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\title{Spontaneous Corrugation of Dipolar Membranes}
\author{Xiuquan Sun and J. Daniel Gezelter}
\email[E-mail:]{gezelter@nd.edu}
\affiliation{Department of Chemistry and Biochemistry,\\
University of Notre Dame, \\
Notre Dame, Indiana 46556}

\date{\today} 

\begin{abstract}
We present a simple model for dipolar elastic membranes that gives
lattice-bound point dipoles complete orientational freedom as well as
translational freedom along one coordinate (out of the plane of the
membrane).  There is an additional harmonic term which binds each of
the dipoles to the six nearest neighbors on either triangular or
distorted lattices.  The translational freedom of the dipoles allows
triangular lattices to find states that break out of the normal
orientational disorder of frustrated configurations and which are
stabilized by long-range antiferroelectric ordering.  In order to
break out of the frustrated states, the dipolar membranes form
corrugated or ``rippled'' phases that make the lattices effectively
non-triangular.  We observe three common features of the corrugated
dipolar membranes: 1) the corrugated phases develop easily when hosted
on triangular lattices, 2) the wave vectors for the surface ripples
are always found to be perpendicular to the dipole director axis, and
3) on triangular lattices, the dipole director axis is found to be
parallel to any of the three equivalent lattice directions.
\end{abstract}

\pacs{68.03.Hj, 82.20.Wt}
\maketitle


\section{Introduction}
\label{Int}

The properties of polymeric membranes are known to depend sensitively
on the details of the internal interactions between the constituent
monomers.  A flexible membrane will always have a competition between
the energy of curvature and the in-plane stretching energy and will be
able to buckle in certain limits of surface tension and
temperature.\cite{Safran94} The buckling can be non-specific and
centered at dislocation~\cite{Seung1988} or grain-boundary
defects,\cite{Carraro1993} or it can be directional and cause long
``roof-tile'' or tube-like structures to appear in
partially-polymerized phospholipid vesicles.\cite{Mutz1991}

One would expect that anisotropic local interactions could lead to
interesting properties of the buckled membrane.  We report here on the
buckling behavior of a membrane composed of harmonically-bound, but
freely-rotating electrostatic dipoles.  The dipoles have strongly
anisotropic local interactions and the membrane exhibits coupling
between the buckling and the long-range ordering of the dipoles.

Buckling behavior in liquid crystalline and biological membranes is a
well-known phenomenon.  Relatively pure phosphatidylcholine (PC)
bilayers are known to form a corrugated or ``rippled'' phase
($P_{\beta'}$) which appears as an intermediate phase between the gel
($L_\beta$) and fluid ($L_{\alpha}$) phases.  The $P_{\beta'}$ phase
has attracted substantial experimental interest over the past 30
years. Most structural information of the ripple phase has been
obtained by the X-ray diffraction~\cite{Sun96,Katsaras00} and
freeze-fracture electron microscopy (FFEM).~\cite{Copeland80,Meyer96}
Recently, Kaasgaard {\it et al.} used atomic force microscopy (AFM) to
observe ripple phase morphology in bilayers supported on
mica.~\cite{Kaasgaard03} The experimental results provide strong
support for a 2-dimensional triangular packing lattice of the lipid
molecules within the ripple phase.  This is a notable change from the
observed lipid packing within the gel phase.~\cite{Cevc87} There have
been a number of theoretical
approaches~\cite{Marder84,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02}
(and some heroic
simulations~\cite{Ayton02,Jiang04,Brannigan04a,deVries05,deJoannis06})
undertaken to try to explain this phase, but to date, none have looked
specifically at the contribution of the dipolar character of the lipid
head groups towards this corrugation.  Lipid chain interdigitation
certainly plays a major role, and the structures of the ripple phase
are highly ordered.  The model we investigate here lacks chain
interdigitation (as well as the chains themselves!) and will not be
detailed enough to rule in favor of (or against) any of these
explanations for the $P_{\beta'}$ phase.

Another interesting properties of elastic membranes containing
electrostatic dipoles is the phenomenon of flexoelectricity,\cite{}
which is the ability of mechanical deformations of the membrane to
result in electrostatic organization of the membrane.  This phenomenon
is a curvature-induced membrane polarization which can lead to
potential differences across a membrane.  Reverse flexoelectric
behavior (in which applied alternating currents affect membrane
curvature) has also been observed.  Explanations of the details of
these effects have typically utilized membrane polarization parallel
to the membrane normal.\cite{}

The problem with using atomistic and even coarse-grained approaches to
study membrane buckling phenomena is that only a relatively small
number of periods of the corrugation (i.e. one or two) can be
realistically simulated given current technology.  Also, simulations
of lipid bilayers are traditionally carried out with periodic boundary
conditions in two or three dimensions and these have the potential to
enhance the periodicity of the system at that wavelength.  To avoid
this pitfall, we are using a model which allows us to have
sufficiently large systems so that we are not causing artificial
corrugation through the use of periodic boundary conditions.

The simplest dipolar membrane is one in which the dipoles are located
on fixed lattice sites. Ferroelectric states (with long-range dipolar
order) can be observed in dipolar systems with non-triangular
packings.  However, {\em triangularly}-packed 2-D dipolar systems are
inherently frustrated and one would expect a dipolar-disordered phase
to be the lowest free energy
configuration.\cite{Toulouse1977,Marland1979} Dipolar lattices already
have rich phase behavior, but in order to allow the membrane to
buckle, a single degree of freedom (translation normal to the membrane
face) must be added to each of the dipoles.  It would also be possible
to allow complete translational freedom.  This approach
is similar in character to a number of elastic Ising models that have
been developed to explain interesting mechanical properties in
magnetic alloys.\cite{Renard1966,Zhu2005,Zhu2006,Jiang2006}

What we present here is an attempt to find the simplest dipolar model
which will exhibit buckling behavior.  We are using a modified XYZ
lattice model; details of the model can be found in section
\ref{sec:model}, results of Monte Carlo simulations using this model
are presented in section
\ref{sec:results}, and section \ref{sec:discussion} contains our conclusions.

\section{2-D Dipolar Membrane}
\label{sec:model}

The point of developing this model was to arrive at the simplest
possible theoretical model which could exhibit spontaneous corrugation
of a two-dimensional dipolar medium.  Since molecules in polymerized
membranes and in in the $P_{\beta'}$ ripple phase have limited
translational freedom, we have chosen a lattice to support the dipoles
in the x-y plane.  The lattice may be either triangular (lattice
constants $a/b =
\sqrt{3}$) or distorted.  However, each dipole has 3 degrees of
freedom.  They may move freely {\em out} of the x-y plane (along the
$z$ axis), and they have complete orientational freedom ($0 <= \theta
<= \pi$, $0 <= \phi < 2
\pi$).  This is essentially a modified X-Y-Z model with translational
freedom along the z-axis.

The potential energy of the system,
\begin{eqnarray}
V = \sum_i & & \left( \sum_{j>i} \frac{|\mu|^2}{4\pi \epsilon_0 r_{ij}^3} \left[
{\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j} -
3({\mathbf{\hat u}_i} \cdot {\mathbf{\hat
r}_{ij}})({\mathbf{\hat u}_j} \cdot {\mathbf{\hat r}_{ij}})\right]
\right. \nonumber \\
 & & \left. + \sum_{j \in NN_i}^6 \frac{k_r}{2}\left(
r_{ij}-\sigma \right)^2 \right)
\label{eq:pot}
\end{eqnarray}


In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector
pointing along the inter-dipole vector $\mathbf{r}_{ij}$.  The entire
potential is governed by three parameters, the dipolar strength
($\mu$), the harmonic spring constant ($k_r$) and the preferred
intermolecular spacing ($\sigma$).  In practice, we set the value of
$\sigma$ to the average inter-molecular spacing from the planar
lattice, yielding a potential model that has only two parameters for a
particular choice of lattice constants $a$ (along the $x$-axis) and
$b$ (along the $y$-axis).  We also define a set of reduced parameters
based on the length scale ($\sigma$) and the energy of the harmonic
potential at a deformation of 2 $\sigma$ ($\epsilon = k_r \sigma^2 /
2$).  Using these two constants, we perform our calculations using
reduced distances, ($r^{*} = r / \sigma$), temperatures ($T^{*} = 2
k_B T / k_r \sigma^2$), densities ($\rho^{*} = N \sigma^2 / L_x L_y$),
and dipole moments ($\mu^{*} = \mu / \sqrt{4 \pi \epsilon_0 \sigma^5
k_r / 2}$). 

To investigate the phase behavior of this model, we have performed a
series of Metropolis Monte Carlo simulations of moderately-sized (34.3
$\sigma$ on a side) patches of membrane hosted on both triangular
($\gamma = a/b = \sqrt{3}$) and distorted ($\gamma \neq \sqrt{3}$)
lattices.  The linear extent of one edge of the monolayer was $20 a$
and the system was kept roughly square. The average distance that
coplanar dipoles were positioned from their six nearest neighbors was
1 $\sigma$ (on both triangular and distorted lattices).  Typical
system sizes were 1360 dipoles for the triangular lattices and
840-2800 dipoles for the distorted lattices.  Two-dimensional periodic
boundary conditions were used, and the cutoff for the dipole-dipole
interaction was set to 4.3 $\sigma$. Since dipole-dipole interactions
decay rapidly with distance, and since the intrinsic three-dimensional
periodicity of the Ewald sum can give artifacts in 2-d systems, we
have chosen not to use it in these calculations.  Although the Ewald
sum has been reformulated to handle 2-D
systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89} these methods
are computationally expensive,\cite{Spohr97,Yeh99} and are not
necessary in this case.  All parameters ($T^{*}$, $\mu^{*}$, and
$\gamma$) were varied systematically to study the effects of these
parameters on the formation of ripple-like phases.

\section{Results and Analysis}
\label{sec:results}
 
\subsection{Dipolar Ordering and Coexistence Temperatures}
The principal method for observing the orientational ordering
transition in dipolar systems is the $P_2$ order parameter (defined as
$1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest
eigenvalue of the matrix,
\begin{equation}
{\mathsf{S}} = \frac{1}{N} \sum_i \left(
\begin{array}{ccc}
        u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\
        u^{y}_i u^{x}_i  & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\
        u^{z}_i u^{x}_i & u^{z}_i u^{y}_i  & u^{z}_i u^{z}_i -\frac{1}{3} 
\end{array} \right).
\label{eq:opmatrix}
\end{equation}
Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector
for dipole $i$.  $P_2$ will be $1.0$ for a perfectly-ordered system
and near $0$ for a randomized system.  Note that this order parameter
is {\em not} equal to the polarization of the system.  For example,
the polarization of the perfect antiferroelectric system is $0$, but
$P_2$ for an antiferroelectric system is $1$.  The eigenvector of
$\mathsf{S}$ corresponding to the largest eigenvalue is familiar as
the director axis, which can be used to determine a privileged dipolar
axis for dipole-ordered systems.  The top panel in Fig. \ref{phase}
shows the values of $P_2$ as a function of temperature for both
triangular ($\gamma = 1.732$) and distorted ($\gamma=1.875$)
lattices.

\begin{figure}
\includegraphics[width=\linewidth]{phase}
\caption{\label{phase} Top panel: The $P_2$ dipolar order parameter as
a function of temperature for both triangular ($\gamma = 1.732$) and
distorted ($\gamma = 1.875$) lattices.  Bottom Panel: The phase
diagram for the dipolar membrane model.  The line denotes the division
between the dipolar ordered (antiferroelectric) and disordered phases.
An enlarged view near the triangular lattice is shown inset.}
\end{figure}

There is a clear order-disorder transition in evidence from this data.
Both the triangular and distorted lattices have dipolar-ordered
low-temperature phases, and orientationally-disordered high
temperature phases.  The coexistence temperature for the triangular
lattice is significantly lower than for the distorted lattices, and
the bulk polarization is approximately $0$ for both dipolar ordered
and disordered phases.  This gives strong evidence that the dipolar
ordered phase is antiferroelectric.  We have verified that this
dipolar ordering transition is not a function of system size by
performing identical calculations with systems twice as large.  The
transition is equally smooth at all system sizes that were studied.
Additionally, we have repeated the Monte Carlo simulations over a wide
range of lattice ratios ($\gamma$) to generate a dipolar
order/disorder phase diagram.  The bottom panel in Fig. \ref{phase}
shows that the triangular lattice is a low-temperature cusp in the
$T^{*}-\gamma$ phase diagram.

This phase diagram is remarkable in that it shows an antiferroelectric
phase near $\gamma=1.732$ where one would expect lattice frustration
to result in disordered phases at all temperatures.  Observations of
the configurations in this phase show clearly that the system has
accomplished dipolar orderering by forming large ripple-like
structures.  We have observed antiferroelectric ordering in all three
of the equivalent directions on the triangular lattice, and the dipoles
have been observed to organize perpendicular to the membrane normal
(in the plane of the membrane).  It is particularly interesting to
note that the ripple-like structures have also been observed to
propagate in the three equivalent directions on the lattice, but the
{\em direction of ripple propagation is always perpendicular to the
dipole director axis}.  A snapshot of a typical antiferroelectric
rippled structure is shown in Fig. \ref{fig:snapshot}.

\begin{figure}
\includegraphics[width=\linewidth]{snapshot}
\caption{\label{fig:snapshot} Top and Side views of a representative
configuration for the dipolar ordered phase supported on the
triangular lattice. Note the antiferroelectric ordering and the long
wavelength buckling of the membrane.  Dipolar ordering has been
observed in all three equivalent directions on the triangular lattice,
and the ripple direction is always perpendicular to the director axis
for the dipoles.}
\end{figure}

Although the snapshot in Fig. \ref{fig:snapshot} gives the appearance
of three-row stair-like structures, these appear to be transient.  On
average, the corrugation of the membrane is a relatively smooth,
long-wavelength phenomenon, with occasional steep drops between
adjacent lines of anti-aligned dipoles.

The height-dipole correlation function ($C(r, \cos \theta)$) makes the
connection between dipolar ordering and the wave vector of the ripple
even more explicit.  $C(r, \cos \theta)$ is an angle-dependent pair
distribution function. The angle ($\theta$) is defined by the
intermolecular vector $\vec{r}_{ij}$ and direction of dipole $i$,
\begin{equation}
C(r, \cos \theta) = \frac{\langle \sum_{i}
\sum_{j} h_i \cdot h_j \delta(r - r_{ij}) \delta(\cos \theta_{ij} -
\cos \theta)\rangle} {\langle h^2 \rangle}
\end{equation}
where $\cos \theta_{ij} = \hat{\mu}_{i} \cdot \hat{r}_{ij}$ and
$\hat{r}_{ij} = \vec{r}_{ij} / r_{ij}$.   Fig. \ref{fig:CrossCorrelation}
shows contours of this correlation function for both anti-ferroelectric, rippled
membranes as well as for the dipole-disordered portion of the phase diagram.  

\begin{figure}
\includegraphics[width=\linewidth]{hdc}
\caption{\label{fig:CrossCorrelation} Contours of the height-dipole
correlation function as a function of the dot product between the
dipole ($\hat{\mu}$) and inter-dipole separation vector ($\hat{r}$)
and the distance ($r$) between the dipoles.  Perfect height
correlation (contours approaching 1) are present in the ordered phase
when the two dipoles are in the same head-to-tail line.
Anti-correlation (contours below 0) is only seen when the inter-dipole
vector is perpendicular to the dipoles.  In the dipole-disordered
portion of the phase diagram, there is only weak correlation in the
dipole direction and this correlation decays rapidly to zero for
intermolecular vectors that are not dipole-aligned.}
\end{figure}

\subsection{Discriminating Ripples from Thermal Undulations}

In order to be sure that the structures we have observed are actually
a rippled phase and not simply thermal undulations, we have computed
the undulation spectrum,
\begin{equation}
h(\vec{q}) = A^{-1/2} \int d\vec{r}
h(\vec{r}) e^{-i \vec{q}\cdot\vec{r}}
\end{equation}
where $h(\vec{r})$ is the height of the membrane at location $\vec{r}
= (x,y)$.~\cite{Safran94,Seifert97} In simple (and more complicated)
elastic continuum models, it can shown that in the $NVT$ ensemble, the
absolute value of the undulation spectrum can be written,
\begin{equation}
\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{k_c q^4 +
\gamma q^2},
\label{eq:fit}
\end{equation}
where $k_c$ is the bending modulus for the membrane, and $\gamma$ is
the mechanical surface tension.~\cite{Safran94} The systems studied in
this paper have essentially zero bending moduli ($k_c$) and relatively
large mechanical surface tensions ($\gamma$), so a much simpler form
can be written,
\begin{equation}
\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{\gamma q^2},
\label{eq:fit2}
\end{equation}

The undulation spectrum is computed by superimposing a rectangular
grid on top of the membrane, and by assigning height ($h(\vec{r})$)
values to the grid from the average of all dipoles that fall within a
given $\vec{r}+d\vec{r}$ grid area.  Empty grid pixels are assigned
height values by interpolation from the nearest neighbor pixels.  A
standard 2-d Fourier transform is then used to obtain $\langle |
h(q)|^2 \rangle$.  Alternatively, since the dipoles sit on a Bravais
lattice, one could use the heights of the lattice points themselves as
the grid for the Fourier transform (without interpolating to a square
grid).  However, if lateral translational freedom is added to this
model (a likely extension), an interpolated grid method for computing
undulation spectra will be required.

As mentioned above, the best fits to our undulation spectra are
obtained by setting the value of $k_c$ to 0.  In Fig. \ref{fig:fit} we
show typical undulation spectra for two different regions of the phase
diagram along with their fits from the Landau free energy approach
(Eq. \ref{eq:fit2}).  In the high-temperature disordered phase, the
Landau fits can be nearly perfect, and from these fits we can estimate
the tension in the surface.  In reduced units, typical values of
$\gamma^{*} = \gamma / \epsilon = 2500$ are obtained for the
disordered phase ($\gamma^{*} = 2551.7$ in the top panel of
Fig. \ref{fig:fit}).

Typical values of $\gamma^{*}$ in the dipolar-ordered phase are much
higher than in the dipolar-disordered phase ($\gamma^{*} = 73,538$ in
the lower panel of Fig. \ref{fig:fit}).  For the dipolar-ordered
triangular lattice near the coexistence temperature, we also observe
long wavelength undulations that are far outliers to the fits.  That
is, the Landau free energy fits are well within error bars for most of
the other points, but can be off by {\em orders of magnitude} for a
few low frequency components.

We interpret these outliers as evidence that these low frequency modes
are {\em non-thermal undulations}.  We take this as evidence that we
are actually seeing a rippled phase developing in this model system.

\begin{figure}
\includegraphics[width=\linewidth]{logFit}
\caption{\label{fig:fit} Evidence that the observed ripples are {\em
not} thermal undulations is obtained from the 2-d fourier transform
$\langle |h^{*}(\vec{q})|^2 \rangle$ of the height profile ($\langle
h^{*}(x,y) \rangle$). Rippled samples show low-wavelength peaks that
are outliers on the Landau free energy fits by an order of magnitude.
Samples exhibiting only thermal undulations fit Eq. \ref{eq:fit}
remarkably well.}
\end{figure}

\subsection{Effects of Potential Parameters on Amplitude and Wavelength}

We have used two different methods to estimate the amplitude and
periodicity of the ripples.  The first method requires projection of
the ripples onto a one dimensional rippling axis. Since the rippling
is always perpendicular to the dipole director axis, we can define a
ripple vector as follows.  The largest eigenvector, $s_1$, of the
$\mathsf{S}$ matrix in Eq. \ref{eq:opmatrix} is projected onto a
planar director axis,
\begin{equation}
\vec{d} = \left(\begin{array}{c}
\vec{s}_1 \cdot \hat{i} \\
\vec{s}_1 \cdot \hat{j} \\
0
\end{array} \right).
\end{equation}
($\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along the $x$,
$y$, and $z$ axes, respectively.)  The rippling axis is in the plane of
the membrane and is perpendicular to the planar director axis,
\begin{equation}
\vec{q}_{\mathrm{rip}} = \vec{d} \times \hat{k}
\end{equation}
We can then find the height profile of the membrane along the ripple
axis by projecting heights of the dipoles to obtain a one-dimensional
height profile, $h(q_{\mathrm{rip}})$. Ripple wavelengths can be
estimated from the largest non-thermal low-frequency component in the
fourier transform of $h(q_{\mathrm{rip}})$.  Amplitudes can be
estimated by measuring peak-to-trough distances in
$h(q_{\mathrm{rip}})$ itself.

A second, more accurate, and simpler method for estimating ripple
shape is to extract the wavelength and height information directly
from the largest non-thermal peak in the undulation spectrum.  For
large-amplitude ripples, the two methods give similar results.  The
one-dimensional projection method is more prone to noise (particularly
in the amplitude estimates for the distorted lattices).  We report
amplitudes and wavelengths taken directly from the undulation spectrum
below.

In the triangular lattice ($\gamma = \sqrt{3}$), the rippling is
observed for temperatures ($T^{*}$) from $61-122$.  The wavelength of
the ripples is remarkably stable at 21.4~$\sigma$ for all but the
temperatures closest to the order-disorder transition.  At $T^{*} =
122$, the wavelength drops to 17.1~$\sigma$.

The dependence of the amplitude on temperature is shown in the top
panel of Fig. \ref{fig:Amplitude}.  The rippled structures shrink
smoothly as the temperature rises towards the order-disorder
transition.  The wavelengths and amplitudes we observe are
surprisingly close to the $\Lambda / 2$ phase observed by Kaasgaard
{\it et al.} in their work on PC-based lipids.\cite{Kaasgaard03}
However, this is coincidental agreement based on a choice of 7~\AA~as
the mean spacing between lipids. 

\begin{figure}
\includegraphics[width=\linewidth]{properties_sq}
\caption{\label{fig:Amplitude} a) The amplitude $A^{*}$ of the ripples
vs. temperature for a triangular lattice. b) The amplitude $A^{*}$ of
the ripples vs. dipole strength ($\mu^{*}$) for both the triangular
lattice (circles) and distorted lattice (squares).  The reduced
temperatures were kept fixed at $T^{*} = 94$ for the triangular
lattice and $T^{*} = 106$ for the distorted lattice (approximately 2/3
of the order-disorder transition temperature for each lattice).}
\end{figure}

The ripples can be made to disappear by increasing the internal
surface tension (i.e. by increasing $k_r$ or equivalently, reducing
the dipole moment).  The amplitude of the ripples depends critically
on the strength of the dipole moments ($\mu^{*}$) in Eq. \ref{eq:pot}.
If the dipoles are weakened substantially (below $\mu^{*}$ = 20) at a
fixed temperature of 94, the membrane loses dipolar ordering
and the ripple structures. The ripples reach a peak amplitude of
3.7~$\sigma$ at a dipolar strength of 25.  We show the dependence
of ripple amplitude on the dipolar strength in
Fig. \ref{fig:Amplitude}.

\subsection{Distorted lattices}

We have also investigated the effect of the lattice geometry by
changing the ratio of lattice constants ($\gamma$) while keeping the
average nearest-neighbor spacing constant. The antiferroelectric state
is accessible for all $\gamma$ values we have used, although the
distorted triangular lattices prefer a particular director axis due to
the anisotropy of the lattice.

Our observation of rippling behavior was not limited to the triangular
lattices.  In distorted lattices the antiferroelectric phase can
develop nearly instantaneously in the Monte Carlo simulations, and
these dipolar-ordered phases tend to be remarkably flat.  Whenever
rippling has been observed in these distorted lattices
(e.g. $\gamma = 1.875$), we see relatively short ripple wavelengths
(14 $\sigma$) and amplitudes of 2.4~$\sigma$.  These ripples are
weakly dependent on dipolar strength (see Fig. \ref{fig:Amplitude}),
although below a dipolar strength of $\mu^{*} = 20$, the membrane
loses dipolar ordering and displays only thermal undulations.

The ripple phase does {\em not} appear at all values of $\gamma$.  We
have only observed non-thermal undulations in the range $1.625 <
\gamma < 1.875$.  Outside this range, the order-disorder transition in
the dipoles remains, but the ordered dipolar phase has only thermal
undulations.  This is one of our strongest pieces of evidence that
rippling is a symmetry-breaking phenomenon for triangular and
nearly-triangular lattices.

\subsection{Effects of System Size}
To evaluate the effect of finite system size, we have performed a
series of simulations on the triangular lattice at a reduced
temperature of 122, which is just below the order-disorder transition
temperature ($T^{*} = 139$).  These conditions are in the
dipole-ordered and rippled portion of the phase diagram.  These are
also the conditions that should be most susceptible to system size
effects.

\begin{figure}
\includegraphics[width=\linewidth]{SystemSize}
\caption{\label{fig:systemsize} The ripple wavelength (top) and
amplitude (bottom) as a function of system size for a triangular
lattice ($\gamma=1.732$) at $T^{*} = 122$.}
\end{figure}

There is substantial dependence on system size for small (less than
29~$\sigma$) periodic boxes.  Notably, there are resonances apparent
in the ripple amplitudes at box lengths of 17.3 and 29.5 $\sigma$.
For larger systems, the behavior of the ripples appears to have
stabilized and is on a trend to slightly smaller amplitudes (and
slightly longer wavelengths) than were observed from the 34.3 $\sigma$
box sizes that were used for most of the calculations.

It is interesting to note that system sizes which are multiples of the
default ripple wavelength can enhance the amplitude of the observed
ripples, but appears to have only a minor effect on the observed
wavelength.  It would, of course, be better to use system sizes that
were many multiples of the ripple wavelength to be sure that the
periodic box is not driving the phenomenon, but at the largest system
size studied (70 $\sigma$ $\times$ 70 $\sigma$), the number of dipoles
(5440) made long Monte Carlo simulations prohibitively expensive. 

\section{Discussion}
\label{sec:discussion} 

We have been able to show that a simple dipolar lattice model which
contains only molecular packing (from the lattice), anisotropy (in the
form of electrostatic dipoles) and a weak surface tension (in the form
of a nearest-neighbor harmonic potential) is capable of exhibiting
stable long-wavelength non-thermal surface corrugations.  The best
explanation for this behavior is that the ability of the dipoles to
translate out of the plane of the membrane is enough to break the
symmetry of the triangular lattice and allow the energetic benefit from
the formation of a bulk antiferroelectric phase.  Were the weak
surface tension absent from our model, it would be possible for the
entire lattice to ``tilt'' using $z$-translation.  Tilting the lattice
in this way would yield an effectively non-triangular lattice which
would avoid dipolar frustration altogether.  With the surface tension
in place, bulk tilt causes a large strain, and the simplest way to
release this strain is along line defects.  Line defects will result
in rippled or sawtooth patterns in the membrane, and allow small
``stripes'' of membrane to form antiferroelectric regions that are
tilted relative to the averaged membrane normal.

Although the dipole-dipole interaction is the major driving force for
the long range orientational ordered state, the formation of the
stable, smooth ripples is a result of the competition between the
surface tension and the dipole-dipole interactions.  This statement is
supported by the variation in $\mu^{*}$.  Substantially weaker dipoles
relative to the surface tension can cause the corrugated phase to
disappear.

The packing of the dipoles into a nearly-triangular lattice is clearly
an important piece of the puzzle.  The dipolar head groups of lipid
molecules are sterically (as well as electrostatically) anisotropic,
and would not be able to pack in triangular arrangements without the
steric interference of adjacent molecular bodies.  Since we only see
rippled phases in the neighborhood of $\gamma=\sqrt{3}$, this implies
that there is a role played by the lipid chains in the organization of
the triangularly ordered phases which support ripples in realistic
lipid bilayers.

The most important prediction we can make using the results from this
simple model is that if dipolar ordering is driving the surface
corrugation, the wave vectors for the ripples should always found to
be {\it perpendicular} to the dipole director axis.  This prediction
should suggest experimental designs which test whether this is really
true in the phosphatidylcholine $P_{\beta'}$ phases.  The dipole
director axis should also be easily computable for the all-atom and
coarse-grained simulations that have been published in the literature.

Our other observation about the ripple and dipolar directionality is
that the dipole director axis can be found to be parallel to any of
the three equivalent lattice vectors in the triangular lattice.
Defects in the ordering of the dipoles can cause the dipole director
(and consequently the surface corrugation) of small regions to be
rotated relative to each other by 120$^{\circ}$.  This is a similar
behavior to the domain rotation seen in the AFM studies of Kaasgaard
{\it et al.}\cite{Kaasgaard03}   

Although our model is simple, it exhibits some rich and unexpected
behaviors.  It would clearly be a closer approximation to the reality
if we allowed greater translational freedom to the dipoles and
replaced the somewhat artificial lattice packing and the harmonic
``surface tension'' with more realistic molecular modeling
potentials.  What we have done is to present an extremely simple model
which exhibits bulk non-thermal corrugation, and our explanation of
this rippling phenomenon will help us design more accurate molecular
models for corrugated membranes and experiments to test whether
rippling is dipole-driven or not.

\begin{acknowledgments}
Support for this project was provided by the National Science
Foundation under grant CHE-0134881.  The authors would like to thank
the reviewers for helpful comments.
\end{acknowledgments}

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