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\begin{document} |
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\title{Symmetry breaking and the Ripple phase} |
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\author{Xiuquan Sun and J. Daniel Gezelter\footnote{Corresponding author. Email: gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry \\ |
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University of Notre Dame \\ |
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\bibliographystyle{pccp} |
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\title{Symmetry breaking and the $P_{\beta'}$ Ripple phase} |
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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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\maketitle |
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\begin{abstract} |
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The ripple phase in phosphatidylcholine (PC) bilayers has never been |
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explained. Our group has developed some simple (XYZ) spin-lattice |
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models that allow spins to vary their elevation as well as their |
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completely explained. We present a simple (XYZ) spin-lattice model |
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that allows spins to vary their elevation as well as their |
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orientation. The extra degree of freedom allows hexagonal lattices of |
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spins to find states that break out of the normally frustrated |
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randomized states and are stabilized by long-range anti-ferroelectric |
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randomized states and are stabilized by long-range antiferroelectric |
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ordering. To break out of the frustrated states, the spins must form |
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``rippled'' phases that make the lattices effectively non-hexagonal. Our |
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XYZ models contain surface tension and dipole-dipole interactions to |
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describe the interaction potential for monolayers and bilayers of |
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model lipid molecules. The existence of the ripples depends primarily |
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on the strength and lattice spacing of the dipoles, while the shape |
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(wavelength and amplitude) of the ripples is only weakly sensitive to |
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the applied surface tension. Additionally, the wave vector for the |
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ripple is always perpendicular to the director axis for the |
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dipoles. Non-hexagonal lattices of dipoles are not inherently |
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frustrated, and are therefore less likely to form ripple phases |
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because they can easily form low-energy anti-ferroelectric states. |
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Indeed, we see that the dipolar order-disorder transition is |
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substantially lower for hexagonal lattices and the ordered phase for |
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this lattice is clearly rippled. |
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``rippled'' phases that make the lattices effectively |
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non-hexagonal. Our XYZ model contains a hydrophobic interaction and |
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dipole-dipole interactions to describe the interaction potential for |
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model lipid molecules. We find non-thermal ripple phases and note |
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that the wave vectors for the ripples are always perpendicular to the |
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director axis for the dipoles. Non-hexagonal lattices of dipoles are |
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not inherently frustrated, and are therefore less likely to form |
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ripple phases because they can easily form low-energy |
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antiferroelectric states. We see that the dipolar order-disorder |
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transition is substantially lower for hexagonal lattices and that the |
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ordered phase for this lattice is clearly rippled. |
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\end{abstract} |
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\maketitle |
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|
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\section{Introduction} |
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\label{Int} |
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Fully hydrated lipids will aggregate spontaneously to form bilayers |
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which exhibit a variety of phases according to temperature and |
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composition. Among these phases, a periodic rippled |
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phase---($P_{\beta'}$) phase is found as an intermediate phase during |
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the phase transition. This ripple phase can be obtained through either |
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cooling the lipids from fluid ($L_{\beta'}$) phase or heating from gel |
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($L_\beta$) phase. The ripple phase attracts lots of researches from |
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chemists in the past 30 years. Most structural information of the |
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ripple phase was obtained by the X-ray diffraction and freeze-fracture |
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electron microscopy |
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(FFEM)\cite{Copeland80,Meyer96,Sun96,Katsaras00}. Recently, atomic |
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force microscopy (AFM) is used as one of these |
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tools\cite{Kaasgaard03}. All these experimental results strongly |
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support a 2-Dimensional hexagonal packing lattice for the ripple phase |
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which is different to the gel phase. Numerous models were built to |
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explain the formation of the ripple |
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phase\cite{Goldstein88,McCullough90,Lubensky93,Tieleman96,Misbah98,Heimburg00,Kubica02,Banerjee02}. However, |
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the origin of the ripple phase is still on debate. The behavior of |
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the dipolar materials in the bulk attracts lots of |
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interests\cite{Luttinger46,Weis92,Ayton95,Ayton97}. The |
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ferroelectric state is observed for this kind of system, however, the |
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frustrated state is found in the 2-D hexagonal lattice of the dipolar |
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materials, the long range orientational ordered state can not be |
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formed in this situation. The experimental results show that the |
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periodicity of the ripples is in the range of 100-600 \AA |
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\cite{Kaasgaard03}, it is a pretty long range ordered state. So, we |
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may ask ourselves: {\it ``How could this long range ordered state be |
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formed in a hexagonal lattice surface?''} We addressed this problem |
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for a dipolar monolayer using Monte Carlo (MC) simulation. |
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which exhibit a variety of phases depending on their temperatures and |
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compositions. Among these phases, a periodic rippled phase |
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($P_{\beta'}$) appears as an intermediate phase between the gel |
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($L_\beta$) and fluid ($L_{\alpha}$) phases for relatively pure |
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phosphatidylcholine (PC) bilayers. The ripple phase has attracted |
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substantial experimental interest over the past 30 years. Most |
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structural information of the ripple phase has been obtained by the |
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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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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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|
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\section{Model and calculation method} |
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\label{Mod} |
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A number of theoretical models have been presented to explain the |
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formation of the ripple phase. 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 continuum |
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models predict higher fluidity in convex regions and that concave |
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portions of the membrane correspond to more solid-like regions. |
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Carlson and Sethna used a packing-competition model (in which head |
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groups and chains have competing packing energetics) to predict the |
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formation of a ripple-like phase. Their model predicted that the |
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high-curvature portions have lower-chain packing and correspond to |
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more fluid-like regions. Goldstein and Leibler used a mean-field |
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approach with a planar model for {\em inter-lamellar} interactions to |
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predict rippling in multilamellar phases.~\cite{Goldstein88} McCullough |
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and Scott proposed that the {\em anisotropy of the nearest-neighbor |
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interactions} coupled to hydrophobic constraining forces which |
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restrict height differences between nearest neighbors is the origin of |
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the ripple phase.~\cite{McCullough90} Lubensky and MacKintosh |
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introduced a Landau theory for tilt order and curvature of a single |
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membrane and concluded that {\em coupling of molecular tilt to membrane |
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curvature} is responsible for the production of |
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ripples.~\cite{Lubensky93} Misbah, Duplat and Houchmandzadeh proposed |
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that {\em inter-layer dipolar interactions} can lead to ripple |
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instabilities.~\cite{Misbah98} Heimburg presented a {\em coexistence |
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model} for ripple formation in which he postulates that fluid-phase |
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line defects cause sharp curvature between relatively flat gel-phase |
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regions.~\cite{Heimburg00} Kubica has suggested that a lattice model of |
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polar head groups could be valuable in trying to understand bilayer |
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phase formation.~\cite{Kubica02} Bannerjee used Monte Carlo simulations |
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of 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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The model used in our simulations is shown in Fig. \ref{fmod1} and |
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Fig. \ref{fmod2}. |
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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} Ayton and Voth have found |
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significant undulations in zero-surface-tension states of membranes |
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simulated via dissipative particle dynamics, but their results are |
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consistent with purely thermal undulations.~\cite{Ayton02} Brannigan, |
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Tamboli and Brown have used a molecular scale model to elucidate the |
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role of molecular shape on membrane phase behavior and |
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elasticity.~\cite{Brannigan04b} They have also observed a buckled hexatic |
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phase with strong tail and moderate alignment attractions.~\cite{Brannigan04a} |
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|
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\begin{figure} |
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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. Concomitantly, 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 al.}).~\cite{Kaasgaard03} |
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|
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The various theoretical models have attributed membrane rippling to |
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various causes which appear contradictory. We are left with a number |
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of open questions: 1) Are inter-layer interactions required to explain |
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the ripple, or can a single bilayer (or even a single leaf) exhibit |
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the rippling? 2) To what degree is the dipolar anisotropy of the head |
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group important in determining the rippling? 3) Is chain fluidity |
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required? (i.e. are the coexistence models necessary to explain the |
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ripple phenomenon?) 4) How could a state with long-range order be |
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formed using a substrate consisting of 2-D hexagonally-packed dipolar |
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molecules? What we present here is an attempt to find the simplest |
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model which will exhibit this phenomenon. We are using a very simple |
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modified XYZ lattice model; details of the model can be found in |
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section \ref{sec:model}, results of Monte Carlo simulations using this |
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model 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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\section{The Web-of-Dipoles Model} |
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\label{sec:model} |
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The model used in our simulations is shown schematically in |
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Figs. \ref{fmod1} and \ref{fmod2}. |
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|
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\begin{figure}[ht] |
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\centering |
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\includegraphics[width=\linewidth]{picture/lattice.eps} |
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\caption{The modified X-Y-Z model in the simulations. The dipoles are |
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represented by the arrows. Dipoles are locked to the lattice points |
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in x-y plane and connect to their nearest neighbors with harmonic |
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potentials.} |
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\caption{The modified X-Y-Z model used in our simulations. Point dipoles are |
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represented as arrows. Dipoles are locked to the lattice points in x-y |
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plane and connect to their nearest neighbors with harmonic |
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potentials. The lattice parameters $a$ and $b$ are indicated above.} |
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\includegraphics[width=\linewidth]{picture/WebOfDipoles.eps} |
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\label{fmod1} |
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\end{figure} |
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|
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\begin{figure} |
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\begin{figure}[ht] |
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\centering |
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\caption{The 6 coordinates describing the state of a 2-dipole system |
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in our extended X-Y-Z model. $z_i$ is the height of dipole $i$ from |
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an arbitrary x-y plane, $\theta_i$ is the angle that the dipole makes |
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with the laboratory z-axis and $\phi_i$ is the angle between |
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the projection of the dipole on x-y plane with the x axis.} |
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\includegraphics[width=\linewidth]{picture/xyz.eps} |
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\caption{The 6 coordinates describing the state of a 2-dipole system in our extended X-Y-Z model. $z_i$ is the height of dipole $i$ from |
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the initial x-y plane, $\theta_i$ is the angle that the dipole is away |
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from the z axis and $\phi_i$ is the angle between the projection of |
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the dipole on x-y plane with the x axis.} |
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\label{fmod2} |
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|
\end{figure} |
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|
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The lipids are represented by the simple point-dipole. During the |
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simulations, dipoles are locked (in the x-y plane) to lattice points |
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of hexagonal (or distorted) lattice. Each dipole can move freely out |
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of the plane and has complete orientational freedom. This is a |
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modified X-Y-Z model with translational freedom along the z-axis. The |
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potential of the system |
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In this model, lipid molecules are represented by point-dipoles (which |
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is a reasonable approximation to the zwitterionic head groups of the |
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phosphatidylcholine head groups). The dipoles are locked in place to |
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their original lattice sites on the x-y plane. The original lattice |
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may be either hexagonal ($a/b = \sqrt{3}$) or non-hexagonal. However, |
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each dipole has 3 degrees of freedom. They may move freely {\em out} of the |
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x-y plane (along the $z$ axis), and they have complete orientational |
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freedom ($0 <= \theta <= \pi$, $0 <= \phi < 2 \pi$). This is a |
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modified X-Y-Z model with translational freedom along the z-axis. |
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|
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The potential energy of the system, |
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|
\begin{equation} |
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V = \sum _i {\sum _{j\in NN_i}^6 {{\frac{k_r}{2}} (r_{ij}-r_0)^2}} + |
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V_{\text |
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{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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\label{tp} |
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V = \sum_i \left[ \sum_{j>i} V^{\mathrm{dd}}_{ij} + \frac{1}{2}\sum_{j |
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\in NN_i}^6 V^{\mathrm{harm}}_{ij} \right] |
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|
\end{equation} |
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where |
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\[ \sum _i {\sum _{j\in NN_i}^6 {{\frac{k_r}{2}} (r_{ij}-r_0)^2}} \] |
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and |
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\[ V_{\text {dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) = \sum _i {\sum _{j>i} {{\frac{|\mu_i||\mu_j|}{4\pi \epsilon_0 r_{ij}^3}} \biggl[ {\boldsymbol{\hat u}_i} \cdot {\boldsymbol{\hat u}_j} - 3({\boldsymbol{\hat u}_i} \cdot {\mathbf{\hat r}_{ij}})({\boldsymbol{\hat u}_j} \cdot {\mathbf{\hat r}_{ij}}) \biggr]}} \] |
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are the surface tension and the dipole-dipole interactions. In our |
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simulation, the surface tension for every dipole is represented by the |
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harmonic potential with its six nearest neighbors. $r_{ij}$ is the |
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distance between dipole $i$ and dipole $j$, $r_0$ is the lattice |
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distance in the x-y plane between dipole $i$ and $j$, $k_r$ is the |
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surface energy and corresponds to $k_BT$, $k_B$ is the Bolzmann's |
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constant. For the dipole-dipole interaction part, $\mathbf{r}_{ij}$ is |
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the vector starting at atom $i$ pointing towards $j$, and |
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$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ are the |
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orientational degrees of freedom for atoms $i$ and $j$ |
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respectively. The magnitude of the dipole moment of atom $i$ is |
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$|\mu_i|$ which is referred as the strength of the dipole $s$, |
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$\boldsymbol{\hat{u}}_i$ is the standard unit orientation vector of |
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$\boldsymbol{\Omega}_i$, and $\mathbf{\hat{r}}_{ij}$ is the unit |
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vector pointing along $\mathbf{r}_{ij}$ |
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($\mathbf{\hat{r}}_{ij}=\mathbf{r}_{ij}/|\mathbf{r}_{ij}|$). The unit |
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of the temperature ($T$) is $kelvin$, the strength of the dipole ($s$) |
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is $Debye$, the surface energy ($k_r$) is $k_B$---Bolzmann's |
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constant. For convenience, we will omit the units in the following |
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discussion. The order parameter $P_2$ is defined as $1.5 \times |
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\lambda_{max}$, where $\lambda_{max}$ is the largest eigenvalue of the |
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matrix $\mathsf S$ |
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The dipolar head groups interact via a traditional point-dipolar |
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electrostatic potential, |
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|
\begin{equation} |
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{\mathsf{S}} = |
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\begin{pmatrix} |
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u_{x}u_{x}-\frac{1}{3} & u_{x}u_{y} & u_{x}u_{z} \\ |
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u_{y}u_{x} & u_{y}u_{y}-\frac{1}{3} & u_{y}u_{z} \\ |
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u_{z}u_{x} & u_{z}u_{y} & u_{z}u_{z}-\frac{1}{3} |
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\end{pmatrix}, |
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\label{opmatrix} |
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V^{\mathrm{dd}}_{ij} = \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}}) |
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\right], |
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\label{eq:vdd} |
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|
\end{equation} |
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and $u_{\alpha}$ is the $\alpha$ element of the dipole moment averaged |
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over all particles and configurations. $P_2$ will be $1.0$ for a |
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perfect ordered system or $0$ for a random one. Note this order |
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parameter is 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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$P_2$ is $1.0$. The eigenvector of this matrix is the direction axis |
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which can detect the direction of the dipoles. The periodicity and |
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amplitude of the ripples is given by the fast Fourier transform (FFT) |
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of the perpendicular axis of the direction axis. To detect the |
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lattice of the system, $\gamma = {aLat}/{bLat}$ is defined, where |
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$aLat$, $bLat$ are the lattice distance in X and Y direction |
| 197 |
< |
respectively. $\gamma = \sqrt 3$ for the hexagonal lattice. The length |
| 198 |
< |
of the monolayer in X axis is $20 \times aLat$ and the system is |
| 199 |
< |
roughly square. The average distance that dipoles are from their six |
| 200 |
< |
nearest neighbors is $7$ \AA. So, for the hexagonal lattice, the size |
| 176 |
< |
of the monolayer is about $250$ \AA $\times$ $250$ \AA \ which is |
| 177 |
< |
large enough for the formation of some types of the ripples. In all |
| 178 |
< |
simulations, $10^8$ Monte Carlo moves are attempted, the results are |
| 179 |
< |
judged by standard Metropolis algorithm. Periodic boundary condition |
| 180 |
< |
are used. The cutoff for the long range dipole-dipole interactions is |
| 181 |
< |
set to 30 \AA. |
| 182 |
< |
%The $P_2$ order parameter allows us to measure the amount of |
| 183 |
< |
%directional ordering that exists in the bodies of the molecules making |
| 184 |
< |
%up the bilayer. Each lipid molecule can be thought of as a cylindrical |
| 185 |
< |
%rod with the head group at the top. If all of the rods are perfectly |
| 186 |
< |
%aligned, the $P_2$ order parameter will be $1.0$. If the rods are |
| 187 |
< |
%completely disordered, the $P_2$ order parameter will be 0. For a |
| 188 |
< |
%collection of unit vectors pointing along the principal axes of the |
| 189 |
< |
%rods, the $P_2$ order parameter can be solved via the following |
| 190 |
< |
%method.\cite{zannoni94} |
| 191 |
< |
% |
| 192 |
< |
%Define an ordering tensor $\overleftrightarrow{\mathsf{Q}}$, such that, |
| 193 |
< |
% |
| 194 |
< |
%where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector |
| 195 |
< |
%$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole |
| 196 |
< |
%collection of unit vectors. This allows the tensor to be written: |
| 197 |
< |
%\begin{equation} |
| 198 |
< |
%\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N \biggl[ |
| 199 |
< |
% \mathbf{\hat{u}}_i \otimes \mathbf{\hat{u}}_i |
| 200 |
< |
% - \frac{1}{3} \cdot \mathsf{1} \biggr]. |
| 201 |
< |
%\label{lipidEq:po2} |
| 202 |
< |
%\end{equation} |
| 203 |
< |
% |
| 204 |
< |
%After constructing the tensor, diagonalizing |
| 205 |
< |
%$\overleftrightarrow{\mathsf{Q}}$ yields three eigenvalues and |
| 206 |
< |
%eigenvectors. The eigenvector associated with the largest eigenvalue, |
| 207 |
< |
%$\lambda_{\text{max}}$, is the director axis for the system of unit |
| 208 |
< |
%vectors. The director axis is the average direction all of the unit vectors |
| 209 |
< |
%are pointing. The $P_2$ order parameter is then simply |
| 210 |
< |
%\begin{equation} |
| 211 |
< |
%\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}. |
| 212 |
< |
%\label{lipidEq:po3} |
| 213 |
< |
%\end{equation} |
| 214 |
< |
% |
| 215 |
< |
%\begin{figure} |
| 216 |
< |
%\begin{center} |
| 217 |
< |
%\includegraphics[scale=0.3]{/home/maul/gezelter/xsun/Documents/ripple/picture/lattice.eps} |
| 218 |
< |
%\caption{ The lattice\label{lat}} |
| 219 |
< |
%\end{center} |
| 220 |
< |
%\end{figure} |
| 186 |
> |
and the hydrophobic interactions are approximated with a nearest |
| 187 |
> |
neighbor sum of harmonic interactions, |
| 188 |
> |
\begin{equation} |
| 189 |
> |
V^{\mathrm{harm}}_{ij} = \frac{k_r}{2} \left(r_{ij}-r_0\right)^2 |
| 190 |
> |
\end{equation} |
| 191 |
> |
In these potentials, $\mathbf{\hat u}_i$ is the unit vector pointing |
| 192 |
> |
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
| 193 |
> |
pointing along the inter-dipole vector $\mathbf{r}_{ij}$. The entire |
| 194 |
> |
potential is governed by three parameters, the dipolar strength |
| 195 |
> |
($\mu$), the harmonic spring constant ($k_r$) and the preferred |
| 196 |
> |
intermolecular spacing ($r_0$). In practice, we set the value of |
| 197 |
> |
$r_0$ to the average inter-molecular spacing from the planar lattice, |
| 198 |
> |
yielding a potential model that has only two parameters for a |
| 199 |
> |
particular choice of lattice constants $a$ (along the $x$-axis) and $b$ |
| 200 |
> |
(along the $y$-axis). |
| 201 |
|
|
| 202 |
< |
\section{Results and discussion} |
| 203 |
< |
\label{Res} |
| 202 |
> |
To investigate the phase behavior of this model, we have performed a |
| 203 |
> |
series of Metropolis Monte Carlo simulations of moderately-sized (24 |
| 204 |
> |
nm on a side) patches of membrane hosted on both hexagonal ($\gamma = |
| 205 |
> |
a/b = \sqrt{3}$) and non-hexagonal ($\gamma \neq \sqrt{3}$) lattices. |
| 206 |
> |
The linear extent of one edge of the monolayer was $20 a$ and the |
| 207 |
> |
system was kept roughly square. The average distance that coplanar |
| 208 |
> |
dipoles were positioned from their six nearest neighbors was $7$ \AA. |
| 209 |
> |
Typical system sizes were 1360 lipids for the hexagonal lattices and |
| 210 |
> |
840-2800 lipids for the non-hexagonal lattices. Periodic boundary |
| 211 |
> |
conditions were used, and the cutoff for the dipole-dipole interaction |
| 212 |
> |
was set to 30 \AA. All parameters ($T$, $\mu$, $k_r$, $\gamma$) were |
| 213 |
> |
varied systematically to study the effects of these parameters on the |
| 214 |
> |
formation of ripple-like phases. |
| 215 |
|
|
| 216 |
< |
\subsection{Hexagonal} |
| 217 |
< |
\label{Hex} |
| 218 |
< |
%Fig. \ref{frip} shows the typical simulation results for the hexagonal system when $T = 300$, $s = 7$, $k_r = 0.1$. |
| 219 |
< |
%\begin{figure} |
| 220 |
< |
%\centering |
| 221 |
< |
%\epsfbox{/home/maul/gezelter/xsun/Documents/ripple/picture/rippletop.eps} |
| 222 |
< |
%\epsfbox{/home/maul/gezelter/xsun/Documents/ripple/picture/rippleside.eps} |
| 223 |
< |
%\caption{A snapshot of our simulation results. The filled circle indicates the position of the dipole, the tail attached on it points out the direction of the dipole. (a)Top view of the monolayer. (b)Side view of the monolayer} |
| 224 |
< |
%\label{frip} |
| 225 |
< |
%\end{figure} |
| 226 |
< |
From the results of the simulation at $T = 300$ for hexagonal lattice, the system is in an antiferroelectric state. Every $3$ or $4$ arrows of the dipoles form a strip whose direction is opposite to its neighbors. $P_2$ is about $0.7$. The ripple is formed clearly. The simulation results shows the ripple has equal opportunity to be formed along different directions, this is due to the isotropic property of the hexagonal lattice. |
| 227 |
< |
We use the last configuration of this simulation as the initial condition to increase the system to $T = 400$ every $10\ kelvin$, at the same time decrease the temperature to $T = 100$ every $10\ kelvin$. The trend that the order parameter varies with temperature is plotted in Fig. \ref{t-op}. |
| 216 |
> |
\section{Results and Analysis} |
| 217 |
> |
\label{sec:results} |
| 218 |
> |
|
| 219 |
> |
\subsection{Dipolar Ordering and Coexistence Temperatures} |
| 220 |
> |
The principal method for observing the orientational ordering |
| 221 |
> |
transition in dipolar systems is the $P_2$ order parameter (defined as |
| 222 |
> |
$1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
| 223 |
> |
eigenvalue of the matrix, |
| 224 |
> |
\begin{equation} |
| 225 |
> |
{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
| 226 |
> |
\begin{array}{ccc} |
| 227 |
> |
u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\ |
| 228 |
> |
u^{y}_i u^{x}_i & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\ |
| 229 |
> |
u^{z}_i u^{x}_i & u^{z}_i u^{y}_i & u^{z}_i u^{z}_i -\frac{1}{3} |
| 230 |
> |
\end{array} \right). |
| 231 |
> |
\label{eq:opmatrix} |
| 232 |
> |
\end{equation} |
| 233 |
> |
Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector |
| 234 |
> |
for dipole $i$. $P_2$ will be $1.0$ for a perfectly-ordered system |
| 235 |
> |
and near $0$ for a randomized system. Note that this order parameter |
| 236 |
> |
is {\em not} equal to the polarization of the system. For example, |
| 237 |
> |
the polarization of the perfect antiferroelectric system is $0$, but |
| 238 |
> |
$P_2$ for an antiferroelectric system is $1$. The eigenvector of |
| 239 |
> |
$\mathsf{S}$ corresponding to the largest eigenvalue is familiar as |
| 240 |
> |
the director axis, which can be used to determine a priveleged dipolar |
| 241 |
> |
axis for dipole-ordered systems. Fig. \ref{t-op} shows the values of |
| 242 |
> |
$P_2$ as a function of temperature for both hexagonal ($\gamma = |
| 243 |
> |
1.732$) and non-hexagonal ($\gamma=1.875$) lattices. |
| 244 |
|
|
| 245 |
< |
\begin{figure} |
| 245 |
> |
\begin{figure}[ht] |
| 246 |
|
\centering |
| 247 |
< |
\includegraphics[width=\linewidth]{picture/hexorderpara.eps} |
| 248 |
< |
\caption{ The orderparameter $P_2$ vs temperature T at hexagonal |
| 249 |
< |
lattice.} |
| 247 |
> |
\caption{The $P_2$ dipolar order parameter as a function of |
| 248 |
> |
temperature for both hexagonal ($\gamma = 1.732$) and non-hexagonal |
| 249 |
> |
($\gamma = 1.875$) lattices} |
| 250 |
> |
\includegraphics[width=\linewidth]{picture/t-orderpara.eps} |
| 251 |
|
\label{t-op} |
| 252 |
|
\end{figure} |
| 253 |
|
|
| 254 |
< |
The $P_2 \approx 0.9$ for $T = 100$ implies that the system is in a |
| 255 |
< |
highly ordered state. As the temperature increases, the order |
| 256 |
< |
parameter is decreasing gradually before $T = 300$, from $T = 310$ the |
| 257 |
< |
order parameter drops dramatically, get to nearly $0$ at $T = |
| 258 |
< |
400$. This means the system reaches a random state from an ordered |
| 259 |
< |
state. The phase transition occurs at $T \approx 340$. At the |
| 260 |
< |
temperature range the ripples formed, the structure is fairly stable |
| 261 |
< |
with the temperature changing, we can say this structure is in one of |
| 262 |
< |
the energy minimum of the energy surface. The amplitude of the ripples |
| 263 |
< |
is around $15$ \AA. With the temperature changing, the amplitude of |
| 264 |
< |
the ripples is stable also. This is contrast with our general |
| 265 |
< |
knowledge that ripples will increase with thermal energy of the system |
| 258 |
< |
increasing. To understand the origin and property of the ripples, we |
| 259 |
< |
need look at the potential of our system, which is $V = V_{\text |
| 260 |
< |
{surface tension}} + V_{\text {dipole}}$. There are two parts of |
| 261 |
< |
it. The intense of the $V_{\text {surface tension}}$ is controlled by |
| 262 |
< |
$k_r$ which is the surface energy, and the intense of the $ V_{\text |
| 263 |
< |
{dipole}}$ is controlled by $s$ which is the strength of the |
| 264 |
< |
dipoles. So, according to adjusting these two parameters, we can get |
| 265 |
< |
the further insight into this problem. At first, we fixed the value |
| 266 |
< |
of $s = 7$, and vary $k_r$, the results are shown in |
| 267 |
< |
Fig. \ref{kr-a-hf}. |
| 254 |
> |
There is a clear order-disorder transition in evidence from this data. |
| 255 |
> |
Both the hexagonal and non-hexagonal lattices have dipolar-ordered |
| 256 |
> |
low-temperature phases, and orientationally-disordered high |
| 257 |
> |
temperature phases. The coexistence temperature for the hexagonal |
| 258 |
> |
lattice is significantly lower than for the non-hexagonal lattices, |
| 259 |
> |
and the bulk polarization is approximately $0$ for both dipolar |
| 260 |
> |
ordered and disordered phases. This gives strong evidence that the |
| 261 |
> |
dipolar ordered phase is antiferroelectric. We have repeated the |
| 262 |
> |
Monte Carlo simulations over a wide range of lattice ratios ($\gamma$) |
| 263 |
> |
to generate a dipolar order/disorder phase diagram. Fig. \ref{phase} |
| 264 |
> |
shows that the hexagonal lattice is a low-temperature cusp in the |
| 265 |
> |
$T-\gamma$ phase diagram. |
| 266 |
|
|
| 267 |
< |
\begin{figure} |
| 267 |
> |
\begin{figure}[ht] |
| 268 |
|
\centering |
| 269 |
< |
\includegraphics[width=\linewidth]{picture/kr_amplitude.eps} |
| 270 |
< |
\caption{ The amplitude $A$ of the ripples at $T = 300$ vs $k_r$ for |
| 271 |
< |
hexagonal lattice. The height fluctuation $h_f$ vs $k_r$ is shown |
| 272 |
< |
inset for the same situation.} |
| 273 |
< |
\label{kr-a-hf} |
| 269 |
> |
\caption{The phase diagram for the web-of-dipoles model. The line |
| 270 |
> |
denotes the division between the dipolar ordered (antiferroelectric) |
| 271 |
> |
and disordered phases. An enlarged view near the hexagonal lattice is |
| 272 |
> |
shown inset.} |
| 273 |
> |
\includegraphics[width=\linewidth]{picture/phase.eps} |
| 274 |
> |
\label{phase} |
| 275 |
|
\end{figure} |
| 276 |
|
|
| 277 |
< |
When $k_r < 0.1$, due to the small surface tension part, the dipoles |
| 278 |
< |
can go far away from their neighbors, lots of noise make the ripples |
| 279 |
< |
undiscernable. So, we start from $k_r = 0.1$. However, even when $k_r |
| 280 |
< |
= 0$, which means the surface tension is turned off, the |
| 281 |
< |
antiferroelectric state still can be reached. This strongly supports |
| 282 |
< |
the dipole-dipole interaction is the major driving force to form the |
| 283 |
< |
long range orietational ordered state. From Fig. \ref{kr-a-hf}, the |
| 284 |
< |
amplitude decreases as the $k_r$ increasing, actually, when |
| 285 |
< |
$k_r > 0.7$, although the FFT results still show the values of amplitudes, |
| 286 |
< |
the ripples disappear. From the inset of the |
| 287 |
< |
Fig. \ref{kr-a-hf}, the trend of the fluctuation of height of the dipoles---$h_f$ |
| 288 |
< |
with $k_r$ is similar to the amplitude. |
| 289 |
< |
Here $h_f = < h^2 > - {< h >}^2$, $h$ is the $z$ |
| 290 |
< |
coordinate of the dipoles, $<>$ means $h$ averaged by all dipoles and |
| 292 |
< |
configurations. The decreasing of the height fluctuation is due to the |
| 293 |
< |
increasing of the surface tension with increasing the $k_r$. |
| 294 |
< |
No ripple is observed |
| 295 |
< |
when $k_r > 0.7$. When $k_r > 0.7$, the surface tension part of the total |
| 296 |
< |
potential of the system dominate the structure of the monolayer, the |
| 297 |
< |
dipoles will be kept as near as possible with their neighbors, the |
| 298 |
< |
whole system is fairly flat under this situation, and the ripples |
| 299 |
< |
disappear. Then we investigate the role of the dipole-dipole |
| 300 |
< |
interactions by fixing the $k_r$ to be $0.1$. This long range |
| 301 |
< |
orientational ordered state is very sensitive to the value of $s$ for |
| 302 |
< |
hexagonal lattice. For $s = 6$, only local orientational ordering |
| 303 |
< |
occurs, when $s$ is even smaller, the system is on a random state. For |
| 304 |
< |
$s \geq 9$, the system enters a frustrated state, the amplitude is |
| 305 |
< |
hard to tell, however, from observation, the amplitude does not change |
| 306 |
< |
too much. We will fully discuss this problem using a distorted |
| 307 |
< |
hexagonal lattice. In brief, asymmetry of the translational freedom |
| 308 |
< |
of the dipoles breaks the symmetry of the hexagonal lattice and allow |
| 309 |
< |
antiferroelectric ordering of the dipoles. The dipole-dipole |
| 310 |
< |
interaction is the major driving force for the long range |
| 311 |
< |
orientational ordered state. The formation of the stable, smooth |
| 312 |
< |
ripples is a result of the competition between surface tension and |
| 313 |
< |
dipole-dipole interaction. |
| 277 |
> |
This phase diagram is remarkable in that it shows an antiferroelectric |
| 278 |
> |
phase near $\gamma=1.732$ where one would expect lattice frustration |
| 279 |
> |
to result in disordered phases at all temperatures. Observations of |
| 280 |
> |
the configurations in this phase show clearly that the system has |
| 281 |
> |
accomplished dipolar orderering by forming large ripple-like |
| 282 |
> |
structures. We have observed antiferroelectric ordering in all three |
| 283 |
> |
of the equivalent directions on the hexagonal lattice, and the dipoles |
| 284 |
> |
have been observed to organize perpendicular to the membrane normal |
| 285 |
> |
(in the plane of the membrane). It is particularly interesting to |
| 286 |
> |
note that the ripple-like structures have also been observed to |
| 287 |
> |
propagate in the three equivalent directions on the lattice, but the |
| 288 |
> |
{\em direction of ripple propagation is always perpendicular to the |
| 289 |
> |
dipole director axis}. A snapshot of a typical antiferroelectric |
| 290 |
> |
rippled structure is shown in Fig. \ref{fig:snapshot}. |
| 291 |
|
|
| 292 |
< |
\subsection{Non-hexagonal} |
| 293 |
< |
\label{Nhe} |
| 294 |
< |
We also investigate the effect of lattice type by changing |
| 295 |
< |
$\gamma$. The antiferroelectric state is accessible for all $\gamma$ |
| 296 |
< |
we use, and will melt with temperature increasing, unlike hexagonal |
| 297 |
< |
lattice, the distorted hexagonal lattices prefer a particular director |
| 298 |
< |
axis due to their anisotropic property. The phase diagram for this |
| 299 |
< |
system is shown in Fig. \ref{phase}. |
| 292 |
> |
\begin{figure}[ht] |
| 293 |
> |
\centering |
| 294 |
> |
\caption{Top and Side views of a representative configuration for the |
| 295 |
> |
dipolar ordered phase supported on the hexagonal lattice. Note the |
| 296 |
> |
antiferroelectric ordering and the long wavelength buckling of the |
| 297 |
> |
membrane. Dipolar ordering has been observed in all three equivalent |
| 298 |
> |
directions on the hexagonal lattice, and the ripple direction is |
| 299 |
> |
always perpendicular to the director axis for the dipoles.} |
| 300 |
> |
\includegraphics[width=\linewidth]{picture/snapshot.eps} |
| 301 |
> |
\label{fig:snapshot} |
| 302 |
> |
\end{figure} |
| 303 |
|
|
| 304 |
< |
\begin{figure} |
| 304 |
> |
\subsection{Discriminating Ripples from Thermal Undulations} |
| 305 |
> |
|
| 306 |
> |
In order to be sure that the structures we have observed are actually |
| 307 |
> |
a rippled phase and not merely thermal undulations, we have computed |
| 308 |
> |
the undulation spectrum, |
| 309 |
> |
\begin{equation} |
| 310 |
> |
h(\vec{q}) = A^{-1/2} \int d\vec{r} |
| 311 |
> |
h(\vec{r}) e^{-i \vec{q}\cdot\vec{r}} |
| 312 |
> |
\end{equation} |
| 313 |
> |
where $h(\vec{r})$ is the height of the membrane at location $\vec{r} |
| 314 |
> |
= (x,y)$.~\cite{Safran94} In simple (and more complicated) elastic |
| 315 |
> |
continuum models, Brannigan {\it et al.} have shown that in the $NVT$ |
| 316 |
> |
ensemble, the absolute value of the undulation spectrum can be |
| 317 |
> |
written, |
| 318 |
> |
\begin{equation} |
| 319 |
> |
\langle | h(q)|^2 \rangle_{NVT} = \frac{k_B T}{k_c |\vec{q}|^4 + |
| 320 |
> |
\tilde{\gamma}|\vec{q}|^2}, |
| 321 |
> |
\label{eq:fit} |
| 322 |
> |
\end{equation} |
| 323 |
> |
where $k_c$ is the bending modulus for the membrane, and |
| 324 |
> |
$\tilde{\gamma}$ is the mechanical surface |
| 325 |
> |
tension.~\cite{Brannigan04b} |
| 326 |
> |
|
| 327 |
> |
The undulation spectrum is computed by superimposing a rectangular |
| 328 |
> |
grid on top of the membrane, and by assigning height ($h(\vec{r})$) |
| 329 |
> |
values to the grid from the average of all dipoles that fall within a |
| 330 |
> |
given $\vec{r}+d\vec{r}$ grid area. Empty grid pixels are assigned |
| 331 |
> |
height values by interpolation from the nearest neighbor pixels. A |
| 332 |
> |
standard 2-d Fourier transform is then used to obtain $\langle | |
| 333 |
> |
h(q)|^2 \rangle$. |
| 334 |
> |
|
| 335 |
> |
The systems studied in this paper have relatively small bending moduli |
| 336 |
> |
($k_c$) and relatively large mechanical surface tensions |
| 337 |
> |
($\tilde{\gamma}$). In practice, the best fits to our undulation |
| 338 |
> |
spectra are obtained by approximating the value of $k_c$ to 0. In |
| 339 |
> |
Fig. \ref{fig:fit} we show typical undulation spectra for two |
| 340 |
> |
different regions of the phase diagram along with their fits from the |
| 341 |
> |
Landau free energy approach (Eq. \ref{eq:fit}). In the |
| 342 |
> |
high-temperature disordered phase, the Landau fits can be nearly |
| 343 |
> |
perfect, and from these fits we can estimate the bending modulus and |
| 344 |
> |
the mechanical surface tension. |
| 345 |
> |
|
| 346 |
> |
For the dipolar-ordered hexagonal lattice near the coexistence |
| 347 |
> |
temperature, however, we observe long wavelength undulations that are |
| 348 |
> |
far outliers to the fits. That is, the Landau free energy fits are |
| 349 |
> |
well within error bars for all other points, but can be off by {\em |
| 350 |
> |
orders of magnitude} for a few (but not all) low frequency |
| 351 |
> |
components. |
| 352 |
> |
|
| 353 |
> |
We interpret these outliers as evidence that these low frequency modes |
| 354 |
> |
are {\em non-thermal undulations} which is clear evidence that we are |
| 355 |
> |
actually seeing a rippled phase developing in this model system. |
| 356 |
> |
|
| 357 |
> |
\begin{figure}[ht] |
| 358 |
|
\centering |
| 359 |
< |
\includegraphics[width=\linewidth]{picture/phase.eps} |
| 360 |
< |
\caption{ The phase diagram with temperature $T$ and lattice variable |
| 361 |
< |
$\gamma$. The enlarged view near the hexagonal lattice is shown |
| 362 |
< |
inset.} |
| 363 |
< |
\label{phase} |
| 359 |
> |
\caption{Evidence that the observed ripples are {\em not} thermal |
| 360 |
> |
undulations is obtained from the 2-d fourier transform $\langle |
| 361 |
> |
|h(\vec{q})|^2 \rangle$ of the height profile ($\langle h(x,y) |
| 362 |
> |
\rangle$). Rippled samples show low-wavelength peaks that are |
| 363 |
> |
outliers on the Landau free energy fits. Samples exhibiting only |
| 364 |
> |
thermal undulations fit Eq. \ref{eq:fit} remarkably well.} |
| 365 |
> |
\includegraphics[width=\linewidth]{picture/fit.eps} |
| 366 |
> |
\label{fig:fit} |
| 367 |
|
\end{figure} |
| 368 |
|
|
| 369 |
< |
$T_c$ is the transition temperature. The hexagonal lattice has the |
| 334 |
< |
lowest $T_c$, and $T_c$ goes up with lattice being more |
| 335 |
< |
distorted. There is only two phases in our diagram. When we do |
| 336 |
< |
annealing for all the system, the antiferroelectric phase is fairly |
| 337 |
< |
stable, although the spin glass is accessible for $\gamma \leq |
| 338 |
< |
\sqrt{3}$ if the simulations is started from the random initial |
| 339 |
< |
configuration. So, we consider the antiferroelectric phase as a local |
| 340 |
< |
minimum energy state even at low temperature. From the inset of |
| 341 |
< |
Fig. \ref{phase}, at the hexagonal lattice, $T_c$ changes |
| 342 |
< |
quickly. $T_c$ increases more quickly for $\gamma$ getting larger than |
| 343 |
< |
$\gamma$ getting smaller. The reason is that: although the average |
| 344 |
< |
distance between dipole and its neighbors is same for all types of |
| 345 |
< |
lattices, $V_\text{dipole} \propto 1/r_{ij}^3$ in our model, the |
| 346 |
< |
change of the lattice spacing in one direction is more effective than |
| 347 |
< |
another in this range of $\gamma$. There is another type of |
| 348 |
< |
antiferroelectric state when the lattice is far away from the |
| 349 |
< |
hexagonal one. Unlike the antiferroelectric state of the hexagonal |
| 350 |
< |
lattice which is composed of the strips that have $3$ or $4$ rows of |
| 351 |
< |
same direction dipoles, the strips in this type of antiferroelectric |
| 352 |
< |
state have $1$, $2$ or $3$ rows of same direction dipoles. In our |
| 353 |
< |
phase diagram, this difference is not shown. However, only when |
| 354 |
< |
$\gamma$ is close to $\sqrt{3}$, the long range spatial |
| 355 |
< |
ordering---ripple is still maintained. The surface is flat when |
| 356 |
< |
$\gamma \ll \sqrt{3}$, and randomly fluctuate due to the appearance of |
| 357 |
< |
another type antiferroelectric state when $\gamma \gg \sqrt{3}$. The |
| 358 |
< |
change of the lattice type changes the contribution of the surface |
| 359 |
< |
tension and the dipole-dipole interaction for the total potential of |
| 360 |
< |
the system. For $\gamma \ll \sqrt{3}$, the total potential is |
| 361 |
< |
dominated by the surface tension part, so, the surface is flat. For |
| 362 |
< |
$\gamma \gg \sqrt{3}$, the total potential is dominated by the |
| 363 |
< |
dipole-dipole interaction part, it is very easy to introduce too much |
| 364 |
< |
noise to make the ripples undiscernable. In our simulations, the |
| 365 |
< |
amplitude of the ripples for distorted hexagonal lattice is larger |
| 366 |
< |
than that for hexagonal lattice in the small range around the |
| 367 |
< |
hexagonal lattice. The reason is still not clear. A possible |
| 368 |
< |
explanation is that the distribution of the dipole-dipole interaction |
| 369 |
< |
through the surface is anisotropic in the distorted hexagonal |
| 370 |
< |
lattice. Another possibility is that the hexagonal lattice has many |
| 371 |
< |
translational local minimum, it has not entered the more rippled state |
| 372 |
< |
for our reasonable simulation period. We investigate the effect of |
| 373 |
< |
the strength of the dipole $s$ to the amplitude of the ripples for |
| 374 |
< |
$\gamma = 1.875$, $k_r = 0.1$, $T = 260$. Under this situation, the |
| 375 |
< |
system reaches the equilibrium very quickly, and the ripples are |
| 376 |
< |
fairly stable. The results are shown in Fig. \ref{samplitude}. |
| 369 |
> |
\subsection{Effects of Parameters on Ripple Amplitude and Wavelength} |
| 370 |
|
|
| 371 |
< |
\begin{figure} |
| 371 |
> |
We have used two different methods to estimate the amplitude and |
| 372 |
> |
periodicity of the ripples. The first method requires projection of |
| 373 |
> |
the ripples onto a one dimensional rippling axis. Since the rippling |
| 374 |
> |
is always perpendicular to the dipole director axis, we can define a |
| 375 |
> |
ripple vector as follows. The largest eigenvector, $s_1$, of the |
| 376 |
> |
$\mathsf{S}$ matrix in Eq. \ref{eq:opmatrix} is projected onto a |
| 377 |
> |
planar director axis, |
| 378 |
> |
\begin{equation} |
| 379 |
> |
\vec{d} = \left(\begin{array}{c} |
| 380 |
> |
\vec{s}_1 \cdot \hat{i} \\ |
| 381 |
> |
\vec{s}_1 \cdot \hat{j} \\ |
| 382 |
> |
0 |
| 383 |
> |
\end{array} \right). |
| 384 |
> |
\end{equation} |
| 385 |
> |
($\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along the $x$, |
| 386 |
> |
$y$, and $z$ axes, respectively.) The rippling axis is in the plane of |
| 387 |
> |
the membrane and is perpendicular to the planar director axis, |
| 388 |
> |
\begin{equation} |
| 389 |
> |
\vec{q}_{\mathrm{rip}} = \vec{d} \times \hat{k} |
| 390 |
> |
\end{equation} |
| 391 |
> |
We can then find the height profile of the membrane along the ripple |
| 392 |
> |
axis by projecting heights of the dipoles to obtain a one-dimensional |
| 393 |
> |
height profile, $h(q_{\mathrm{rip}})$. Ripple wavelengths can be |
| 394 |
> |
estimated from the largest non-thermal low-frequency component in the |
| 395 |
> |
fourier transform of $h(q_{\mathrm{rip}})$. Amplitudes can be |
| 396 |
> |
estimated by measuring peak-to-trough distances in |
| 397 |
> |
$h(q_{\mathrm{rip}})$ itself. |
| 398 |
> |
|
| 399 |
> |
A second, more accurate, and simpler method for estimating ripple |
| 400 |
> |
shape is to extract the wavelength and height information directly |
| 401 |
> |
from the largest non-thermal peak in the undulation spectrum. For |
| 402 |
> |
large-amplitude ripples, the two methods give similar results. The |
| 403 |
> |
one-dimensional projection method is more prone to noise (particularly |
| 404 |
> |
in the amplitude estimates for the non-hexagonal lattices). We report |
| 405 |
> |
amplitudes and wavelengths taken directly from the undulation spectrum |
| 406 |
> |
below. |
| 407 |
> |
|
| 408 |
> |
In the hexagonal lattice ($\gamma = \sqrt{3}$), the rippling is |
| 409 |
> |
observed from $150-300$ K. The wavelength of the ripples is |
| 410 |
> |
remarkably stable at 150~\AA~for all but the temperatures closest to |
| 411 |
> |
the order-disorder transition. At 300 K, the wavelength drops to 120 |
| 412 |
> |
\AA. |
| 413 |
> |
|
| 414 |
> |
The dependence of the amplitude on temperature is shown in |
| 415 |
> |
Fig. \ref{fig:t-a}. The rippled structures shrink smoothly as the |
| 416 |
> |
temperature rises towards the order-disorder transition. The |
| 417 |
> |
wavelengths and amplitudes we observe are surprisingly close to the |
| 418 |
> |
$\Lambda / 2$ phase observed by Kaasgaard {\it et al.} in their work |
| 419 |
> |
on PC-based lipids,\cite{Kaasgaard03} although this may be |
| 420 |
> |
coincidental agreement given our choice of parameters. |
| 421 |
> |
|
| 422 |
> |
\begin{figure}[ht] |
| 423 |
|
\centering |
| 424 |
< |
\includegraphics[width=\linewidth]{picture/samplitude.eps} |
| 425 |
< |
\caption{ The amplitude of ripples $A$ at $T = 260$ vs strength of |
| 426 |
< |
dipole $s$ for $\gamma = 1.875$. The orderparameter $P_2$ vs $s$ is |
| 427 |
< |
shown inset at the same situation.} |
| 384 |
< |
\label{samplitude} |
| 424 |
> |
\caption{ The amplitude $A$ of the ripples vs. temperature for a |
| 425 |
> |
hexagonal lattice.} |
| 426 |
> |
\includegraphics[width=\linewidth]{picture/t-a-error.eps} |
| 427 |
> |
\label{fig:t-a} |
| 428 |
|
\end{figure} |
| 429 |
|
|
| 430 |
< |
For small $s$, there is no long range ordering in the system, so, we |
| 431 |
< |
start from $s = 7$, and we use the rippled state as the initial |
| 432 |
< |
configuration for all the simulations to reduce the noise. There is no |
| 433 |
< |
considerable change of the amplitude in our simulations. At first, the |
| 434 |
< |
system is under the competition of the surface tension and |
| 435 |
< |
dipole-dipole interactions, increasing $s$ will make the dipole-dipole |
| 436 |
< |
interactions more contribute to the total potential and the amplitude |
| 437 |
< |
of the ripples is increased a little bit. After the total potential is |
| 395 |
< |
totally dominated by the dipole-dipole interactions, the amplitude |
| 396 |
< |
does not change too much. This result indicates that the ripples are |
| 397 |
< |
the natural property of the dipolar system, the existence of the |
| 398 |
< |
ripples does not depend on the surface tension. The orderparameter |
| 399 |
< |
increases with increasing the strength of the dipole. |
| 430 |
> |
The ripples can be made to disappear by increasing the internal |
| 431 |
> |
surface tension (i.e. by increasing $k_r$). In Fig. \ref{fig:kr-a} |
| 432 |
> |
we show the ripple amplitude as a function of the internal spring |
| 433 |
> |
constant for non-dipolar part of the lipid interaction potential. |
| 434 |
> |
Weaker ``hydrophobic'' interactions allow the lipid structure to be |
| 435 |
> |
dominated by the dipoles, and stronger ``hydrophobic'' interactions |
| 436 |
> |
result in much flatter membranes. Section \ref{sec:discussion} |
| 437 |
> |
contains further discussion of this effect. |
| 438 |
|
|
| 439 |
< |
\section{Conclusion} |
| 440 |
< |
\label{Con} |
| 441 |
< |
In conclusion, the molecular explanation of the origin of the long |
| 442 |
< |
range ordering of the hexagonal lattice is given by our |
| 443 |
< |
simulations. Asymmetry of the translational freedom of the dipoles |
| 444 |
< |
breaks the symmetry of the hexagonal lattice and allow |
| 445 |
< |
antiferroelectric ordering of the dipoles. The simulation results |
| 446 |
< |
demonstrate that the dipole-dipole interaction is the major driving |
| 447 |
< |
force for the long range orientational ordered state. According to |
| 410 |
< |
the study of the effect of the surface tension and the dipole-dipole |
| 411 |
< |
interaction, we find ripples are the natural property of the dipolar |
| 412 |
< |
system. Its existence does not depend on the surface tension, however, |
| 413 |
< |
a stable, smooth ripple phase is a result of the competition between |
| 414 |
< |
surface tension and dipole-dipole interaction, and when surface |
| 415 |
< |
tension is large enough to dominate the total potential, the amplitude |
| 416 |
< |
of the ripples can be determined by it. The ripple phase can only be |
| 417 |
< |
reached near the hexagonal lattice. Under same condition, the |
| 418 |
< |
amplitude of the ripples for hexagonal lattice is smaller than that |
| 419 |
< |
for distorted hexagonal lattice. The reason is not clear, however, we |
| 420 |
< |
think it is a result of the anisotropic distribution of the |
| 421 |
< |
dipole-dipole interaction through the surface in the distorted |
| 422 |
< |
hexagonal lattice. From the phase diagram, the reason of the |
| 423 |
< |
existence of the ripple phase in organism is elucidated. To melt at |
| 424 |
< |
the body temperature and perform its bio-function, the lipid bilayer |
| 425 |
< |
must have a relative low transition temperature which can be realized |
| 426 |
< |
near the hexagonal lattice, and the ripple phase is a natural phase |
| 427 |
< |
for dipolar system at the hexagonal lattice. So, with the temperature |
| 428 |
< |
increasing, the lipid bilayer undergoes a translational adjustment to |
| 429 |
< |
enter the ripple phase to lower the transition temperature for the |
| 430 |
< |
gel-liquid phase transition, then it can enter the liquid phase even |
| 431 |
< |
at a low temperature. |
| 439 |
> |
\begin{figure}[ht] |
| 440 |
> |
\centering |
| 441 |
> |
\caption{The amplitude $A$ of the ripples vs. the harmonic binding |
| 442 |
> |
constant $k_r$ for both the hexagonal lattice (circles) and |
| 443 |
> |
non-hexagonal lattice (squares). In both simulations the dipole |
| 444 |
> |
strength ($\mu$) was 7 Debye and the temperature was 260K.} |
| 445 |
> |
\includegraphics[width=\linewidth]{picture/k-a-error.eps} |
| 446 |
> |
\label{fig:kr-a} |
| 447 |
> |
\end{figure} |
| 448 |
|
|
| 449 |
< |
\newpage |
| 450 |
< |
\bibliographystyle{jcp} |
| 451 |
< |
\bibliography{ripple.bib} |
| 449 |
> |
The amplitude of the ripples depends critically on the strength of the |
| 450 |
> |
dipole moments ($\mu$) in Eq. \ref{eq:vdd}. If the dipoles are |
| 451 |
> |
weakened substantially (below $\mu$ = 5 Debye) at a fixed temperature |
| 452 |
> |
of 230 K, the membrane loses dipolar ordering and the ripple |
| 453 |
> |
structures. The ripples reach a peak amplitude |
| 454 |
> |
of 26~\AA~at a dipolar strength of 9 Debye. We show the dependence of |
| 455 |
> |
ripple amplitude on the dipolar strength in Fig. \ref{fig:s-a}. |
| 456 |
> |
|
| 457 |
> |
\begin{figure}[ht] |
| 458 |
> |
\centering |
| 459 |
> |
\caption{The amplitude $A$ of the ripples vs. dipole strength ($\mu$) |
| 460 |
> |
for both the hexagonal lattice (circles) and non-hexagonal lattice |
| 461 |
> |
(squares). In both simulations the dipole |
| 462 |
> |
strength ($k_r$) was kept constant at a value of $1.987 \times |
| 463 |
> |
10^{-4}$ kcal mol$^{-1}$ \AA$^{-2}$. The temperatures were also kept |
| 464 |
> |
fixed at 230K for the hexagonal lattice and 260K for the non-hexagonal |
| 465 |
> |
lattice (approximately 2/3 of the order-disorder transition |
| 466 |
> |
temperature for each lattice).} |
| 467 |
> |
\includegraphics[width=\linewidth]{picture/A-s.eps} |
| 468 |
> |
\label{fig:s-a} |
| 469 |
> |
\end{figure} |
| 470 |
> |
|
| 471 |
> |
\subsection{Non-hexagonal lattices} |
| 472 |
> |
|
| 473 |
> |
We have also investigated the effect of the lattice geometry by |
| 474 |
> |
changing the ratio of lattice constants ($\gamma$) while keeping the |
| 475 |
> |
average nearest-neighbor spacing constant. The antiferroelectric state |
| 476 |
> |
is accessible for all $\gamma$ values we have used, although the |
| 477 |
> |
distorted hexagonal lattices prefer a particular director axis due to |
| 478 |
> |
the anisotropy of the lattice. |
| 479 |
> |
|
| 480 |
> |
Our observation of rippling behavior was not limited to the hexagonal |
| 481 |
> |
lattices. In non-hexagonal lattices the antiferroelectric phase can |
| 482 |
> |
develop nearly instantaneously in the Monte Carlo simulations, and |
| 483 |
> |
these dipolar-ordered phases tend to be remarkably flat. Whenever |
| 484 |
> |
rippling has been observed in these non-hexagonal lattices |
| 485 |
> |
(e.g. $\gamma = 1.875$), we see relatively short ripple wavelengths |
| 486 |
> |
(98 \AA) and amplitudes of 17 \AA. These ripples are weakly dependent |
| 487 |
> |
on dipolar strength (see Fig. \ref{fig:s-a}), although below a dipolar |
| 488 |
> |
strength of 5.5 Debye, the membrane loses dipolar ordering and |
| 489 |
> |
displays only thermal undulations. |
| 490 |
> |
|
| 491 |
> |
The rippling in non-hexagonal lattices also shows a strong dependence |
| 492 |
> |
on the internal surface tension ($k_r$). It is possible to make the |
| 493 |
> |
ripples disappear by increasing the internal tension. The low-tension |
| 494 |
> |
limit appears to result in somewhat smaller ripples than in the |
| 495 |
> |
hexagonal lattice (see Fig. \ref{fig:kr-a}). |
| 496 |
> |
|
| 497 |
> |
The ripple phase does {\em not} appear at all values of $\gamma$. We |
| 498 |
> |
have only observed non-thermal undulations in the range $1.625 < |
| 499 |
> |
\gamma < 1.875$. Outside this range, the order-disorder transition in |
| 500 |
> |
the dipoles remains, but the ordered dipolar phase has only thermal |
| 501 |
> |
undulations. This is one of our strongest pieces of evidence that |
| 502 |
> |
rippling is a symmetry-breaking phenomenon for hexagonal and |
| 503 |
> |
nearly-hexagonal lattices. |
| 504 |
> |
|
| 505 |
> |
\subsection{Effects of System Size} |
| 506 |
> |
To evaluate the effect of finite system size, we have performed a |
| 507 |
> |
series of simulations on the hexagonal lattice at a temperature of 300K, |
| 508 |
> |
which is just below the order-disorder transition temperature (340K). |
| 509 |
> |
These conditions are in the dipole-ordered and rippled portion of the phase |
| 510 |
> |
diagram. These are also the conditions that should be most susceptible to |
| 511 |
> |
system size effects. The wavelength and amplitude of the observed |
| 512 |
> |
ripples as a function of system size are shown in Fig. \ref{fig:systemsize}. |
| 513 |
> |
|
| 514 |
> |
\begin{figure}[ht] |
| 515 |
> |
\centering |
| 516 |
> |
\caption{The ripple wavelength (top) and amplitude (bottom) as a function of |
| 517 |
> |
system size for a hexagonal lattice ($\gamma=1.732$) at 300K.} |
| 518 |
> |
\includegraphics[width=\linewidth]{picture/SystemSize.eps} |
| 519 |
> |
\label{fig:systemsize} |
| 520 |
> |
\end{figure} |
| 521 |
> |
|
| 522 |
> |
There is substantial dependence on system size for small (less than 200 \AA) |
| 523 |
> |
periodic boxes. Notably, there are resonances apparent in the ripple |
| 524 |
> |
amplitudes at box lengths of 121 \AA and 206 \AA. For larger systems, |
| 525 |
> |
the behavior of the ripples appears to have stabilized and is on a trend to |
| 526 |
> |
slightly smaller amplitudes (and slightly longer wavelenghts) than were |
| 527 |
> |
observed from the 240 \AA box sizes that were used for most of the calculations. |
| 528 |
> |
|
| 529 |
> |
It is interesting to note that system sizes which are multiples of the |
| 530 |
> |
default ripple wavelength can enhance the amplitude of the observed ripples, |
| 531 |
> |
but appears to have only a minor effect on the observed wavelength. It would, |
| 532 |
> |
of course, be better to use system sizes that were many multiples of the ripple |
| 533 |
> |
wavelength to be sure that the periodic box is not driving the phenomenon, but at |
| 534 |
> |
the largest system size studied (485 \AA $\times$ 485 \AA), the number of |
| 535 |
> |
molecules (5440) made long Monte Carlo simulations prohibitively expensive. |
| 536 |
> |
We recognize this as a possible flaw of our model for bilayer rippling, but |
| 537 |
> |
it is a flaw that will plague any molecular-scale computational model for |
| 538 |
> |
this phenomenon. |
| 539 |
> |
|
| 540 |
> |
\section{Discussion} |
| 541 |
> |
\label{sec:discussion} |
| 542 |
> |
|
| 543 |
> |
We have been able to show that a simple lattice model for membranes |
| 544 |
> |
which contains only molecular packing (from the lattice), head-group |
| 545 |
> |
anisotropy (in the form of electrostatic dipoles) and ``hydrophobic'' |
| 546 |
> |
interactions (in the form of a nearest-neighbor harmonic potential) is |
| 547 |
> |
capable of exhibiting stable long-wavelength non-thermal ripple |
| 548 |
> |
structures. The best explanation for this behavior is that the |
| 549 |
> |
ability of the molecules to translate out of the plane of the membrane |
| 550 |
> |
is enough to break the symmetry of the hexagonal lattice and allow the |
| 551 |
> |
enormous energetic benefit from the formation of a bulk |
| 552 |
> |
antiferroelectric phase. Were the hydrophobic interactions absent |
| 553 |
> |
from our model, it would be possible for the entire lattice to |
| 554 |
> |
``tilt'' using $z$-translation. Tilting the lattice in this way would |
| 555 |
> |
yield an effectively non-hexagonal lattice which would avoid dipolar |
| 556 |
> |
frustration altogether. With the hydrophobic interactions, bulk tilt |
| 557 |
> |
would cause a large strain, and the simplest way to release this |
| 558 |
> |
strain is along line defects. Line defects will result in rippled or |
| 559 |
> |
sawtooth patterns in the membrane, and allow small ``stripes'' of |
| 560 |
> |
membrane to form antiferroelectric regions that are tilted relative to |
| 561 |
> |
the averaged membrane normal. |
| 562 |
> |
|
| 563 |
> |
Although the dipole-dipole interaction is the major driving force for |
| 564 |
> |
the long range orientational ordered state, the formation of the |
| 565 |
> |
stable, smooth ripples is a result of the competition between the |
| 566 |
> |
hydrophobic and dipole-dipole interactions. This statement is |
| 567 |
> |
supported by the variations in both $\mu$ and $k_r$. Substantially |
| 568 |
> |
weaker dipoles or stronger hydrophobic forces can both cause the |
| 569 |
> |
ripple phase to disappear. |
| 570 |
> |
|
| 571 |
> |
Molecular packing also plays a role in the formation of the ripple |
| 572 |
> |
phase. It would be surprising if strongly anisotropic head groups |
| 573 |
> |
would be able to pack in hexagonal lattices without the underlying |
| 574 |
> |
steric interactions between the rest of the molecular bodies. Since |
| 575 |
> |
we only see rippled phases in the neighborhood of $\gamma=\sqrt{3}$, |
| 576 |
> |
this implies that there is a role played by the lipid chains in the |
| 577 |
> |
organization of the hexagonally ordered phases which support ripples. |
| 578 |
> |
|
| 579 |
> |
Our simple model would clearly be a closer approximation to reality if |
| 580 |
> |
we allowed greater translational freedom to the dipoles and replaced |
| 581 |
> |
the somewhat artificial lattice packing and the harmonic mimic of the |
| 582 |
> |
hydrophobic interaction with more realistic molecular modelling |
| 583 |
> |
potentials. What we have done is to present an extremely simple model |
| 584 |
> |
which exhibits bulk non-thermal rippling, and our explanation of the |
| 585 |
> |
rippling phenomenon will help us design more accurate molecular models |
| 586 |
> |
for the rippling phenomenon. |
| 587 |
> |
|
| 588 |
> |
\clearpage |
| 589 |
> |
|
| 590 |
> |
\bibliography{ripple} |
| 591 |
> |
|
| 592 |
> |
\clearpage |
| 593 |
> |
|
| 594 |
|
\end{document} |
