| 1 |
< |
\chapter{\label{chap:mc}Spontaneous Corrugation of Dipolar Membranes} |
| 1 |
> |
\chapter{\label{chap:mc}SPONTANEOUS CORRUGATION OF DIPOLAR MEMBRANES} |
| 2 |
|
|
| 3 |
|
\section{Introduction} |
| 4 |
|
\label{mc:sec:Int} |
| 26 |
|
bilayers form a corrugated or ``rippled'' phase ($P_{\beta'}$) which |
| 27 |
|
appears as an intermediate phase between the gel ($L_\beta$) and fluid |
| 28 |
|
($L_{\alpha}$) phases. The $P_{\beta'}$ phase has attracted |
| 29 |
< |
substantial experimental interest over the past 30 years. Most |
| 30 |
< |
structural information of the ripple phase has been obtained by the |
| 31 |
< |
X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron |
| 32 |
< |
microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it |
| 33 |
< |
et al.} used atomic force microscopy (AFM) to observe ripple phase |
| 34 |
< |
morphology in bilayers supported on mica.~\cite{Kaasgaard03} The |
| 35 |
< |
experimental results provide strong support for a 2-dimensional |
| 36 |
< |
triangular packing lattice of the lipid molecules within the ripple |
| 37 |
< |
phase. This is a notable change from the observed lipid packing |
| 38 |
< |
within the gel phase.~\cite{Cevc87} There have been a number of |
| 39 |
< |
theoretical |
| 40 |
< |
approaches~\cite{Marder84,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02} |
| 29 |
> |
substantial experimental interest over the past 30 |
| 30 |
> |
years,~\cite{Sun96,Katsaras00,Copeland80,Meyer96,Kaasgaard03} and |
| 31 |
> |
there have been a number of theoretical |
| 32 |
> |
approaches~\cite{Marder84,Carlson87,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02} |
| 33 |
|
(and some heroic |
| 34 |
|
simulations~\cite{Ayton02,Jiang04,Brannigan04a,deVries05,deJoannis06}) |
| 35 |
|
undertaken to try to explain this phase, but to date, none have looked |
| 106 |
|
freedom along the z-axis. |
| 107 |
|
|
| 108 |
|
The potential energy of the system, |
| 109 |
< |
\begin{eqnarray} |
| 110 |
< |
V = \sum_i & & \left( \sum_{j>i} \frac{|\mu|^2}{4\pi \epsilon_0 r_{ij}^3} \left[ |
| 109 |
> |
\begin{equation} |
| 110 |
> |
\begin{split} |
| 111 |
> |
V = \sum_i &\left( \sum_{j>i} \frac{|\mu|^2}{4\pi \epsilon_0 r_{ij}^3} \left[ |
| 112 |
|
{\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j} - |
| 113 |
|
3({\mathbf{\hat u}_i} \cdot {\mathbf{\hat |
| 114 |
< |
r}_{ij}})({\mathbf{\hat u}_j} \cdot {\mathbf{\hat r}_{ij}})\right] |
| 115 |
< |
\right. \nonumber \\ |
| 123 |
< |
& & \left. + \sum_{j \in NN_i}^6 \frac{k_r}{2}\left( |
| 114 |
> |
r}_{ij}})({\mathbf{\hat u}_j} \cdot {\mathbf{\hat r}_{ij}})\right] \right. \\ |
| 115 |
> |
& \left. + \sum_{j \in NN_i}^6 \frac{k_r}{2}\left( |
| 116 |
|
r_{ij}-\sigma \right)^2 \right) |
| 117 |
+ |
\end{split} |
| 118 |
|
\label{mceq:pot} |
| 119 |
< |
\end{eqnarray} |
| 119 |
> |
\end{equation} |
| 120 |
|
|
| 121 |
|
In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing |
| 122 |
|
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
| 166 |
|
|
| 167 |
|
\subsection{Dipolar Ordering and Coexistence Temperatures} |
| 168 |
|
The principal method for observing the orientational ordering |
| 169 |
< |
transition in dipolar systems is the $P_2$ order parameter (defined as |
| 170 |
< |
$1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
| 171 |
< |
eigenvalue of the matrix, |
| 169 |
> |
transition in dipolar or liquid crystalline systems is the $P_2$ order |
| 170 |
> |
parameter (defined as $1.5 \times \lambda_{max}$, where |
| 171 |
> |
$\lambda_{max}$ is the largest eigenvalue of the matrix, |
| 172 |
|
\begin{equation} |
| 173 |
|
{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
| 174 |
|
\begin{array}{ccc} |
| 326 |
|
large mechanical surface tensions ($\gamma$), so a much simpler form |
| 327 |
|
can be written, |
| 328 |
|
\begin{equation} |
| 329 |
< |
\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{\gamma q^2}, |
| 329 |
> |
\langle | h(q) |^2 \rangle_{NVT} = \frac{k_B T}{\gamma q^2}. |
| 330 |
|
\label{mceq:fit2} |
| 331 |
|
\end{equation} |
| 332 |
|
|
