| 465 |
|
|
| 466 |
|
It is reasonable to ask how well the parameters we used can produce |
| 467 |
|
bilayer properties that match experimentally known values for real |
| 468 |
< |
lipid bilayers. Using a value of $l = 13.8$ \AA for the ellipsoidal |
| 468 |
> |
lipid bilayers. Using a value of $l = 13.8$ \AA~for the ellipsoidal |
| 469 |
|
tails and the fixed ellipsoidal aspect ratio of 3, our values for the |
| 470 |
|
area per lipid ($A$) and inter-layer spacing ($D_{HH}$) depend |
| 471 |
|
entirely on the size of the head bead relative to the molecular body. |
| 526 |
|
different direction from the upper leaf.\label{mdfig:topView}} |
| 527 |
|
\end{figure} |
| 528 |
|
|
| 529 |
< |
The principal method for observing orientational ordering in dipolar |
| 530 |
< |
or liquid crystalline systems is the $P_2$ order parameter (defined |
| 531 |
< |
as $1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
| 532 |
< |
eigenvalue of the matrix, |
| 533 |
< |
\begin{equation} |
| 534 |
< |
{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
| 535 |
< |
\begin{array}{ccc} |
| 536 |
< |
u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\ |
| 537 |
< |
u^{y}_i u^{x}_i & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\ |
| 538 |
< |
u^{z}_i u^{x}_i & u^{z}_i u^{y}_i & u^{z}_i u^{z}_i -\frac{1}{3} |
| 539 |
< |
\end{array} \right). |
| 540 |
< |
\label{mdeq:opmatrix} |
| 541 |
< |
\end{equation} |
| 542 |
< |
Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector |
| 543 |
< |
for molecule $i$. (Here, $\hat{\bf u}_i$ can refer either to the |
| 529 |
> |
The orientational ordering in the system is observed by $P_2$ order |
| 530 |
> |
parameter, which is calculated from Eq.~\ref{mceq:opmatrix} |
| 531 |
> |
in Ch.~\ref{chap:mc}. Here, $\hat{\bf u}_i$ can refer either to the |
| 532 |
|
principal axis of the molecular body or to the dipole on the head |
| 533 |
< |
group of the molecule.) $P_2$ will be $1.0$ for a perfectly-ordered |
| 534 |
< |
system and near $0$ for a randomized system. Note that this order |
| 535 |
< |
parameter is {\em not} equal to the polarization of the system. For |
| 536 |
< |
example, the polarization of a perfect anti-ferroelectric arrangement |
| 549 |
< |
of point dipoles is $0$, but $P_2$ for the same system is $1$. The |
| 550 |
< |
eigenvector of $\mathsf{S}$ corresponding to the largest eigenvalue is |
| 551 |
< |
familiar as the director axis, which can be used to determine a |
| 552 |
< |
privileged axis for an orientationally-ordered system. Since the |
| 553 |
< |
molecular bodies are perpendicular to the head group dipoles, it is |
| 554 |
< |
possible for the director axes for the molecular bodies and the head |
| 555 |
< |
groups to be completely decoupled from each other. |
| 533 |
> |
group of the molecule. Since the molecular bodies are perpendicular to |
| 534 |
> |
the head group dipoles, it is possible for the director axes for the |
| 535 |
> |
molecular bodies and the head groups to be completely decoupled from |
| 536 |
> |
each other. |
| 537 |
|
|
| 538 |
|
Figure \ref{mdfig:topView} shows snapshots of bird's-eye views of the |
| 539 |
|
flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.35, 1.41 d$) |
