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\begin{document} |
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%\bibliographystyle{aps} |
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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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\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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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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\end{abstract} |
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\pacs{68.03.Hj, 82.20.Wt} |
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\maketitle |
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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 |
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even biological membranes. Computer simulations have provided useful |
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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 |
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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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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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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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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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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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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 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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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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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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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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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 |
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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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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 |
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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 |
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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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\section{2-D Dipolar Membrane} |
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\label{sec:model} |
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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 |
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phase have limited translational freedom, we have chosen a lattice to |
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support the dipoles in the x-y plane. The lattice may be either |
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triangular (lattice constants $a/b = \sqrt{3}$) or distorted. |
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However, each dipole has 3 degrees of freedom. They may move freely |
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{\em out} of the x-y plane (along the $z$ axis), and they have |
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complete orientational freedom ($0 <= \theta <= \pi$, $0 <= \phi < 2 |
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\pi$). This is essentially a modified X-Y-Z model with translational |
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freedom along the z-axis. |
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The potential energy of the system, |
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\begin{eqnarray} |
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V = \sum_i & & \left( \sum_{j>i} \frac{|\mu|^2}{4\pi \epsilon_0 r_{ij}^3} \left[ |
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{\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j} - |
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3({\mathbf{\hat u}_i} \cdot {\mathbf{\hat |
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r}_{ij}})({\mathbf{\hat u}_j} \cdot {\mathbf{\hat r}_{ij}})\right] |
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\right. \nonumber \\ |
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& & \left. + \sum_{j \in NN_i}^6 \frac{k_r}{2}\left( |
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r_{ij}-\sigma \right)^2 \right) |
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\label{eq:pot} |
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\end{eqnarray} |
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In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing |
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along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
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pointing along the inter-dipole vector $\mathbf{r}_{ij}$. The entire |
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potential is governed by three parameters, the dipolar strength |
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($\mu$), the harmonic spring constant ($k_r$) and the preferred |
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intermolecular spacing ($\sigma$). In practice, we set the value of |
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$\sigma$ to the average inter-molecular spacing from the planar |
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lattice, yielding a potential model that has only two parameters for a |
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particular choice of lattice constants $a$ (along the $x$-axis) and |
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$b$ (along the $y$-axis). We also define a set of reduced parameters |
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based on the length scale ($\sigma$) and the energy of the harmonic |
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potential at a deformation of 2 $\sigma$ ($\epsilon = k_r \sigma^2 / |
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2$). Using these two constants, we perform our calculations using |
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reduced distances, ($r^{*} = r / \sigma$), temperatures ($T^{*} = 2 |
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k_B T / k_r \sigma^2$), densities ($\rho^{*} = N \sigma^2 / L_x L_y$), |
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and dipole moments ($\mu^{*} = \mu / \sqrt{4 \pi \epsilon_0 \sigma^5 |
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k_r / 2}$). |
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To investigate the phase behavior of this model, we have performed a |
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series of Metropolis Monte Carlo simulations of moderately-sized (34.3 |
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$\sigma$ on a side) patches of membrane hosted on both triangular |
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($\gamma = a/b = \sqrt{3}$) and distorted ($\gamma \neq \sqrt{3}$) |
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lattices. The linear extent of one edge of the monolayer was $20 a$ |
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and the system was kept roughly square. The average distance that |
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coplanar dipoles were positioned from their six nearest neighbors was |
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1 $\sigma$ (on both triangular and distorted lattices). Typical |
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system sizes were 1360 dipoles for the triangular lattices and |
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840-2800 dipoles for the distorted lattices. Two-dimensional periodic |
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boundary conditions were used, and the cutoff for the dipole-dipole |
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interaction was set to 4.3 $\sigma$. Since dipole-dipole interactions |
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decay rapidly with distance, and since the intrinsic three-dimensional |
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periodicity of the Ewald sum can give artifacts in 2-d systems, we |
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have chosen not to use it in these calculations. Although the Ewald |
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sum has been reformulated to handle 2-D |
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systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89} these methods |
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are computationally expensive,\cite{Spohr97,Yeh99} and are not |
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necessary in this case. All parameters ($T^{*}$, $\mu^{*}$, and |
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$\gamma$) were varied systematically to study the effects of these |
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parameters on the formation of ripple-like phases. |
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\section{Results and Analysis} |
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\label{sec:results} |
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\subsection{Dipolar Ordering and Coexistence Temperatures} |
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The principal method for observing the orientational ordering |
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transition in dipolar systems is the $P_2$ order parameter (defined as |
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$1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
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eigenvalue of the matrix, |
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\begin{equation} |
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{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
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\begin{array}{ccc} |
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u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\ |
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u^{y}_i u^{x}_i & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\ |
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u^{z}_i u^{x}_i & u^{z}_i u^{y}_i & u^{z}_i u^{z}_i -\frac{1}{3} |
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\end{array} \right). |
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\label{eq:opmatrix} |
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\end{equation} |
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Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector |
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for dipole $i$. $P_2$ will be $1.0$ for a perfectly-ordered system |
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and near $0$ for a randomized system. Note that this order parameter |
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is {\em not} equal to the polarization of the system. For example, |
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the polarization of the perfect antiferroelectric system is $0$, but |
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$P_2$ for an antiferroelectric system is $1$. The eigenvector of |
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$\mathsf{S}$ corresponding to the largest eigenvalue is familiar as |
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the director axis, which can be used to determine a privileged dipolar |
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axis for dipole-ordered systems. The top panel in Fig. \ref{phase} |
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shows the values of $P_2$ as a function of temperature for both |
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triangular ($\gamma = 1.732$) and distorted ($\gamma=1.875$) |
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lattices. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{phase.pdf} |
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\caption{\label{phase} Top panel: The $P_2$ dipolar order parameter as |
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a function of temperature for both triangular ($\gamma = 1.732$) and |
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distorted ($\gamma = 1.875$) lattices. Bottom Panel: The phase |
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diagram for the dipolar membrane model. The line denotes the division |
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between the dipolar ordered (antiferroelectric) and disordered phases. |
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An enlarged view near the triangular lattice is shown inset.} |
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\end{figure} |
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There is a clear order-disorder transition in evidence from this data. |
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Both the triangular and distorted lattices have dipolar-ordered |
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low-temperature phases, and orientationally-disordered high |
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temperature phases. The coexistence temperature for the triangular |
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lattice is significantly lower than for the distorted lattices, and |
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the bulk polarization is approximately $0$ for both dipolar ordered |
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and disordered phases. This gives strong evidence that the dipolar |
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ordered phase is antiferroelectric. We have verified that this |
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dipolar ordering transition is not a function of system size by |
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performing identical calculations with systems twice as large. The |
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transition is equally smooth at all system sizes that were studied. |
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Additionally, we have repeated the Monte Carlo simulations over a wide |
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range of lattice ratios ($\gamma$) to generate a dipolar |
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order/disorder phase diagram. The bottom panel in Fig. \ref{phase} |
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shows that the triangular lattice is a low-temperature cusp in the |
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$T^{*}-\gamma$ phase diagram. |
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This phase diagram is remarkable in that it shows an antiferroelectric |
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phase near $\gamma=1.732$ where one would expect lattice frustration |
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to result in disordered phases at all temperatures. Observations of |
317 |
|
|
the configurations in this phase show clearly that the system has |
318 |
|
|
accomplished dipolar orderering by forming large ripple-like |
319 |
|
|
structures. We have observed antiferroelectric ordering in all three |
320 |
|
|
of the equivalent directions on the triangular lattice, and the dipoles |
321 |
|
|
have been observed to organize perpendicular to the membrane normal |
322 |
|
|
(in the plane of the membrane). It is particularly interesting to |
323 |
|
|
note that the ripple-like structures have also been observed to |
324 |
|
|
propagate in the three equivalent directions on the lattice, but the |
325 |
|
|
{\em direction of ripple propagation is always perpendicular to the |
326 |
|
|
dipole director axis}. A snapshot of a typical antiferroelectric |
327 |
|
|
rippled structure is shown in Fig. \ref{fig:snapshot}. |
328 |
|
|
|
329 |
|
|
\begin{figure} |
330 |
|
|
\includegraphics[width=\linewidth]{snapshot.pdf} |
331 |
|
|
\caption{\label{fig:snapshot} Top and Side views of a representative |
332 |
|
|
configuration for the dipolar ordered phase supported on the |
333 |
|
|
triangular lattice. Note the antiferroelectric ordering and the long |
334 |
|
|
wavelength buckling of the membrane. Dipolar ordering has been |
335 |
|
|
observed in all three equivalent directions on the triangular lattice, |
336 |
|
|
and the ripple direction is always perpendicular to the director axis |
337 |
|
|
for the dipoles.} |
338 |
|
|
\end{figure} |
339 |
|
|
|
340 |
|
|
Although the snapshot in Fig. \ref{fig:snapshot} gives the appearance |
341 |
|
|
of three-row stair-like structures, these appear to be transient. On |
342 |
|
|
average, the corrugation of the membrane is a relatively smooth, |
343 |
|
|
long-wavelength phenomenon, with occasional steep drops between |
344 |
|
|
adjacent lines of anti-aligned dipoles. |
345 |
|
|
|
346 |
|
|
The height-dipole correlation function ($C(r, \cos \theta)$) makes the |
347 |
|
|
connection between dipolar ordering and the wave vector of the ripple |
348 |
|
|
even more explicit. $C(r, \cos \theta)$ is an angle-dependent pair |
349 |
|
|
distribution function. The angle ($\theta$) is defined by the |
350 |
|
|
intermolecular vector $\vec{r}_{ij}$ and direction of dipole $i$, |
351 |
|
|
\begin{equation} |
352 |
|
|
C(r, \cos \theta) = \frac{\langle \sum_{i} |
353 |
|
|
\sum_{j} h_i \cdot h_j \delta(r - r_{ij}) \delta(\cos \theta_{ij} - |
354 |
|
|
\cos \theta)\rangle} {\langle h^2 \rangle} |
355 |
|
|
\end{equation} |
356 |
|
|
where $\cos \theta_{ij} = \hat{\mu}_{i} \cdot \hat{r}_{ij}$ and |
357 |
|
|
$\hat{r}_{ij} = \vec{r}_{ij} / r_{ij}$. Fig. \ref{fig:CrossCorrelation} |
358 |
|
|
shows contours of this correlation function for both anti-ferroelectric, rippled |
359 |
|
|
membranes as well as for the dipole-disordered portion of the phase diagram. |
360 |
|
|
|
361 |
|
|
\begin{figure} |
362 |
|
|
\includegraphics[width=\linewidth]{height-dipole-correlation.pdf} |
363 |
|
|
\caption{\label{fig:CrossCorrelation} Contours of the height-dipole |
364 |
|
|
correlation function as a function of the dot product between the |
365 |
|
|
dipole ($\hat{\mu}$) and inter-dipole separation vector ($\hat{r}$) |
366 |
|
|
and the distance ($r$) between the dipoles. Perfect height |
367 |
|
|
correlation (contours approaching 1) are present in the ordered phase |
368 |
|
|
when the two dipoles are in the same head-to-tail line. |
369 |
|
|
Anti-correlation (contours below 0) is only seen when the inter-dipole |
370 |
|
|
vector is perpendicular to the dipoles. In the dipole-disordered |
371 |
|
|
portion of the phase diagram, there is only weak correlation in the |
372 |
|
|
dipole direction and this correlation decays rapidly to zero for |
373 |
|
|
intermolecular vectors that are not dipole-aligned.} |
374 |
|
|
\end{figure} |
375 |
|
|
|
376 |
|
|
\subsection{Discriminating Ripples from Thermal Undulations} |
377 |
|
|
|
378 |
|
|
In order to be sure that the structures we have observed are actually |
379 |
|
|
a rippled phase and not simply thermal undulations, we have computed |
380 |
|
|
the undulation spectrum, |
381 |
|
|
\begin{equation} |
382 |
|
|
h(\vec{q}) = A^{-1/2} \int d\vec{r} |
383 |
|
|
h(\vec{r}) e^{-i \vec{q}\cdot\vec{r}} |
384 |
|
|
\end{equation} |
385 |
|
|
where $h(\vec{r})$ is the height of the membrane at location $\vec{r} |
386 |
|
|
= (x,y)$.~\cite{Safran94,Seifert97} In simple (and more complicated) |
387 |
|
|
elastic continuum models, it can shown that in the $NVT$ ensemble, the |
388 |
|
|
absolute value of the undulation spectrum can be written, |
389 |
|
|
\begin{equation} |
390 |
|
|
\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{k_c q^4 + |
391 |
|
|
\gamma q^2}, |
392 |
|
|
\label{eq:fit} |
393 |
|
|
\end{equation} |
394 |
|
|
where $k_c$ is the bending modulus for the membrane, and $\gamma$ is |
395 |
|
|
the mechanical surface tension.~\cite{Safran94} The systems studied in |
396 |
|
|
this paper have essentially zero bending moduli ($k_c$) and relatively |
397 |
|
|
large mechanical surface tensions ($\gamma$), so a much simpler form |
398 |
|
|
can be written, |
399 |
|
|
\begin{equation} |
400 |
|
|
\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{\gamma q^2}, |
401 |
|
|
\label{eq:fit2} |
402 |
|
|
\end{equation} |
403 |
|
|
|
404 |
|
|
The undulation spectrum is computed by superimposing a rectangular |
405 |
|
|
grid on top of the membrane, and by assigning height ($h(\vec{r})$) |
406 |
|
|
values to the grid from the average of all dipoles that fall within a |
407 |
|
|
given $\vec{r}+d\vec{r}$ grid area. Empty grid pixels are assigned |
408 |
|
|
height values by interpolation from the nearest neighbor pixels. A |
409 |
|
|
standard 2-d Fourier transform is then used to obtain $\langle | |
410 |
|
|
h(q)|^2 \rangle$. Alternatively, since the dipoles sit on a Bravais |
411 |
|
|
lattice, one could use the heights of the lattice points themselves as |
412 |
|
|
the grid for the Fourier transform (without interpolating to a square |
413 |
|
|
grid). However, if lateral translational freedom is added to this |
414 |
|
|
model (a likely extension), an interpolated grid method for computing |
415 |
|
|
undulation spectra will be required. |
416 |
|
|
|
417 |
|
|
As mentioned above, the best fits to our undulation spectra are |
418 |
|
|
obtained by setting the value of $k_c$ to 0. In Fig. \ref{fig:fit} we |
419 |
|
|
show typical undulation spectra for two different regions of the phase |
420 |
|
|
diagram along with their fits from the Landau free energy approach |
421 |
|
|
(Eq. \ref{eq:fit2}). In the high-temperature disordered phase, the |
422 |
|
|
Landau fits can be nearly perfect, and from these fits we can estimate |
423 |
|
|
the tension in the surface. In reduced units, typical values of |
424 |
|
|
$\gamma^{*} = \gamma / \epsilon = 2500$ are obtained for the |
425 |
|
|
disordered phase ($\gamma^{*} = 2551.7$ in the top panel of |
426 |
|
|
Fig. \ref{fig:fit}). |
427 |
|
|
|
428 |
|
|
Typical values of $\gamma^{*}$ in the dipolar-ordered phase are much |
429 |
|
|
higher than in the dipolar-disordered phase ($\gamma^{*} = 73,538$ in |
430 |
|
|
the lower panel of Fig. \ref{fig:fit}). For the dipolar-ordered |
431 |
|
|
triangular lattice near the coexistence temperature, we also observe |
432 |
|
|
long wavelength undulations that are far outliers to the fits. That |
433 |
|
|
is, the Landau free energy fits are well within error bars for most of |
434 |
|
|
the other points, but can be off by {\em orders of magnitude} for a |
435 |
|
|
few low frequency components. |
436 |
|
|
|
437 |
|
|
We interpret these outliers as evidence that these low frequency modes |
438 |
|
|
are {\em non-thermal undulations}. We take this as evidence that we |
439 |
|
|
are actually seeing a rippled phase developing in this model system. |
440 |
|
|
|
441 |
|
|
\begin{figure} |
442 |
|
|
\includegraphics[width=\linewidth]{logFit.pdf} |
443 |
|
|
\caption{\label{fig:fit} Evidence that the observed ripples are {\em |
444 |
|
|
not} thermal undulations is obtained from the 2-d fourier transform |
445 |
|
|
$\langle |h^{*}(\vec{q})|^2 \rangle$ of the height profile ($\langle |
446 |
|
|
h^{*}(x,y) \rangle$). Rippled samples show low-wavelength peaks that |
447 |
|
|
are outliers on the Landau free energy fits by an order of magnitude. |
448 |
|
|
Samples exhibiting only thermal undulations fit Eq. \ref{eq:fit} |
449 |
|
|
remarkably well.} |
450 |
|
|
\end{figure} |
451 |
|
|
|
452 |
|
|
\subsection{Effects of Potential Parameters on Amplitude and Wavelength} |
453 |
|
|
|
454 |
|
|
We have used two different methods to estimate the amplitude and |
455 |
|
|
periodicity of the ripples. The first method requires projection of |
456 |
|
|
the ripples onto a one dimensional rippling axis. Since the rippling |
457 |
|
|
is always perpendicular to the dipole director axis, we can define a |
458 |
|
|
ripple vector as follows. The largest eigenvector, $s_1$, of the |
459 |
|
|
$\mathsf{S}$ matrix in Eq. \ref{eq:opmatrix} is projected onto a |
460 |
|
|
planar director axis, |
461 |
|
|
\begin{equation} |
462 |
|
|
\vec{d} = \left(\begin{array}{c} |
463 |
|
|
\vec{s}_1 \cdot \hat{i} \\ |
464 |
|
|
\vec{s}_1 \cdot \hat{j} \\ |
465 |
|
|
0 |
466 |
|
|
\end{array} \right). |
467 |
|
|
\end{equation} |
468 |
|
|
($\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along the $x$, |
469 |
|
|
$y$, and $z$ axes, respectively.) The rippling axis is in the plane of |
470 |
|
|
the membrane and is perpendicular to the planar director axis, |
471 |
|
|
\begin{equation} |
472 |
|
|
\vec{q}_{\mathrm{rip}} = \vec{d} \times \hat{k} |
473 |
|
|
\end{equation} |
474 |
|
|
We can then find the height profile of the membrane along the ripple |
475 |
|
|
axis by projecting heights of the dipoles to obtain a one-dimensional |
476 |
|
|
height profile, $h(q_{\mathrm{rip}})$. Ripple wavelengths can be |
477 |
|
|
estimated from the largest non-thermal low-frequency component in the |
478 |
|
|
fourier transform of $h(q_{\mathrm{rip}})$. Amplitudes can be |
479 |
|
|
estimated by measuring peak-to-trough distances in |
480 |
|
|
$h(q_{\mathrm{rip}})$ itself. |
481 |
|
|
|
482 |
|
|
A second, more accurate, and simpler method for estimating ripple |
483 |
|
|
shape is to extract the wavelength and height information directly |
484 |
|
|
from the largest non-thermal peak in the undulation spectrum. For |
485 |
|
|
large-amplitude ripples, the two methods give similar results. The |
486 |
|
|
one-dimensional projection method is more prone to noise (particularly |
487 |
|
|
in the amplitude estimates for the distorted lattices). We report |
488 |
|
|
amplitudes and wavelengths taken directly from the undulation spectrum |
489 |
|
|
below. |
490 |
|
|
|
491 |
|
|
In the triangular lattice ($\gamma = \sqrt{3}$), the rippling is |
492 |
|
|
observed for temperatures ($T^{*}$) from $61-122$. The wavelength of |
493 |
|
|
the ripples is remarkably stable at 21.4~$\sigma$ for all but the |
494 |
|
|
temperatures closest to the order-disorder transition. At $T^{*} = |
495 |
|
|
122$, the wavelength drops to 17.1~$\sigma$. |
496 |
|
|
|
497 |
|
|
The dependence of the amplitude on temperature is shown in the top |
498 |
|
|
panel of Fig. \ref{fig:Amplitude}. The rippled structures shrink |
499 |
|
|
smoothly as the temperature rises towards the order-disorder |
500 |
|
|
transition. The wavelengths and amplitudes we observe are |
501 |
|
|
surprisingly close to the $\Lambda / 2$ phase observed by Kaasgaard |
502 |
|
|
{\it et al.} in their work on PC-based lipids.\cite{Kaasgaard03} |
503 |
|
|
However, this is coincidental agreement based on a choice of 7~\AA~as |
504 |
|
|
the mean spacing between lipids. |
505 |
|
|
|
506 |
|
|
\begin{figure} |
507 |
|
|
\includegraphics[width=\linewidth]{properties_sq.pdf} |
508 |
|
|
\caption{\label{fig:Amplitude} a) The amplitude $A^{*}$ of the ripples |
509 |
|
|
vs. temperature for a triangular lattice. b) The amplitude $A^{*}$ of |
510 |
|
|
the ripples vs. dipole strength ($\mu^{*}$) for both the triangular |
511 |
|
|
lattice (circles) and distorted lattice (squares). The reduced |
512 |
|
|
temperatures were kept fixed at $T^{*} = 94$ for the triangular |
513 |
|
|
lattice and $T^{*} = 106$ for the distorted lattice (approximately 2/3 |
514 |
|
|
of the order-disorder transition temperature for each lattice).} |
515 |
|
|
\end{figure} |
516 |
|
|
|
517 |
|
|
The ripples can be made to disappear by increasing the internal |
518 |
|
|
surface tension (i.e. by increasing $k_r$ or equivalently, reducing |
519 |
|
|
the dipole moment). The amplitude of the ripples depends critically |
520 |
|
|
on the strength of the dipole moments ($\mu^{*}$) in Eq. \ref{eq:pot}. |
521 |
|
|
If the dipoles are weakened substantially (below $\mu^{*}$ = 20) at a |
522 |
|
|
fixed temperature of 94, the membrane loses dipolar ordering |
523 |
|
|
and the ripple structures. The ripples reach a peak amplitude of |
524 |
|
|
3.7~$\sigma$ at a dipolar strength of 25. We show the dependence |
525 |
|
|
of ripple amplitude on the dipolar strength in |
526 |
|
|
Fig. \ref{fig:Amplitude}. |
527 |
|
|
|
528 |
|
|
\subsection{Distorted lattices} |
529 |
|
|
|
530 |
|
|
We have also investigated the effect of the lattice geometry by |
531 |
|
|
changing the ratio of lattice constants ($\gamma$) while keeping the |
532 |
|
|
average nearest-neighbor spacing constant. The antiferroelectric state |
533 |
|
|
is accessible for all $\gamma$ values we have used, although the |
534 |
|
|
distorted triangular lattices prefer a particular director axis due to |
535 |
|
|
the anisotropy of the lattice. |
536 |
|
|
|
537 |
|
|
Our observation of rippling behavior was not limited to the triangular |
538 |
|
|
lattices. In distorted lattices the antiferroelectric phase can |
539 |
|
|
develop nearly instantaneously in the Monte Carlo simulations, and |
540 |
|
|
these dipolar-ordered phases tend to be remarkably flat. Whenever |
541 |
|
|
rippling has been observed in these distorted lattices |
542 |
|
|
(e.g. $\gamma = 1.875$), we see relatively short ripple wavelengths |
543 |
|
|
(14 $\sigma$) and amplitudes of 2.4~$\sigma$. These ripples are |
544 |
|
|
weakly dependent on dipolar strength (see Fig. \ref{fig:Amplitude}), |
545 |
|
|
although below a dipolar strength of $\mu^{*} = 20$, the membrane |
546 |
|
|
loses dipolar ordering and displays only thermal undulations. |
547 |
|
|
|
548 |
|
|
The ripple phase does {\em not} appear at all values of $\gamma$. We |
549 |
|
|
have only observed non-thermal undulations in the range $1.625 < |
550 |
|
|
\gamma < 1.875$. Outside this range, the order-disorder transition in |
551 |
|
|
the dipoles remains, but the ordered dipolar phase has only thermal |
552 |
|
|
undulations. This is one of our strongest pieces of evidence that |
553 |
|
|
rippling is a symmetry-breaking phenomenon for triangular and |
554 |
|
|
nearly-triangular lattices. |
555 |
|
|
|
556 |
|
|
\subsection{Effects of System Size} |
557 |
|
|
To evaluate the effect of finite system size, we have performed a |
558 |
|
|
series of simulations on the triangular lattice at a reduced |
559 |
|
|
temperature of 122, which is just below the order-disorder transition |
560 |
|
|
temperature ($T^{*} = 139$). These conditions are in the |
561 |
|
|
dipole-ordered and rippled portion of the phase diagram. These are |
562 |
|
|
also the conditions that should be most susceptible to system size |
563 |
|
|
effects. |
564 |
|
|
|
565 |
|
|
\begin{figure} |
566 |
|
|
\includegraphics[width=\linewidth]{SystemSize.pdf} |
567 |
|
|
\caption{\label{fig:systemsize} The ripple wavelength (top) and |
568 |
|
|
amplitude (bottom) as a function of system size for a triangular |
569 |
|
|
lattice ($\gamma=1.732$) at $T^{*} = 122$.} |
570 |
|
|
\end{figure} |
571 |
|
|
|
572 |
|
|
There is substantial dependence on system size for small (less than |
573 |
|
|
29~$\sigma$) periodic boxes. Notably, there are resonances apparent |
574 |
|
|
in the ripple amplitudes at box lengths of 17.3 and 29.5 $\sigma$. |
575 |
|
|
For larger systems, the behavior of the ripples appears to have |
576 |
|
|
stabilized and is on a trend to slightly smaller amplitudes (and |
577 |
|
|
slightly longer wavelengths) than were observed from the 34.3 $\sigma$ |
578 |
|
|
box sizes that were used for most of the calculations. |
579 |
|
|
|
580 |
|
|
It is interesting to note that system sizes which are multiples of the |
581 |
|
|
default ripple wavelength can enhance the amplitude of the observed |
582 |
|
|
ripples, but appears to have only a minor effect on the observed |
583 |
|
|
wavelength. It would, of course, be better to use system sizes that |
584 |
|
|
were many multiples of the ripple wavelength to be sure that the |
585 |
|
|
periodic box is not driving the phenomenon, but at the largest system |
586 |
|
|
size studied (70 $\sigma$ $\times$ 70 $\sigma$), the number of dipoles |
587 |
|
|
(5440) made long Monte Carlo simulations prohibitively expensive. |
588 |
|
|
|
589 |
|
|
\section{Discussion} |
590 |
|
|
\label{sec:discussion} |
591 |
|
|
|
592 |
|
|
We have been able to show that a simple dipolar lattice model which |
593 |
|
|
contains only molecular packing (from the lattice), anisotropy (in the |
594 |
|
|
form of electrostatic dipoles) and a weak surface tension (in the form |
595 |
|
|
of a nearest-neighbor harmonic potential) is capable of exhibiting |
596 |
|
|
stable long-wavelength non-thermal surface corrugations. The best |
597 |
|
|
explanation for this behavior is that the ability of the dipoles to |
598 |
|
|
translate out of the plane of the membrane is enough to break the |
599 |
|
|
symmetry of the triangular lattice and allow the energetic benefit from |
600 |
|
|
the formation of a bulk antiferroelectric phase. Were the weak |
601 |
|
|
surface tension absent from our model, it would be possible for the |
602 |
|
|
entire lattice to ``tilt'' using $z$-translation. Tilting the lattice |
603 |
|
|
in this way would yield an effectively non-triangular lattice which |
604 |
|
|
would avoid dipolar frustration altogether. With the surface tension |
605 |
|
|
in place, bulk tilt causes a large strain, and the simplest way to |
606 |
|
|
release this strain is along line defects. Line defects will result |
607 |
|
|
in rippled or sawtooth patterns in the membrane, and allow small |
608 |
|
|
``stripes'' of membrane to form antiferroelectric regions that are |
609 |
|
|
tilted relative to the averaged membrane normal. |
610 |
|
|
|
611 |
|
|
Although the dipole-dipole interaction is the major driving force for |
612 |
|
|
the long range orientational ordered state, the formation of the |
613 |
|
|
stable, smooth ripples is a result of the competition between the |
614 |
|
|
surface tension and the dipole-dipole interactions. This statement is |
615 |
|
|
supported by the variation in $\mu^{*}$. Substantially weaker dipoles |
616 |
|
|
relative to the surface tension can cause the corrugated phase to |
617 |
|
|
disappear. |
618 |
|
|
|
619 |
|
|
The packing of the dipoles into a nearly-triangular lattice is clearly |
620 |
|
|
an important piece of the puzzle. The dipolar head groups of lipid |
621 |
|
|
molecules are sterically (as well as electrostatically) anisotropic, |
622 |
|
|
and would not be able to pack in triangular arrangements without the |
623 |
|
|
steric interference of adjacent molecular bodies. Since we only see |
624 |
|
|
rippled phases in the neighborhood of $\gamma=\sqrt{3}$, this implies |
625 |
|
|
that there is a role played by the lipid chains in the organization of |
626 |
|
|
the triangularly ordered phases which support ripples in realistic |
627 |
|
|
lipid bilayers. |
628 |
|
|
|
629 |
|
|
The most important prediction we can make using the results from this |
630 |
|
|
simple model is that if dipolar ordering is driving the surface |
631 |
|
|
corrugation, the wave vectors for the ripples should always found to |
632 |
|
|
be {\it perpendicular} to the dipole director axis. This prediction |
633 |
|
|
should suggest experimental designs which test whether this is really |
634 |
|
|
true in the phosphatidylcholine $P_{\beta'}$ phases. The dipole |
635 |
|
|
director axis should also be easily computable for the all-atom and |
636 |
|
|
coarse-grained simulations that have been published in the literature. |
637 |
|
|
|
638 |
|
|
Our other observation about the ripple and dipolar directionality is |
639 |
|
|
that the dipole director axis can be found to be parallel to any of |
640 |
|
|
the three equivalent lattice vectors in the triangular lattice. |
641 |
|
|
Defects in the ordering of the dipoles can cause the dipole director |
642 |
|
|
(and consequently the surface corrugation) of small regions to be |
643 |
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rotated relative to each other by 120$^{\circ}$. This is a similar |
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behavior to the domain rotation seen in the AFM studies of Kaasgaard |
645 |
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{\it et al.}\cite{Kaasgaard03} |
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|
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Although our model is simple, it exhibits some rich and unexpected |
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behaviors. It would clearly be a closer approximation to the reality |
649 |
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|
if we allowed greater translational freedom to the dipoles and |
650 |
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replaced the somewhat artificial lattice packing and the harmonic |
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``surface tension'' with more realistic molecular modeling |
652 |
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|
potentials. What we have done is to present an extremely simple model |
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which exhibits bulk non-thermal corrugation, and our explanation of |
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this rippling phenomenon will help us design more accurate molecular |
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models for corrugated membranes and experiments to test whether |
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rippling is dipole-driven or not. |
657 |
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|
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\begin{acknowledgments} |
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Support for this project was provided by the National Science |
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Foundation under grant CHE-0134881. The authors would like to thank |
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the reviewers for helpful comments. |
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\end{acknowledgments} |
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|
664 |
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\bibliography{ripple} |
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\end{document} |