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\chapter{\label{chap:mc}Spontaneous Corrugation of Dipolar Membranes} |
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\chapter{\label{chap:mc}SPONTANEOUS CORRUGATION OF DIPOLAR MEMBRANES} |
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\section{Introduction} |
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\label{mc:sec:Int} |
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substantial experimental interest over the past 30 years. Most |
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structural information of the ripple phase has been obtained by the |
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X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron |
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microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it |
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et al.} used atomic force microscopy (AFM) to observe ripple phase |
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morphology in bilayers supported on mica.~\cite{Kaasgaard03} The |
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microscopy (FFEM).~\cite{Copeland80,Meyer96} The X-ray diffraction |
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work by Katsaras {\it et al.} showed that a rich phase diagram |
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exhibiting both {\it asymmetric} and {\it symmetric} ripples is |
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possible for lecithin bilayers.\cite{Katsaras00} Recently, Kaasgaard |
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{\it et al.} used atomic force microscopy (AFM) to observe ripple |
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phase morphology in bilayers supported on mica.~\cite{Kaasgaard03} The |
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experimental results provide strong support for a 2-dimensional |
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triangular packing lattice of the lipid molecules within the ripple |
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phase. This is a notable change from the observed lipid packing |
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within the gel phase.~\cite{Cevc87} There have been a number of |
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within the gel phase,~\cite{Cevc87} although Tenchov {\it et al.} have |
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recently observed near-hexagonal packing in some phosphatidylcholine |
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(PC) gel phases.~\cite{Tenchov2001} There have been a number of |
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theoretical |
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approaches~\cite{Marder84,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02} |
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approaches~\cite{Marder84,Carlson87,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02} |
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(and some heroic |
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simulations~\cite{Ayton02,Jiang04,Brannigan04a,deVries05,deJoannis06}) |
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undertaken to try to explain this phase, but to date, none have looked |
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freedom along the z-axis. |
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The potential energy of the system, |
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\begin{eqnarray} |
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V = \sum_i & & \left( \sum_{j>i} \frac{|\mu|^2}{4\pi \epsilon_0 r_{ij}^3} \left[ |
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\begin{equation} |
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\begin{split} |
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V = \sum_i &\left( \sum_{j>i} \frac{|\mu|^2}{4\pi \epsilon_0 r_{ij}^3} \left[ |
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{\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j} - |
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3({\mathbf{\hat u}_i} \cdot {\mathbf{\hat |
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r}_{ij}})({\mathbf{\hat u}_j} \cdot {\mathbf{\hat r}_{ij}})\right] |
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\right. \nonumber \\ |
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& & \left. + \sum_{j \in NN_i}^6 \frac{k_r}{2}\left( |
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r}_{ij}})({\mathbf{\hat u}_j} \cdot {\mathbf{\hat r}_{ij}})\right] \right. \\ |
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& \left. + \sum_{j \in NN_i}^6 \frac{k_r}{2}\left( |
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r_{ij}-\sigma \right)^2 \right) |
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+ |
\end{split} |
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\label{mceq:pot} |
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\end{eqnarray} |
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\end{equation} |
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In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing |
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along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
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\subsection{Dipolar Ordering and Coexistence Temperatures} |
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The principal method for observing the orientational ordering |
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transition in dipolar systems is the $P_2$ order parameter (defined as |
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$1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
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eigenvalue of the matrix, |
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transition in dipolar or liquid crystalline systems is the $P_2$ order |
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parameter (defined as $1.5 \times \lambda_{max}$, where |
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$\lambda_{max}$ is the largest eigenvalue of the matrix, |
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\begin{equation} |
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{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
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\begin{array}{ccc} |
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large mechanical surface tensions ($\gamma$), so a much simpler form |
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can be written, |
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\begin{equation} |
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\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{\gamma q^2}, |
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\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{\gamma q^2}. |
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\label{mceq:fit2} |
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\end{equation} |
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