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\chapter{\label{chap:intro}INTRODUCTION AND BACKGROUND} |
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\section{Background on the Problem\label{In:sec:pro}} |
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Phospholipid molecules are chosen to be studied in this dissertation |
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because of their critical role as a foundation of the bio-membrane |
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construction. The self assembled bilayer of the lipids when dispersed |
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in water is the micro structure of the membrane. The phase behavior of |
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lipid bilayer is explored experimentally~\cite{Cevc87}, however, fully |
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understanding on the mechanism is far beyond accomplished. |
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Phospholipid molecules are the primary topic of this dissertation |
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because of their critical role as the foundation of biological |
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membranes. Lipids, when dispersed in water, self assemble into bilayer |
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structures. The phase behavior of lipid bilayers have been explored |
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experimentally~\cite{Cevc87}, however, a complete understanding of the |
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mechanism for self-assembly has not been achieved. |
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\subsection{Ripple Phase\label{In:ssec:ripple}} |
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The {\it ripple phase} $P_{\beta'}$ of lipid bilayers, named from the |
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The $P_{\beta'}$ {\it ripple phase} of lipid bilayers, named from the |
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periodic buckling of the membrane, is an intermediate phase which is |
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developed either from heating the gel phase $L_{\beta'}$ or cooling |
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the fluid phase $L_\alpha$. Although the ripple phase is observed in |
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different |
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the fluid phase $L_\alpha$. Although the ripple phase has been |
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observed in several |
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experiments~\cite{Sun96,Katsaras00,Copeland80,Meyer96,Kaasgaard03}, |
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the mechanism of the formation of the ripple phase has never been |
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the physical mechanism for the formation of the ripple phase has never been |
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explained and the microscopic structure of the ripple phase has never |
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been elucidated by experiments. Computational simulation is a perfect |
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tool to study the microscopic properties for a system, however, the |
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long range dimension of the ripple structure and the long time scale |
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tool to study the microscopic properties for a system. However, the |
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large length scale the ripple structure and the long time scale |
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of the formation of the ripples are crucial obstacles to performing |
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the actual work. The idea to break through this dilemma forks into: |
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the actual work. The principal ideas explored in this dissertation are |
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attempts to break the computational task up by |
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\begin{itemize} |
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\item Simplify the lipid model. |
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\item Improve the integrating algorithm. |
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\item Simplifying the lipid model. |
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\item Improving algorithm for integrating the equations of motion. |
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\end{itemize} |
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In Ch.~\ref{chap:mc} and~\ref{chap:md}, we use a simple point dipole |
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model and a coarse-grained model to perform the Monte Carlo and |
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Molecular Dynamics simulations respectively, and in Ch.~\ref{chap:ld}, |
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we implement a Langevin Dynamics algorithm to exclude the explicit |
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solvent to improve the efficiency of the simulations. |
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In chapters~\ref{chap:mc} and~\ref{chap:md}, we use a simple point |
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dipole model and a coarse-grained model to perform the Monte Carlo and |
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Molecular Dynamics simulations respectively, and in |
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chapter~\ref{chap:ld}, we develop a Langevin Dynamics algorithm which |
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excludes the explicit solvent to improve the efficiency of the |
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simulations. |
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\subsection{Lattice Model\label{In:ssec:model}} |
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The gel like characteristic of the ripple phase ensures the feasiblity |
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of applying the lattice model to study the system. It is claimed that |
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the packing of the lipid molecules in ripple phase is |
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The gel-like characteristic of the ripple phase makes it feasible to |
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apply a lattice model to study the system. It is claimed that the |
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packing of the lipid molecules in ripple phase is |
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hexagonal~\cite{Cevc87}. The popular $2$ dimensional lattice models, |
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{\it i.e.}, Ising model, Heisenberg model and $X-Y$ model, show |
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{\it frustration} on triangular lattice. |
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{\it i.e.}, the Ising, $X-Y$, and Heisenberg models, show {\it |
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frustration} on triangular lattices. The Hamiltonian of the systems |
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are given by |
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\begin{equation} |
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H = |
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\begin{cases} |
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-J \sum_n \sum_{n'} s_n s_n' & \text{Ising}, \\ |
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-J \sum_n \sum_{n'} \vec s_n \cdot \vec s_{n'} & \text{$X-Y$ and |
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Heisenberg}, |
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\end{cases} |
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\end{equation} |
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where $J$ has non zero value only when spins $s_n$ ($\vec s_n$) and |
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$s_{n'}$ ($\vec s_{n'}$) are the nearest neighbours. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{./figures/inFrustration.pdf} |
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\caption{Sketch to illustrate the frustration on triangular |
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lattice. Spins are represented by arrows, no matter which direction |
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the spin on the top of triangle points to, the Hamiltonian of the |
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system is the same, hence there are infinite possibilities for the |
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packing of the spins.} |
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\caption{Frustration on a triangular lattice, the spins are |
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represented by arrows. No matter which direction the spin on the top |
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of triangle points to, the Hamiltonain of the system goes up.} |
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\label{Infig:frustration} |
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\end{figure} |
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Figure~\ref{Infig:frustration} shows an illustration of the frustration |
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on a triangular lattice. The direction of the spin on top of the |
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triangle has no effects on the Hamiltonian of the system, therefore |
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infinite possibilities for the packing of spins induce the frustration |
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of the lattice. |
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Figure~\ref{Infig:frustration} shows an illustration of the |
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frustration on a triangular lattice. When $J < 0$, the spins want to |
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be anti-aligned, The direction of the spin on top of the triangle will |
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make the energy go up no matter which direction it picks, therefore |
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infinite possibilities for the packing of spins induce what is known |
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as ``complete regular frustration'' which leads to disordered low |
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temperature phases. |
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The lack of translational degree of freedom in lattice models prevents |
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their utilization on investigating the emergence of the surface |
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buckling which is the imposition of the ripple formation. In this |
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dissertation, a modified lattice model is introduced to this specific |
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situation in Ch.~\ref{chap:mc}. |
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their utilization in models for surface buckling which would |
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correspond to ripple formation. In chapter~\ref{chap:mc}, a modified |
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lattice model is introduced to tackle this specific situation. |
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\section{Overview of Classical Statistical Mechanics\label{In:sec:SM}} |
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Statistical mechanics provides a way to calculate the macroscopic |
