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%\documentclass[aps,pre,twocolumn,amssymb,showpacs,floatfix]{revtex4} |
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\documentclass[aps,pre,preprint,amssymb,showpacs]{revtex4} |
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\usepackage{amsmath} |
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\usepackage{amssymb} |
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\usepackage{graphicx} |
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\begin{document} |
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%\bibliographystyle{aps} |
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\title{Dipolar Ordering of the Ripple Phase in Lipid Membranes} |
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\title{Dipolar Ordering in Molecular-scale Models of the Ripple Phase |
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in Lipid 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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University of Notre Dame, \\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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interdigitated. The mechanism for the formation of the ripple phase |
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suggested by their work is a packing competition between the head |
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groups and the tails of the lipid molecules.\cite{Carlson87} Recently, |
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the ripple phase has also been studied by the XXX group using Monte |
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the ripple phase has also been studied by Lenz and Schmid using Monte |
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Carlo simulations.\cite{Lenz07} Their structures are similar to the De |
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Vries {\it et al.} structures except that the connection between the |
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two leaves of the bilayer is a narrow interdigitated line instead of |
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this was evident in the ordering of the dipole director axis |
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perpendicular to the wave vector of the surface ripples. A similiar |
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phenomenon has also been observed by Tsonchev {\it et al.} in their |
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work on the spontaneous formation of dipolar molecules into curved |
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nano-structures.\cite{Tsonchev04} |
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work on the spontaneous formation of dipolar peptide chains into |
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curved nano-structures.\cite{Tsonchev04,Tsonchev04II} |
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In this paper, we construct a somewhat more realistic molecular-scale |
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lipid model than our previous ``web of dipoles'' and use molecular |
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\section{Computational Model} |
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\label{sec:method} |
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|
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=4in]{lipidModels} |
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\caption{Three different representations of DPPC lipid molecules, |
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including the chemical structure, an atomistic model, and the |
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head-body ellipsoidal coarse-grained model used in this |
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work.\label{fig:lipidModels}} |
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\end{figure} |
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|
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Our simple molecular-scale lipid model for studying the ripple phase |
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is based on two facts: one is that the most essential feature of lipid |
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molecules is their amphiphilic structure with polar head groups and |
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nearly perpendicular to the tail, so we have fixed the direction of |
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the point dipole rigidly in this orientation. |
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|
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{lipidModels} |
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\caption{Three different representations of DPPC lipid molecules, |
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including the chemical structure, an atomistic model, and the |
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head-body ellipsoidal coarse-grained model used in this |
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work.\label{fig:lipidModels}} |
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\end{figure} |
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The ellipsoidal portions of the model interact via the Gay-Berne |
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potential which has seen widespread use in the liquid crystal |
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community. In its original form, the Gay-Berne potential was a single |
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site model for the interactions of rigid ellipsoidal |
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community. Ayton and Voth have also used Gay-Berne ellipsoids for |
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modelling large length-scale properties of lipid |
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bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential |
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was a single site model for the interactions of rigid ellipsoidal |
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molecules.\cite{Gay81} It can be thought of as a modification of the |
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Gaussian overlap model originally described by Berne and |
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Pechukas.\cite{Berne72} The potential is constructed in the familiar |
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\label{eq:gb} |
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\end{eqnarray*} |
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|
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– |
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– |
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The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
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\hat{r}}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
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\hat{u}}_{j},{\bf \hat{r}}))$ parameters |
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\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
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\hat{u}}_{j},{\bf \hat{r}}_{ij}))$ parameters |
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are dependent on the relative orientations of the two molecules (${\bf |
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\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
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intermolecular separation (${\bf \hat{r}}$). The functional forms for |
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$\sigma({\bf |
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\hat{u}}_{i},{\bf |
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\hat{u}}_{j},{\bf \hat{r}})$ and $\epsilon({\bf \hat{u}}_{i},{\bf |
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\hat{u}}_{j},{\bf \hat{r}}))$ are given elsewhere |
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and will not be repeated here. However, $\epsilon$ and $\sigma$ are |
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governed by two anisotropy parameters, |
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intermolecular separation (${\bf \hat{r}}_{ij}$). $\sigma$ and |
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$\sigma_0$ are also governed by shape mixing and anisotropy variables, |
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\begin {equation} |
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\begin{array}{rcl} |
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\chi & = & \frac |
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{(\sigma_{e}/\sigma_{s})^2-1}{(\sigma_{e}/\sigma_{s})^2+1} \\ \\ |
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\chi\prime & = & \frac {1-(\epsilon_{e}/\epsilon_{s})^{1/\mu}}{1+(\epsilon_{e}/ |
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\epsilon_{s})^{1/\mu}} |
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\sigma_0 & = & \sqrt{d_i^2 + d_j^2} \\ \\ |
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\chi & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 - |
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d_j^2 \right)}{\left( l_j^2 + d_i^2 \right) \left(l_i^2 + |
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d_j^2 \right)}\right]^{1/2} \\ \\ |
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\alpha^2 & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 + |
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d_i^2 \right)}{\left( l_j^2 - d_j^2 \right) \left(l_i^2 + |
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d_j^2 \right)}\right]^{1/2}, |
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\end{array} |
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\end{equation} |
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In these equations, $\sigma$ and $\epsilon$ refer to the point of |
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closest contact and the depth of the well in different orientations of |
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the two molecules. The subscript $s$ refers to the {\it side-by-side} |
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configuration where $\sigma$ has it's smallest value, |
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$\sigma_{s}$, and where the potential well is $\epsilon_{s}$ deep. |
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The subscript $e$ refers to the {\it end-to-end} configuration where |
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$\sigma$ is at it's largest value, $\sigma_{e}$ and where the well |
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depth, $\epsilon_{e}$ is somewhat smaller than in the side-by-side |
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configuration. For the prolate ellipsoids we are using, we have |
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where $l$ and $d$ describe the length and width of each uniaxial |
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ellipsoid. These shape anisotropy parameters can then be used to |
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calculate the range function, |
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\begin {equation} |
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\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \sigma_{0} |
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\left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2 |
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\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\} |
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\right]^{-1/2} |
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\end{equation} |
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|
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Gay-Berne ellipsoids also have an energy scaling parameter, |
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$\epsilon^s$, which describes the well depth for two identical |
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ellipsoids in a {\it side-by-side} configuration. Additionaly, a well |
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depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, describes |
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the ratio between the well depths in the {\it end-to-end} and |
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side-by-side configurations. As in the range parameter, a set of |
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mixing and anisotropy variables can be used to describe the well |
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depths for dissimilar particles, |
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\begin {eqnarray*} |
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\epsilon_0 & = & \sqrt{\epsilon^s_i * \epsilon^s_j} \\ \\ |
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\epsilon^r & = & \sqrt{\epsilon^r_i * \epsilon^r_j} \\ \\ |
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\chi' & = & \frac{1 - (\epsilon^r)^{1/\mu}}{1 + (\epsilon^r)^{1/\mu}} |
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\\ \\ |
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\alpha'^2 & = & \left[1 + (\epsilon^r)^{1/\mu}\right]^{-1} |
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\end{eqnarray*} |
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The form of the strength function is somewhat complicated, |
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\begin {eqnarray*} |
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\epsilon({\bf \hat{u}}_{i}, {\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & = & |
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\epsilon_{0} \epsilon_{1}^{\nu}({\bf \hat{u}}_{i}.{\bf \hat{u}}_{j}) |
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\epsilon_{2}^{\mu}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
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\hat{r}}_{ij}) \\ \\ |
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\epsilon_{1}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j}) & = & |
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\left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf |
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\hat{u}}_{j})^{2}\right]^{-1/2} \\ \\ |
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\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & |
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= & |
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1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi'^2 |
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\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}, |
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\end {eqnarray*} |
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although many of the quantities and derivatives are identical with |
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those obtained for the range parameter. Ref. \onlinecite{Luckhurst90} |
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has a particularly good explanation of the choice of the Gay-Berne |
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parameters $\mu$ and $\nu$ for modeling liquid crystal molecules. An |
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excellent overview of the computational methods that can be used to |
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efficiently compute forces and torques for this potential can be found |
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in Ref. \onlinecite{Golubkov06} |
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|
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The choices of parameters we have used in this study correspond to a |
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shape anisotropy of 3 for the chain portion of the molecule. In |
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principle, this could be varied to allow for modeling of longer or |
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shorter chain lipid molecules. For these prolate ellipsoids, we have: |
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\begin{equation} |
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\begin{array}{rcl} |
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\sigma_{s} & < & \sigma_{e} \\ |
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\epsilon_{s} & > & \epsilon_{e} |
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d & < & l \\ |
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\epsilon^{r} & < & 1 |
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\end{array} |
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\end{equation} |
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Ref. \onlinecite{Luckhurst90} has a particularly good explanation of the |
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choice of the Gay-Berne parameters $\mu$ and $\nu$ for modeling liquid |
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crystal molecules. |
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|
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The breadth and length of tail are $\sigma_0$, $3\sigma_0$, |
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corresponding to a shape anisotropy of 3 for the chain portion of the |
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molecule. In principle, this could be varied to allow for modeling of |
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longer or shorter chain lipid molecules. |
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=4in]{2lipidModel} |
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\caption{The parameters defining the behavior of the lipid |
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models. $l / d$ is the ratio of the head group to body diameter. |
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Molecular bodies had a fixed aspect ratio of 3.0. The solvent model |
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was a simplified 4-water bead ($\sigma_w \approx d$) that has been |
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used in other coarse-grained (DPD) simulations. The dipolar strength |
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(and the temperature and pressure) were the only other parameters that |
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were varied systematically.\label{fig:lipidModel}} |
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\end{figure} |
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|
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To take into account the permanent dipolar interactions of the |
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zwitterionic head groups, we place fixed dipole moments $\mu_{i}$ at |
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one end of the Gay-Berne particles. The dipoles will be oriented at |
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an angle $\theta = \pi / 2$ relative to the major axis. These dipoles |
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one end of the Gay-Berne particles. The dipoles are oriented at an |
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angle $\theta = \pi / 2$ relative to the major axis. These dipoles |
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are protected by a head ``bead'' with a range parameter which we have |
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varied between $1.20\sigma_0$ and $1.41\sigma_0$. The head groups |
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interact with each other using a combination of Lennard-Jones, |
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varied between $1.20 d$ and $1.41 d$. The head groups interact with |
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each other using a combination of Lennard-Jones, |
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\begin{eqnarray*} |
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V_{ij} = 4\epsilon \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
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\left(\frac{\sigma_h}{r_{ij}}\right)^6\right], |
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\end{eqnarray*} |
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and dipole, |
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and dipole-dipole, |
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\begin{eqnarray*} |
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V_{ij} = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
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\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
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potentials. |
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In these potentials, $\mathbf{\hat u}_i$ 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 $\mathbf{r}_{ij}$. |
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pointing along the inter-dipole vector $\mathbf{r}_{ij}$. |
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|
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For the interaction between nonequivalent uniaxial ellipsoids (in this |
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case, between spheres and ellipsoids), the range parameter is |
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generalized as\cite{Cleaver96} |
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\begin{eqnarray*} |
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\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
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{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{(\alpha{\mathbf{\hat r}_{ij}} |
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\cdot {\mathbf{\hat u}_i} + \alpha^{-1}{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
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u}_j})^2}{1+\chi({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
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\frac{(\alpha{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - \alpha^{-1}{\mathbf{\hat r}_{ij}} |
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\cdot {\mathbf{\hat u}_j})^2}{1-\chi({\mathbf{\hat u}_i} \cdot |
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{\mathbf{\hat u}_j})} \right] \right)^{-\frac{1}{2}} |
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\end{eqnarray*} |
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where $\alpha$ is given by |
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\begin{eqnarray*} |
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\alpha^2 = |
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\left[\frac{\left({\sigma_e}_i^2-{\sigma_s}_i^2\right)\left({\sigma_e}_j^2+{\sigma_s}_i^2\right)}{\left({\sigma_e}_j^2-{\sigma_s}_j^2\right)\left({\sigma_e}_i^2+{\sigma_s}_j^2\right)} |
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\right]^{\frac{1}{2}} |
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\end{eqnarray*} |
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the strength parameter has been adjusted as suggested by Cleaver {\it |
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et al.}\cite{Cleaver96} A switching function has been applied to all |
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potentials to smoothly turn off the interactions between a range of $22$ and $25$ \AA. |
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case, between spheres and ellipsoids), the spheres are treated as |
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ellipsoids with an aspect ratio of 1 ($d = l$) and with an well depth |
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ratio of 1 ($\epsilon^e = \epsilon^s$). The form of the Gay-Berne |
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potential we are using was generalized by Cleaver {\it et al.} and is |
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appropriate for dissimilar uniaxial ellipsoids.\cite{Cleaver96} |
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|
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The solvent model in our simulations is identical to one used by XXX |
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in their dissipative particle dynamics (DPD) simulation of lipid |
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bilayers.]cite{XXX} This solvent bead is a single site that represents |
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four water molecules (m = 72 amu) and has comparable density and |
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diffusive behavior to liquid water. However, since there are no |
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electrostatic sites on these beads, this solvent model cannot |
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replicate the dielectric properties of water. |
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The solvent model in our simulations is identical to one used by |
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Marrink {\it et al.} in their dissipative particle dynamics (DPD) |
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simulation of lipid bilayers.\cite{Marrink04} This solvent bead is a single |
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site that represents four water molecules (m = 72 amu) and has |
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comparable density and diffusive behavior to liquid water. However, |
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since there are no electrostatic sites on these beads, this solvent |
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model cannot replicate the dielectric properties of water. |
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|
\begin{table*} |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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\caption{} |
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\begin{tabular}{lccc} |
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\caption{Potential parameters used for molecular-scale coarse-grained |
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lipid simulations} |
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\begin{tabular}{llccc} |
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|
\hline |
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N/A & Head & Chain & Solvent \\ |
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& & Head & Chain & Solvent \\ |
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\hline |
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$\sigma_0$ (\AA) & varied & 4.6 & 4.7 \\ |
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l (aspect ratio) & N/A & 3 & N/A \\ |
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$\epsilon_0$ (kcal/mol) & 0.185 & 1.0 & 0.8 \\ |
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$\epsilon_e$ (aspect ratio) & N/A & 0.2 & N/A \\ |
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M (amu) & 196 & 760 & 72.06112 \\ |
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$I_{xx}$, $I_{yy}$, $I_{zz}$ (amu \AA$^2$) & 1125, 1125, 0 & 45000, 45000, 9000 & N/A \\ |
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$\mu$ (Debye) & varied & N/A & N/A \\ |
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$d$ (\AA) & & varied & 4.6 & 4.7 \\ |
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$l$ (\AA) & & 1 & 3 & 1 \\ |
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$\epsilon^s$ (kcal/mol) & & 0.185 & 1.0 & 0.8 \\ |
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$\epsilon_r$ (well-depth aspect ratio)& & 1 & 0.2 & 1 \\ |
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$m$ (amu) & & 196 & 760 & 72.06112 \\ |
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$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$) & & & & \\ |
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\multicolumn{2}c{$I_{xx}$} & 1125 & 45000 & N/A \\ |
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\multicolumn{2}c{$I_{yy}$} & 1125 & 45000 & N/A \\ |
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\multicolumn{2}c{$I_{zz}$} & 0 & 9000 & N/A \\ |
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$\mu$ (Debye) & & varied & 0 & 0 \\ |
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\end{tabular} |
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\label{tab:parameters} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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|
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{2lipidModel} |
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\caption{The parameters defining the behavior of the lipid |
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models. $\sigma_h / \sigma_0$ is the ratio of the head group to body |
| 309 |
< |
diameter. Molecular bodies had a fixed aspect ratio of 3.0. The |
| 310 |
< |
solvent model was a simplified 4-water bead ($\sigma_w = 1.02 |
| 311 |
< |
\sigma_0$) that has been used in other coarse-grained (DPD) simulations. |
| 312 |
< |
The dipolar strength (and the temperature and pressure) were the only |
| 313 |
< |
other parameters that were varied |
| 314 |
< |
systematically.\label{fig:lipidModel}} |
| 315 |
< |
\end{figure} |
| 348 |
> |
A switching function has been applied to all potentials to smoothly |
| 349 |
> |
turn off the interactions between a range of $22$ and $25$ \AA. |
| 350 |
|
|
| 351 |
|
\section{Experimental Methodology} |
| 352 |
|
\label{sec:experiment} |
| 392 |
|
non-thermal. |
| 393 |
|
\begin{figure}[htb] |
| 394 |
|
\centering |
| 395 |
< |
\includegraphics[width=\linewidth]{phaseCartoon} |
| 395 |
> |
\includegraphics[width=4in]{phaseCartoon} |
| 396 |
|
\caption{A sketch to discribe the structure of the phases observed in |
| 397 |
|
our simulations.\label{fig:phaseCartoon}} |
| 398 |
|
\end{figure} |
| 444 |
|
|
| 445 |
|
\begin{figure}[htb] |
| 446 |
|
\centering |
| 447 |
< |
\includegraphics[width=\linewidth]{topDown} |
| 447 |
> |
\includegraphics[width=4in]{topDown} |
| 448 |
|
\caption{Top views of the flat (upper), asymmetric ripple (middle), |
| 449 |
|
and symmetric ripple (lower) phases. Note that the head-group dipoles |
| 450 |
|
have formed head-to-tail chains in all three of these phases, but in |
| 500 |
|
normal of the surface,(making this structure remindcent of the |
| 501 |
|
$L_{\beta}$ phase. Increasing the size of the heads, results in |
| 502 |
|
rapidly decreasing $P_2$ ordering for the molecular bodies. |
| 503 |
+ |
|
| 504 |
|
\begin{figure}[htb] |
| 505 |
|
\centering |
| 506 |
|
\includegraphics[width=\linewidth]{rP2} |
| 583 |
|
\section{Discussion} |
| 584 |
|
\label{sec:discussion} |
| 585 |
|
|
| 586 |
+ |
\newpage |
| 587 |
|
\bibliography{mdripple} |
| 588 |
|
\end{document} |
