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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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\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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hexagonal or distorted-hexagonal lattices. The translational freedom |
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of the dipoles allows hexagonal 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-hexagonal. 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 hexagonal 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 hexagonal 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 anti-ferroelectric 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} |
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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 |
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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 |
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intermediate phase between the gel ($L_\beta$) and fluid |
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($L_{\alpha}$) phases for relatively pure phosphatidylcholine (PC) |
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bilayers. The $P_{\beta'}$ phase has attracted substantial |
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experimental interest over the past 30 years. Most structural |
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information of the ripple phase has been obtained by the X-ray |
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diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron |
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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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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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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 form a corrugated or ``rippled'' phase ($P_{\beta'}$) which |
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appears as an intermediate phase between the gel ($L_\beta$) and fluid |
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($L_{\alpha}$) phases. The $P_{\beta'}$ phase has attracted |
| 70 |
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substantial experimental interest over the past 30 years. Most |
| 71 |
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structural information of the ripple phase has been obtained by the |
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X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron |
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microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it |
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et al.} used atomic force microscopy (AFM) to observe ripple phase |
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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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hexagonal packing lattice of the lipid molecules within the ripple |
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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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within the gel phase.~\cite{Cevc87} There have been a number of |
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theoretical |
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approaches~\cite{Marder84,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02} |
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(and some heroic |
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simulations~\cite{Ayton02,Jiang04,Brannigan04a,deVries05,deJoannis06}) |
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undertaken to try to explain this phase, but to date, none have looked |
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specifically at the contribution of the dipolar character of the lipid |
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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 |
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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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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 |
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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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|
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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 |
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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} |
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Heimburg presented a {\em coexistence model} for ripple formation in |
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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 hexagonal 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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|
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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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Membranes containing electrostatic dipoles can also exhibit the |
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flexoelectric effect,\cite{Todorova2004,Harden2006,Petrov2006} which |
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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 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-hexagonal packings. However, {\em |
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hexagonally}-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 |
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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 |
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of a two-dimensional dipolar medium. Since molecules in the ripple |
| 146 |
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phase have limited translational freedom, we have chosen a lattice to |
| 147 |
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support the dipoles in the x-y plane. The lattice may be either |
| 148 |
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hexagonal (lattice constants $a/b = \sqrt{3}$) or non-hexagonal. |
| 149 |
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However, each dipole has 3 degrees of freedom. They may move freely |
| 150 |
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{\em out} of the x-y plane (along the $z$ axis), and they have |
| 151 |
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complete orientational freedom ($0 <= \theta <= \pi$, $0 <= \phi < 2 |
| 145 |
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of a two-dimensional dipolar medium. Since molecules in polymerized |
| 146 |
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membranes and in the $P_{\beta'}$ ripple phase have limited |
| 147 |
> |
translational freedom, we have chosen a lattice to support the dipoles |
| 148 |
> |
in the x-y plane. The lattice may be either triangular (lattice |
| 149 |
> |
constants $a/b = |
| 150 |
> |
\sqrt{3}$) or distorted. However, each dipole has 3 degrees of |
| 151 |
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freedom. They may move freely {\em out} of the x-y plane (along the |
| 152 |
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$z$ axis), and they have complete orientational freedom ($0 <= \theta |
| 153 |
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<= \pi$, $0 <= \phi < 2 |
| 154 |
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\pi$). This is essentially a modified X-Y-Z model with translational |
| 155 |
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freedom along the z-axis. |
| 156 |
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|
| 157 |
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The potential energy of the system, |
| 158 |
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\begin{equation} |
| 159 |
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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[ |
| 158 |
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\begin{eqnarray} |
| 159 |
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V = \sum_i & & \left( \sum_{j>i} \frac{|\mu|^2}{4\pi \epsilon_0 r_{ij}^3} \left[ |
| 160 |
|
{\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j} - |
| 161 |
|
3({\mathbf{\hat u}_i} \cdot {\mathbf{\hat |
| 162 |
|
r}_{ij}})({\mathbf{\hat u}_j} \cdot {\mathbf{\hat r}_{ij}})\right] |
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\right) |
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\right. \nonumber \\ |
| 164 |
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& & \left. + \sum_{j \in NN_i}^6 \frac{k_r}{2}\left( |
| 165 |
> |
r_{ij}-\sigma \right)^2 \right) |
| 166 |
|
\label{eq:pot} |
| 167 |
< |
\end{equation} |
| 167 |
> |
\end{eqnarray} |
| 168 |
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|
| 169 |
|
In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing |
| 170 |
|
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
| 182 |
|
reduced distances, ($r^{*} = r / \sigma$), temperatures ($T^{*} = 2 |
| 183 |
|
k_B T / k_r \sigma^2$), densities ($\rho^{*} = N \sigma^2 / L_x L_y$), |
| 184 |
|
and dipole moments ($\mu^{*} = \mu / \sqrt{4 \pi \epsilon_0 \sigma^5 |
| 185 |
< |
k_r / 2}$). |
| 185 |
> |
k_r / 2}$). It should be noted that the density ($\rho^{*}$) depends |
| 186 |
> |
only on the mean particle spacing in the $x-y$ plane; the lattice is |
| 187 |
> |
fully populated. |
| 188 |
|
|
| 189 |
|
To investigate the phase behavior of this model, we have performed a |
| 190 |
|
series of Metropolis Monte Carlo simulations of moderately-sized (34.3 |
| 191 |
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$\sigma$ on a side) patches of membrane hosted on both hexagonal |
| 192 |
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($\gamma = a/b = \sqrt{3}$) and non-hexagonal ($\gamma \neq \sqrt{3}$) |
| 191 |
> |
$\sigma$ on a side) patches of membrane hosted on both triangular |
| 192 |
> |
($\gamma = a/b = \sqrt{3}$) and distorted ($\gamma \neq \sqrt{3}$) |
| 193 |
|
lattices. The linear extent of one edge of the monolayer was $20 a$ |
| 194 |
|
and the system was kept roughly square. The average distance that |
| 195 |
|
coplanar dipoles were positioned from their six nearest neighbors was |
| 196 |
< |
1 $\sigma$ (on both hexagonal and non-hexagonal lattices). Typical |
| 197 |
< |
system sizes were 1360 dipoles for the hexagonal lattices and 840-2800 |
| 198 |
< |
dipoles for the non-hexagonal lattices. Periodic boundary conditions |
| 199 |
< |
were used, and the cutoff for the dipole-dipole interaction was set to |
| 200 |
< |
4.3 $\sigma$. All parameters ($T^{*}$, $\mu^{*}$, and $\gamma$) were |
| 201 |
< |
varied systematically to study the effects of these parameters on the |
| 202 |
< |
formation of ripple-like phases. |
| 196 |
> |
1 $\sigma$ (on both triangular and distorted lattices). Typical |
| 197 |
> |
system sizes were 1360 dipoles for the triangular lattices and |
| 198 |
> |
840-2800 dipoles for the distorted lattices. Two-dimensional periodic |
| 199 |
> |
boundary conditions were used, and the cutoff for the dipole-dipole |
| 200 |
> |
interaction was set to 4.3 $\sigma$. This cutoff is roughly 2.5 times |
| 201 |
> |
the typical real-space electrostatic cutoff for molecular systems. |
| 202 |
> |
Since dipole-dipole interactions decay rapidly with distance, and |
| 203 |
> |
since the intrinsic three-dimensional periodicity of the Ewald sum can |
| 204 |
> |
give artifacts in 2-d systems, we have chosen not to use it in these |
| 205 |
> |
calculations. Although the Ewald sum has been reformulated to handle |
| 206 |
> |
2-D systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89} these |
| 207 |
> |
methods are computationally expensive,\cite{Spohr97,Yeh99} and are not |
| 208 |
> |
necessary in this case. All parameters ($T^{*}$, $\mu^{*}$, and |
| 209 |
> |
$\gamma$) were varied systematically to study the effects of these |
| 210 |
> |
parameters on the formation of ripple-like phases. |
| 211 |
|
|
| 212 |
|
\section{Results and Analysis} |
| 213 |
|
\label{sec:results} |
| 230 |
|
for dipole $i$. $P_2$ will be $1.0$ for a perfectly-ordered system |
| 231 |
|
and near $0$ for a randomized system. Note that this order parameter |
| 232 |
|
is {\em not} equal to the polarization of the system. For example, |
| 233 |
< |
the polarization of the perfect antiferroelectric system is $0$, but |
| 234 |
< |
$P_2$ for an antiferroelectric system is $1$. The eigenvector of |
| 233 |
> |
the polarization of the perfect anti-ferroelectric system is $0$, but |
| 234 |
> |
$P_2$ for an anti-ferroelectric system is $1$. The eigenvector of |
| 235 |
|
$\mathsf{S}$ corresponding to the largest eigenvalue is familiar as |
| 236 |
|
the director axis, which can be used to determine a privileged dipolar |
| 237 |
|
axis for dipole-ordered systems. The top panel in Fig. \ref{phase} |
| 238 |
|
shows the values of $P_2$ as a function of temperature for both |
| 239 |
< |
hexagonal ($\gamma = 1.732$) and non-hexagonal ($\gamma=1.875$) |
| 239 |
> |
triangular ($\gamma = 1.732$) and distorted ($\gamma=1.875$) |
| 240 |
|
lattices. |
| 241 |
|
|
| 242 |
< |
\begin{figure}[ht] |
| 243 |
< |
\centering |
| 244 |
< |
\caption{Top panel: The $P_2$ dipolar order parameter as a function of |
| 245 |
< |
temperature for both hexagonal ($\gamma = 1.732$) and non-hexagonal |
| 246 |
< |
($\gamma = 1.875$) lattices. Bottom Panel: The phase diagram for the |
| 247 |
< |
dipolar membrane model. The line denotes the division between the |
| 248 |
< |
dipolar ordered (antiferroelectric) and disordered phases. An |
| 249 |
< |
enlarged view near the hexagonal lattice is shown inset.} |
| 285 |
< |
\includegraphics[width=\linewidth]{phase.pdf} |
| 286 |
< |
\label{phase} |
| 242 |
> |
\begin{figure} |
| 243 |
> |
\includegraphics[width=\linewidth]{phase} |
| 244 |
> |
\caption{\label{phase} Top panel: The $P_2$ dipolar order parameter as |
| 245 |
> |
a function of temperature for both triangular ($\gamma = 1.732$) and |
| 246 |
> |
distorted ($\gamma = 1.875$) lattices. Bottom Panel: The phase |
| 247 |
> |
diagram for the dipolar membrane model. The line denotes the division |
| 248 |
> |
between the dipolar ordered (anti-ferroelectric) and disordered phases. |
| 249 |
> |
An enlarged view near the triangular lattice is shown inset.} |
| 250 |
|
\end{figure} |
| 251 |
|
|
| 252 |
|
There is a clear order-disorder transition in evidence from this data. |
| 253 |
< |
Both the hexagonal and non-hexagonal lattices have dipolar-ordered |
| 253 |
> |
Both the triangular and distorted lattices have dipolar-ordered |
| 254 |
|
low-temperature phases, and orientationally-disordered high |
| 255 |
< |
temperature phases. The coexistence temperature for the hexagonal |
| 256 |
< |
lattice is significantly lower than for the non-hexagonal lattices, |
| 257 |
< |
and the bulk polarization is approximately $0$ for both dipolar |
| 258 |
< |
ordered and disordered phases. This gives strong evidence that the |
| 259 |
< |
dipolar ordered phase is antiferroelectric. We have repeated the |
| 260 |
< |
Monte Carlo simulations over a wide range of lattice ratios ($\gamma$) |
| 261 |
< |
to generate a dipolar order/disorder phase diagram. The bottom panel |
| 262 |
< |
in Fig. \ref{phase} shows that the hexagonal lattice is a |
| 263 |
< |
low-temperature cusp in the $T^{*}-\gamma$ phase diagram. |
| 255 |
> |
temperature phases. The coexistence temperature for the triangular |
| 256 |
> |
lattice is significantly lower than for the distorted lattices, and |
| 257 |
> |
the bulk polarization is approximately $0$ for both dipolar ordered |
| 258 |
> |
and disordered phases. This gives strong evidence that the dipolar |
| 259 |
> |
ordered phase is anti-ferroelectric. We have verified that this |
| 260 |
> |
dipolar ordering transition is not a function of system size by |
| 261 |
> |
performing identical calculations with systems twice as large. The |
| 262 |
> |
transition is equally smooth at all system sizes that were studied. |
| 263 |
> |
Additionally, we have repeated the Monte Carlo simulations over a wide |
| 264 |
> |
range of lattice ratios ($\gamma$) to generate a dipolar |
| 265 |
> |
order/disorder phase diagram. The bottom panel in Fig. \ref{phase} |
| 266 |
> |
shows that the triangular lattice is a low-temperature cusp in the |
| 267 |
> |
$T^{*}-\gamma$ phase diagram. |
| 268 |
|
|
| 269 |
< |
This phase diagram is remarkable in that it shows an antiferroelectric |
| 270 |
< |
phase near $\gamma=1.732$ where one would expect lattice frustration |
| 271 |
< |
to result in disordered phases at all temperatures. Observations of |
| 272 |
< |
the configurations in this phase show clearly that the system has |
| 273 |
< |
accomplished dipolar orderering by forming large ripple-like |
| 274 |
< |
structures. We have observed antiferroelectric ordering in all three |
| 275 |
< |
of the equivalent directions on the hexagonal lattice, and the dipoles |
| 276 |
< |
have been observed to organize perpendicular to the membrane normal |
| 277 |
< |
(in the plane of the membrane). It is particularly interesting to |
| 278 |
< |
note that the ripple-like structures have also been observed to |
| 279 |
< |
propagate in the three equivalent directions on the lattice, but the |
| 280 |
< |
{\em direction of ripple propagation is always perpendicular to the |
| 281 |
< |
dipole director axis}. A snapshot of a typical antiferroelectric |
| 282 |
< |
rippled structure is shown in Fig. \ref{fig:snapshot}. |
| 269 |
> |
This phase diagram is remarkable in that it shows an |
| 270 |
> |
anti-ferroelectric phase near $\gamma=1.732$ where one would expect |
| 271 |
> |
lattice frustration to result in disordered phases at all |
| 272 |
> |
temperatures. Observations of the configurations in this phase show |
| 273 |
> |
clearly that the system has accomplished dipolar ordering by forming |
| 274 |
> |
large ripple-like structures. We have observed anti-ferroelectric |
| 275 |
> |
ordering in all three of the equivalent directions on the triangular |
| 276 |
> |
lattice, and the dipoles have been observed to organize perpendicular |
| 277 |
> |
to the membrane normal (in the plane of the membrane). It is |
| 278 |
> |
particularly interesting to note that the ripple-like structures have |
| 279 |
> |
also been observed to propagate in the three equivalent directions on |
| 280 |
> |
the lattice, but the {\em direction of ripple propagation is always |
| 281 |
> |
perpendicular to the dipole director axis}. A snapshot of a typical |
| 282 |
> |
anti-ferroelectric rippled structure is shown in |
| 283 |
> |
Fig. \ref{fig:snapshot}. |
| 284 |
|
|
| 285 |
< |
\begin{figure}[ht] |
| 286 |
< |
\centering |
| 287 |
< |
\caption{Top and Side views of a representative configuration for the |
| 288 |
< |
dipolar ordered phase supported on the hexagonal lattice. Note the |
| 289 |
< |
antiferroelectric ordering and the long wavelength buckling of the |
| 290 |
< |
membrane. Dipolar ordering has been observed in all three equivalent |
| 291 |
< |
directions on the hexagonal lattice, and the ripple direction is |
| 292 |
< |
always perpendicular to the director axis for the dipoles.} |
| 293 |
< |
\includegraphics[width=5.5in]{snapshot.pdf} |
| 326 |
< |
\label{fig:snapshot} |
| 285 |
> |
\begin{figure} |
| 286 |
> |
\includegraphics[width=\linewidth]{snapshot} |
| 287 |
> |
\caption{\label{fig:snapshot} Top and Side views of a representative |
| 288 |
> |
configuration for the dipolar ordered phase supported on the |
| 289 |
> |
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 |
| 293 |
> |
for the dipoles.} |
| 294 |
|
\end{figure} |
| 295 |
|
|
| 296 |
+ |
Although the snapshot in Fig. \ref{fig:snapshot} gives the appearance |
| 297 |
+ |
of three-row stair-like structures, these appear to be transient. On |
| 298 |
+ |
average, the corrugation of the membrane is a relatively smooth, |
| 299 |
+ |
long-wavelength phenomenon, with occasional steep drops between |
| 300 |
+ |
adjacent lines of anti-aligned dipoles. |
| 301 |
+ |
|
| 302 |
+ |
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. $C_{\textrm{hd}}(r, \cos |
| 305 |
+ |
\theta)$ is an angle-dependent pair distribution function. The angle |
| 306 |
+ |
($\theta$) is the angle between the intermolecular vector |
| 307 |
+ |
$\vec{r}_{ij}$ and direction of dipole $i$, |
| 308 |
+ |
\begin{equation} |
| 309 |
+ |
C_{\textrm{hd}} = \frac{\langle \frac{1}{n(r)} \sum_{i}\sum_{j>i} |
| 310 |
+ |
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 |
+ |
$\hat{r}_{ij} = \vec{r}_{ij} / r_{ij}$. $n(r)$ is the number of |
| 315 |
+ |
dipoles found in a cylindrical shell between $r$ and $r+\delta r$ of |
| 316 |
+ |
the central particle. Fig. \ref{fig:CrossCorrelation} shows contours |
| 317 |
+ |
of this correlation function for both anti-ferroelectric, rippled |
| 318 |
+ |
membranes as well as for the dipole-disordered portion of the phase |
| 319 |
+ |
diagram. |
| 320 |
+ |
|
| 321 |
+ |
\begin{figure} |
| 322 |
+ |
\includegraphics[width=\linewidth]{hdc} |
| 323 |
+ |
\caption{\label{fig:CrossCorrelation} Contours of the height-dipole |
| 324 |
+ |
correlation function as a function of the dot product between the |
| 325 |
+ |
dipole ($\hat{\mu}$) and inter-dipole separation vector ($\hat{r}$) |
| 326 |
+ |
and the distance ($r$) between the dipoles. Perfect height |
| 327 |
+ |
correlation (contours approaching 1) are present in the ordered phase |
| 328 |
+ |
when the two dipoles are in the same head-to-tail line. |
| 329 |
+ |
Anti-correlation (contours below 0) is only seen when the inter-dipole |
| 330 |
+ |
vector is perpendicular to the dipoles. In the dipole-disordered |
| 331 |
+ |
portion of the phase diagram, there is only weak correlation in the |
| 332 |
+ |
dipole direction and this correlation decays rapidly to zero for |
| 333 |
+ |
intermolecular vectors that are not dipole-aligned.} |
| 334 |
+ |
\end{figure} |
| 335 |
+ |
|
| 336 |
+ |
The height-dipole correlation function gives a map of how the topology |
| 337 |
+ |
of the membrane surface varies with angular deviation around a given |
| 338 |
+ |
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 |
+ |
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 |
|
|
| 355 |
|
In order to be sure that the structures we have observed are actually |
| 360 |
|
h(\vec{r}) e^{-i \vec{q}\cdot\vec{r}} |
| 361 |
|
\end{equation} |
| 362 |
|
where $h(\vec{r})$ is the height of the membrane at location $\vec{r} |
| 363 |
< |
= (x,y)$.~\cite{Safran94} In simple (and more complicated) elastic |
| 364 |
< |
continuum models, Brannigan {\it et al.} have shown that in the $NVT$ |
| 365 |
< |
ensemble, the absolute value of the undulation spectrum can be |
| 342 |
< |
written, |
| 363 |
> |
= (x,y)$.~\cite{Safran94,Seifert97} In simple (and more complicated) |
| 364 |
> |
elastic continuum models, it can shown that in the $NVT$ ensemble, the |
| 365 |
> |
absolute value of the undulation spectrum can be written, |
| 366 |
|
\begin{equation} |
| 367 |
< |
\langle | h(q)|^2 \rangle_{NVT} = \frac{k_B T}{k_c |\vec{q}|^4 + |
| 368 |
< |
\tilde{\gamma}|\vec{q}|^2}, |
| 367 |
> |
\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{k_c q^4 + |
| 368 |
> |
\gamma q^2}, |
| 369 |
|
\label{eq:fit} |
| 370 |
|
\end{equation} |
| 371 |
< |
where $k_c$ is the bending modulus for the membrane, and |
| 372 |
< |
$\tilde{\gamma}$ is the mechanical surface |
| 373 |
< |
tension.~\cite{Brannigan04b} |
| 371 |
> |
where $k_c$ is the bending modulus for the membrane, and $\gamma$ is |
| 372 |
> |
the mechanical surface tension.~\cite{Safran94} The systems studied in |
| 373 |
> |
this paper have essentially zero bending moduli ($k_c$) and relatively |
| 374 |
> |
large mechanical surface tensions ($\gamma$), so a much simpler form |
| 375 |
> |
can be written, |
| 376 |
> |
\begin{equation} |
| 377 |
> |
\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{\gamma q^2}, |
| 378 |
> |
\label{eq:fit2} |
| 379 |
> |
\end{equation} |
| 380 |
|
|
| 381 |
|
The undulation spectrum is computed by superimposing a rectangular |
| 382 |
|
grid on top of the membrane, and by assigning height ($h(\vec{r})$) |
| 384 |
|
given $\vec{r}+d\vec{r}$ grid area. Empty grid pixels are assigned |
| 385 |
|
height values by interpolation from the nearest neighbor pixels. A |
| 386 |
|
standard 2-d Fourier transform is then used to obtain $\langle | |
| 387 |
< |
h(q)|^2 \rangle$. |
| 387 |
> |
h(q)|^2 \rangle$. Alternatively, since the dipoles sit on a Bravais |
| 388 |
> |
lattice, one could use the heights of the lattice points themselves as |
| 389 |
> |
the grid for the Fourier transform (without interpolating to a square |
| 390 |
> |
grid). However, if lateral translational freedom is added to this |
| 391 |
> |
model (a likely extension), an interpolated grid method for computing |
| 392 |
> |
undulation spectra will be required. |
| 393 |
|
|
| 394 |
< |
The systems studied in this paper have relatively small bending moduli |
| 395 |
< |
($k_c$) and relatively large mechanical surface tensions |
| 396 |
< |
($\tilde{\gamma}$). In practice, the best fits to our undulation |
| 397 |
< |
spectra are obtained by approximating the value of $k_c$ to 0. In |
| 398 |
< |
Fig. \ref{fig:fit} we show typical undulation spectra for two |
| 399 |
< |
different regions of the phase diagram along with their fits from the |
| 400 |
< |
Landau free energy approach (Eq. \ref{eq:fit}). In the |
| 401 |
< |
high-temperature disordered phase, the Landau fits can be nearly |
| 402 |
< |
perfect, and from these fits we can estimate the bending modulus and |
| 403 |
< |
the mechanical surface tension. |
| 394 |
> |
As mentioned above, the best fits to our undulation spectra are |
| 395 |
> |
obtained by setting the value of $k_c$ to 0. In Fig. \ref{fig:fit} we |
| 396 |
> |
show typical undulation spectra for two different regions of the phase |
| 397 |
> |
diagram along with their fits from the Landau free energy approach |
| 398 |
> |
(Eq. \ref{eq:fit2}). In the high-temperature disordered phase, the |
| 399 |
> |
Landau fits can be nearly perfect, and from these fits we can estimate |
| 400 |
> |
the tension in the surface. In reduced units, typical values of |
| 401 |
> |
$\gamma^{*} = \gamma / \epsilon = 2500$ are obtained for the |
| 402 |
> |
disordered phase ($\gamma^{*} = 2551.7$ in the top panel of |
| 403 |
> |
Fig. \ref{fig:fit}). |
| 404 |
|
|
| 405 |
< |
For the dipolar-ordered hexagonal lattice near the coexistence |
| 406 |
< |
temperature, however, we observe long wavelength undulations that are |
| 407 |
< |
far outliers to the fits. That is, the Landau free energy fits are |
| 408 |
< |
well within error bars for all other points, but can be off by {\em |
| 409 |
< |
orders of magnitude} for a few low frequency components. |
| 405 |
> |
Typical values of $\gamma^{*}$ in the dipolar-ordered phase are much |
| 406 |
> |
higher than in the dipolar-disordered phase ($\gamma^{*} = 73,538$ in |
| 407 |
> |
the lower panel of Fig. \ref{fig:fit}). For the dipolar-ordered |
| 408 |
> |
triangular lattice near the coexistence temperature, we also observe |
| 409 |
> |
long wavelength undulations that are far outliers to the fits. That |
| 410 |
> |
is, the Landau free energy fits are well within error bars for most of |
| 411 |
> |
the other points, but can be off by {\em orders of magnitude} for a |
| 412 |
> |
few low frequency components. |
| 413 |
|
|
| 414 |
|
We interpret these outliers as evidence that these low frequency modes |
| 415 |
|
are {\em non-thermal undulations}. We take this as evidence that we |
| 416 |
|
are actually seeing a rippled phase developing in this model system. |
| 417 |
|
|
| 418 |
< |
\begin{figure}[ht] |
| 419 |
< |
\centering |
| 420 |
< |
\caption{Evidence that the observed ripples are {\em not} thermal |
| 421 |
< |
undulations is obtained from the 2-d fourier transform $\langle |
| 422 |
< |
|h^{*}(\vec{q})|^2 \rangle$ of the height profile ($\langle h^{*}(x,y) |
| 423 |
< |
\rangle$). Rippled samples show low-wavelength peaks that are |
| 424 |
< |
outliers on the Landau free energy fits. Samples exhibiting only |
| 425 |
< |
thermal undulations fit Eq. \ref{eq:fit} remarkably well.} |
| 426 |
< |
\includegraphics[width=5.5in]{fit.pdf} |
| 390 |
< |
\label{fig:fit} |
| 418 |
> |
\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 |
| 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. |
| 425 |
> |
Samples exhibiting only thermal undulations fit Eq. \ref{eq:fit} |
| 426 |
> |
remarkably well.} |
| 427 |
|
\end{figure} |
| 428 |
|
|
| 429 |
|
\subsection{Effects of Potential Parameters on Amplitude and Wavelength} |
| 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 |
|
|
| 423 |
– |
\begin{figure}[ht] |
| 424 |
– |
\centering |
| 425 |
– |
\caption{Contours of the height-dipole correlation function as a function |
| 426 |
– |
of the dot product between the dipole ($\hat{\mu}$) and inter-dipole |
| 427 |
– |
separation vector ($\hat{r}$) and the distance ($r$) between the dipoles. |
| 428 |
– |
Perfect height correlation (contours approaching 1) are present in the |
| 429 |
– |
ordered phase when the two dipoles are in the same head-to-tail line. |
| 430 |
– |
Anti-correlation (contours below 0) is only seen when the inter-dipole |
| 431 |
– |
vector is perpendicular to the dipoles. } |
| 432 |
– |
\includegraphics[width=\linewidth]{height-dipole-correlation.pdf} |
| 433 |
– |
\label{fig:CrossCorrelation} |
| 434 |
– |
\end{figure} |
| 435 |
– |
|
| 459 |
|
A second, more accurate, and simpler method for estimating ripple |
| 460 |
|
shape is to extract the wavelength and height information directly |
| 461 |
|
from the largest non-thermal peak in the undulation spectrum. For |
| 462 |
|
large-amplitude ripples, the two methods give similar results. The |
| 463 |
|
one-dimensional projection method is more prone to noise (particularly |
| 464 |
< |
in the amplitude estimates for the non-hexagonal lattices). We report |
| 464 |
> |
in the amplitude estimates for the distorted lattices). We report |
| 465 |
|
amplitudes and wavelengths taken directly from the undulation spectrum |
| 466 |
|
below. |
| 467 |
|
|
| 468 |
< |
In the hexagonal lattice ($\gamma = \sqrt{3}$), the rippling is |
| 468 |
> |
In the triangular lattice ($\gamma = \sqrt{3}$), the rippling is |
| 469 |
|
observed for temperatures ($T^{*}$) from $61-122$. The wavelength of |
| 470 |
|
the ripples is remarkably stable at 21.4~$\sigma$ for all but the |
| 471 |
|
temperatures closest to the order-disorder transition. At $T^{*} = |
| 480 |
|
However, this is coincidental agreement based on a choice of 7~\AA~as |
| 481 |
|
the mean spacing between lipids. |
| 482 |
|
|
| 483 |
< |
\begin{figure}[ht] |
| 484 |
< |
\centering |
| 485 |
< |
\caption{a) The amplitude $A^{*}$ of the ripples vs. temperature for a |
| 486 |
< |
hexagonal lattice. b) The amplitude $A^{*}$ of the ripples vs. dipole |
| 487 |
< |
strength ($\mu^{*}$) for both the hexagonal lattice (circles) and |
| 488 |
< |
non-hexagonal lattice (squares). The reduced temperatures were kept |
| 489 |
< |
fixed at $T^{*} = 94$ for the hexagonal lattice and $T^{*} = 106$ for |
| 490 |
< |
the non-hexagonal lattice (approximately 2/3 of the order-disorder |
| 491 |
< |
transition temperature for each lattice).} |
| 469 |
< |
\includegraphics[width=\linewidth]{properties_sq.pdf} |
| 470 |
< |
\label{fig:Amplitude} |
| 483 |
> |
\begin{figure} |
| 484 |
> |
\includegraphics[width=\linewidth]{properties_sq} |
| 485 |
> |
\caption{\label{fig:Amplitude} a) The amplitude $A^{*}$ of the ripples |
| 486 |
> |
vs. temperature for a triangular lattice. b) The amplitude $A^{*}$ of |
| 487 |
> |
the ripples vs. dipole strength ($\mu^{*}$) for both the triangular |
| 488 |
> |
lattice (circles) and distorted lattice (squares). The reduced |
| 489 |
> |
temperatures were kept fixed at $T^{*} = 94$ for the triangular |
| 490 |
> |
lattice and $T^{*} = 106$ for the distorted lattice (approximately 2/3 |
| 491 |
> |
of the order-disorder transition temperature for each lattice).} |
| 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 |
| 502 |
|
of ripple amplitude on the dipolar strength in |
| 503 |
|
Fig. \ref{fig:Amplitude}. |
| 504 |
|
|
| 505 |
< |
\subsection{Non-hexagonal lattices} |
| 505 |
> |
\subsection{Distorted lattices} |
| 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 hexagonal lattices prefer a particular director axis due to |
| 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 hexagonal |
| 515 |
< |
lattices. In non-hexagonal lattices the antiferroelectric phase can |
| 514 |
> |
Our observation of rippling behavior was not limited to the triangular |
| 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 non-hexagonal lattices |
| 518 |
> |
rippling has been observed in these distorted lattices |
| 519 |
|
(e.g. $\gamma = 1.875$), we see relatively short ripple wavelengths |
| 520 |
|
(14 $\sigma$) and amplitudes of 2.4~$\sigma$. These ripples are |
| 521 |
|
weakly dependent on dipolar strength (see Fig. \ref{fig:Amplitude}), |
| 527 |
|
\gamma < 1.875$. Outside this range, the order-disorder transition in |
| 528 |
|
the dipoles remains, but the ordered dipolar phase has only thermal |
| 529 |
|
undulations. This is one of our strongest pieces of evidence that |
| 530 |
< |
rippling is a symmetry-breaking phenomenon for hexagonal and |
| 531 |
< |
nearly-hexagonal lattices. |
| 530 |
> |
rippling is a symmetry-breaking phenomenon for triangular and |
| 531 |
> |
nearly-triangular lattices. |
| 532 |
|
|
| 533 |
|
\subsection{Effects of System Size} |
| 534 |
|
To evaluate the effect of finite system size, we have performed a |
| 535 |
< |
series of simulations on the hexagonal lattice at a reduced |
| 535 |
> |
series of simulations on the triangular lattice at a reduced |
| 536 |
|
temperature of 122, which is just below the order-disorder transition |
| 537 |
|
temperature ($T^{*} = 139$). These conditions are in the |
| 538 |
|
dipole-ordered and rippled portion of the phase diagram. These are |
| 539 |
|
also the conditions that should be most susceptible to system size |
| 540 |
|
effects. |
| 541 |
|
|
| 542 |
< |
\begin{figure}[ht] |
| 543 |
< |
\centering |
| 544 |
< |
\caption{The ripple wavelength (top) and amplitude (bottom) as a |
| 545 |
< |
function of system size for a hexagonal lattice ($\gamma=1.732$) at $T^{*} = |
| 546 |
< |
122$.} |
| 526 |
< |
\includegraphics[width=\linewidth]{SystemSize.pdf} |
| 527 |
< |
\label{fig:systemsize} |
| 542 |
> |
\begin{figure} |
| 543 |
> |
\includegraphics[width=\linewidth]{SystemSize} |
| 544 |
> |
\caption{\label{fig:systemsize} The ripple wavelength (top) and |
| 545 |
> |
amplitude (bottom) as a function of system size for a triangular |
| 546 |
> |
lattice ($\gamma=1.732$) at $T^{*} = 122$.} |
| 547 |
|
\end{figure} |
| 548 |
|
|
| 549 |
|
There is substantial dependence on system size for small (less than |
| 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 hexagonal 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-hexagonal 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. |
| 580 |
> |
in this way would yield an effectively non-triangular lattice which |
| 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. |
| 596 |
|
|
| 597 |
< |
The packing of the dipoles into a nearly-hexagonal lattice is clearly |
| 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 hexagonally without the steric |
| 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 |
< |
there is a role played by the lipid chains in the organization of the |
| 604 |
< |
hexagonally ordered phases which support ripples in realistic lipid |
| 605 |
< |
bilayers. |
| 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 |
| 619 |
|
|
| 620 |
|
Our other observation about the ripple and dipolar directionality is |
| 621 |
|
that the dipole director axis can be found to be parallel to any of |
| 622 |
< |
the three equivalent lattice vectors in the hexagonal lattice. |
| 622 |
> |
the three equivalent lattice vectors in the triangular lattice. |
| 623 |
|
Defects in the ordering of the dipoles can cause the dipole director |
| 624 |
|
(and consequently the surface corrugation) of small regions to be |
| 625 |
|
rotated relative to each other by 120$^{\circ}$. This is a similar |
| 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. |
| 639 |
< |
\clearpage |
| 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 |
| 642 |
> |
Foundation under grant CHE-0134881. The authors would like to thank |
| 643 |
> |
the reviewers for helpful comments. |
| 644 |
> |
\end{acknowledgments} |
| 645 |
> |
|
| 646 |
|
\bibliography{ripple} |
| 617 |
– |
\printfigures |
| 647 |
|
\end{document} |
