--- trunk/ripple2/ripple.tex	2006/12/27 22:13:09	3097
+++ trunk/ripple2/ripple.tex	2006/12/28 21:55:59	3098
@@ -1,5 +1,6 @@
-\documentclass[aps,pre,endfloats*,preprint,amssymb,showpacs]{revtex4}
-\usepackage{epsfig}
+\documentclass[aps,pre,twocolumn,amssymb,showpacs]{revtex4}
+%\documentclass[aps,pre,preprint,amssymb,showpacs]{revtex4}
+\usepackage{graphicx}
 
 \begin{document}
 \renewcommand{\thefootnote}{\fnsymbol{footnote}}
@@ -9,7 +10,7 @@
 
 \title{Spontaneous Corrugation of Dipolar Membranes}
 \author{Xiuquan Sun and J. Daniel Gezelter}
-\email[]{E-mail: gezelter@nd.edu}
+\email[E-mail:]{gezelter@nd.edu}
 \affiliation{Department of Chemistry and Biochemistry,\\
 University of Notre Dame, \\
 Notre Dame, Indiana 46556}
@@ -17,23 +18,23 @@ We present a simple model for dipolar membranes that g
 \date{\today} 
 
 \begin{abstract}
-We present a simple model for dipolar membranes that gives
+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 surface tension 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.
+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}
@@ -42,145 +43,93 @@ There has been intense recent interest in the phase be
 
 \section{Introduction}
 \label{Int}
-There has been intense recent interest in the phase behavior of
-dipolar
-fluids.\cite{Tlusty00,Teixeira00,Tavares02,Duncan04,Holm05,Duncan06}
-Due to the anisotropic interactions between dipoles, dipolar fluids
-can present anomalous phase behavior.  Examples of condensed-phase
-dipolar systems include ferrofluids, electro-rheological fluids, and
-even biological membranes.  Computer simulations have provided useful
-information on the structural features and phase transition of the
-dipolar fluids. Simulation results indicate that at low densities,
-these fluids spontaneously organize into head-to-tail dipolar
-``chains''.\cite{Teixeira00,Holm05} At low temperatures, these chains
-and rings prevent the occurrence of a liquid-gas phase transition.
-However, Tlusty and Safran showed that there is a defect-induced phase
-separation into a low-density ``chain'' phase and a higher density
-Y-defect phase.\cite{Tlusty00} Recently, inspired by experimental
-studies on monolayers of dipolar fluids, theoretical models using
-two-dimensional dipolar soft spheres have appeared in the literature.
-Tavares {\it et al.} tested their theory for chain and ring length
-distributions in two dimensions and carried out Monte Carlo
-simulations in the low-density phase.\cite{Tavares02} Duncan and Camp
-performed dynamical simulations on two-dimensional dipolar fluids to
-study transport and orientational dynamics in these
-systems.\cite{Duncan04} They have recently revisited two-dimensional
-systems to study the kinetic conditions for the defect-induced
-condensation into the Y-defect phase.\cite{Duncan06}
 
-Although they are not traditionally classified as 2-dimensional
-dipolar fluids, hydrated lipids aggregate spontaneously to form
-bilayers which exhibit a variety of phases depending on their
-temperatures and compositions.  At high temperatures, the fluid
-($L_{\alpha}$) phase of Phosphatidylcholine (PC) lipids closely
-resembles a dipolar fluid.  However, at lower temperatures, packing of
-the molecules becomes important, and the translational freedom of
-lipid molecules is thought to be substantially restricted.  A
-corrugated or ``rippled'' phase ($P_{\beta'}$) appears as an
-intermediate phase between the gel ($L_\beta$) and fluid
-($L_{\alpha}$) phases for relatively pure phosphatidylcholine (PC)
-bilayers.  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}
+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}
 
-Although the results of dipolar fluid simulations can not be directly
-mapped onto the phases of lipid bilayers, the rich behaviors exhibited
-by simple dipolar models can give us some insight into the corrugation
-phenomenon of the $P_{\beta'}$ phase.  There have been a number of
-theoretical approaches (and some heroic simulations) 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.  Before we present our simple model, we will
-briefly survey the previous theoretical work on this topic.
+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.
 
-The theoretical models that have been put forward to explain the
-formation of the $P_{\beta'}$ phase have presented a number of
-conflicting but intriguing explanations. Marder {\it et al.} used a
-curvature-dependent Landau-de Gennes free-energy functional to predict
-a rippled phase.~\cite{Marder84} This model and other related
-continuum models predict higher fluidity in convex regions and that
-concave portions of the membrane correspond to more solid-like
-regions.  Carlson and Sethna used a packing-competition model (in
-which head groups and chains have competing packing energetics) to
-predict the formation of a ripple-like phase.  Their model predicted
-that the high-curvature portions have lower-chain packing and
-correspond to more fluid-like regions.  Goldstein and Leibler used a
-mean-field approach with a planar model for {\em inter-lamellar}
-interactions to predict rippling in multilamellar
-phases.~\cite{Goldstein88} McCullough and Scott proposed that the {\em
-anisotropy of the nearest-neighbor interactions} coupled to
-hydrophobic constraining forces which restrict height differences
-between nearest neighbors is the origin of the ripple
-phase.~\cite{McCullough90} Lubensky and MacKintosh introduced a Landau
-theory for tilt order and curvature of a single membrane and concluded
-that {\em coupling of molecular tilt to membrane curvature} is
-responsible for the production of ripples.~\cite{Lubensky93} Misbah,
-Duplat and Houchmandzadeh proposed that {\em inter-layer dipolar
-interactions} can lead to ripple instabilities.~\cite{Misbah98}
-Heimburg presented a {\em coexistence model} for ripple formation in
-which he postulates that fluid-phase line defects cause sharp
-curvature between relatively flat gel-phase regions.~\cite{Heimburg00}
-Kubica has suggested that a lattice model of polar head groups could
-be valuable in trying to understand bilayer phase
-formation.~\cite{Kubica02} Bannerjee used Monte Carlo simulations of
-lamellar stacks of triangular lattices to show that large headgroups
-and molecular tilt with respect to the membrane normal vector can
-cause bulk rippling.~\cite{Bannerjee02}
+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.
 
-Large-scale molecular dynamics simulations have also been performed on
-rippled phases using united atom as well as molecular scale
-models. De~Vries {\it et al.} studied the structure of lecithin ripple
-phases via molecular dynamics and their simulations seem to support
-the coexistence models (i.e. fluid-like chain dynamics was observed in
-the kink regions).~\cite{deVries05} A similar coarse-grained approach
-has been used to study the line tension of bilayer
-edges.\cite{Jiang04,deJoannis06} Ayton and Voth have found significant
-undulations in zero-surface-tension states of membranes simulated via
-dissipative particle dynamics, but their results are consistent with
-purely thermal undulations.~\cite{Ayton02} Brannigan, Tamboli and
-Brown have used a molecular scale model to elucidate the role of
-molecular shape on membrane phase behavior and
-elasticity.~\cite{Brannigan04b} They have also observed a buckled
-hexatic phase with strong tail and moderate alignment
-attractions.~\cite{Brannigan04a} 
+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 this phenomenon 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
+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.
 
-At the other extreme in density from the traditional simulations of
-dipolar fluids is the behavior of dipoles locked on regular lattices.
-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.  Therefore, it would seem unlikely that a
-frustrated lattice in a dipolar-disordered state could exhibit the
-long-range periodicity in the range of 100-600 \AA (as exhibited in
-the ripple phases studied by Kaasgard {\it et
-al.}).~\cite{Kaasgaard03}
+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}
 
-Is there an intermediate model between the low-density dipolar fluids
-and the rigid lattice models which has the potential to exhibit the
-corrugation phenomenon of the $P_{\beta'}$ phase?  What we present
-here is an attempt to find a simple dipolar model which will exhibit
-this behavior.  We are using a modified XYZ lattice model; details of
-the model can be found in section
+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.
@@ -190,27 +139,31 @@ of a two-dimensional dipolar medium.  Since molecules 
 
 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 the 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
+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{equation}
-V = \sum_i \left( \sum_{j \in NN_i}^6
-\frac{k_r}{2}\left( r_{ij}-\sigma \right)^2  +  \sum_{j>i} \frac{|\mu|^2}{4\pi \epsilon_0 r_{ij}^3} \left[
+\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)
+\right. \nonumber \\
+ & & \left. + \sum_{j \in NN_i}^6 \frac{k_r}{2}\left(
+r_{ij}-\sigma \right)^2 \right)
 \label{eq:pot}
-\end{equation}
+\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
@@ -281,16 +234,14 @@ lattices.
 triangular ($\gamma = 1.732$) and distorted ($\gamma=1.875$)
 lattices.
 
-\begin{figure}[ht]
-\centering
-\caption{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.}
-\includegraphics[width=\linewidth]{phase.pdf}
-\label{phase}
+\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.
@@ -325,16 +276,15 @@ rippled structure is shown in Fig. \ref{fig:snapshot}.
 dipole director axis}.  A snapshot of a typical antiferroelectric
 rippled structure is shown in Fig. \ref{fig:snapshot}.
 
-\begin{figure}[ht]
-\centering
-\caption{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.}
-\includegraphics[width=5.5in]{snapshot.pdf}
-\label{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
@@ -347,30 +297,30 @@ intermolecular vector $\vec{r}_{ij}$ and dipolar-axis 
 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 dipolar-axis of atom $i$,
+intermolecular vector $\vec{r}_{ij}$ and direction of dipole $i$,
 \begin{equation}
-C(r, \cos \theta) = \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
+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}[ht]
-\centering
-\caption{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.
+\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.}
-\includegraphics[width=\linewidth]{height-dipole-correlation.pdf}
-\label{fig:CrossCorrelation}
+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}
@@ -387,17 +337,17 @@ absolute value of the undulation spectrum can be writt
 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 |\vec{q}|^4 +
-\tilde{\gamma}|\vec{q}|^2},
+\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
-$\tilde{\gamma}$ 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
-($\tilde{\gamma}$), so a much simpler form can be written,
+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}{\tilde{\gamma}|\vec{q}|^2},
+\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{\gamma q^2},
 \label{eq:fit2}
 \end{equation}
 
@@ -411,38 +361,42 @@ model, an interpolated method for computing undulation
 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, an interpolated method for computing undulation spectra will be
-required.
+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 approximating 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.
+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}).
 
-For the dipolar-ordered triangular lattice near the coexistence
-temperature, however, we 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.
+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}[ht]
-\centering
-\caption{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.  Samples exhibiting only
-thermal undulations fit Eq. \ref{eq:fit} remarkably well.}
-\includegraphics[width=5.5in]{logFit.pdf}
-\label{fig:fit}
+\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}
@@ -499,17 +453,15 @@ the mean spacing between lipids. 
 However, this is coincidental agreement based on a choice of 7~\AA~as
 the mean spacing between lipids. 
 
-\begin{figure}[ht]
-\centering
-\caption{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).}
-\includegraphics[width=\linewidth]{properties_sq.pdf}
-\label{fig:Amplitude}
+\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
@@ -560,13 +512,11 @@ effects.
 also the conditions that should be most susceptible to system size
 effects.
 
-\begin{figure}[ht]
-\centering
-\caption{The ripple wavelength (top) and amplitude (bottom) as a
-function of system size for a triangular lattice ($\gamma=1.732$) at $T^{*} =
-122$.}
-\includegraphics[width=\linewidth]{SystemSize.pdf}
-\label{fig:systemsize}
+\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
@@ -654,7 +604,12 @@ rippling is dipole-driven or not.
 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.
-\clearpage
+
+\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}
+
 \bibliography{ripple}
-\printfigures
 \end{document}