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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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\usepackage{graphicx} |
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\begin{document} |
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\renewcommand{\thefootnote}{\fnsymbol{footnote}} |
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\title{Spontaneous Corrugation of Dipolar Membranes} |
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\author{Xiuquan Sun and J. Daniel Gezelter} |
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\email[]{E-mail: gezelter@nd.edu} |
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\email[E-mail:]{gezelter@nd.edu} |
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\affiliation{Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame, \\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\begin{abstract} |
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We present a simple model for dipolar membranes that gives |
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We present a simple model for dipolar elastic membranes that gives |
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lattice-bound point dipoles complete orientational freedom as well as |
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translational freedom along one coordinate (out of the plane of the |
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membrane). There is an additional harmonic surface tension which |
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binds each of the dipoles to the six nearest neighbors on either |
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triangular or distorted lattices. The translational freedom |
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of the dipoles allows triangular lattices to find states that break out |
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of the normal orientational disorder of frustrated configurations and |
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which are stabilized by long-range antiferroelectric ordering. In |
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order to break out of the frustrated states, the dipolar membranes |
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form corrugated or ``rippled'' phases that make the lattices |
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effectively non-triangular. We observe three common features of the |
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corrugated dipolar membranes: 1) the corrugated phases develop easily |
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when hosted on triangular lattices, 2) the wave vectors for the surface |
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ripples are always found to be perpendicular to the dipole director |
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axis, and 3) on triangular lattices, the dipole director axis is found |
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to be parallel to any of the three equivalent lattice directions. |
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membrane). There is an additional harmonic term which binds each of |
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the dipoles to the six nearest neighbors on either triangular or |
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distorted lattices. The translational freedom of the dipoles allows |
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triangular lattices to find states that break out of the normal |
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orientational disorder of frustrated configurations and which are |
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stabilized by long-range antiferroelectric ordering. In order to |
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break out of the frustrated states, the dipolar membranes form |
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corrugated or ``rippled'' phases that make the lattices effectively |
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non-triangular. We observe three common features of the corrugated |
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dipolar membranes: 1) the corrugated phases develop easily when hosted |
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on triangular lattices, 2) the wave vectors for the surface ripples |
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are always found to be perpendicular to the dipole director axis, and |
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3) on triangular lattices, the dipole director axis is found to be |
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parallel to any of the three equivalent lattice directions. |
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\end{abstract} |
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\pacs{68.03.Hj, 82.20.Wt} |
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\section{Introduction} |
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\label{Int} |
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There has been intense recent interest in the phase behavior of |
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dipolar |
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fluids.\cite{Tlusty00,Teixeira00,Tavares02,Duncan04,Holm05,Duncan06} |
| 48 |
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Due to the anisotropic interactions between dipoles, dipolar fluids |
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can present anomalous phase behavior. Examples of condensed-phase |
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dipolar systems include ferrofluids, electro-rheological fluids, and |
| 51 |
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even biological membranes. Computer simulations have provided useful |
| 52 |
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information on the structural features and phase transition of the |
| 53 |
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dipolar fluids. Simulation results indicate that at low densities, |
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these fluids spontaneously organize into head-to-tail dipolar |
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``chains''.\cite{Teixeira00,Holm05} At low temperatures, these chains |
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and rings prevent the occurrence of a liquid-gas phase transition. |
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However, Tlusty and Safran showed that there is a defect-induced phase |
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separation into a low-density ``chain'' phase and a higher density |
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Y-defect phase.\cite{Tlusty00} Recently, inspired by experimental |
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studies on monolayers of dipolar fluids, theoretical models using |
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two-dimensional dipolar soft spheres have appeared in the literature. |
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Tavares {\it et al.} tested their theory for chain and ring length |
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distributions in two dimensions and carried out Monte Carlo |
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simulations in the low-density phase.\cite{Tavares02} Duncan and Camp |
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performed dynamical simulations on two-dimensional dipolar fluids to |
| 66 |
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study transport and orientational dynamics in these |
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systems.\cite{Duncan04} They have recently revisited two-dimensional |
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systems to study the kinetic conditions for the defect-induced |
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condensation into the Y-defect phase.\cite{Duncan06} |
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|
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Although they are not traditionally classified as 2-dimensional |
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dipolar fluids, hydrated lipids aggregate spontaneously to form |
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bilayers which exhibit a variety of phases depending on their |
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temperatures and compositions. At high temperatures, the fluid |
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($L_{\alpha}$) phase of Phosphatidylcholine (PC) lipids closely |
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resembles a dipolar fluid. However, at lower temperatures, packing of |
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the molecules becomes important, and the translational freedom of |
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lipid molecules is thought to be substantially restricted. A |
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corrugated or ``rippled'' phase ($P_{\beta'}$) appears as an |
| 56 |
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intermediate phase between the gel ($L_\beta$) and fluid |
| 81 |
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($L_{\alpha}$) phases for relatively pure phosphatidylcholine (PC) |
| 82 |
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bilayers. The $P_{\beta'}$ phase has attracted substantial |
| 83 |
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experimental interest over the past 30 years. Most structural |
| 84 |
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information of the ripple phase has been obtained by the X-ray |
| 85 |
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diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron |
| 86 |
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microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it |
| 87 |
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et al.} used atomic force microscopy (AFM) to observe ripple phase |
| 88 |
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morphology in bilayers supported on mica.~\cite{Kaasgaard03} The |
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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 |
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phase. This is a notable change from the observed lipid packing |
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within the gel phase.~\cite{Cevc87} |
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The properties of polymeric membranes are known to depend sensitively |
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on the details of the internal interactions between the constituent |
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monomers. A flexible membrane will always have a competition between |
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the energy of curvature and the in-plane stretching energy and will be |
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able to buckle in certain limits of surface tension and |
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temperature.\cite{Safran94} The buckling can be non-specific and |
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centered at dislocation~\cite{Seung1988} or grain-boundary |
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defects,\cite{Carraro1993} or it can be directional and cause long |
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``roof-tile'' or tube-like structures to appear in |
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partially-polymerized phospholipid vesicles.\cite{Mutz1991} |
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|
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Although the results of dipolar fluid simulations can not be directly |
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mapped onto the phases of lipid bilayers, the rich behaviors exhibited |
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by simple dipolar models can give us some insight into the corrugation |
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phenomenon of the $P_{\beta'}$ phase. There have been a number of |
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theoretical approaches (and some heroic simulations) undertaken to try |
| 63 |
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to explain this phase, but to date, none have looked specifically at |
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the contribution of the dipolar character of the lipid head groups |
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towards this corrugation. Before we present our simple model, we will |
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briefly survey the previous theoretical work on this topic. |
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One would expect that anisotropic local interactions could lead to |
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interesting properties of the buckled membrane. We report here on the |
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buckling behavior of a membrane composed of harmonically-bound, but |
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freely-rotating electrostatic dipoles. The dipoles have strongly |
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anisotropic local interactions and the membrane exhibits coupling |
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between the buckling and the long-range ordering of the dipoles. |
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|
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The theoretical models that have been put forward to explain the |
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formation of the $P_{\beta'}$ phase have presented a number of |
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conflicting but intriguing explanations. Marder {\it et al.} used a |
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curvature-dependent Landau-de Gennes free-energy functional to predict |
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a rippled phase.~\cite{Marder84} This model and other related |
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continuum models predict higher fluidity in convex regions and that |
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concave portions of the membrane correspond to more solid-like |
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regions. Carlson and Sethna used a packing-competition model (in |
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which head groups and chains have competing packing energetics) to |
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predict the formation of a ripple-like phase. Their model predicted |
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that the high-curvature portions have lower-chain packing and |
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correspond to more fluid-like regions. Goldstein and Leibler used a |
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mean-field approach with a planar model for {\em inter-lamellar} |
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interactions to predict rippling in multilamellar |
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phases.~\cite{Goldstein88} McCullough and Scott proposed that the {\em |
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anisotropy of the nearest-neighbor interactions} coupled to |
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hydrophobic constraining forces which restrict height differences |
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between nearest neighbors is the origin of the ripple |
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phase.~\cite{McCullough90} Lubensky and MacKintosh introduced a Landau |
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theory for tilt order and curvature of a single membrane and concluded |
| 85 |
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that {\em coupling of molecular tilt to membrane curvature} is |
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responsible for the production of ripples.~\cite{Lubensky93} Misbah, |
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Duplat and Houchmandzadeh proposed that {\em inter-layer dipolar |
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interactions} can lead to ripple instabilities.~\cite{Misbah98} |
| 89 |
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Heimburg presented a {\em coexistence model} for ripple formation in |
| 90 |
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which he postulates that fluid-phase line defects cause sharp |
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curvature between relatively flat gel-phase regions.~\cite{Heimburg00} |
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Kubica has suggested that a lattice model of polar head groups could |
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be valuable in trying to understand bilayer phase |
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formation.~\cite{Kubica02} Bannerjee used Monte Carlo simulations of |
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lamellar stacks of triangular lattices to show that large headgroups |
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and molecular tilt with respect to the membrane normal vector can |
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cause bulk rippling.~\cite{Bannerjee02} |
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Buckling behavior in liquid crystalline and biological membranes is a |
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well-known phenomenon. Relatively pure phosphatidylcholine (PC) |
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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 |
| 70 |
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has attracted substantial experimental interest over the past 30 |
| 71 |
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years. Most structural information of the ripple phase has been |
| 72 |
> |
obtained by the X-ray diffraction~\cite{Sun96,Katsaras00} and |
| 73 |
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freeze-fracture electron microscopy (FFEM).~\cite{Copeland80,Meyer96} |
| 74 |
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Recently, Kaasgaard {\it et al.} used atomic force microscopy (AFM) to |
| 75 |
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observe ripple phase morphology in bilayers supported on |
| 76 |
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mica.~\cite{Kaasgaard03} The experimental results provide strong |
| 77 |
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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 |
| 80 |
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been a number of theoretical |
| 81 |
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approaches~\cite{Marder84,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02} |
| 82 |
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(and some heroic |
| 83 |
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simulations~\cite{Ayton02,Jiang04,Brannigan04a,deVries05,deJoannis06}) |
| 84 |
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undertaken to try to explain this phase, but to date, none have looked |
| 85 |
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specifically at the contribution of the dipolar character of the lipid |
| 86 |
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head groups towards this corrugation. Lipid chain interdigitation |
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certainly plays a major role, and the structures of the ripple phase |
| 88 |
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are highly ordered. The model we investigate here lacks chain |
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interdigitation (as well as the chains themselves!) and will not be |
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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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Large-scale molecular dynamics simulations have also been performed on |
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rippled phases using united atom as well as molecular scale |
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models. De~Vries {\it et al.} studied the structure of lecithin ripple |
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phases via molecular dynamics and their simulations seem to support |
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the coexistence models (i.e. fluid-like chain dynamics was observed in |
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the kink regions).~\cite{deVries05} A similar coarse-grained approach |
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has been used to study the line tension of bilayer |
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edges.\cite{Jiang04,deJoannis06} Ayton and Voth have found significant |
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undulations in zero-surface-tension states of membranes simulated via |
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dissipative particle dynamics, but their results are consistent with |
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purely thermal undulations.~\cite{Ayton02} Brannigan, Tamboli and |
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Brown have used a molecular scale model to elucidate the role of |
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molecular shape on membrane phase behavior and |
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elasticity.~\cite{Brannigan04b} They have also observed a buckled |
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hexatic phase with strong tail and moderate alignment |
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attractions.~\cite{Brannigan04a} |
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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 |
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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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|
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The problem with using atomistic and even coarse-grained approaches to |
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study this phenomenon is that only a relatively small number of |
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periods of the corrugation (i.e. one or two) can be realistically |
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simulated given current technology. Also, simulations of lipid |
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bilayers are traditionally carried out with periodic boundary |
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study membrane buckling phenomena is that only a relatively small |
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number of periods of the corrugation (i.e. one or two) can be |
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realistically simulated given current technology. Also, simulations |
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of lipid bilayers are traditionally carried out with periodic boundary |
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conditions in two or three dimensions and these have the potential to |
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enhance the periodicity of the system at that wavelength. To avoid |
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this pitfall, we are using a model which allows us to have |
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sufficiently large systems so that we are not causing artificial |
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corrugation through the use of periodic boundary conditions. |
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|
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At the other extreme in density from the traditional simulations of |
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dipolar fluids is the behavior of dipoles locked on regular lattices. |
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Ferroelectric states (with long-range dipolar order) can be observed |
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in dipolar systems with non-triangular packings. However, {\em |
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triangularly}-packed 2-D dipolar systems are inherently frustrated and |
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one would expect a dipolar-disordered phase to be the lowest free |
| 121 |
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energy configuration. Therefore, it would seem unlikely that a |
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frustrated lattice in a dipolar-disordered state could exhibit the |
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long-range periodicity in the range of 100-600 \AA (as exhibited in |
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the ripple phases studied by Kaasgard {\it et |
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al.}).~\cite{Kaasgaard03} |
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The simplest dipolar membrane is one in which the dipoles are located |
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on fixed lattice sites. Ferroelectric states (with long-range dipolar |
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order) can be observed in dipolar systems with non-triangular |
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packings. However, {\em triangularly}-packed 2-D dipolar systems are |
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inherently frustrated and one would expect a dipolar-disordered phase |
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to be the lowest free energy |
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configuration.\cite{Toulouse1977,Marland1979} Dipolar lattices already |
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have rich phase behavior, but in order to allow the membrane to |
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buckle, a single degree of freedom (translation normal to the membrane |
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face) must be added to each of the dipoles. It would also be possible |
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to allow complete translational freedom. This approach |
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is similar in character to a number of elastic Ising models that have |
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been developed to explain interesting mechanical properties in |
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magnetic alloys.\cite{Renard1966,Zhu2005,Zhu2006,Jiang2006} |
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|
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Is there an intermediate model between the low-density dipolar fluids |
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and the rigid lattice models which has the potential to exhibit the |
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corrugation phenomenon of the $P_{\beta'}$ phase? What we present |
| 181 |
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here is an attempt to find a simple dipolar model which will exhibit |
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this behavior. We are using a modified XYZ lattice model; details of |
| 183 |
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the model can be found in section |
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What we present here is an attempt to find the simplest dipolar model |
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which will exhibit buckling behavior. We are using a modified XYZ |
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lattice model; details of the model can be found in section |
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\ref{sec:model}, results of Monte Carlo simulations using this model |
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are presented in section |
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\ref{sec:results}, and section \ref{sec:discussion} contains our conclusions. |
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|
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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 |
| 142 |
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of a two-dimensional dipolar medium. Since molecules in the ripple |
| 143 |
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phase have limited translational freedom, we have chosen a lattice to |
| 144 |
< |
support the dipoles in the x-y plane. The lattice may be either |
| 145 |
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triangular (lattice constants $a/b = \sqrt{3}$) or distorted. |
| 146 |
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However, each dipole has 3 degrees of freedom. They may move freely |
| 147 |
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{\em out} of the x-y plane (along the $z$ axis), and they have |
| 148 |
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complete orientational freedom ($0 <= \theta <= \pi$, $0 <= \phi < 2 |
| 142 |
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of a two-dimensional dipolar medium. Since molecules in polymerized |
| 143 |
> |
membranes and in in the $P_{\beta'}$ ripple phase have limited |
| 144 |
> |
translational freedom, we have chosen a lattice to support the dipoles |
| 145 |
> |
in the x-y plane. The lattice may be either triangular (lattice |
| 146 |
> |
constants $a/b = |
| 147 |
> |
\sqrt{3}$) or distorted. However, each dipole has 3 degrees of |
| 148 |
> |
freedom. They may move freely {\em out} of the x-y plane (along the |
| 149 |
> |
$z$ axis), and they have complete orientational freedom ($0 <= \theta |
| 150 |
> |
<= \pi$, $0 <= \phi < 2 |
| 151 |
|
\pi$). This is essentially a modified X-Y-Z model with translational |
| 152 |
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freedom along the z-axis. |
| 153 |
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|
| 154 |
|
The potential energy of the system, |
| 155 |
< |
\begin{equation} |
| 156 |
< |
V = \sum_i \left( \sum_{j \in NN_i}^6 |
| 206 |
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\frac{k_r}{2}\left( r_{ij}-\sigma \right)^2 + \sum_{j>i} \frac{|\mu|^2}{4\pi \epsilon_0 r_{ij}^3} \left[ |
| 155 |
> |
\begin{eqnarray} |
| 156 |
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V = \sum_i & & \left( \sum_{j>i} \frac{|\mu|^2}{4\pi \epsilon_0 r_{ij}^3} \left[ |
| 157 |
|
{\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j} - |
| 158 |
|
3({\mathbf{\hat u}_i} \cdot {\mathbf{\hat |
| 159 |
|
r}_{ij}})({\mathbf{\hat u}_j} \cdot {\mathbf{\hat r}_{ij}})\right] |
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< |
\right) |
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> |
\right. \nonumber \\ |
| 161 |
> |
& & \left. + \sum_{j \in NN_i}^6 \frac{k_r}{2}\left( |
| 162 |
> |
r_{ij}-\sigma \right)^2 \right) |
| 163 |
|
\label{eq:pot} |
| 164 |
< |
\end{equation} |
| 164 |
> |
\end{eqnarray} |
| 165 |
|
|
| 166 |
+ |
|
| 167 |
|
In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing |
| 168 |
|
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
| 169 |
|
pointing along the inter-dipole vector $\mathbf{r}_{ij}$. The entire |
| 234 |
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triangular ($\gamma = 1.732$) and distorted ($\gamma=1.875$) |
| 235 |
|
lattices. |
| 236 |
|
|
| 237 |
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\begin{figure}[ht] |
| 238 |
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\centering |
| 239 |
< |
\caption{Top panel: The $P_2$ dipolar order parameter as a function of |
| 240 |
< |
temperature for both triangular ($\gamma = 1.732$) and distorted |
| 241 |
< |
($\gamma = 1.875$) lattices. Bottom Panel: The phase diagram for the |
| 242 |
< |
dipolar membrane model. The line denotes the division between the |
| 243 |
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dipolar ordered (antiferroelectric) and disordered phases. An |
| 244 |
< |
enlarged view near the triangular lattice is shown inset.} |
| 292 |
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\includegraphics[width=\linewidth]{phase.pdf} |
| 293 |
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\label{phase} |
| 237 |
> |
\begin{figure} |
| 238 |
> |
\includegraphics[width=\linewidth]{phase} |
| 239 |
> |
\caption{\label{phase} Top panel: The $P_2$ dipolar order parameter as |
| 240 |
> |
a function of temperature for both triangular ($\gamma = 1.732$) and |
| 241 |
> |
distorted ($\gamma = 1.875$) lattices. Bottom Panel: The phase |
| 242 |
> |
diagram for the dipolar membrane model. The line denotes the division |
| 243 |
> |
between the dipolar ordered (antiferroelectric) and disordered phases. |
| 244 |
> |
An enlarged view near the triangular lattice is shown inset.} |
| 245 |
|
\end{figure} |
| 246 |
|
|
| 247 |
|
There is a clear order-disorder transition in evidence from this data. |
| 276 |
|
dipole director axis}. A snapshot of a typical antiferroelectric |
| 277 |
|
rippled structure is shown in Fig. \ref{fig:snapshot}. |
| 278 |
|
|
| 279 |
< |
\begin{figure}[ht] |
| 280 |
< |
\centering |
| 281 |
< |
\caption{Top and Side views of a representative configuration for the |
| 282 |
< |
dipolar ordered phase supported on the triangular lattice. Note the |
| 283 |
< |
antiferroelectric ordering and the long wavelength buckling of the |
| 284 |
< |
membrane. Dipolar ordering has been observed in all three equivalent |
| 285 |
< |
directions on the triangular lattice, and the ripple direction is |
| 286 |
< |
always perpendicular to the director axis for the dipoles.} |
| 287 |
< |
\includegraphics[width=5.5in]{snapshot.pdf} |
| 337 |
< |
\label{fig:snapshot} |
| 279 |
> |
\begin{figure} |
| 280 |
> |
\includegraphics[width=\linewidth]{snapshot} |
| 281 |
> |
\caption{\label{fig:snapshot} Top and Side views of a representative |
| 282 |
> |
configuration for the dipolar ordered phase supported on the |
| 283 |
> |
triangular lattice. Note the antiferroelectric ordering and the long |
| 284 |
> |
wavelength buckling of the membrane. Dipolar ordering has been |
| 285 |
> |
observed in all three equivalent directions on the triangular lattice, |
| 286 |
> |
and the ripple direction is always perpendicular to the director axis |
| 287 |
> |
for the dipoles.} |
| 288 |
|
\end{figure} |
| 289 |
|
|
| 290 |
|
Although the snapshot in Fig. \ref{fig:snapshot} gives the appearance |
| 297 |
|
connection between dipolar ordering and the wave vector of the ripple |
| 298 |
|
even more explicit. $C(r, \cos \theta)$ is an angle-dependent pair |
| 299 |
|
distribution function. The angle ($\theta$) is defined by the |
| 300 |
< |
intermolecular vector $\vec{r}_{ij}$ and dipolar-axis of atom $i$, |
| 300 |
> |
intermolecular vector $\vec{r}_{ij}$ and direction of dipole $i$, |
| 301 |
|
\begin{equation} |
| 302 |
< |
C(r, \cos \theta) = \langle \sum_{i} |
| 303 |
< |
\sum_{j} h_i \cdot h_j \delta(r - r_{ij}) \delta(\cos \theta_{ij} - \cos \theta)\rangle / \langle h^2 \rangle |
| 302 |
> |
C(r, \cos \theta) = \frac{\langle \sum_{i} |
| 303 |
> |
\sum_{j} h_i \cdot h_j \delta(r - r_{ij}) \delta(\cos \theta_{ij} - |
| 304 |
> |
\cos \theta)\rangle} {\langle h^2 \rangle} |
| 305 |
|
\end{equation} |
| 306 |
|
where $\cos \theta_{ij} = \hat{\mu}_{i} \cdot \hat{r}_{ij}$ and |
| 307 |
|
$\hat{r}_{ij} = \vec{r}_{ij} / r_{ij}$. Fig. \ref{fig:CrossCorrelation} |
| 308 |
|
shows contours of this correlation function for both anti-ferroelectric, rippled |
| 309 |
|
membranes as well as for the dipole-disordered portion of the phase diagram. |
| 310 |
|
|
| 311 |
< |
\begin{figure}[ht] |
| 312 |
< |
\centering |
| 313 |
< |
\caption{Contours of the height-dipole correlation function as a function |
| 314 |
< |
of the dot product between the dipole ($\hat{\mu}$) and inter-dipole |
| 315 |
< |
separation vector ($\hat{r}$) and the distance ($r$) between the dipoles. |
| 316 |
< |
Perfect height correlation (contours approaching 1) are present in the |
| 317 |
< |
ordered phase when the two dipoles are in the same head-to-tail line. |
| 311 |
> |
\begin{figure} |
| 312 |
> |
\includegraphics[width=\linewidth]{hdc} |
| 313 |
> |
\caption{\label{fig:CrossCorrelation} Contours of the height-dipole |
| 314 |
> |
correlation function as a function of the dot product between the |
| 315 |
> |
dipole ($\hat{\mu}$) and inter-dipole separation vector ($\hat{r}$) |
| 316 |
> |
and the distance ($r$) between the dipoles. Perfect height |
| 317 |
> |
correlation (contours approaching 1) are present in the ordered phase |
| 318 |
> |
when the two dipoles are in the same head-to-tail line. |
| 319 |
|
Anti-correlation (contours below 0) is only seen when the inter-dipole |
| 320 |
< |
vector is perpendicular to the dipoles. In the dipole-disordered portion |
| 321 |
< |
of the phase diagram, there is only weak correlation in the dipole direction |
| 322 |
< |
and this correlation decays rapidly to zero for intermolecular vectors that are |
| 323 |
< |
not dipole-aligned.} |
| 372 |
< |
\includegraphics[width=\linewidth]{height-dipole-correlation.pdf} |
| 373 |
< |
\label{fig:CrossCorrelation} |
| 320 |
> |
vector is perpendicular to the dipoles. In the dipole-disordered |
| 321 |
> |
portion of the phase diagram, there is only weak correlation in the |
| 322 |
> |
dipole direction and this correlation decays rapidly to zero for |
| 323 |
> |
intermolecular vectors that are not dipole-aligned.} |
| 324 |
|
\end{figure} |
| 325 |
|
|
| 326 |
|
\subsection{Discriminating Ripples from Thermal Undulations} |
| 337 |
|
elastic continuum models, it can shown that in the $NVT$ ensemble, the |
| 338 |
|
absolute value of the undulation spectrum can be written, |
| 339 |
|
\begin{equation} |
| 340 |
< |
\langle | h(q)|^2 \rangle_{NVT} = \frac{k_B T}{k_c |\vec{q}|^4 + |
| 341 |
< |
\tilde{\gamma}|\vec{q}|^2}, |
| 340 |
> |
\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{k_c q^4 + |
| 341 |
> |
\gamma q^2}, |
| 342 |
|
\label{eq:fit} |
| 343 |
|
\end{equation} |
| 344 |
< |
where $k_c$ is the bending modulus for the membrane, and |
| 345 |
< |
$\tilde{\gamma}$ is the mechanical surface tension.~\cite{Safran94} |
| 346 |
< |
The systems studied in this paper have essentially zero bending moduli |
| 347 |
< |
($k_c$) and relatively large mechanical surface tensions |
| 348 |
< |
($\tilde{\gamma}$), so a much simpler form can be written, |
| 344 |
> |
where $k_c$ is the bending modulus for the membrane, and $\gamma$ is |
| 345 |
> |
the mechanical surface tension.~\cite{Safran94} The systems studied in |
| 346 |
> |
this paper have essentially zero bending moduli ($k_c$) and relatively |
| 347 |
> |
large mechanical surface tensions ($\gamma$), so a much simpler form |
| 348 |
> |
can be written, |
| 349 |
|
\begin{equation} |
| 350 |
< |
\langle | h(q)|^2 \rangle_{NVT} = \frac{k_B T}{\tilde{\gamma}|\vec{q}|^2}, |
| 350 |
> |
\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{\gamma q^2}, |
| 351 |
|
\label{eq:fit2} |
| 352 |
|
\end{equation} |
| 353 |
|
|
| 361 |
|
lattice, one could use the heights of the lattice points themselves as |
| 362 |
|
the grid for the Fourier transform (without interpolating to a square |
| 363 |
|
grid). However, if lateral translational freedom is added to this |
| 364 |
< |
model, an interpolated method for computing undulation spectra will be |
| 365 |
< |
required. |
| 364 |
> |
model (a likely extension), an interpolated grid method for computing |
| 365 |
> |
undulation spectra will be required. |
| 366 |
|
|
| 367 |
|
As mentioned above, the best fits to our undulation spectra are |
| 368 |
< |
obtained by approximating the value of $k_c$ to 0. In |
| 369 |
< |
Fig. \ref{fig:fit} we show typical undulation spectra for two |
| 370 |
< |
different regions of the phase diagram along with their fits from the |
| 371 |
< |
Landau free energy approach (Eq. \ref{eq:fit2}). In the |
| 372 |
< |
high-temperature disordered phase, the Landau fits can be nearly |
| 373 |
< |
perfect, and from these fits we can estimate the tension in the |
| 374 |
< |
surface. |
| 368 |
> |
obtained by setting the value of $k_c$ to 0. In Fig. \ref{fig:fit} we |
| 369 |
> |
show typical undulation spectra for two different regions of the phase |
| 370 |
> |
diagram along with their fits from the Landau free energy approach |
| 371 |
> |
(Eq. \ref{eq:fit2}). In the high-temperature disordered phase, the |
| 372 |
> |
Landau fits can be nearly perfect, and from these fits we can estimate |
| 373 |
> |
the tension in the surface. In reduced units, typical values of |
| 374 |
> |
$\gamma^{*} = \gamma / \epsilon = 2500$ are obtained for the |
| 375 |
> |
disordered phase ($\gamma^{*} = 2551.7$ in the top panel of |
| 376 |
> |
Fig. \ref{fig:fit}). |
| 377 |
|
|
| 378 |
< |
For the dipolar-ordered triangular lattice near the coexistence |
| 379 |
< |
temperature, however, we observe long wavelength undulations that are |
| 380 |
< |
far outliers to the fits. That is, the Landau free energy fits are |
| 381 |
< |
well within error bars for most of the other points, but can be off by |
| 382 |
< |
{\em orders of magnitude} for a few low frequency components. |
| 378 |
> |
Typical values of $\gamma^{*}$ in the dipolar-ordered phase are much |
| 379 |
> |
higher than in the dipolar-disordered phase ($\gamma^{*} = 73,538$ in |
| 380 |
> |
the lower panel of Fig. \ref{fig:fit}). For the dipolar-ordered |
| 381 |
> |
triangular lattice near the coexistence temperature, we also observe |
| 382 |
> |
long wavelength undulations that are far outliers to the fits. That |
| 383 |
> |
is, the Landau free energy fits are well within error bars for most of |
| 384 |
> |
the other points, but can be off by {\em orders of magnitude} for a |
| 385 |
> |
few low frequency components. |
| 386 |
|
|
| 387 |
|
We interpret these outliers as evidence that these low frequency modes |
| 388 |
|
are {\em non-thermal undulations}. We take this as evidence that we |
| 389 |
|
are actually seeing a rippled phase developing in this model system. |
| 390 |
|
|
| 391 |
< |
\begin{figure}[ht] |
| 392 |
< |
\centering |
| 393 |
< |
\caption{Evidence that the observed ripples are {\em not} thermal |
| 394 |
< |
undulations is obtained from the 2-d fourier transform $\langle |
| 395 |
< |
|h^{*}(\vec{q})|^2 \rangle$ of the height profile ($\langle h^{*}(x,y) |
| 396 |
< |
\rangle$). Rippled samples show low-wavelength peaks that are |
| 397 |
< |
outliers on the Landau free energy fits. Samples exhibiting only |
| 398 |
< |
thermal undulations fit Eq. \ref{eq:fit} remarkably well.} |
| 399 |
< |
\includegraphics[width=5.5in]{logFit.pdf} |
| 445 |
< |
\label{fig:fit} |
| 391 |
> |
\begin{figure} |
| 392 |
> |
\includegraphics[width=\linewidth]{logFit} |
| 393 |
> |
\caption{\label{fig:fit} Evidence that the observed ripples are {\em |
| 394 |
> |
not} thermal undulations is obtained from the 2-d fourier transform |
| 395 |
> |
$\langle |h^{*}(\vec{q})|^2 \rangle$ of the height profile ($\langle |
| 396 |
> |
h^{*}(x,y) \rangle$). Rippled samples show low-wavelength peaks that |
| 397 |
> |
are outliers on the Landau free energy fits by an order of magnitude. |
| 398 |
> |
Samples exhibiting only thermal undulations fit Eq. \ref{eq:fit} |
| 399 |
> |
remarkably well.} |
| 400 |
|
\end{figure} |
| 401 |
|
|
| 402 |
|
\subsection{Effects of Potential Parameters on Amplitude and Wavelength} |
| 453 |
|
However, this is coincidental agreement based on a choice of 7~\AA~as |
| 454 |
|
the mean spacing between lipids. |
| 455 |
|
|
| 456 |
< |
\begin{figure}[ht] |
| 457 |
< |
\centering |
| 458 |
< |
\caption{a) The amplitude $A^{*}$ of the ripples vs. temperature for a |
| 459 |
< |
triangular lattice. b) The amplitude $A^{*}$ of the ripples vs. dipole |
| 460 |
< |
strength ($\mu^{*}$) for both the triangular lattice (circles) and |
| 461 |
< |
distorted lattice (squares). The reduced temperatures were kept |
| 462 |
< |
fixed at $T^{*} = 94$ for the triangular lattice and $T^{*} = 106$ for |
| 463 |
< |
the distorted lattice (approximately 2/3 of the order-disorder |
| 464 |
< |
transition temperature for each lattice).} |
| 511 |
< |
\includegraphics[width=\linewidth]{properties_sq.pdf} |
| 512 |
< |
\label{fig:Amplitude} |
| 456 |
> |
\begin{figure} |
| 457 |
> |
\includegraphics[width=\linewidth]{properties_sq} |
| 458 |
> |
\caption{\label{fig:Amplitude} a) The amplitude $A^{*}$ of the ripples |
| 459 |
> |
vs. temperature for a triangular lattice. b) The amplitude $A^{*}$ of |
| 460 |
> |
the ripples vs. dipole strength ($\mu^{*}$) for both the triangular |
| 461 |
> |
lattice (circles) and distorted lattice (squares). The reduced |
| 462 |
> |
temperatures were kept fixed at $T^{*} = 94$ for the triangular |
| 463 |
> |
lattice and $T^{*} = 106$ for the distorted lattice (approximately 2/3 |
| 464 |
> |
of the order-disorder transition temperature for each lattice).} |
| 465 |
|
\end{figure} |
| 466 |
|
|
| 467 |
|
The ripples can be made to disappear by increasing the internal |
| 512 |
|
also the conditions that should be most susceptible to system size |
| 513 |
|
effects. |
| 514 |
|
|
| 515 |
< |
\begin{figure}[ht] |
| 516 |
< |
\centering |
| 517 |
< |
\caption{The ripple wavelength (top) and amplitude (bottom) as a |
| 518 |
< |
function of system size for a triangular lattice ($\gamma=1.732$) at $T^{*} = |
| 519 |
< |
122$.} |
| 568 |
< |
\includegraphics[width=\linewidth]{SystemSize.pdf} |
| 569 |
< |
\label{fig:systemsize} |
| 515 |
> |
\begin{figure} |
| 516 |
> |
\includegraphics[width=\linewidth]{SystemSize} |
| 517 |
> |
\caption{\label{fig:systemsize} The ripple wavelength (top) and |
| 518 |
> |
amplitude (bottom) as a function of system size for a triangular |
| 519 |
> |
lattice ($\gamma=1.732$) at $T^{*} = 122$.} |
| 520 |
|
\end{figure} |
| 521 |
|
|
| 522 |
|
There is substantial dependence on system size for small (less than |
| 604 |
|
this rippling phenomenon will help us design more accurate molecular |
| 605 |
|
models for corrugated membranes and experiments to test whether |
| 606 |
|
rippling is dipole-driven or not. |
| 607 |
< |
\clearpage |
| 607 |
> |
|
| 608 |
> |
\begin{acknowledgments} |
| 609 |
> |
Support for this project was provided by the National Science |
| 610 |
> |
Foundation under grant CHE-0134881. The authors would like to thank |
| 611 |
> |
the reviewers for helpful comments. |
| 612 |
> |
\end{acknowledgments} |
| 613 |
> |
|
| 614 |
|
\bibliography{ripple} |
| 659 |
– |
\printfigures |
| 615 |
|
\end{document} |
