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\documentclass[aps,pre,twocolumn,amssymb,showpacs]{revtex4} |
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%\documentclass[aps,pre,preprint,amssymb,showpacs]{revtex4} |
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%\documentclass[aps,pre,twocolumn,amssymb,showpacs,floatfix]{revtex4} |
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\documentclass[aps,pre,preprint,amssymb,showpacs]{revtex4} |
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\usepackage{graphicx} |
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\begin{document} |
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distorted lattices. The translational freedom of the dipoles allows |
| 27 |
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triangular lattices to find states that break out of the normal |
| 28 |
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orientational disorder of frustrated configurations and which are |
| 29 |
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stabilized by long-range antiferroelectric ordering. In order to |
| 29 |
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stabilized by long-range anti-ferroelectric ordering. In order to |
| 30 |
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break out of the frustrated states, the dipolar membranes form |
| 31 |
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corrugated or ``rippled'' phases that make the lattices effectively |
| 32 |
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non-triangular. We observe three common features of the corrugated |
| 64 |
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|
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Buckling behavior in liquid crystalline and biological membranes is a |
| 66 |
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well-known phenomenon. Relatively pure phosphatidylcholine (PC) |
| 67 |
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bilayers are known to form a corrugated or ``rippled'' phase |
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($P_{\beta'}$) which appears as an intermediate phase between the gel |
| 69 |
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($L_\beta$) and fluid ($L_{\alpha}$) phases. The $P_{\beta'}$ phase |
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has attracted substantial experimental interest over the past 30 |
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years. Most structural information of the ripple phase has been |
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obtained by the X-ray diffraction~\cite{Sun96,Katsaras00} and |
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freeze-fracture electron microscopy (FFEM).~\cite{Copeland80,Meyer96} |
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Recently, Kaasgaard {\it et al.} used atomic force microscopy (AFM) to |
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observe ripple phase morphology in bilayers supported on |
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mica.~\cite{Kaasgaard03} The experimental results provide strong |
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support for a 2-dimensional triangular packing lattice of the lipid |
| 78 |
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molecules within the ripple phase. This is a notable change from the |
| 79 |
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observed lipid packing within the gel phase.~\cite{Cevc87} There have |
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been a number of theoretical |
| 67 |
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bilayers form a corrugated or ``rippled'' phase ($P_{\beta'}$) which |
| 68 |
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appears as an intermediate phase between the gel ($L_\beta$) and fluid |
| 69 |
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($L_{\alpha}$) phases. The $P_{\beta'}$ phase has attracted |
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substantial experimental interest over the past 30 years. Most |
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structural information of the ripple phase has been obtained by the |
| 72 |
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X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron |
| 73 |
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microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it |
| 74 |
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et al.} used atomic force microscopy (AFM) to observe ripple phase |
| 75 |
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morphology in bilayers supported on mica.~\cite{Kaasgaard03} The |
| 76 |
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experimental results provide strong support for a 2-dimensional |
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triangular packing lattice of the lipid molecules within the ripple |
| 78 |
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phase. This is a notable change from the observed lipid packing |
| 79 |
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within the gel phase.~\cite{Cevc87} There have been a number of |
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theoretical |
| 81 |
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approaches~\cite{Marder84,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02} |
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(and some heroic |
| 83 |
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simulations~\cite{Ayton02,Jiang04,Brannigan04a,deVries05,deJoannis06}) |
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detailed enough to rule in favor of (or against) any of these |
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explanations for the $P_{\beta'}$ phase. |
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|
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Another interesting properties of elastic membranes containing |
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electrostatic dipoles is the phenomenon of flexoelectricity,\cite{} |
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which is the ability of mechanical deformations of the membrane to |
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result in electrostatic organization of the membrane. This phenomenon |
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is a curvature-induced membrane polarization which can lead to |
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potential differences across a membrane. Reverse flexoelectric |
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behavior (in which applied alternating currents affect membrane |
| 100 |
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curvature) has also been observed. Explanations of the details of |
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these effects have typically utilized membrane polarization parallel |
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to the membrane normal.\cite{} |
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Membranes containing electrostatic dipoles can also exhibit the |
| 94 |
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flexoelectric effect,\cite{Todorova2004,Harden2006,Petrov2006} which |
| 95 |
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is the ability of mechanical deformations to result in electrostatic |
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organization of the membrane. This phenomenon is a curvature-induced |
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membrane polarization which can lead to potential differences across a |
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membrane. Reverse flexoelectric behavior (in which applied currents |
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effect membrane curvature) has also been observed. Explanations of |
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the details of these effects have typically utilized membrane |
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polarization perpendicular to the face of the |
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membrane,\cite{Petrov2006} and the effect has been observed in both |
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biological,\cite{Raphael2000} bent-core liquid |
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crystalline,\cite{Harden2006} and polymer-dispersed liquid crystalline |
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membranes.\cite{Todorova2004} |
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|
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The problem with using atomistic and even coarse-grained approaches to |
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study membrane buckling phenomena is that only a relatively small |
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The point of developing this model was to arrive at the simplest |
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possible theoretical model which could exhibit spontaneous corrugation |
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of a two-dimensional dipolar medium. Since molecules in polymerized |
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membranes and in in the $P_{\beta'}$ ripple phase have limited |
| 146 |
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membranes and in the $P_{\beta'}$ ripple phase have limited |
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translational freedom, we have chosen a lattice to support the dipoles |
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in the x-y plane. The lattice may be either triangular (lattice |
| 149 |
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constants $a/b = |
| 166 |
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\label{eq:pot} |
| 167 |
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\end{eqnarray} |
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|
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– |
|
| 169 |
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In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing |
| 170 |
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along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
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pointing along the inter-dipole vector $\mathbf{r}_{ij}$. The entire |
| 182 |
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reduced distances, ($r^{*} = r / \sigma$), temperatures ($T^{*} = 2 |
| 183 |
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k_B T / k_r \sigma^2$), densities ($\rho^{*} = N \sigma^2 / L_x L_y$), |
| 184 |
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and dipole moments ($\mu^{*} = \mu / \sqrt{4 \pi \epsilon_0 \sigma^5 |
| 185 |
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k_r / 2}$). |
| 185 |
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k_r / 2}$). It should be noted that the density ($\rho^{*}$) depends |
| 186 |
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only on the mean particle spacing in the $x-y$ plane; the lattice is |
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fully populated. |
| 188 |
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|
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To investigate the phase behavior of this model, we have performed a |
| 190 |
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series of Metropolis Monte Carlo simulations of moderately-sized (34.3 |
| 197 |
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system sizes were 1360 dipoles for the triangular lattices and |
| 198 |
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840-2800 dipoles for the distorted lattices. Two-dimensional periodic |
| 199 |
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boundary conditions were used, and the cutoff for the dipole-dipole |
| 200 |
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interaction was set to 4.3 $\sigma$. Since dipole-dipole interactions |
| 201 |
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decay rapidly with distance, and since the intrinsic three-dimensional |
| 202 |
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periodicity of the Ewald sum can give artifacts in 2-d systems, we |
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have chosen not to use it in these calculations. Although the Ewald |
| 204 |
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sum has been reformulated to handle 2-D |
| 205 |
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systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89} these methods |
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are computationally expensive,\cite{Spohr97,Yeh99} and are not |
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interaction was set to 4.3 $\sigma$. This cutoff is roughly 2.5 times |
| 201 |
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the typical real-space electrostatic cutoff for molecular systems. |
| 202 |
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Since dipole-dipole interactions decay rapidly with distance, and |
| 203 |
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since the intrinsic three-dimensional periodicity of the Ewald sum can |
| 204 |
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give artifacts in 2-d systems, we have chosen not to use it in these |
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calculations. Although the Ewald sum has been reformulated to handle |
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2-D systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89} these |
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methods are computationally expensive,\cite{Spohr97,Yeh99} and are not |
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necessary in this case. All parameters ($T^{*}$, $\mu^{*}$, and |
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$\gamma$) were varied systematically to study the effects of these |
| 210 |
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parameters on the formation of ripple-like phases. |
| 230 |
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for dipole $i$. $P_2$ will be $1.0$ for a perfectly-ordered system |
| 231 |
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and near $0$ for a randomized system. Note that this order parameter |
| 232 |
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is {\em not} equal to the polarization of the system. For example, |
| 233 |
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the polarization of the perfect antiferroelectric system is $0$, but |
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$P_2$ for an antiferroelectric system is $1$. The eigenvector of |
| 233 |
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the polarization of the perfect anti-ferroelectric system is $0$, but |
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$P_2$ for an anti-ferroelectric system is $1$. The eigenvector of |
| 235 |
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$\mathsf{S}$ corresponding to the largest eigenvalue is familiar as |
| 236 |
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the director axis, which can be used to determine a privileged dipolar |
| 237 |
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axis for dipole-ordered systems. The top panel in Fig. \ref{phase} |
| 245 |
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a function of temperature for both triangular ($\gamma = 1.732$) and |
| 246 |
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distorted ($\gamma = 1.875$) lattices. Bottom Panel: The phase |
| 247 |
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diagram for the dipolar membrane model. The line denotes the division |
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between the dipolar ordered (antiferroelectric) and disordered phases. |
| 248 |
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between the dipolar ordered (anti-ferroelectric) and disordered phases. |
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An enlarged view near the triangular lattice is shown inset.} |
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\end{figure} |
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|
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lattice is significantly lower than for the distorted lattices, and |
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the bulk polarization is approximately $0$ for both dipolar ordered |
| 258 |
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and disordered phases. This gives strong evidence that the dipolar |
| 259 |
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ordered phase is antiferroelectric. We have verified that this |
| 259 |
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ordered phase is anti-ferroelectric. We have verified that this |
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dipolar ordering transition is not a function of system size by |
| 261 |
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performing identical calculations with systems twice as large. The |
| 262 |
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transition is equally smooth at all system sizes that were studied. |
| 266 |
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shows that the triangular lattice is a low-temperature cusp in the |
| 267 |
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$T^{*}-\gamma$ phase diagram. |
| 268 |
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|
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This phase diagram is remarkable in that it shows an antiferroelectric |
| 270 |
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phase near $\gamma=1.732$ where one would expect lattice frustration |
| 271 |
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to result in disordered phases at all temperatures. Observations of |
| 272 |
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the configurations in this phase show clearly that the system has |
| 273 |
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accomplished dipolar orderering by forming large ripple-like |
| 274 |
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structures. We have observed antiferroelectric ordering in all three |
| 275 |
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of the equivalent directions on the triangular lattice, and the dipoles |
| 276 |
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have been observed to organize perpendicular to the membrane normal |
| 277 |
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(in the plane of the membrane). It is particularly interesting to |
| 278 |
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note that the ripple-like structures have also been observed to |
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propagate in the three equivalent directions on the lattice, but the |
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{\em direction of ripple propagation is always perpendicular to the |
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dipole director axis}. A snapshot of a typical antiferroelectric |
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rippled structure is shown in Fig. \ref{fig:snapshot}. |
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This phase diagram is remarkable in that it shows an |
| 270 |
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anti-ferroelectric phase near $\gamma=1.732$ where one would expect |
| 271 |
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lattice frustration to result in disordered phases at all |
| 272 |
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temperatures. Observations of the configurations in this phase show |
| 273 |
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clearly that the system has accomplished dipolar ordering by forming |
| 274 |
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large ripple-like structures. We have observed anti-ferroelectric |
| 275 |
> |
ordering in all three of the equivalent directions on the triangular |
| 276 |
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lattice, and the dipoles have been observed to organize perpendicular |
| 277 |
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to the membrane normal (in the plane of the membrane). It is |
| 278 |
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particularly interesting to note that the ripple-like structures have |
| 279 |
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also been observed to propagate in the three equivalent directions on |
| 280 |
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the lattice, but the {\em direction of ripple propagation is always |
| 281 |
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perpendicular to the dipole director axis}. A snapshot of a typical |
| 282 |
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anti-ferroelectric rippled structure is shown in |
| 283 |
> |
Fig. \ref{fig:snapshot}. |
| 284 |
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|
| 285 |
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\begin{figure} |
| 286 |
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\includegraphics[width=\linewidth]{snapshot} |
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\caption{\label{fig:snapshot} Top and Side views of a representative |
| 288 |
|
configuration for the dipolar ordered phase supported on the |
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triangular lattice. Note the antiferroelectric ordering and the long |
| 289 |
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triangular lattice. Note the anti-ferroelectric ordering and the long |
| 290 |
|
wavelength buckling of the membrane. Dipolar ordering has been |
| 291 |
|
observed in all three equivalent directions on the triangular lattice, |
| 292 |
|
and the ripple direction is always perpendicular to the director axis |
| 299 |
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long-wavelength phenomenon, with occasional steep drops between |
| 300 |
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adjacent lines of anti-aligned dipoles. |
| 301 |
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|
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< |
The height-dipole correlation function ($C(r, \cos \theta)$) makes the |
| 303 |
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connection between dipolar ordering and the wave vector of the ripple |
| 304 |
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even more explicit. $C(r, \cos \theta)$ is an angle-dependent pair |
| 305 |
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distribution function. The angle ($\theta$) is defined by the |
| 306 |
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intermolecular vector $\vec{r}_{ij}$ and direction of dipole $i$, |
| 302 |
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The height-dipole correlation function ($C_{\textrm{hd}}(r, \cos |
| 303 |
> |
\theta)$) makes the connection between dipolar ordering and the wave |
| 304 |
> |
vector of the ripple even more explicit. $C_{\textrm{hd}}(r, \cos |
| 305 |
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\theta)$ is an angle-dependent pair distribution function. The angle |
| 306 |
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($\theta$) is the angle between the intermolecular vector |
| 307 |
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$\vec{r}_{ij}$ and direction of dipole $i$, |
| 308 |
|
\begin{equation} |
| 309 |
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C(r, \cos \theta) = \frac{\langle \sum_{i} |
| 310 |
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\sum_{j} h_i \cdot h_j \delta(r - r_{ij}) \delta(\cos \theta_{ij} - |
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C_{\textrm{hd}} = \frac{\langle \frac{1}{n(r)} \sum_{i}\sum_{j>i} |
| 310 |
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h_i \cdot h_j \delta(r - r_{ij}) \delta(\cos \theta_{ij} - |
| 311 |
|
\cos \theta)\rangle} {\langle h^2 \rangle} |
| 312 |
|
\end{equation} |
| 313 |
|
where $\cos \theta_{ij} = \hat{\mu}_{i} \cdot \hat{r}_{ij}$ and |
| 314 |
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$\hat{r}_{ij} = \vec{r}_{ij} / r_{ij}$. Fig. \ref{fig:CrossCorrelation} |
| 315 |
< |
shows contours of this correlation function for both anti-ferroelectric, rippled |
| 316 |
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membranes as well as for the dipole-disordered portion of the phase diagram. |
| 314 |
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$\hat{r}_{ij} = \vec{r}_{ij} / r_{ij}$. $n(r)$ is the number of |
| 315 |
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dipoles found in a cylindrical shell between $r$ and $r+\delta r$ of |
| 316 |
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the central particle. Fig. \ref{fig:CrossCorrelation} shows contours |
| 317 |
> |
of this correlation function for both anti-ferroelectric, rippled |
| 318 |
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membranes as well as for the dipole-disordered portion of the phase |
| 319 |
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diagram. |
| 320 |
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|
| 321 |
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\begin{figure} |
| 322 |
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\includegraphics[width=\linewidth]{hdc} |
| 333 |
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intermolecular vectors that are not dipole-aligned.} |
| 334 |
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\end{figure} |
| 335 |
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|
| 336 |
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The height-dipole correlation function gives a map of how the topology |
| 337 |
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of the membrane surface varies with angular deviation around a given |
| 338 |
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dipole. The upper panel of Fig. \ref{fig:CrossCorrelation} shows that |
| 339 |
+ |
in the anti-ferroelectric phase, the dipole heights are strongly |
| 340 |
+ |
correlated for dipoles in head-to-tail arrangements, and this |
| 341 |
+ |
correlation persists for very long distances (up to 15 $\sigma$). For |
| 342 |
+ |
portions of the membrane located perpendicular to a given dipole, the |
| 343 |
+ |
membrane height becomes anti-correlated at distances of 10 $\sigma$. |
| 344 |
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The correlation function is relatively smooth; there are no steep |
| 345 |
+ |
jumps or steps, so the stair-like structures in |
| 346 |
+ |
Fig. \ref{fig:snapshot} are indeed transient and disappear when |
| 347 |
+ |
averaged over many configurations. In the dipole-disordered phase, |
| 348 |
+ |
the height-dipole correlation function is relatively flat (and hovers |
| 349 |
+ |
near zero). The only significant height correlations are for axial |
| 350 |
+ |
dipoles at very short distances ($r \approx |
| 351 |
+ |
\sigma$). |
| 352 |
+ |
|
| 353 |
|
\subsection{Discriminating Ripples from Thermal Undulations} |
| 354 |
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|
| 355 |
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In order to be sure that the structures we have observed are actually |
| 418 |
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\begin{figure} |
| 419 |
|
\includegraphics[width=\linewidth]{logFit} |
| 420 |
|
\caption{\label{fig:fit} Evidence that the observed ripples are {\em |
| 421 |
< |
not} thermal undulations is obtained from the 2-d fourier transform |
| 421 |
> |
not} thermal undulations is obtained from the 2-d Fourier transform |
| 422 |
|
$\langle |h^{*}(\vec{q})|^2 \rangle$ of the height profile ($\langle |
| 423 |
|
h^{*}(x,y) \rangle$). Rippled samples show low-wavelength peaks that |
| 424 |
|
are outliers on the Landau free energy fits by an order of magnitude. |
| 452 |
|
axis by projecting heights of the dipoles to obtain a one-dimensional |
| 453 |
|
height profile, $h(q_{\mathrm{rip}})$. Ripple wavelengths can be |
| 454 |
|
estimated from the largest non-thermal low-frequency component in the |
| 455 |
< |
fourier transform of $h(q_{\mathrm{rip}})$. Amplitudes can be |
| 455 |
> |
Fourier transform of $h(q_{\mathrm{rip}})$. Amplitudes can be |
| 456 |
|
estimated by measuring peak-to-trough distances in |
| 457 |
|
$h(q_{\mathrm{rip}})$ itself. |
| 458 |
|
|
| 492 |
|
\end{figure} |
| 493 |
|
|
| 494 |
|
The ripples can be made to disappear by increasing the internal |
| 495 |
< |
surface tension (i.e. by increasing $k_r$ or equivalently, reducing |
| 495 |
> |
elastic tension (i.e. by increasing $k_r$ or equivalently, reducing |
| 496 |
|
the dipole moment). The amplitude of the ripples depends critically |
| 497 |
|
on the strength of the dipole moments ($\mu^{*}$) in Eq. \ref{eq:pot}. |
| 498 |
|
If the dipoles are weakened substantially (below $\mu^{*}$ = 20) at a |
| 506 |
|
|
| 507 |
|
We have also investigated the effect of the lattice geometry by |
| 508 |
|
changing the ratio of lattice constants ($\gamma$) while keeping the |
| 509 |
< |
average nearest-neighbor spacing constant. The antiferroelectric state |
| 509 |
> |
average nearest-neighbor spacing constant. The anti-ferroelectric state |
| 510 |
|
is accessible for all $\gamma$ values we have used, although the |
| 511 |
|
distorted triangular lattices prefer a particular director axis due to |
| 512 |
|
the anisotropy of the lattice. |
| 513 |
|
|
| 514 |
|
Our observation of rippling behavior was not limited to the triangular |
| 515 |
< |
lattices. In distorted lattices the antiferroelectric phase can |
| 515 |
> |
lattices. In distorted lattices the anti-ferroelectric phase can |
| 516 |
|
develop nearly instantaneously in the Monte Carlo simulations, and |
| 517 |
|
these dipolar-ordered phases tend to be remarkably flat. Whenever |
| 518 |
|
rippling has been observed in these distorted lattices |
| 568 |
|
|
| 569 |
|
We have been able to show that a simple dipolar lattice model which |
| 570 |
|
contains only molecular packing (from the lattice), anisotropy (in the |
| 571 |
< |
form of electrostatic dipoles) and a weak surface tension (in the form |
| 571 |
> |
form of electrostatic dipoles) and a weak elastic tension (in the form |
| 572 |
|
of a nearest-neighbor harmonic potential) is capable of exhibiting |
| 573 |
|
stable long-wavelength non-thermal surface corrugations. The best |
| 574 |
|
explanation for this behavior is that the ability of the dipoles to |
| 575 |
|
translate out of the plane of the membrane is enough to break the |
| 576 |
< |
symmetry of the triangular lattice and allow the energetic benefit from |
| 577 |
< |
the formation of a bulk antiferroelectric phase. Were the weak |
| 578 |
< |
surface tension absent from our model, it would be possible for the |
| 576 |
> |
symmetry of the triangular lattice and allow the energetic benefit |
| 577 |
> |
from the formation of a bulk anti-ferroelectric phase. Were the weak |
| 578 |
> |
elastic tension absent from our model, it would be possible for the |
| 579 |
|
entire lattice to ``tilt'' using $z$-translation. Tilting the lattice |
| 580 |
|
in this way would yield an effectively non-triangular lattice which |
| 581 |
< |
would avoid dipolar frustration altogether. With the surface tension |
| 582 |
< |
in place, bulk tilt causes a large strain, and the simplest way to |
| 583 |
< |
release this strain is along line defects. Line defects will result |
| 584 |
< |
in rippled or sawtooth patterns in the membrane, and allow small |
| 585 |
< |
``stripes'' of membrane to form antiferroelectric regions that are |
| 586 |
< |
tilted relative to the averaged membrane normal. |
| 581 |
> |
would avoid dipolar frustration altogether. With the elastic tension |
| 582 |
> |
in place, bulk tilt causes a large strain, and the least costly way to |
| 583 |
> |
release this strain is between two rows of anti-aligned dipoles. |
| 584 |
> |
These ``breaks'' will result in rippled or sawtooth patterns in the |
| 585 |
> |
membrane, and allow small stripes of membrane to form |
| 586 |
> |
anti-ferroelectric regions that are tilted relative to the averaged |
| 587 |
> |
membrane normal. |
| 588 |
|
|
| 589 |
|
Although the dipole-dipole interaction is the major driving force for |
| 590 |
|
the long range orientational ordered state, the formation of the |
| 591 |
|
stable, smooth ripples is a result of the competition between the |
| 592 |
< |
surface tension and the dipole-dipole interactions. This statement is |
| 592 |
> |
elastic tension and the dipole-dipole interactions. This statement is |
| 593 |
|
supported by the variation in $\mu^{*}$. Substantially weaker dipoles |
| 594 |
|
relative to the surface tension can cause the corrugated phase to |
| 595 |
|
disappear. |
| 597 |
|
The packing of the dipoles into a nearly-triangular lattice is clearly |
| 598 |
|
an important piece of the puzzle. The dipolar head groups of lipid |
| 599 |
|
molecules are sterically (as well as electrostatically) anisotropic, |
| 600 |
< |
and would not be able to pack in triangular arrangements without the |
| 601 |
< |
steric interference of adjacent molecular bodies. Since we only see |
| 602 |
< |
rippled phases in the neighborhood of $\gamma=\sqrt{3}$, this implies |
| 603 |
< |
that there is a role played by the lipid chains in the organization of |
| 604 |
< |
the triangularly ordered phases which support ripples in realistic |
| 605 |
< |
lipid bilayers. |
| 600 |
> |
and would not pack in triangular arrangements without the steric |
| 601 |
> |
interference of adjacent molecular bodies. Since we only see rippled |
| 602 |
> |
phases in the neighborhood of $\gamma=\sqrt{3}$, this implies that |
| 603 |
> |
even if this dipolar mechanism is the correct explanation for the |
| 604 |
> |
ripple phase in realistic bilayers, there would still be a role played |
| 605 |
> |
by the lipid chains in the in-plane organization of the triangularly |
| 606 |
> |
ordered phases which could support ripples. The present model is |
| 607 |
> |
certainly not detailed enough to answer exactly what drives the |
| 608 |
> |
formation of the $P_{\beta'}$ phase in real lipids, but suggests some |
| 609 |
> |
avenues for further experiments. |
| 610 |
|
|
| 611 |
|
The most important prediction we can make using the results from this |
| 612 |
|
simple model is that if dipolar ordering is driving the surface |
| 630 |
|
behaviors. It would clearly be a closer approximation to the reality |
| 631 |
|
if we allowed greater translational freedom to the dipoles and |
| 632 |
|
replaced the somewhat artificial lattice packing and the harmonic |
| 633 |
< |
``surface tension'' with more realistic molecular modeling |
| 634 |
< |
potentials. What we have done is to present an extremely simple model |
| 635 |
< |
which exhibits bulk non-thermal corrugation, and our explanation of |
| 636 |
< |
this rippling phenomenon will help us design more accurate molecular |
| 637 |
< |
models for corrugated membranes and experiments to test whether |
| 638 |
< |
rippling is dipole-driven or not. |
| 633 |
> |
elastic tension with more realistic molecular modeling potentials. |
| 634 |
> |
What we have done is to present a simple model which exhibits bulk |
| 635 |
> |
non-thermal corrugation, and our explanation of this rippling |
| 636 |
> |
phenomenon will help us design more accurate molecular models for |
| 637 |
> |
corrugated membranes and experiments to test whether rippling is |
| 638 |
> |
dipole-driven or not. |
| 639 |
|
|
| 640 |
|
\begin{acknowledgments} |
| 641 |
|
Support for this project was provided by the National Science |
