--- trunk/ripple2/ripple.tex	2006/12/26 21:50:50	3091
+++ trunk/ripple2/ripple.tex	2006/12/27 22:13:09	3097
@@ -22,17 +22,17 @@ hexagonal or distorted-hexagonal lattices.  The transl
 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
-hexagonal or distorted-hexagonal lattices.  The translational freedom
-of the dipoles allows hexagonal lattices to find states that break out
+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-hexagonal.  We observe three common features of the
+effectively non-triangular.  We observe three common features of the
 corrugated dipolar membranes: 1) the corrugated phases develop easily
-when hosted on hexagonal lattices, 2) the wave vectors for the surface
+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 hexagonal lattices, the dipole director axis is found
+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}
 
@@ -87,7 +87,7 @@ hexagonal packing lattice of the lipid molecules withi
 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
-hexagonal packing lattice of the lipid molecules within the ripple
+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}
 
@@ -131,7 +131,7 @@ lamellar stacks of hexagonal lattices to show that lar
 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 hexagonal lattices to show that large headgroups
+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}
 
@@ -166,8 +166,8 @@ in dipolar systems with non-hexagonal packings.  Howev
 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-hexagonal packings.  However, {\em
-hexagonally}-packed 2-D dipolar systems are inherently frustrated and
+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
@@ -193,7 +193,7 @@ hexagonal (lattice constants $a/b = \sqrt{3}$) or non-
 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
-hexagonal (lattice constants $a/b = \sqrt{3}$) or non-hexagonal.
+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
@@ -231,18 +231,25 @@ $\sigma$ on a side) patches of membrane hosted on both
 
 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 hexagonal
-($\gamma = a/b = \sqrt{3}$) and non-hexagonal ($\gamma \neq \sqrt{3}$)
+$\sigma$ 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 hexagonal and non-hexagonal lattices).  Typical
-system sizes were 1360 dipoles for the hexagonal lattices and 840-2800
-dipoles for the non-hexagonal lattices.  Periodic boundary conditions
-were used, and the cutoff for the dipole-dipole interaction was set to
-4.3 $\sigma$.  All parameters ($T^{*}$, $\mu^{*}$, and $\gamma$) were
-varied systematically to study the effects of these parameters on the
-formation of ripple-like phases.
+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}
@@ -271,33 +278,37 @@ hexagonal ($\gamma = 1.732$) and non-hexagonal ($\gamm
 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
-hexagonal ($\gamma = 1.732$) and non-hexagonal ($\gamma=1.875$)
+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 hexagonal ($\gamma = 1.732$) and non-hexagonal
+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 hexagonal lattice is shown inset.}
+enlarged view near the triangular lattice is shown inset.}
 \includegraphics[width=\linewidth]{phase.pdf}
 \label{phase}
 \end{figure}
 
 There is a clear order-disorder transition in evidence from this data.
-Both the hexagonal and non-hexagonal lattices have dipolar-ordered
+Both the triangular and distorted lattices have dipolar-ordered
 low-temperature phases, and orientationally-disordered high
-temperature phases.  The coexistence temperature for the hexagonal
-lattice is significantly lower than for the non-hexagonal 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 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 hexagonal lattice is a
-low-temperature cusp in the $T^{*}-\gamma$ phase diagram.
+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
@@ -305,7 +316,7 @@ of the equivalent directions on the hexagonal lattice,
 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 hexagonal lattice, and the dipoles
+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
@@ -317,15 +328,51 @@ dipolar ordered phase supported on the hexagonal latti
 \begin{figure}[ht]
 \centering
 \caption{Top and Side views of a representative configuration for the
-dipolar ordered phase supported on the hexagonal lattice. Note 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 hexagonal lattice, and the ripple direction is
+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}
 \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 dipolar-axis of atom $i$,
+\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
+\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.
+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}
+\end{figure}
+
 \subsection{Discriminating Ripples from Thermal Undulations}
 
 In order to be sure that the structures we have observed are actually
@@ -336,18 +383,23 @@ where $h(\vec{r})$ is the height of the membrane at lo
 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} In simple (and more complicated) elastic
-continuum models, Brannigan {\it et al.} have shown that in the $NVT$
-ensemble, the absolute value of the undulation spectrum can be
-written,
+= (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 |\vec{q}|^4 +
 \tilde{\gamma}|\vec{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{Brannigan04b}
+$\tilde{\gamma}$ 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,
+\begin{equation}
+\langle | h(q)|^2 \rangle_{NVT} = \frac{k_B T}{\tilde{\gamma}|\vec{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 ($h(\vec{r})$)
@@ -355,24 +407,27 @@ h(q)|^2 \rangle$.
 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 $\langle |
-h(q)|^2 \rangle$.
+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, an interpolated method for computing undulation spectra will be
+required.
 
-The systems studied in this paper have relatively small bending moduli
-($k_c$) and relatively large mechanical surface tensions
-($\tilde{\gamma}$).  In practice, the best fits to our undulation
-spectra are obtained by approximating the value of $k_c$ to 0.  In
+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:fit}).  In 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 bending modulus and
-the mechanical surface tension.
+perfect, and from these fits we can estimate the tension in the
+surface.
 
-For the dipolar-ordered hexagonal lattice near the coexistence
+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 all other points, but can be off by {\em
-orders of magnitude} for a few low frequency components.
+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
@@ -386,7 +441,7 @@ thermal undulations fit Eq. \ref{eq:fit} remarkably we
 \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]{fit.pdf}
+\includegraphics[width=5.5in]{logFit.pdf}
 \label{fig:fit}
 \end{figure}
 
@@ -420,29 +475,16 @@ $h(q_{\mathrm{rip}})$ itself.
 estimated by measuring peak-to-trough distances in
 $h(q_{\mathrm{rip}})$ itself.
 
-\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.
-Anti-correlation (contours below 0) is only seen when the inter-dipole
-vector is perpendicular to the dipoles. }
-\includegraphics[width=\linewidth]{height-dipole-correlation.pdf}
-\label{fig:CrossCorrelation}
-\end{figure}
-
 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 non-hexagonal lattices).  We report
+in the amplitude estimates for the distorted lattices).  We report
 amplitudes and wavelengths taken directly from the undulation spectrum
 below.
 
-In the hexagonal lattice ($\gamma = \sqrt{3}$), the rippling is
+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^{*} =
@@ -460,11 +502,11 @@ hexagonal lattice. b) The amplitude $A^{*}$ of the rip
 \begin{figure}[ht]
 \centering
 \caption{a) The amplitude $A^{*}$ of the ripples vs. temperature for a 
-hexagonal lattice. b) The amplitude $A^{*}$ of the ripples vs. dipole
-strength ($\mu^{*}$) for both the hexagonal lattice (circles) and
-non-hexagonal lattice (squares).  The reduced temperatures were kept
-fixed at $T^{*} = 94$ for the hexagonal lattice and $T^{*} = 106$ for
-the non-hexagonal lattice (approximately 2/3 of the order-disorder
+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}
@@ -481,20 +523,20 @@ Fig. \ref{fig:Amplitude}.
 of ripple amplitude on the dipolar strength in
 Fig. \ref{fig:Amplitude}.
 
-\subsection{Non-hexagonal lattices}
+\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 hexagonal lattices prefer a particular director axis due to
+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 hexagonal
-lattices.  In non-hexagonal lattices the antiferroelectric phase can
+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 non-hexagonal lattices
+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}),
@@ -506,12 +548,12 @@ rippling is a symmetry-breaking phenomenon for hexagon
 \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 hexagonal and
-nearly-hexagonal lattices.
+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 hexagonal lattice at a reduced
+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
@@ -521,7 +563,7 @@ function of system size for a hexagonal lattice ($\gam
 \begin{figure}[ht]
 \centering
 \caption{The ripple wavelength (top) and amplitude (bottom) as a
-function of system size for a hexagonal lattice ($\gamma=1.732$) at $T^{*} =
+function of system size for a triangular lattice ($\gamma=1.732$) at $T^{*} =
 122$.}
 \includegraphics[width=\linewidth]{SystemSize.pdf}
 \label{fig:systemsize}
@@ -554,11 +596,11 @@ symmetry of the hexagonal lattice and allow the energe
 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 hexagonal lattice and allow the energetic benefit from
+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-hexagonal lattice which
+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
@@ -574,15 +616,15 @@ The packing of the dipoles into a nearly-hexagonal lat
 relative to the surface tension can cause the corrugated phase to
 disappear.
 
-The packing of the dipoles into a nearly-hexagonal lattice is clearly
+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 hexagonally 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
-hexagonally ordered phases which support ripples in realistic lipid
-bilayers.
+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
@@ -595,7 +637,7 @@ the three equivalent lattice vectors in the hexagonal 
 
 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 hexagonal lattice.
+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
