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\begin{document} |
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\bibliographystyle{apsrev} |
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\title{Bilayer phase and antiphase formation in polar liquid crystals} |
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\author{J. Saha\footnote{Current Address: Department of Physics, Visva-Bharati |
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University, Santiniketan - 731235, West Bengal, India} and |
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J. Daniel Gezelter\footnote{e-mail: gezelter@nd.edu}} |
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\affiliation{Department of Chemistry and Biochemistry, University of |
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Notre Dame, Notre Dame, Indiana 46556} |
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\date{\today} |
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\begin{abstract} |
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We present results of a series of Molecular Dynamics simulations on |
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the molecular organization of systems of ellipsoidal Gay-Berne |
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molecules containing two fixed dipole moments. The effects of relative |
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dipolar orientations and positions on the generation of bilayer |
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lamellar phase, antiphases and monolayer phases has been studied. We |
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report on the structural features of the phases formed by molecules of |
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both liquid crystalline and biological interest. |
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\end{abstract} |
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\pacs{61.30} |
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\keywords{Gay-Berne molecule, two dipoles, molecular dynamics, bilayer |
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phase, antiphase} |
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\maketitle |
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\newpage |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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\label{sec:intro} |
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Long range orientational order is the most fundamental property of |
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liquid crystal mesophases, while positional order is limited or |
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absent. This orientational anisotropy of the macroscopic phases |
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originates in the anisotropy of the constituent molecules. For the |
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existence of orientational ordering, these molecules typically have |
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highly non-spherical structure with some degree of rigidity. Liquid |
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crystalline compounds typically possess cylindrically symmetric rod or |
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disc-like rigid core structures and usually have flexible substituents |
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associated with these central regions. In nematic phases, rod-like |
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molecules are orientationally ordered with isotropic distributions of |
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molecular centers of mass. In smectic phases, the molecules arrange |
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themselves into layers with their long (symmetry) axis normal |
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($S_{A}$) or tilted ($S_{C}$) with respect to the layer planes. The |
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layers themselves are two dimensional liquids. |
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\subsection{Previous Experimental Work} |
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Experimental and theoretical studies of smectic liquid crystals |
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include a range of polymorphic variations on the basic smectic |
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organization. The behaviour of the $S_{A}$ phase can be explained with |
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theoretical models mainly based on geometric factors and van der Waals |
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interactions. However, these simple models are insufficient to |
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describe liquid crystal phases which exhibit more complex polymorphic |
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nature. X-ray diffraction studies have shown that the ratio between |
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lamellar spacing ($s$) and molecular length ($l$) can take |
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significantly different values.\cite{Leadbetter77,Gray84} Typical |
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$S_{A}$ phases have $s/l$ ratios on the order of $0.8$ , whereas for |
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some compounds e.g. 4-alkyl-4'-cyanobiphenyls the $s/l$ ratio is on |
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the order of $1.4$. Experiments show that depending on dipole |
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delocalization within the molecules, $s$ can take values ranging from |
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the length of a single molecule to twice the molecular |
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length~\cite{Leadbetter77,Hardouin80}. Extensive experimental studies |
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reveal that compounds of the $S_{A}$ type which show a variety of |
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phases like monolayers ($S_{A1}$), uniform bilayers ($S_{A2}$), |
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partial bilayers ($S_{\tilde A}$) and interdigitated bilayers |
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($S_{A_{d}}$), usually have a terminal cyano or nitro group. These |
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classes of liquid crystal and in particular, lyotropic liquid crystals |
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(those exhibiting liquid crystal phase transition as a function of |
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water concentration) which form surfactant and lipid membranes, often |
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have polar head groups or zwitterionic charge separated groups that |
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result in strong dipolar interactions.\cite{Collings97} Apart from the |
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uniaxial $S_{A}$ phases mentioned above, compounds having permanent |
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dipole moments can also give rise to reentrant smectic, biaxial |
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($S_{C}$) and ferroelectric phases. Because of their versatile |
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polymorphic nature, these liquid crystalline materials have important |
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technological applications in addition to their immense relevance to |
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biological systems.\cite{Collings97} |
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Experimental studies by Levelut {\it et |
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al.}~\cite{Levelut81a,Levelut81b} revealed that terminal cyano or |
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nitro groups usually induce permanent longitudinal dipole moments on |
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the molecules. Strong lateral dipole $(C=O)$ and terminal transverse |
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dipoles can also effect the phase behaviour considerably. Many |
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liquid-crystal forming molecules of biological interest |
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(e.g. phospholipids) could be modelled more accurately with {\em |
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transverse} or angled dipole moments at the terminus of the molecule. |
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In particular, the dipole moment of phosphatidylcholine (PC) head |
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groups are typically oriented perpendicular to the molecular axis, |
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while the dipole of phosphatidylethanolamine (PE) head groups are |
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tilted relative to the molecular axis. Moreover, there is strong |
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indication from the molecular structure of liquid crystalline |
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molecules that form these phases, that in adddition to the terminal |
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dipole, it is advantageous to have the presence of aromatic $\pi$ |
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bonds and/or other dipoles near the core which can be easily polarized |
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by the strong electron withdrawing properties of the terminal group, |
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resulting in the various $S_{A}$ |
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phases.~\cite{Gray84,Jeu83,Levelut81a,Levelut81b} |
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\subsection{Previous theoretical work} |
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Theoretical models of polar smectics using generic molecular field |
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descriptions were studied by Photinos, Saupe and |
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others.~\cite{Photinos76,Vanakaras98} Prost coupled two different |
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order parameters (the density wave and dipolar ordering of molecules |
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along the director) and using a mean field approch showed that |
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competition between them was responsible for polar liquid crystal |
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behaviour.~\cite{Prost84,Prost80} Comparing some polar liquid crystal |
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compounds, de~Jeu inferred that variation of dipole correlation with |
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molecular structure could affect the phase behaviour of polar liquid |
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crystals.~\cite{Jeu83} A number of other analytical approaches to |
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these phases have been presented in the |
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literature,~\cite{Meyer76,Dowell85,Indekeu86,Baus89,Netz92} but most |
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have been too simplified to confirm any more than the qualitative |
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behavior of these phases. Therefore, it seems that a molecular-scale |
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simulation approach will be required for a more complete understanding |
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of bilayer and antiphases. |
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Levesque {\em et al.} presented a hard rod model which exhibited |
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monolayers and further indicated that the very symmetric nature of the |
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potential was responsible for their failure to generate true |
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bilayer.~\cite{Levesque93} Reproduction of small domains of bilayers |
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was reported for transverse dipoles but these were not always present |
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in their simulations. |
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The Gay-Berne potential has seen widespread use in the liquid crystal |
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community to describe this anisotropic phase |
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behavior.~\cite{Gay81,Berne72,Kushick76,Luckhurst90,Perram96} It is an |
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appropriate model for simulation of these systems because fairly rigid |
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liquid crystal-forming molecules maintain their rod-like or disc-like |
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shapes, which produce interactions that favor local alignment. In its |
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original form, the Gay-Berne potential was a computationally efficient |
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{\em 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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form of the Lennard-Jones function using orientation-dependent |
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$\sigma$ and $\epsilon$ parameters. The functional form for the |
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potential is given in section 2. Luckhurst has given a particularly |
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good explanation of the choice of the Gay-Berne parameters $\mu$ and |
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$\nu$ for modeling non-polar liquid crystal molecules. |
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Although there have been a large number of studies of the phase |
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behaviour of the {\em non-polar} Gay-Berne potential using Monte Carlo |
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and Molecular Dynamics techniques,~\cite{Zannonibook2000} there has |
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been comparatively little work done on an important class of liquid |
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crystal forming molecules which have {\em dipolar} interactions. |
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There have been some preliminary Monte Carlo studies of the Gay-Berne |
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potential with fixed longitudinal dipoles (i.e. pointed along the |
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principal axis of rotation).~\cite{Berardi96,Satoh96} There have also |
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been some recent molecular dynamics simulations on polar Gay-Berne |
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models by Pasterny {\em et al.}~\cite{Pasterny2000} Zannoni's group |
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has studied the phase behavior of Gay-Berne ellipsoids with |
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longitudinal and transverse dipoles both at the midpoint and terminus |
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of the molecule.~\cite{Berardi99} Their exhaustive simulation with a |
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model potential comprising both the attractive-repulsive G-B |
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interaction and a single dipolar interaction exhibited partial striped |
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bilayer structures. |
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MISSING: AYTON AND VOTH WORK ON GB with terminal LJ spheres. |
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Simulations of small domains of local bilayers for nCB {WHAT IS NCB} |
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have been reported recently.~\cite{Fukunaga2004} |
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Since none of these simulations, which considered molecules with |
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single dipoles and studied the effects of either the terminal or the |
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central dipoles seperately, were able to reproduce perfect bilayer |
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arrangement the molecular origin of these liquid crystal phases is not |
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very well understood. There is, of course, a vast literature of |
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all-atom, or coarse-grained simulation models for lipid bilayers, but |
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typical system sizes with these models allow only relatively small |
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patches of single or double bilayers to be studied. In this work, we |
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present a model that is simple enough to allow us to probe the |
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equilibrated phase behavior at a number of different conditions, while |
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still maintaining enough molecular-scale realism to be useful as a |
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predictive tool. |
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To mimic the terminal dipolar interaction coupled with polar cores we |
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considered systems comprising Gay-Berne particles with an embedded |
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terminal dipole and another weaker central dipole. Performing a series |
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of molecular dynamics simulations, we studied the structural |
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properties of these phases in systems of prolate ellipsoidal particles |
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each having relatively two dipole moments oriented perpendicularly |
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with respect to their respective molecular symmetry axes. In this |
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paper we report the generation of bilayer, monolayer and wavy |
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antiphase structures. To our knowledge, the present simulation work |
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is the first of its kind which could generate the bilayer liquid |
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crystalline phase successfully along with other important |
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experimentally-observed phases. |
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\section{Model} |
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\label{sec:model} |
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In this work, rod-like polar molecules are modelled as prolate |
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ellipsoidal Gay-Berne (GB) particles. The GB interaction potential |
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used to mimic the apolar characteristics of liquid crystal molecules |
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takes the familiar form of Lennard-Jones function with orientation and |
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position dependent range ($\sigma$) and well depth ($\epsilon$) |
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parameters. It can can be expressed as, |
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\begin{equation} |
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\begin{array}{ll} |
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V^{GB}_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}) = |
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4 \epsilon({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}) |
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& \left[ \left( |
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\frac{\sigma_{o}}{r - \sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
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\hat{r}})+\sigma_{o}} \right)^{12} \right. \\ \\ |
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& - \left. \left( |
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\frac{\sigma_{o}}{r - \sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
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\hat{r}})+\sigma_{o}} |
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\right)^{6} |
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\right], \\ |
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\end{array} |
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\label{eq:gb} |
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\end {equation} |
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where ${\bf \hat{u}_{i},\hat{u}_{j}}$ are unit vectors specifying the |
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orientation of two molecules $i$ and $j$ separated by intermolecular |
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vector ${\bf r}$. ${\bf \hat{r}}$ is the unit vector along the |
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intermolecular vector. |
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The functional form for $\sigma$ is given by |
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\begin {equation} |
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\begin{array}{ll} |
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\sigma ({\bf \hat{u}_{i}, \hat{u}_{j},\hat{r}}) = \sigma_{0} |
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& \left[ 1- \frac {\chi}{2} \left( \frac{({\bf \hat{r}}.{\bf \hat{u}_{i}}+{\bf \hat{r}}. |
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{\bf \hat {u}_{j}})^2}{1+\chi |
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({\bf \hat{u}_{i}}.{\bf \hat{u}_{j}})} \right. \right. \\ \\ |
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& \left.\left. + \frac {({\bf \hat{r}}.{\bf \hat{u}_{i |
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}}-{\bf \hat{r}}. |
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{\bf \hat{u}_{j}})^2}{1-\chi ({\bf \hat{u}_{i}}.{\bf \hat{u}_{j}})} \right) |
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\right ]^{-1/2} |
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\end{array} |
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\end {equation} |
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The aspect ratio of the particles is governed by shape anisotropy |
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parameter |
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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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\end{array} |
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\label{eq:chi} |
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\end {equation} |
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Here, the subscript $s$ indicates the {\it side-by-side} configuration |
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where $\sigma$ has its minimum value, $\sigma_{s}$, and where the |
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potential well is $\epsilon_{s}$ deep. The subscript $e$ refers to |
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the {\it end-to-end} configuration where $\sigma$ has its maximum |
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value, $\sigma_{e}$, and where the well depth, $\epsilon_{e}$ is |
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somewhat smaller than in the side-by-side configuration. For prolate |
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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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\end{array} |
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\end{equation} |
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where, $\sigma_{e}$ is the measure of the length and $\sigma_{s}$ is |
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the breadth of a molecule. The shape anisotropy parameter $\chi$ has |
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a functional dependence on the length to breadth ratio (i.e. the |
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aspect ratio of the particles.). |
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The functional form of well depth is |
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\begin {equation} |
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\epsilon({\bf \hat{u}_{i},\hat{u}_{j},\hat{r}}) = \epsilon_{0} |
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\epsilon^{\nu}({\bf \hat{u}_{i}.\hat{u}_{j}}) |
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\epsilon^{\prime\mu}({\bf \hat{u}_{i},\hat{u}_{j},\hat{r}}) |
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\end {equation} |
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where $\epsilon_{0}$ is a constant term and |
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\begin {equation} |
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\epsilon ({\bf \hat{u}_{i},\hat{u}_{j}}) = |
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[1-\chi^{2}({\bf \hat{u}_{i}.\hat{u}_{j}})^{2}]^{-1/2} |
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\end {equation} |
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and |
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\begin {equation} |
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\epsilon^{\prime} ({\bf \hat{u}_{i}, \hat{u}_{j},\hat{r}}) |
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= 1- \frac {\chi^{'}}{2} \left |
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( \frac {({\bf \hat{r}}.{\bf \hat{u}_{i}}+{\bf\hat{r}}. |
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{\bf \hat{u}_{j}})^2}{1+\chi^{\prime} |
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({\bf \hat{u}_{i}}.{\bf \hat{u}_{j}})} +\frac {({\bf\hat{ r}}.{\bf\hat{ |
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u}_{i}}-{\bf \hat{r}}. |
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{\bf \hat{u}_{j}})^2}{1-\chi^{\prime} ({\bf \hat{u}_{i}}.{\bf |
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\hat{u}_{j}})} \right) |
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\end {equation} |
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where the well depth anisotropy parameter $\chi\prime$ can be expressed as |
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\begin {equation} |
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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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\end {equation} |
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As the molecules have two dipoles, for each pair of them (4 pairs for |
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two interacting molecules), there should be an electrostatic |
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interaction term of the form |
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\begin {equation} |
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U_{dd} = \frac { \mu^{*}_{id} \mu^{*}_{jd}}{r_{d}^{3}} |
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\left [({\bf \hat{u}_{id}.\hat{u}_{jd}}) - 3 ({\bf \hat{u}_{id}. \hat{r}_{d}})( |
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{\bf \hat{u}_{jd}. \hat{r}_{d}}) \right ] |
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\end {equation} |
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|
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where reduced dipole moment |
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\begin{equation} |
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\mu_{d}^{*} = \frac {\mu_{d}^{2}} |
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{(4 \pi {\it \epsilon} \epsilon_{s} \sigma_{s}^{3})^{1/2}} |
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\end{equation} |
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$\it {\epsilon}$ is the permitivity of the free space and $r_{d}$ is |
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the unit vector along the vector joining the two dipoles. $\bf \hat{u}_{id}$ |
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and |
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$\bf \hat{u}_{jd}$ are the direction of the unit vectors along the direction of |
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the dipoles situated on molecules i and j respectively. |
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|
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|
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So the total interaction potential for a pair of polar molecules is |
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the sum of attractive-repulsive term and dipole-dople interaction term, |
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which can be expressed as |
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|
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\begin {equation} |
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|
U_{ij} = U_{GB} + (U_{dd})_{1st~ dipole} + (U_{dd})_{2nd~ dipole} |
335 |
|
|
\end {equation} |
336 |
|
|
|
337 |
|
|
To simulate systems of dipolar Gay-Berne |
338 |
|
|
particles with different relative dipolar orientations and positions, we |
339 |
|
|
used this model potential. |
340 |
|
|
|
341 |
|
|
\begin{figure} |
342 |
|
|
\begin{center} |
343 |
|
|
\epsfxsize=3in |
344 |
|
|
\epsfbox{system_sketch.eps} |
345 |
|
|
\end{center} |
346 |
|
|
\caption{The molecular models studied in this work. All are prolate |
347 |
|
|
Gay-Berne ellipsoids which have point dipoles embedded centrally or |
348 |
|
|
terminally within the molecular bodies. System A is a molecular-scale |
349 |
|
|
model for XXXX, System B is a model of YYYY, and System C could be |
350 |
|
|
considered a model for phosphatidylcholine (PC) lipids. Details on |
351 |
|
|
the dipolar locations and strengths are given in the text.} |
352 |
|
|
\label{fig:gbdp} |
353 |
|
|
\end{figure} |
354 |
|
|
|
355 |
|
|
|
356 |
|
|
\section{Computational Methodology} |
357 |
|
|
We performed a series of extensive Molecular Dynamics (MD) simulations to |
358 |
|
|
study |
359 |
|
|
the phase behaviour of a family of polar liquid crystals. |
360 |
|
|
|
361 |
|
|
In each simulation, rod-like polar |
362 |
|
|
molecules have been represented by polar ellipsoidal |
363 |
|
|
Gay-Berne (GB) particles. The four parameters characterizing G-B |
364 |
|
|
potential were taken as $\mu = 1,~ \nu = 2, ~\epsilon_{e}/\epsilon_{s} |
365 |
|
|
= 1/5$ and $\sigma_{e}/\sigma_{s} = 3$. The components of the |
366 |
|
|
scaled moment of inertia $(I^{*} = I/m \sigma_{s}^{2})$ along |
367 |
|
|
the major and minor axes were $I_{z}^{*} = 0.2$ and $I_{\perp}^{*} |
368 |
|
|
= 1.0$. We used the |
369 |
|
|
reduced dipole |
370 |
|
|
moments $ \mu^{*} = \mu/(4 |
371 |
|
|
\pi \epsilon_{fs} \sigma_{0}^{3})^{1/2}= 1.0$ for terminal dipole and |
372 |
|
|
$ \mu^{*} = \mu/(4 |
373 |
|
|
\pi \epsilon_{fs} \sigma_{0}^{3})^{1/2}= 0.5$ for second dipole, |
374 |
|
|
where $\epsilon_{fs}$ |
375 |
|
|
was the permitivitty of free space. For all simulations the position of the |
376 |
|
|
terminal dipole |
377 |
|
|
has been kept |
378 |
|
|
at a fixed distance $d^{*} = d/\sigma_{s} = 1.0 $ from the |
379 |
|
|
centre of mass on the molecular symmetry axis. The second dipole |
380 |
|
|
takes $d^{*} = d/\sigma_{s} = 0.0 $ |
381 |
|
|
i.e. it is on the centre of mass. To investigate |
382 |
|
|
the molecular organization behaviour due to different dipolar |
383 |
|
|
orientation with respect to the symmetry axis, we selected dipolar |
384 |
|
|
angle $\alpha_{d} = 0$ to model terminal outward longitudinal |
385 |
|
|
dipole and $\alpha_{d} = \pi/2$ to model transverse outward dipole where |
386 |
|
|
the second |
387 |
|
|
dipole takes relative anti |
388 |
|
|
antiparallel orientation with respect to the first. System of molecules |
389 |
|
|
having a single transverse terminal dipole has also been studied. We ran |
390 |
|
|
a series of |
391 |
|
|
simulations to investigate the effect of dipoles on molecular organization. |
392 |
|
|
|
393 |
|
|
In each of the simulations 864 molecules were confined in a cubic box with |
394 |
|
|
periodic boundary conditions. The run started from a density |
395 |
|
|
$\rho^{*} = \rho \sigma_{0}^{3}$ = 0.01 with nonpolar molecules |
396 |
|
|
loacted on the sites of FCC lattice and having parallel |
397 |
|
|
orientation. This structure was not a stable structure at this density |
398 |
|
|
and it was melted at a reduced temperature $T^{*} = k_{B}T/ |
399 |
|
|
\epsilon_{0} = 4.0$ . |
400 |
|
|
We used this isotropic |
401 |
|
|
configuration which was both orientationally and translationally |
402 |
|
|
disordered, as the initial configuration for each simulation. The dipoles |
403 |
|
|
were also switched on from this point. |
404 |
|
|
Initial translational and angular velocities were assigned |
405 |
|
|
from the gaussian distribution of velocities. |
406 |
|
|
|
407 |
|
|
To get the ordered structure for each system of particular dipolar |
408 |
|
|
angles we increased the density from |
409 |
|
|
$\rho^{*} = 0.01$ to $\rho_{*} = 0.3$ with an increament size |
410 |
|
|
of 0.002 upto $\rho^{*} = 0.1$ and 0.01 for the rest at some higher |
411 |
|
|
temperature. Temperature was |
412 |
|
|
then lowered in finer steps to avoid ending up with disordered glass phase |
413 |
|
|
and thus to help the molecules set with more order. |
414 |
|
|
For each system this process required altogether $5 \times 10^{6}$ MC cycles |
415 |
|
|
for equilibration. |
416 |
|
|
|
417 |
|
|
The torques and forces were calculated using |
418 |
|
|
velocity verlet algorithm. The time step size $\delta t^{*} = |
419 |
|
|
\delta t/(m \sigma_{0}^{2} / \epsilon_{0})^{1/2}$ was set at 0.0012 during |
420 |
|
|
the process. The orientations of molecules were described by quaternions |
421 |
|
|
instead of Eulerian angles to get the singularity-free orientational |
422 |
|
|
equations of motion. |
423 |
|
|
|
424 |
|
|
The interaction potential was truncated at a cut-off radius |
425 |
|
|
$r_{c} = 3.8 \sigma_{0}$. The long range dipole-dipole interaction potential |
426 |
|
|
and torque were handled by the application of reaction field method |
427 |
|
|
~\cite{Allen87}. |
428 |
|
|
|
429 |
|
|
To investigate the phase structure of the model liquid |
430 |
|
|
crystal family we calculated |
431 |
|
|
the orientational order parameter, correlation functions. |
432 |
|
|
To identify a particular phase we took configurational snapshots |
433 |
|
|
at the onset of each layered phase. |
434 |
|
|
|
435 |
|
|
The orientational order parameter for uniaxial phase was calculated |
436 |
|
|
from the largest eigen value obtained by diagonalization of the order |
437 |
|
|
parameter tensor |
438 |
|
|
|
439 |
|
|
\begin{equation} |
440 |
|
|
\begin{array}{lr} |
441 |
|
|
Q_{\alpha \beta} = \frac{1}{2 N} \sum(3 e_{i \alpha} e_{i \beta} |
442 |
|
|
- \delta_{\alpha \beta}) & \alpha, \beta = x,y,z \\ |
443 |
|
|
\end{array} |
444 |
|
|
\end{equation} |
445 |
|
|
|
446 |
|
|
where $e_{i \alpha}$ was the $\alpha$ th component of the unit vector |
447 |
|
|
$e_{i}$ along the symmetry axis of the i th molecule. Corresponding |
448 |
|
|
eigenvector gave the director which defines the average direction |
449 |
|
|
of molecular alignment. |
450 |
|
|
|
451 |
|
|
The density correlation along the director is $g(z) = < \delta |
452 |
|
|
(z-z_{ij})>_{ij} / \pi R^{2} \rho $, where $z_{ij} = r_{ij} cos |
453 |
|
|
\beta_{r_{ij}}$ was measured in the director frame and $R$ is the |
454 |
|
|
radius of the cylindrical sampling region. |
455 |
|
|
|
456 |
|
|
|
457 |
|
|
\section{Results and Conclusion} |
458 |
|
|
\label{sec:results and conclusion} |
459 |
|
|
|
460 |
|
|
Analysis of the simulation results shows that relative dipolar orientation |
461 |
|
|
angle of the molecules can give rise to rich polymorphism |
462 |
|
|
of polar mesophases. |
463 |
|
|
|
464 |
|
|
The correlation function g(z) shows layering along perpendicular |
465 |
|
|
direction to the plane for a system of G-B molecules with two |
466 |
|
|
transverse outward pointing dipoles in fig. \ref{fig:1}. Both the |
467 |
|
|
correlation plot and the snapshot (fig. \ref{fig:4}) of their |
468 |
|
|
organization indicate a bilayer phase. Snapshot for larger system of |
469 |
|
|
1372 molecules also confirms bilayer structure (Fig. \ref{fig:7}). |
470 |
|
|
Fig. \ref{fig:2} shows g(z) for a system of molecules having two |
471 |
|
|
antiparallel longitudinal dipoles and the snapshot of their |
472 |
|
|
organization shows a monolayer phase |
473 |
|
|
(Fig. \ref{fig:5}). Fig. \ref{fig:3} gives g(z) for a system of G-B |
474 |
|
|
molecules with single transverse outward pointing dipole and |
475 |
|
|
fig. \ref{fig:6} gives the snapshot. Their organization is like a wavy |
476 |
|
|
antiphase (stripe domain). Fig. \ref{fig:8} gives the snapshot for |
477 |
|
|
1372 molecules with single transverse dipole near the end of the |
478 |
|
|
molecule. |
479 |
|
|
|
480 |
|
|
\begin{figure} |
481 |
|
|
\begin{center} |
482 |
|
|
\epsfxsize=3in |
483 |
|
|
\epsfbox{fig1.ps} |
484 |
|
|
\end{center} |
485 |
|
|
\caption { Density projection of molecular centres (solid) and terminal dipoles (broken) with respect to the director g(z) |
486 |
|
|
for a system of G-B molecules with two transverse |
487 |
|
|
outward pointing dipoles, the first dipole having $d^{*}=1.0$, |
488 |
|
|
$\mu^{*}=1.0$ and the second dipole having $d^{*}=0.0$, |
489 |
|
|
$\mu^{*}=0.5$} |
490 |
|
|
\label{fig:1} |
491 |
|
|
\end{figure} |
492 |
|
|
|
493 |
|
|
|
494 |
|
|
\begin{figure} |
495 |
|
|
\begin{center} |
496 |
|
|
\epsfxsize=3in |
497 |
|
|
\epsfbox{fig2.ps} |
498 |
|
|
\end{center} |
499 |
|
|
\caption { Density projection of molecular centres (solid) and terminal |
500 |
|
|
dipoles (broken) with respect to the director |
501 |
|
|
g(z) for a system of G-B molecules with two antiparallel |
502 |
|
|
longitudinal dipoles, the first outward pointing dipole having $d^{*}=1.0$, |
503 |
|
|
$\mu^{*}=1.0$ and the second dipole having $d^{*}=0.0$, |
504 |
|
|
$\mu^{*}=0.5$} |
505 |
|
|
\label{fig:2} |
506 |
|
|
\end{figure} |
507 |
|
|
|
508 |
|
|
\begin{figure} |
509 |
|
|
\begin{center} |
510 |
|
|
\epsfxsize=3in |
511 |
|
|
\epsfbox{fig3.ps} |
512 |
|
|
\end{center} |
513 |
|
|
\caption {Density projection of molecular centres (solid) and terminal |
514 |
|
|
dipoles (broken) with respect to the director g(z) |
515 |
|
|
for a system of G-B molecules with single transverse |
516 |
|
|
outward pointing dipole, having $d^{*}=1.0$, |
517 |
|
|
$\mu^{*}=1.0$} |
518 |
|
|
\label{fig:3} |
519 |
|
|
\end{figure} |
520 |
|
|
|
521 |
|
|
\begin{figure} |
522 |
|
|
\centering |
523 |
|
|
\epsfxsize=2.5in |
524 |
|
|
\epsfbox{fig4.eps} |
525 |
|
|
\caption{Typical configuration for a system of 864 G-B molecules |
526 |
|
|
with two transverse dipoles, the first dipole having $d^{*}=1.0$, |
527 |
|
|
$\mu^{*}=1.0$ and the second dipole having $d^{*}=0.0$, |
528 |
|
|
$\mu^{*}=0.5$. The white caps indicate the location of the terminal |
529 |
|
|
dipole, while the orientation of the dipoles is indicated by the |
530 |
|
|
blue/gold coloring.} |
531 |
|
|
\label{fig:4} |
532 |
|
|
\end{figure} |
533 |
|
|
|
534 |
|
|
\begin{figure} |
535 |
|
|
\begin{center} |
536 |
|
|
\epsfxsize=3in |
537 |
|
|
\epsfbox{fig5.ps} |
538 |
|
|
\end{center} |
539 |
|
|
\caption {Snapshot of molecular configuration for a system of 864 G-B molecules with |
540 |
|
|
two antiparallel longitudinal dipoles, the first outward pointing dipole |
541 |
|
|
having $d^{*}=1.0$, $\mu^{*}=1.0$ and the second dipole having $d^{*}=0.0$, |
542 |
|
|
$\mu^{*}=0.5$ (fine lines are molecular symmetry axes and small thick lines |
543 |
|
|
show terminal dipolar direction, central dipoles are not shown).} |
544 |
|
|
\label{fig:5} |
545 |
|
|
\end{figure} |
546 |
|
|
|
547 |
|
|
|
548 |
|
|
\begin{figure} |
549 |
|
|
\begin{center} |
550 |
|
|
\epsfxsize=3in |
551 |
|
|
\epsfbox{fig6.ps} |
552 |
|
|
\end{center} |
553 |
|
|
\caption {Snapshot of molecular configuration for a system of 864 G-B molecules with |
554 |
|
|
single transverse outward pointing dipole, having $d^{*}=1.0$, $\mu^{*}=1.0$ |
555 |
|
|
(fine lines are molecular symmetry axes and small thick lines |
556 |
|
|
show terminal dipolar direction).} |
557 |
|
|
\label{fig:6} |
558 |
|
|
\end{figure} |
559 |
|
|
|
560 |
|
|
\begin{figure} |
561 |
|
|
\begin{center} |
562 |
|
|
\epsfxsize=3in |
563 |
|
|
\epsfbox{fig7.ps} |
564 |
|
|
\end{center} |
565 |
|
|
\caption {Snapshot of molecular configuration for a system of 1372 G-B molecules |
566 |
|
|
with two transverse outward pointing dipoles, the first dipole having |
567 |
|
|
$d^{*}=1.0$, $\mu^{*}=1.0$ and the second dipole having $d^{*}=0.0$, |
568 |
|
|
$\mu^{*}=0.5$(fine lines are molecular symmetry axes and small thick lines |
569 |
|
|
show terminal dipolar direction, central dipoles are not shown).} |
570 |
|
|
\label{fig:7} |
571 |
|
|
\end{figure} |
572 |
|
|
|
573 |
|
|
\begin{figure} |
574 |
|
|
\begin{center} |
575 |
|
|
\epsfxsize=3in |
576 |
|
|
\epsfbox{fig8.ps} |
577 |
|
|
\end{center} |
578 |
|
|
\caption {Snapshot of molecular configuration for a system of 1372 G-B molecules with |
579 |
|
|
single transverse outward pointing dipole, having $d^{*}=1.0$, $\mu^{*}=1.0$ |
580 |
|
|
(fine lines are molecular symmetry axes and small thick lines |
581 |
|
|
show terminal dipolar direction).} |
582 |
|
|
\label{fig:8} |
583 |
|
|
\end{figure} |
584 |
|
|
|
585 |
|
|
Starting from an isotropic configuaration of polar Gay-Berne molecules, |
586 |
|
|
we could successfully simulate perfect bilayer, antiphase and monolayer |
587 |
|
|
structure. To break the up-down symmetry i.e. the nonequivalence of |
588 |
|
|
directions ${\bf \hat {n}}$ and ${ -\bf \hat{n}}$, the molecules should have permanent |
589 |
|
|
electric or magnetic dipoles. Longitudinal electric dipole interaction could |
590 |
|
|
not form |
591 |
|
|
polar nematic phase as orientationally disordered phase with larger entropy |
592 |
|
|
is stabler than polarly ordered phase. In fact, stronger central dipole moment |
593 |
|
|
opposes polar nematic ordering more effectively in case of rod-like |
594 |
|
|
molecules. However, polar ordering like bilayer $A_{2}$, interdigitated |
595 |
|
|
$A_{d}$, and wavy $\tilde A$ in smectic layers can be achieved, where adjacent |
596 |
|
|
layers with opposite polarities makes bulk phase a-polar. More so, lyotropic |
597 |
|
|
liquid crystals and bilayer bio-membranes can have polar layers. These |
598 |
|
|
arrangements appear to get favours with the shifting of longitudinal dipole |
599 |
|
|
moment to the molecular terminus, so that they can have |
600 |
|
|
anti-ferroelectric |
601 |
|
|
dipolar arrangement giving rise to local (within the sublayer) breaking of |
602 |
|
|
up-down symmetry along the director. Transverse polarity breaks two-fold |
603 |
|
|
rotational symmetry, which favours more in-plane polar order. However, the |
604 |
|
|
molecular origin of these phases requires something more which are apparent |
605 |
|
|
from the earlier simulation results. We have shown that to get perfect bilayer |
606 |
|
|
structure in a G-B system, alongwith transverse terminal dipole, another |
607 |
|
|
central dipole (or |
608 |
|
|
a polarizable core) is required so that polar head and a-polar tail |
609 |
|
|
of Gay-Berne molecules go to opposite directions within a bilayer. This |
610 |
|
|
gives some kind of clipping interactions which forbid the molecular |
611 |
|
|
tail go in other way. |
612 |
|
|
Moreover, |
613 |
|
|
we could simulate other varieties of polar smectic phases e.g. monolayer |
614 |
|
|
$A_{1}$, |
615 |
|
|
antiphase $\tilde A$ |
616 |
|
|
successfully. |
617 |
|
|
Apart from guiding chemical synthesization of ferroelectric, |
618 |
|
|
antiferroelectric liquid crystals for technological applications, the present |
619 |
|
|
study will be of scientific interest in understanding molecular level |
620 |
|
|
interactions of lyotropic liquid crystals as well as nature-designed |
621 |
|
|
bio-membranes. |
622 |
|
|
|
623 |
|
|
\begin{acknowledgments} |
624 |
|
|
Support for this project was provided by the National Science |
625 |
|
|
Foundation under grant CHE-0134881. Computation time was provided by |
626 |
|
|
the Notre Dame High Performance Computing Cluster and the Notre Dame |
627 |
|
|
Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). |
628 |
|
|
\end{acknowledgments} |
629 |
|
|
|
630 |
|
|
|
631 |
|
|
\bibliography{bilayer} |
632 |
|
|
|
633 |
|
|
\pagebreak |
634 |
|
|
|
635 |
|
|
\end {document} |
636 |
|
|
|
637 |
|
|
|
638 |
|
|
|
639 |
|
|
|