cap}}$ in §3 depended on the assumption
that the capacitance of the cell was _constant_. In our system, however, the
capacitance changes with changes in the director field in the cell, and we
must modify this calculation accordingly.

As before, we start from the fact that the increment of work $\delta W$ done
in moving an increment of charge $\delta Q$ from one location to another
location of potential difference $\Delta U$ is $\delta W=\delta Q\Delta U$,
giving (as before)

$W_{\text{cap}}=\int_{0}^{Q_{\text{f}}}\\!\\!\Delta
U\,\text{d}{Q}=\int_{0}^{Q_{\text{f}}}\\!\frac{Q}{C}\,\text{d}{Q}.$

Now, however, $C$ depends on $Q$ (in a way that could be complicated and not
easy to express)—as charge is added to the upper electrode (depleted from the
lower electrode), the electric field in the cell will eventually become strong
enough to distort the director field (and change the capacitance). An
integration by parts provides a simple way to assess the new situation:

$W_{\text{cap}}=\int_{0}^{Q_{\text{f}}}\\!\\!\frac{Q}{C(Q)}\,\text{d}{Q}=\int_{0}^{Q_{\text{f}}}\\!\\!\frac{1}{C(Q)}\,\text{d}\Bigl{(}\frac{Q^{2}}{2}\Bigr{)}\\\
=\frac{1}{2}\frac{Q_{\text{f}}^{2}}{C(Q_{\text{f}})}+\frac{1}{2}\int_{0}^{Q_{\text{f}}}\\!\\!\frac{Q^{2}}{C(Q)^{2}}\,\frac{\text{d}C}{\text{d}{Q}}\,\text{d}{Q}=\frac{1}{2}\frac{Q_{\text{f}}^{2}}{C(Q_{\text{f}})}+\frac{1}{2}\int_{C(0)}^{C(Q_{\text{f}})}\\!\\!(\Delta
U)^{2}\,\text{d}C.$

The first term on the right-hand side above is one form of the electrostatic
energy $\mathcal{E}_{\text{cap}}$ of a capacitor with a constant capacitance
$C=C(Q_{\text{f}})$, as in (15). The second term above can be interpreted as
follows.

The increment of work $\delta W$ done in changing the capacitance by an
incremental amount $\delta C$ is $\delta W=\frac{1}{2}\delta C(\Delta U)^{2}$,
which can be seen from the alternate form of the potential energy of the
capacitor $\mathcal{E}_{\text{cap}}=\frac{1}{2}C(\Delta U)^{2}$. Thus the
integral term on the right-hand side above is the reversible work done in
changing the capacitance of the cell from its initial value at $Q=0$ to its
final value at $Q=Q_{\text{f}}$. In our system, the only way the capacitance
can be changed is by distorting the director field in $\Omega$, and the work
function for that process is the distortional elastic energy
$\mathcal{F}_{\text{e}}$. Thus the integral in question gives the change in
distortional elastic energy from its value at $Q=0$ to its value at
$Q=Q_{\text{f}}$:

$\frac{1}{2}\int_{C(0)}^{C(Q_{\text{f}})}\\!\\!(\Delta
U)^{2}\,\text{d}C=\mathcal{F}_{\text{e}}\bigr{|}_{Q=Q_{\text{f}}}-\mathcal{F}_{\text{e}}\bigr{|}_{Q=0}=\mathcal{F}_{\text{e}}\bigr{|}_{Q=Q_{\text{f}}},$

since $\mathcal{F}_{\text{e}}|_{Q=0}=0$ in our system. Under more general
circumstances, one could have $\mathcal{F}_{\text{e}}|_{Q=0}\not=0$, but that
term would just add a constant to the potential energy and could be ignored.

We see that the potential energy of the liquid-crystal cell in our system is
given by

$W_{\text{cap}}=\mathcal{E}_{\text{cap}}+\mathcal{F}_{\text{e}}[\bm{n}],\quad\mathcal{E}_{\text{cap}}=\frac{1}{2}Q\Delta
U=\frac{1}{2}C[\bm{n}]^{-1}Q^{2}=\frac{1}{2}C[\bm{n}](\Delta U)^{2},$

where $\bm{n}$, $Q$, and $\Delta U$ are the instantaneous values of these
state variables. This expression captures, in a clean and decoupled way, both
the work done in moving charge and that done by inducing change in the
capacitance (by distorting the director field). We note that we continue to
use the notation $\mathcal{E}_{\text{cap}}$ for any of the equivalent formulas
for the electrostatic energy of a capacitor of constant capacitance (15), with
the understanding that the value of $C$ is always taken to be that associated
with the current state of the director field: $C=C[\bm{n}]$.

###  4.3 Potential energy of the system and coupled equilibrium equations

The total potential energy of the system $\mathcal{G}$ is thus given by

$\mathcal{G}=\mathcal{F}_{\text{e}}+\mathcal{E}_{\text{cap}}+\mathcal{E}_{\text{bat}}=\mathcal{F}_{\text{e}}+\frac{1}{2}Q\Delta
U-QV.$ (16)

The first two terms represent the potential energy of the liquid-crystal cell,
as derived in §4.2, while the third term is the potential energy associated
with the battery. The expression for $\mathcal{G}$ can be related to more
familiar forms by noting that for static electric fields,
$\mathcal{E}_{\text{cap}}$ can be expressed in terms of field intensities
inside the cell:

$\frac{1}{2}Q\Delta
U=\frac{1}{2}\int_{\Omega}(\bm{D}\cdot\bm{E})\,\text{d}{V},$ (17)

where $\bm{D}$ and $\bm{E}$ are the fields associated with the electrostatic
problem $\operatorname{div}\bm{D}=0$ in $\Omega$ with a potential difference
$\Delta U$ between the electrodes. This can be established as follows. Using
the relations $\bm{D}\cdot\bm{\nu}=-\sigma_{\text{f}}$ (with $\bm{\nu}$ the
outward normal from $\Omega$), $\operatorname{div}\bm{D}=0$, and
$\bm{E}=-\nabla U$, we obtain the following relation for a general surface
charge density $\sigma_{\text{f}}$ and electric potential $U$:

$\int_{\partial\Omega}\\!\sigma_{\text{f}}\,U\,\text{d}{S}=-\int_{\partial\Omega}(U\bm{D})\cdot\bm{\nu}\,\text{d}{S}=-\int_{\Omega}\operatorname{div}(U\bm{D})\,\text{d}{V}\\\
=-\int_{\Omega}(U\operatorname{div}\bm{D}+\nabla
U\cdot\bm{D})\,\text{d}{V}=\int_{\Omega}(\bm{D}\cdot\bm{E})\,\text{d}{V}.$

See, for example, [16, §6.3] or [20, §2.8]. In our system, the surface charge
density is supported on the upper and lower boundary electrodes, and the
electric potential $U$ is constant on each electrode ($-\Delta U/2$ on the
lower, $\Delta U/2$ on the upper), giving

$\int_{\partial\Omega}\\!\sigma_{\text{f}}\,U\,\text{d}{S}=\int_{\Gamma_{1}}\\!\sigma_{\text{f}}\,U\,\text{d}{S}+\int_{\Gamma_{2}}\\!\sigma_{\text{f}}\,U\,\text{d}{S}=-\frac{\Delta
U}{2}\int_{\Gamma_{1}}\\!\sigma_{\text{f}}\,\text{d}{S}+\frac{\Delta
U}{2}\int_{\Gamma_{2}}\\!\sigma_{\text{f}}\,\text{d}{S}\\\ =-\frac{\Delta
U}{2}(-Q)+\frac{\Delta U}{2}Q=Q\Delta U.$

Here $\Gamma_{1}$ and $\Gamma_{2}$ are the lower and upper boundary electrode
interfaces. Combining these two calculations establishes the validity of (17).
Note that this argument does not require $\sigma_{\text{f}}$ to be constant on
$\Gamma_{1}$ or $\Gamma_{2}$.

Concerning these equivalent formulas for electrostatic energy

$\frac{1}{2}\int_{\partial\Omega}\\!\sigma_{\text{f}}\,U\,\text{d}{S}=\frac{1}{2}\int_{\Omega}(\bm{D}\cdot\bm{E})\,\text{d}{V},$

the left-hand side is in fact the more primitive expression (how electrostatic
energy associated with surface charge distributions is often first derived in
electromagnetics textbooks)—see, for example, [16, §6.2] or [20, §2.7]. We
note that when there is no current flowing in the circuit, then $\Delta U=V$,
and the combination $\mathcal{E}_{\text{cap}}+\mathcal{E}_{\text{bat}}$
satisfies

$\mathcal{E}_{\text{cap}}+\mathcal{E}_{\text{bat}}=\frac{1}{2}QV-
QV=-\frac{1}{2}QV=-\frac{1}{2}\int_{\Omega}(\bm{D}\cdot\bm{E})\,\text{d}{V},$
(18)

which is the appropriate contribution to the free energy to be used in
modeling liquid crystal equilibrium states in a setting such as ours. We
emphasize that in order for (18) to be valid, one requires equilibrium
conditions in the electrical circuit ($I=0$, $\Delta U=V$) and also
equilibrium of the electric field in the cell
($\operatorname{curl}\bm{E}=\bm{0}$, $\bm{E}=-\nabla U$)—with time-varying
electric fields, the time-dependent Maxwell equations have
$\operatorname{curl}\bm{E}\not=\bm{0}$, in general, and $\bm{E}$ cannot be
expressed as the gradient of a scalar potential.

The expression for $\mathcal{G}$ in (16) can be written in different forms,
depending upon the choice of state variable for the circuit:

$\mathcal{G}[\bm{n},\Delta
U]=\mathcal{F}_{\text{e}}[\bm{n}]+C[\bm{n}]\Bigl{[}\frac{1}{2}(\Delta
U)^{2}-V\Delta U\Bigr{]}$ (19a) or
$\mathcal{G}[\bm{n},Q]=\mathcal{F}_{\text{e}}[\bm{n}]+\frac{1}{2}C[\bm{n}]^{-1}Q^{2}-VQ.$
(19b)

Here $\bm{n}$ is the current state of the director field in $\Omega$,
$\mathcal{F}_{\text{e}}[\bm{n}]$ is the distortional elastic energy of that
state (as in (2)), and $C[\bm{n}]$ is the capacitance of the cell in that
state (given by (14) for our model problem). The above expressions apply more
generally than just to our model problem; they are valid, for example, even if
the fields in the cell depend on more than one space variable (though in that
case, one would not have an explicit formula for $C[\bm{n}]$, in general).

Stable equilibrium states of the system correspond to minimizers of the total
potential energy $\mathcal{G}$ with respect to $(\bm{n},\Delta U)$ or
$(\bm{n},Q)$, depending on the choice of state variables. The capacitance is
positive; so

$\min\mathcal{G}~{}\Rightarrow~{}\Delta U=V,~{}Q=CV,$

and the following minimization problem for $\bm{n}$ results:

$\min_{\bm{n}}\Bigl{\\{}\mathcal{F}_{\text{e}}[\bm{n}]-\frac{1}{2}C[\bm{n}]V^{2}\Bigr{\\}}.$

Using (17) and the equilibrium condition $Q=C[\bm{n}]V$, the objective
functional above can be written

$\mathcal{F}_{\text{e}}[\bm{n}]-\frac{1}{2}C[\bm{n}]V^{2}=\mathcal{F}_{\text{e}}[\bm{n}]-\frac{1}{2}\int_{\Omega}(\bm{D}\cdot\bm{E})\,\text{d}{V}.$

Here the terms $\bm{D}$ and $\bm{E}$ in the last integral correspond to the
solution of the electrostatics problem in $\Omega$ with the dielectric tensor
associated with the current director field,
${{\bm{\epsilon}}}={{\bm{\epsilon}}}(\bm{n})$, and with a potential difference
of $V$ across the cell. The minimization with respect to $\bm{n}$ is subject
to the pointwise constraint $|\bm{n}|=1$ and appropriate boundary conditions.
The functional
$\mathcal{F}_{\text{e}}-\frac{1}{2}\int_{\Omega}(\bm{D}\cdot\bm{E})\,\text{d}{V}$
is the correct form of the free energy for equilibrium states of $\bm{n}$ in
an electric field, as in (1) (with the identification
$\bm{D}={\bm{\epsilon}}(\bm{n})\bm{E}$, $\bm{E}=-\nabla U$).

All of these expressions can be put in explicit forms for our model problem,
for which it is convenient to use the formulation in terms of $Q$. The state
variables are $\bm{n}=\bm{n}(z,t)$ (the director field in the cell) and
$\sigma=\sigma(t)$ (the charge density on the upper electrode). Scaling our
energies and capacitance by the cross-sectional area of the cell,

$\widetilde{\mathcal{G}}:=\mathcal{G}/A,\quad\widetilde{\mathcal{F}}_{\text{e}}:=\mathcal{F}_{\text{e}}/A,\quad\widetilde{C}:=C/A,$

and using $Q=\sigma A$, we obtain

$\displaystyle\widetilde{\mathcal{G}}[\bm{n},\sigma]$
$\displaystyle=\widetilde{\mathcal{F}}_{\text{e}}[\bm{n}]+\frac{1}{2}\widetilde{C}[\bm{n}]^{-1}\sigma^{2}-\sigma
V$
$\displaystyle=\frac{1}{2}\int_{0}^{d}\\!\bigl{(}K_{1}n_{z,z}^{2}+K_{3}n_{x,z}^{2}\bigr{)}\,\text{d}{z}+\frac{1}{2}\frac{\,\,\sigma^{2}}{\,\epsilon_{0}}\\!\int_{0}^{d}\\!\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}n_{x}^{2}+\epsilon_{\scriptscriptstyle\parallel}n_{z}^{2}}-\sigma
V.$

Here we have used (2), (3), and (14) to provide the expressions for
$\mathcal{F}_{\text{e}}[\bm{n}]$ and $C[\bm{n}]$. We shall work with this
mostly in terms of the tilt-angle representation
$\bm{n}=\cos\theta\,\bm{e}_{x}+\sin\theta\,\bm{e}_{z}$:

$\widetilde{\mathcal{G}}[\theta,\sigma]=\frac{1}{2}\int_{0}^{d}\bigl{(}K_{1}\cos^{2}\\!\theta+K_{3}\SS\bigr{)}\theta_{z}^{2}\,\text{d}{z}+\frac{1}{2}\frac{\,\,\sigma^{2}}{\,\epsilon_{0}}\\!\int_{0}^{d}\\!\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS}-\sigma
V.$ (20)

The associated coupled equilibrium equations from
$\delta_{\theta}\widetilde{\mathcal{G}}=0$ and
$\partial\widetilde{\mathcal{G}}/\partial\sigma=0$ are

$\displaystyle\frac{\text{d}}{\text{d}{z}}\bigl{[}\bigl{(}K_{1}\cos^{2}\\!\theta+K_{3}\SS\bigr{)}\theta_{z}\bigr{]}=\sin\theta\cos\theta\biggl{[}(K_{3}-K_{1})\theta_{z}^{2}-\epsilon_{\text{a}}\frac{\,\,\sigma^{2}}{\,\epsilon_{0}}\frac{1}{(\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS)^{2}}\biggr{]},$
$\displaystyle\frac{\,\sigma}{\,\epsilon_{0}}\int_{0}^{d}\\!\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS}=V.$

The latter equation is equivalent to $Q=CV$, which is the correct equilibrium
condition for the charge. Substituting the equilibrium value for $\sigma$
above into (20) and into the equilibrium ODE for $\theta$ above correctly
gives the reduced free energy (10) and equilibrium equation (9), in agreement
with [19, (3.221), (3.226)].

###  4.4 Dissipation and dynamics

We wish to use a dissipation principle to obtain a dynamical system involving
the coupled state variables $\bm{n}$ and $\sigma$. The simplest expression for
dissipation associated with the dynamics of the director field is usually
given in terms of the single rotational viscosity parameter $\gamma_{1}$ via
the Rayleigh function

$\frac{1}{2}\gamma_{1}\\!\int_{\Omega}\,\Bigl{|}\frac{\partial\bm{n}}{\partial
t}\Bigr{|}^{2}\text{d}{V}=\frac{1}{2}\gamma_{1}A\\!\int_{0}^{d}\Bigl{(}\frac{\partial\theta}{\partial
t}\Bigr{)}^{\\!2}\text{d}{z}=:\mathcal{R}_{\theta}.$

Per unit area, then, we have

$\widetilde{\mathcal{R}}_{\theta}:=\frac{1}{A}\mathcal{R}_{\theta}=\frac{1}{2}\gamma_{1}\\!\int_{0}^{d}\Bigl{(}\frac{\partial\theta}{\partial
t}\Bigr{)}^{\\!2}\text{d}{z}.$

We have already seen in (13) that the Rayleigh dissipation function for the
electric circuit is

$\frac{1}{2}R\Bigl{(}\frac{\text{d}{Q}}{\text{d}{t}}\Bigr{)}^{\\!2}=\frac{1}{2}RA^{2}\Bigl{(}\frac{\text{d}\sigma}{\text{d}{t}}\Bigr{)}^{\\!2}=:\mathcal{R}_{\sigma},$

which leads to

$\widetilde{\mathcal{R}}_{\sigma}:=\frac{1}{A}\mathcal{R}_{\sigma}=\frac{1}{2}RA\Bigl{(}\frac{\text{d}\sigma}{\text{d}{t}}\Bigr{)}^{\\!2}.$

Combining these gives

$\widetilde{\mathcal{R}}:=\widetilde{\mathcal{R}}_{\theta}+\widetilde{\mathcal{R}}_{\sigma}=\frac{1}{2}\gamma_{1}\\!\int_{0}^{d}\Bigl{(}\frac{\partial\theta}{\partial
t}\Bigr{)}^{\\!2}\text{d}{z}+\frac{1}{2}RA\Bigl{(}\frac{\text{d}\sigma}{\text{d}{t}}\Bigr{)}^{\\!2},$
(21)

which is the form we shall adopt for our combined dissipation function.

The total potential energy (20) and Rayleigh function (21) produce the correct
circuit dynamics:

$\frac{\partial\widetilde{\mathcal{G}}}{\partial\sigma}+\frac{\partial\widetilde{\mathcal{R}}}{\partial\dot{\sigma}}=0~{}\Rightarrow~{}\frac{\,\sigma}{\,\epsilon_{0}}\int_{0}^{d}\\!\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS}-V+RA\frac{\text{d}\sigma}{\text{d}{t}}=0~{}\Rightarrow~{}R\frac{\text{d}{Q}}{\text{d}{t}}+\frac{1}{C}Q=V,$

using (14) and $Q=\sigma A$—this agrees with (11). The dynamics for the
director angle results from the analogous expression for $\theta$:

$\delta_{\theta}\widetilde{\mathcal{G}}+\delta_{\dot{\theta}}\widetilde{\mathcal{R}}=0~{}\Rightarrow~{}\frac{\partial
W}{\partial\theta}-\frac{\partial}{\partial z}\Bigl{(}\frac{\partial
W}{\partial\theta_{z}}\Bigr{)}+\gamma_{1}\frac{\partial\theta}{\partial t}=0,$

with $W$ the free-energy density that results from combining the integrated
terms in (20). In expanded form, the PDE for $\theta$ is given by

$\gamma_{1}\frac{\partial\theta}{\partial t}=\frac{\partial}{\partial
z}\Bigl{[}\bigl{(}K_{1}\cos^{2}\\!\theta+K_{3}\SS\bigr{)}\frac{\partial\theta}{\partial
z}\Bigr{]}\\\
{}-\sin\theta\cos\theta\biggl{[}(K_{3}-K_{1})\Bigl{(}\frac{\partial\theta}{\partial
z}\Bigr{)}^{\\!2}-\epsilon_{\text{a}}\frac{\,\,\sigma^{2}}{\,\epsilon_{0}}\frac{1}{(\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS)^{2}}\biggr{]}.$
(22a) Our dynamical system, then, consists of the PDE (22a) above for
$\theta$, supplemented by auxiliary conditions
$\theta(0,t)=\theta(d,t)=0,\quad\theta(z,0)=\theta_{0}(z),$ (22b) and the ODE
IVP for $\sigma$
$RA\frac{\text{d}\sigma}{\text{d}{t}}+\frac{\,\sigma}{\,\epsilon_{0}}\int_{0}^{d}\\!\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS}=V,\quad\sigma(0)=\sigma_{0},$
(22c)

with the initial states $\theta_{0}$ and $\sigma_{0}$ prescribed. The
variational approach used here to obtain the director dynamics equation (22a)
is similar to that given in [19, §4.3]—compare (22a) with [19, (4.152)]. See
also [19, §5.9], where the same ideas are used to model the dynamics of
certain Fréedericksz transitions.

Some of the terms above can be related to more familiar expressions. The term
involving $\sigma^{2}$ in (22a) corresponds to the “dielectric torque”
$\bm{D}\times\bm{E}$, the couple per unit volume exerted by the electric field
on the director field. In our system, as in (4) and (5), we have

$\bm{E}=E\bm{e}_{z},\quad\bm{D}={\bm{\epsilon}}(\bm{n})\bm{E}=\epsilon_{0}E\bigl{[}\epsilon_{\text{a}}n_{x}n_{z}\bm{e}_{x}+\bigl{(}\epsilon_{\scriptscriptstyle\perp}n_{x}^{2}+\epsilon_{\scriptscriptstyle\parallel}n_{z}^{2}\bigr{)}\bm{e}_{z}\bigr{]}.$

In terms of $\theta$, then,

$\bm{D}\times\bm{E}=-\epsilon_{0}\epsilon_{\text{a}}E^{2}\sin\theta\cos\theta\,\bm{e}_{y}.$

The connection between $E^{2}$ and $\sigma^{2}$ follows from

$\sigma=-D_{z}~{}\Rightarrow~{}\sigma^{2}=\epsilon_{0}^{2}E^{2}(\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS)^{2},$

which gives

$\bm{D}\times\bm{E}=-\epsilon_{\text{a}}\frac{\,\,\sigma^{2}}{\,\epsilon_{0}}\frac{\sin\theta\cos\theta}{(\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS)^{2}}\bm{e}_{y},$

as in (22a). Also, using (14) we have

$C[\theta]^{-1}=\frac{1}{\epsilon_{0}A}\int_{0}^{d}\\!\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS},$

and (22c) can be written

$RA\frac{\text{d}\sigma}{\text{d}{t}}+C[\theta]^{-1}\\!A\sigma=V~{}~{}\text{or}~{}~{}R\frac{\text{d}Q}{\text{d}{t}}+C[\theta]^{-1}Q=V,$

as in (11).

As we have already noted (and as we shall see in §4.6), the time scale for
(22c) is often faster than that for (22a). If one were to choose to model the
charge distribution as adjusting instantaneously to changes in the director
field, then one would have $\text{d}\sigma/\text{d}{t}=0$ in (22c), and that
equation would collapse to the equilibrium condition for $\sigma$ in the
previous section. Substituting the equilibrium value for $\sigma$ into (22a)
would give the director-dynamics equation for the reduced free energy
$\widetilde{\mathcal{F}}_{\text{r}}$ in (10) (with $\Delta U=V$), that is, the
dynamical version of (9) (again with $\Delta U=V$).

###  4.5 Time-varying $\bm{D}$ versus $\bm{E}$

With an electric field that varies in space and in time
($\bm{E}=\bm{E}(\bm{x},t)$), the linear relationship between $\bm{D}$ and
$\bm{E}$ is in general nonlocal in space and in time; however, for macroscopic
models of pure dielectrics, nonlocality in space can generally be ignored [13,
§I.4], [15, §77, §78, §103]. Nonlocality in time can be an issue in some
circumstances, such as fast-switching experiments with electric fields from
electric pulses of large voltage and short duration [11]. If the dielectric
properties of the medium remain constant, spatial dispersion is ignored, and
the electric field is time harmonic, then the induced polarization will be
time harmonic with the same frequency and hence so will $\bm{D}$ (via
$\bm{D}=\epsilon_{0}\bm{E}+\bm{P}$):

$\bm{E}(\bm{x},t)=e^{-i\omega_{0}t}\bm{E}_{0}(\bm{x})~{}~{}\Rightarrow~{}~{}\bm{D}(\bm{x},t)=e^{-i\omega_{0}t}{\bm{\epsilon}}(\bm{x},\omega_{0})\bm{E}_{0}(\bm{x}).$

See [13, §I.4], [15, §77]. This includes the special case of a static electric
field ($\omega_{0}=0$):
$\bm{D}(\bm{x})={{\bm{\epsilon}}}(\bm{x})\bm{E}(\bm{x})$.

The relationship between $\bm{D}$ and $\bm{E}$ in the time-harmonic case is
_frequency dependent_ , however
(${{\bm{\epsilon}}}={{\bm{\epsilon}}}(\bm{x},\omega)$), and so for a general
time-varying electric field (again in a medium of constant dielectric
properties and no spatial dispersion), one has

$\bm{D}(\bm{x},t)=\epsilon_{0}\bm{E}(\bm{x},t)+\epsilon_{0}\\!\int_{-\infty}^{t}\\!{\bm{\alpha}}(\bm{x},t-t^{\prime})\bm{E}(\bm{x},t^{\prime})\,\text{d}t^{\prime},$
(23)

as written in [11]. The tensor field ${{\bm{\alpha}}}$ comes from the inverse
Fourier transform of the dispersion relation, that is,

${\bm{\epsilon}}(\bm{x},\omega)=\epsilon_{0}\Bigl{[}\mathbf{I}+\\!\int_{0}^{\infty}\\!e^{i\omega
t}{\bm{\alpha}}(\bm{x},t)\,\text{d}{t}\Bigr{]}.$

An illustrative example (for a homogeneous, isotropic medium) is given in [13,
§7.10], where a “one-resonance dispersion model” is employed (a complex
rational function of $\omega$). The poles of the dispersion model are in the
lower half of the complex $\omega$ plane, which gives rise to a one-sided
inverse Fourier transform and an appropriately causal relationship of the form
(23). From a physical point of view, at high frequencies, changes in the
induced polarization can lag changes in the electric field. An electric pulse
resolves into Fourier modes of arbitrarily high frequency, and because of
this, these nonlocal-in-time effects (“dielectric relaxation”) can become
important. In [11] and [17], the term “Dielectric Memory Effects” is used to
describe them.

The dynamics of liquid-crystal systems induced by time-varying electric fields
is even more complicated, since in that case, the dielectric properties of the
medium are changing in time as well, and the convolution representation (23)
is no longer valid. A research program described in the review article [17]
(which contains references to earlier works) addressed this issue and
developed a generalization of (23) to the case of a time-varying director
field $\bm{n}=\bm{n}(\bm{x},t)$. The theory was used to model successfully the
experiments with fast-switching (pulse-driven) dynamics of [21] and others.

These issues are beyond the scope of our investigation here. Our assessment is
that the local relationship $\bm{D}={{\bm{\epsilon}}}\bm{E}$ is valid in
static equilibrium and more generally if $\bm{E}$ is time harmonic and the
dielectric properties of the medium are constant. The relationship
$\bm{D}={{\bm{\epsilon}}}\bm{E}$ is a good approximation if $\bm{E}$ contains
only low-frequency content and the director dynamics are relatively slow. For
our purposes here, we accept and adopt the local relationship
$\bm{D}={{\bm{\epsilon}}}\bm{E}$ and acknowledge the limitations and
shortcomings of that in some settings.

###  4.6 Time scales

The dynamic processes associated with a system such as the one under
consideration here exhibit several different time scales, including those for
director dynamics, circuit dynamics, and the evolution of the electric field
in the cell. The time scales for the changes in the circuit and for the
dynamics of the director field can be gleaned from rescalings of (22c) and
(22a). First, it is convenient to relate the integral expression in (22c) to
the capacitance (14) as follows:

$C[\theta]=\epsilon_{0}A\biggl{[}\int_{0}^{d}\\!\\!\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS}\biggr{]}^{-1}=\frac{\epsilon_{0}A}{d}\epsilon_{\text{r}}[\theta],$

where $\epsilon_{\text{r}}[\theta]$ is the _effective relative dielectric
constant_ of the cell, given by

$\epsilon_{\text{r}}[\theta]^{-1}=\frac{1}{d}\int_{0}^{d}\\!\\!\frac{\text{d}{z}}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS}.$

This quantity is dimensionless, $O(1)$, and satisfies

$\epsilon_{\scriptscriptstyle\perp}\leq\epsilon_{\text{r}}[\theta]\leq\epsilon_{\scriptscriptstyle\parallel},$

since
$\epsilon_{\scriptscriptstyle\perp}<\epsilon_{\scriptscriptstyle\parallel}$
for our system. In terms of it, (22c) can be written

$\tau_{\sigma}\frac{\text{d}\bar{\sigma}}{\text{d}{t}}+\epsilon_{\text{r}}[\theta]^{-1}\bar{\sigma}=1,\quad\tau_{\sigma}:=R\frac{\epsilon_{0}A}{d},\quad\bar{\sigma}:=\frac{\sigma}{\epsilon_{0}V/d}.$
(24a) Note that $\epsilon_{0}A/d$ would be the capacitance of the cell and
$\epsilon_{0}V/d$ would be the surface charge density on the upper electrode
if there were a vacuum between the electrodes. Note also that in steady state,
we have $\text{d}\bar{\sigma}/\text{d}{t}=0$ and
$\bar{\sigma}=\epsilon_{\text{r}}[\theta]$.

For a fairly typical experimental setup, the “RC load” $\tau_{\sigma}$ is
generally in the sub-microsecond range. For example, with $R=100\,\Omega$,
$\epsilon_{0}=8.854\times 10^{-12}\,\text{F/m}$, $A=10\,\text{cm}^{2}$, and
$d=5\,\mu\text{m}$, we obtain $\tau_{\sigma}\doteq 1.77\times
10^{-7}\,\text{s}$. In [14], $\tau_{\sigma}$ is estimated to be
$2\,\mu\text{s}$ and is a limiting factor in the experiment presented there.
In the fast-switching experiments discussed in [11] and [21], measures are
taken to minimize $\tau_{\sigma}$ (including the use of gold connectors to
minimize resistance and cells of small area to minimize capacitance), and
values for $\tau_{\sigma}$ in the nanosecond range are reported.

If one scales $z$ by the cell gap ($\bar{z}=z/d$), then (22a) can be put in
the form

$\frac{\partial\theta}{\partial
t}=\frac{1}{\,\tau\\!\raisebox{-2.58334pt}{\tiny{$K$}}}\biggl{\\{}\frac{\partial}{\partial\bar{z}}\Bigl{[}(\bar{K}_{1}\cos^{2}\\!\theta+\bar{K}_{3}\SS)\frac{\partial\theta}{\partial\bar{z}}\Bigr{]}-\sin\theta\cos\theta(\bar{K}_{3}-\bar{K}_{1})\Bigl{(}\frac{\partial\theta}{\partial\bar{z}}\Bigr{)}^{\\!2}\biggr{\\}}\\\
{}+\frac{1}{\,\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}}\bar{\sigma}^{2}\frac{\sin\theta\cos\theta}{(\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS)^{2}},$
(24b)

where

$\tau\\!\raisebox{-2.58334pt}{\tiny{$K$}}:=\frac{\gamma_{1}d^{2}}{K},\quad\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}:=\frac{\gamma_{1}d^{2}}{\epsilon_{0}\epsilon_{\text{a}}V^{2}},\quad\bar{K}_{1}:=\frac{\,K_{1}}{K},\quad\bar{K}_{3}:=\frac{\,K_{3}}{K},$
(24c)

with $K$ a representative value for $K_{1}$ and $K_{3}$. This exposes two
times scales associated with director reorientation:
$\tau\\!\raisebox{-2.58334pt}{\tiny{$K$}}$ and
$\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}$. These correspond to “switch off”
and “switch on” times, which are usually written

$\tau_{\text{off}}=\frac{\gamma_{1}d^{2}}{K\pi^{2}},\quad\tau_{\text{on}}=\frac{\gamma_{1}}{\epsilon_{0}\epsilon_{\text{a}}E^{2}}.$

See [19, §5.9].

The switch-off time $\tau_{\text{off}}$ is the time scale for the slowest
decaying mode of $\gamma_{1}\theta_{t}=K\theta_{zz}$, $\theta(0)=\theta(d)=0$.
It gives the time it takes for a distorted director field to relax back to its
ground state under the influence of only distortional elastic forces. This is
a relatively slow process, usually estimated to be in the range of 10s of
milliseconds for typical cells and materials (e.g., with
$\gamma_{1}=0.1\,\text{Pa\,s}$, $d=5\,\mu\text{m}$, and $K=10\,\text{pN}$, we
obtain $\tau_{\text{off}}\doteq 2.53\times 10^{-2}\,\text{s}$).

The switch-on time $\tau_{\text{on}}=\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}$
(with $E=V/d$) corresponds to the time it takes to align the director field
with an applied electric field. It can be made quite small by using large
voltages. For example, with $\gamma_{1}=0.1\,\text{Pa\,s}$,
$d=5\,\mu\text{m}$, $\epsilon_{0}=8.854\times 10^{-12}\,\text{F/m}$,
$\epsilon_{\text{a}}=10$, and $V=100\,\text{volts}$, we have
$\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}\doteq 2.82\times 10^{-6}\,\text{s}$.
For small voltages near the Fréedericksz threshold, on the other hand (e.g.,
with $V=1\,\text{volt}$), we have
$\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}\doteq 2.82\times 10^{-2}\,\text{s}$.
In the fast-switching experiments in [9, 11, 21], using cells of narrow gaps
and electrical pulses of 100s of volts,
$\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}$’s of the order of 10s of
nanoseconds are reported.

By comparison, the evolution of the electric field in the cell is governed by
the time-dependent Maxwell equations, and the time scale associated with this
type of wave equation (which we denote
$\tau\\!\raisebox{-2.58334pt}{\tiny{$E$}}$) corresponds to the width of the
cell gap divided by the speed of light in the medium. For a cell gap
$d=10\,\mu\text{m}$ with relative dielectric permittivity
$\epsilon_{\text{r}}=10$ and relative magnetic permeability
$\mu_{\text{r}}=1$, we obtain $\tau\\!\raisebox{-2.58334pt}{\tiny{$E$}}\doteq
1.05\times 10^{-13}\,\text{s}$, which is four orders of magnitude faster than
the smallest $\tau_{\sigma}$’s and
$\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}$’s we have found in published
results on experiments on fast switching of liquid crystal cells. This
justifies treating the electric field as adjusting instantaneously to any
changes in the circuit and cell.

Thus, for the kinds of experimental systems that we have in mind, we consider
time scales in the following ranges:

$10^{-9}\,\text{s}\lesssim\tau_{\sigma}\lesssim
10^{-6}\,\text{s},\quad\tau\\!\raisebox{-2.58334pt}{\tiny{$K$}}\approx
10^{-2}\,\text{s},\quad
10^{-8}\,\text{s}\lesssim\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}\lesssim
10^{-2}\,\text{s},\quad\tau\\!\raisebox{-2.58334pt}{\tiny{$E$}}\approx
10^{-13}\,\text{s}.$

In the numerical examples discussed in the next section, we have taken
$\tau_{\sigma}=10^{-6}\,\text{s}$,
$\tau\\!\raisebox{-2.58334pt}{\tiny{$K$}}=10^{-2}\,\text{s}$, and
$\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}=10^{-7}\,\text{s}$,
$10^{-6}\,\text{s}$, and $10^{-5}\,\text{s}$, for purposes of illustration,
with the electric field taken to adjust instantaneously
($\tau\\!\raisebox{-2.58334pt}{\tiny{$E$}}=0$, in essence, as we have assumed
throughout).

##  5 Numerical illustration

We illustrate the coupled dynamics of (22) by the numerical modeling of a
simple “switch on” experiment. For simplicity, we assume equal elastic
constants: $K_{1}=K_{3}=K$. Starting from the partially scaled system (24), we
express times in units of $\tau_{\sigma}$ to obtain

$\displaystyle\frac{\text{d}\bar{\sigma}}{\text{d}{\bar{t}}}+\epsilon_{\text{r}}[\theta]^{-1}\bar{\sigma}=1,\quad\epsilon_{\text{r}}[\theta]^{-1}=\int_{0}^{1}\\!\\!\frac{\text{d}{\bar{z}}}{\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS},$
(25a)
$\displaystyle\frac{\partial\theta}{\partial\bar{t}}=\frac{1}{\,\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$K$}}}\frac{\partial^{2}\theta}{\partial\bar{z}^{2}}+\frac{1}{\,\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$V$}}}\bar{\sigma}^{2}\frac{\sin\theta\cos\theta}{(\epsilon_{\scriptscriptstyle\perp}\\!\cos^{2}\\!\theta+\epsilon_{\scriptscriptstyle\parallel}\SS)^{2}},\quad
0<\bar{z}<1,$ (25b) where
$\bar{t}:=\frac{t}{\,\tau_{\sigma}},\quad\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$K$}}:=\frac{\tau\\!\raisebox{-2.58334pt}{\tiny{$K$}}}{\,\tau_{\sigma}},\quad\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$V$}}:=\frac{\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}}{\,\tau_{\sigma}}.$
We add a small pretilt to the boundary conditions on $\theta$, in order to
bias the director to rotate counter clockwise when the electric field is
switched on, and we take the initial state of the director field to be
parallel to this. There is assumed to be no excess charge on the electrodes
when the circuit is closed at time $\bar{t}=0$. The boundary and initial
conditions are thus given by
$\theta(0,\bar{t})=\theta(1,\bar{t})=0.1,\quad\theta(\bar{z},0)=0.1,\quad\bar{\sigma}(0)=0.$
(25c)

For the relative dielectric permittivities, we use
$\epsilon_{\scriptscriptstyle\perp}=5$ and
$\epsilon_{\scriptscriptstyle\parallel}=15$, which are comparable to the
values for the typical liquid crystal 5CB—see [19, Appendix D, Table D.3].
Based upon the discussion of time scales in the previous section, we choose
the following values for our numerical experiment:

$\tau\\!\raisebox{-2.58334pt}{\tiny{$K$}}=10^{-2}\,\text{s},\quad\tau_{\sigma}=10^{-6}\,\text{s},\quad\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}=10^{-7}\,\text{s},~{}10^{-6}\,\text{s},~{}10^{-5}\,\text{s},$

giving

$\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$K$}}=10^{4},\quad\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$V$}}=10^{-1},~{}10^{0},~{}10^{1}.$

The three values of $\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$V$}}$ cover the
situations when the director switching dynamics are faster than, equal to, and
slower than the circuit dynamics.

Standard finite differences were employed to discretize our model (25):
explicit Euler for (25a), Forward Time Centered Space (FTCS) for (25b). The
same time step was used for both equations ($\Delta\bar{t}=0.1$, 10 times
steps per unit $\tau_{\sigma}$), with a spatial grid of 128 uniform cells in
$\bar{z}$. A Composite Trapezoid Rule (on the same spatial grid) was used to
approximate the integral in the functional $\epsilon_{\text{r}}[\theta]$ in
(25a). For each of the values
$\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$V$}}=10^{-1}$, $10^{0}$, and
$10^{1}$, $\bar{\sigma}$ is plotted against $\bar{t}$, along with “time
snapshots” of $\theta$ versus $\bar{z}$ for every 16th time step. In all three
cases, the same number of time steps per snapshot (16) and the same number of
total time steps (1024) were used, in order to facilitate comparison. The
results are presented in figure 2.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Coupled dynamics (25). Figures 2a, 2c, 2e: electrode charge density
versus time ($\bar{\sigma}=\sigma/(\epsilon_{0}V/d)$,
$\bar{t}=t/\tau_{\sigma}$). Figures 2b, 2d, 2f: time snapshots of director
tilt angle $\theta$ versus position ($\bar{z}=z/d$, 16 time steps per
snapshot, time step $\Delta\bar{t}=0.1$). Parameters:
$\epsilon_{\scriptscriptstyle\perp}=5$,
$\epsilon_{\scriptscriptstyle\parallel}=15$,
$\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$K$}}=\tau\\!\raisebox{-2.58334pt}{\tiny{$K$}}/\tau_{\sigma}=10^{4}$,
$\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$V$}}=\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}/\tau_{\sigma}=10^{-1}$
(figures 2a, 2b), $10^{0}$ (figures 2c, 2d), $10^{1}$ (figures 2e, 2f).

The results are as one would anticipate. The case
$\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$V$}}=10^{-1}$ corresponds to a high
voltage and fast switching: the director quickly saturates (after 5–6
snapshots), and the charge density behaves accordingly
($\bar{\sigma}\approx\epsilon_{\scriptscriptstyle\parallel}(1-e^{-\bar{t}})$).
We note that in the steady state limit $\bar{t}\rightarrow\infty$, we have
$\bar{\sigma}\rightarrow\epsilon_{\text{r}}[\theta_{\infty}]\approx\epsilon_{\scriptscriptstyle\parallel}$,
since $\theta_{\infty}\approx\pi/2$ (except for boundary layers near
$\bar{z}=0$ and $\bar{z}=1$). The case
$\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$V$}}=10^{0}$ corresponds to a
moderate voltage and switching
($\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}=\tau_{\sigma}$): $\theta$ saturates
after 15–16 snapshots, and one starts to see an inflection in $\bar{\sigma}$.
The last case, $\bar{\tau}\\!\raisebox{-2.58334pt}{\tiny{$V$}}=10^{1}$, is
associated with a low voltage and slow switching: it takes 50–60 snapshots for
the director to align with the electric field, and one sees a pronounced
inflection in $\bar{\sigma}$ versus $\bar{t}$. This is because the time scale
for the $\bar{\sigma}$ dynamics is approximately
$R\epsilon_{0}\epsilon_{\scriptscriptstyle\perp}A/d$ in the early stages (when
$\theta\approx 0$ and
$\epsilon_{\text{r}}[\theta]\approx\epsilon_{\scriptscriptstyle\perp}$), but
the time scale is approximately
$R\epsilon_{0}\epsilon_{\scriptscriptstyle\parallel}A/d$ in the later stages
(when $\theta\approx\pi/2$ and
$\epsilon_{\text{r}}[\theta]\approx\epsilon_{\scriptscriptstyle\parallel}$).
Since $\epsilon_{\scriptscriptstyle\perp}=5$ and
$\epsilon_{\scriptscriptstyle\parallel}=15$, these time scales differ by a
factor of three, and this leads to the change in the rate of approach of
$\bar{\sigma}$ to its limiting value. We emphasize that this is merely an
illustration of the coupled dynamics that emerge from our simple model; in
order to model carefully the dynamics in an actual fast-switching experiment,
for example, one would need to take into account other influences, such as
“dielectric relaxation” [11, 17].

##  6 Conclusions

We have modeled a nematic-liquid-crystal cell subject to an electric field
created by electrodes held at constant potential as a variable capacitor in an
RC circuit. The general model couples the state of the liquid-crystal director
field $\bm{n}$ in the cell with the state of the electric circuit,
characterized in terms of either the total charge on the upper electrode, $Q$,
or the potential difference between the electrodes, $\Delta U$. A dynamical
system was derived for an example in the splay-Fréedericksz geometry, subject
to several simplifying assumptions: no fluid flow in the cell, an electric
field that adjusts instantaneously to changes in $\bm{n}$, fields in the cell
that are functions of one space variable only, and a single rotational
viscosity for energy dissipation associated with $\partial\bm{n}/\partial t$.
The dynamical system, given in (22), involves a PDE for director dynamics
($\bm{n}=\bm{n}(z,t)$) coupled to an ODE for charge dynamics
($\sigma=\sigma(t)$, where $\sigma$ is the density of free charge on the
surface of the upper electrode). We have produced estimates for the time
scales of the various dynamic processes and have provided numerical examples
illustrating the coupled dynamics for three different cases relating the time
scale for director dynamics to that of the circuit dynamics. We have made an
effort to show consistency with established results, where possible.

The original motivation for this exercise was to understand better the
dynamical characterization of local stability of equilibrium states of such
systems, which can be characterized as stationary points of a free-energy
functional of the form (1):

$\mathcal{F}[\bm{n},U]=\int_{\Omega}\Bigl{[}W_{\text{e}}(\bm{n},\nabla\bm{n})-\frac{1}{2}{\bm{\epsilon}}(\bm{n})\nabla
U\cdot\nabla U\Bigr{]}\,\text{d}{V}.$

Here ${{\bm{\epsilon}}}$ is the (positive definite) dielectric tensor and $U$
is the electric potential (related to the electric field $\bm{E}$ via
$\bm{E}=-\nabla U$). The minimax nature of the critical points of
$\mathcal{F}$ is at odds with the expected picture of an out-of-equilibrium
pair $(\bm{n},U)$ relaxing to a locally stable state by minimizing free
energy. This confusion, as we have seen from our analysis, stems from the fact
that the free-energy functional above presumes either a static electric field
or an electric field that adjusts instantaneously to changes in $\bm{n}$.

The more primitive expression for the potential energy of the system is
$\mathcal{G}$ as in (16):

$\mathcal{G}=\mathcal{F}_{\text{e}}+\mathcal{E}_{\text{cap}}+\mathcal{E}_{\text{bat}}=\mathcal{F}_{\text{e}}+\frac{1}{2}Q\Delta
U-QV.$

Here $\mathcal{F}_{\text{e}}$ is the distortional elasticity, as in (2) and
(3), and $V$ is the voltage of the battery. The combination
$\mathcal{F}_{\text{e}}+\mathcal{E}_{\text{cap}}$ gives the potential energy
of the cell (associated with the work done in distorting the director field
plus work done in moving charge on/off the electrodes); while
$\mathcal{E}_{\text{bat}}$ is the potential energy associated with the
battery. The total potential energy for the system can be expressed in
different forms depending the choice of state variable for the circuit, as in
(19a) and (19b):

$\mathcal{G}[\bm{n},\Delta
U]=\mathcal{F}_{\text{e}}[\bm{n}]+C[\bm{n}]\Bigl{[}\frac{1}{2}(\Delta
U)^{2}-V\Delta
U\Bigr{]},\quad\mathcal{G}[\bm{n},Q]=\mathcal{F}_{\text{e}}[\bm{n}]+\frac{1}{2}C[\bm{n}]^{-1}Q^{2}-VQ.$

In either case, equilibria are locally minimizing with respect to the pair
$(\bm{n},\Delta U)$ or $(\bm{n},Q)$.

In order to transform $\mathcal{G}$ into the form $\mathcal{F}$ above, one
must assume (1) either a static electric field or an electric field that
adjusts instantaneously to any changes in $\bm{n}$ (so that
$\operatorname{curl}\bm{E}=\bm{0}$ and $\bm{E}=-\nabla U$) and (2) either
equilibrium conditions in the circuit (no current) or a circuit that adjusts
instantaneously to any changes in the capacitance of the cell (so that $\Delta
U=V$ and $Q=CV$ at all times). In §4.3, we have described how $\mathcal{G}$
above can be transformed to $\mathcal{F}$ if one makes these assumptions (and
also employs the constitutive assumption
$\bm{D}={{\bm{\epsilon}}}(\bm{n})\bm{E}$), the main points being that
$\bm{E}=-\nabla U$ and $\operatorname{div}\bm{D}=0$ imply

$Q\Delta U=\int_{\Omega}(\bm{D}\cdot\bm{E})\,\text{d}{V},$

$\Delta U=V$ gives

$\mathcal{E}_{\text{cap}}+\mathcal{E}_{\text{bat}}=\frac{1}{2}QV-
QV=-\frac{1}{2}QV=-\frac{1}{2}\int_{\Omega}(\bm{D}\cdot\bm{E})\,\text{d}{V},$

and $\bm{D}={{\bm{\epsilon}}}(\bm{n})\bm{E}$ and $\bm{E}=-\nabla U$ give

$-\frac{1}{2}\int_{\Omega}(\bm{D}\cdot\bm{E})\,\text{d}{V}=-\frac{1}{2}\int_{\Omega}\bigl{[}{\bm{\epsilon}}(\bm{n})\nabla
U\cdot\nabla U\bigr{]}\,\text{d}{V}.$

Thus $\mathcal{F}$ is not the appropriate free energy for modeling dynamics
that include the coupled evolution of the electric field in the cell or the
charge dynamics of the electric circuit. The potential energy $\mathcal{G}$,
on the other hand, is valid in the absence of any of these equilibrium or
instantaneous-adjustment assumptions.

The time scales for the various dynamic processes in our system vary widely
and depend on details of any specific experiment being modeled. At one end of
the spectrum is the switch-off time
($\tau\\!\raisebox{-2.58334pt}{\tiny{$K$}}$ in our notation, the slowest time
scale, of the order of $10^{-2}$ s). At the other end is the time scale for
the evolution of the electric field in the cell,
$\tau\\!\raisebox{-2.58334pt}{\tiny{$E$}}$, governed by the time-dependent
Maxwell equations, which is of the order of $10^{-13}$ s for the kinds of
systems of interest to us. In between these extremes are the switch-on time,
$\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}$, and the time scale for the
dynamics of the electric circuit, $\tau_{\sigma}$. The switch-on time
$\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}$ is proportional to $1/V^{2}$ (where
$V$ is the applied voltage) and can vary from values comparable to
$\tau\\!\raisebox{-2.58334pt}{\tiny{$K$}}$ to values of the order of $10^{-8}$
s. There is some overlap between values that
$\tau\\!\raisebox{-2.58334pt}{\tiny{$V$}}$ can take and those that
$\tau_{\sigma}$ can take, and our modeling has been concerned with such
situations. The evolution of the electric field in the cell is several orders
of magnitude faster than any of these, and its response has been treated as
instantaneous, as is always assumed in modeling such systems.

For the example problem that we have analyzed in detail (a splay-Fréedericksz
cell), the final form of the dynamical system (22) is quite clean, with
everything given in simple explicit analytical expressions. This is, of
course, a consequence of our modeling assumptions. The assumption that fields
in the cell depend on only one space variable buys one a lot. It gives
$D_{z}=\text{const}$ and $\sigma=\text{const}$ (in equilibrium) or
$\sigma=\sigma(t)$ (in dynamics), and it enables us to write an explicit
analytical expression for the capacitance of the cell, $C[\bm{n}]$ in (14),
and for the electric field in the cell, as in (5) and (8). If one abandons
this assumption and allows fields in the cell to depend on more than one space
variable, then all these consequences are lost. The assumption of fields
depending on one space dimension is very common (and appropriate) in modeling
thin-film liquid-crystal systems.

The interplay between electric fields and liquid crystals has been of interest
and importance ever since the discovery of the electro-optic effect in the
1970s—for an early review, see [10] and references therein. Bringing the
electric circuit into the picture (as we have done here) has illuminated the
role of the battery in producing the term
$-\frac{1}{2}{{\bm{\epsilon}}}(\bm{n})\nabla U\cdot\nabla U$ in the free
energy (1), and it has highlighted what assumptions must be made to express
the free energy in that form. If one merely wanted to obtain a coupled
dynamical system such as (22), then one could have proceeded more directly,
starting with appropriate equations for director dynamics and circuit dynamics

$\gamma_{1}\frac{\partial\bm{n}}{\partial
t}=\operatorname{div}\Bigl{(}\frac{\partial
W_{\text{e}}}{\partial\nabla\bm{n}}\Bigr{)}-\frac{\partial
W_{\text{e}}}{\partial\bm{n}}+\lambda\bm{n}+\epsilon_{0}\epsilon_{\text{a}}(\bm{E}\cdot\bm{n})\bm{E},\quad
R\frac{\text{d}{Q}}{\text{d}{t}}+\frac{1}{C}Q=V$

and coupling them via appropriate expressions for the electric field (which
depends on $Q=\sigma A$, as in (5) and (8)) and the capacitance (which depends
on $\bm{n}$, as in (14)). Our more elaborated approach was motivated by a
desire to see the “full picture” in terms of the potential energies, where
they come from, and the assumptions needed for various simplifications and
reductions.

## Acknowledgments

The impetus for this analysis came from exchanges with John Ball and Nigel
Mottram, and we are grateful to both of them for their thoughts, suggestions,
and feedback on versions of this note. We are also grateful to Lev
Truskinovsky for discussions much earlier that planted the seeds for some of
these ideas, and for the reference [7], which illuminates some related
concepts.

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