| 140 |
|
Pechukas.\cite{Berne72} The potential is constructed in the familiar |
| 141 |
|
form of the Lennard-Jones function using orientation-dependent |
| 142 |
|
$\sigma$ and $\epsilon$ parameters, |
| 143 |
< |
\begin{equation} |
| 144 |
< |
\begin{split} |
| 143 |
> |
\begin{multline} |
| 144 |
|
V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 145 |
< |
r}_{ij}}) = & 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 145 |
> |
r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 146 |
|
{\mathbf{\hat r}_{ij}})\left[ \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
| 147 |
< |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} \right.\\ |
| 148 |
< |
&\left. -\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
| 147 |
> |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
| 148 |
> |
\right. \\ |
| 149 |
> |
\left. - \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
| 150 |
|
{\mathbf{\hat u}_j}, {\mathbf{\hat |
| 151 |
|
r}_{ij}})+\sigma_0}\right)^6\right] |
| 152 |
– |
\end{split} |
| 152 |
|
\label{mdeq:gb} |
| 153 |
< |
\end{equation} |
| 153 |
> |
\end{multline} |
| 154 |
|
|
| 155 |
|
The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 156 |
|
\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
| 171 |
|
where $l$ and $d$ describe the length and width of each uniaxial |
| 172 |
|
ellipsoid. These shape anisotropy parameters can then be used to |
| 173 |
|
calculate the range function, |
| 174 |
< |
\begin{equation} |
| 175 |
< |
\begin{split} |
| 176 |
< |
& \sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = |
| 178 |
< |
\sigma_{0} \times \\ |
| 179 |
< |
& \left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
| 174 |
> |
\begin{multline} |
| 175 |
> |
\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \\ |
| 176 |
> |
\sigma_0 \left[ 1 - \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
| 177 |
|
\hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf |
| 178 |
|
\hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf |
| 179 |
|
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
| 180 |
|
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2 |
| 181 |
|
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\} |
| 182 |
< |
\right]^{-1/2} |
| 183 |
< |
\end{split} |
| 187 |
< |
\end{equation} |
| 182 |
> |
\right]^{-1/2} |
| 183 |
> |
\end{multline} |
| 184 |
|
|
| 185 |
|
Gay-Berne ellipsoids also have an energy scaling parameter, |
| 186 |
|
$\epsilon^s$, which describes the well depth for two identical |
| 207 |
|
\left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf |
| 208 |
|
\hat{u}}_{j})^{2}\right]^{-1/2} |
| 209 |
|
\end{eqnarray*} |
| 210 |
< |
\begin{equation*} |
| 211 |
< |
\begin{split} |
| 212 |
< |
& \epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) |
| 213 |
< |
= 1 - \\ |
| 218 |
< |
& \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
| 210 |
> |
\begin{multline*} |
| 211 |
> |
\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) |
| 212 |
> |
= \\ |
| 213 |
> |
1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
| 214 |
|
\hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf |
| 215 |
|
\hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf |
| 216 |
|
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
| 217 |
|
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi'^2 |
| 218 |
|
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}, |
| 219 |
< |
\end{split} |
| 225 |
< |
\end{equation*} |
| 219 |
> |
\end{multline*} |
| 220 |
|
although many of the quantities and derivatives are identical with |
| 221 |
|
those obtained for the range parameter. Ref. \citen{Luckhurst90} |
| 222 |
|
has a particularly good explanation of the choice of the Gay-Berne |
| 696 |
|
We have computed translational diffusion constants for lipid molecules |
| 697 |
|
from the mean-square displacement, |
| 698 |
|
\begin{equation} |
| 699 |
< |
D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
| 699 |
> |
D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf |
| 700 |
> |
r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
| 701 |
> |
\label{mdeq:msdisplacement} |
| 702 |
|
\end{equation} |
| 703 |
|
of the lipid bodies. Translational diffusion constants for the |
| 704 |
|
different head-to-tail size ratios (all at 300 K) are shown in table |
