| 62 |
|
site, as well as a point dipole. A tetrahedral potential is added to |
| 63 |
|
correct for hydrogen bonding. The following equation describes the |
| 64 |
|
interaction between two water molecules: |
| 65 |
< |
\[ |
| 65 |
> |
\begin{equation} |
| 66 |
|
V_{SSD} = V_{LJ} (r_{ij} ) + V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) |
| 67 |
|
+ V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) |
| 68 |
< |
\] |
| 69 |
< |
where $r_{ij}$ is the vector between molecule $i$ and molecule $j$, |
| 68 |
> |
\label{lipidSection:ssdEquation} |
| 69 |
> |
\end{equation} |
| 70 |
> |
where$r_{ij}$ is the vector between molecule $i$ and molecule $j$, |
| 71 |
|
$\Omega _i$ and $\Omega _j$ are the orientational degrees of freedom |
| 72 |
< |
for molecule $i$ and molecule $j$ respectively. |
| 73 |
< |
\begin{eqnarray*} |
| 72 |
> |
for molecule $i$ and molecule $j$ respectively. The potential terms |
| 73 |
> |
in Eq.~\ref{lipidSection:ssdEquation} are given by : |
| 74 |
> |
\begin{eqnarray} |
| 75 |
|
V_{LJ} (r_{ij} ) &= &4\varepsilon _{ij} \left[ {\left( |
| 76 |
|
{\frac{{\sigma _{ij} }}{{r_{ij} }}} \right)^{12} - \left( |
| 77 |
|
{\frac{{\sigma _{ij} |
| 82 |
|
(\hat{u}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr],\\ |
| 83 |
|
V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) &=& v_0 [s(r_{ij} |
| 84 |
|
)w(r_{ij} ,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i |
| 85 |
< |
,\Omega _j )].\\ |
| 86 |
< |
\end{eqnarray*} |
| 85 |
> |
,\Omega _j )] |
| 86 |
> |
\end{eqnarray} |
| 87 |
|
where $v_0$ is a strength parameter, $s$ and $s'$ are cubic |
| 88 |
|
switching functions, while $w$ and $w'$ are responsible for the |
| 89 |
|
tetrahedral potential and the short-range correction to the dipolar |
| 90 |
< |
interaction respectively. |
| 91 |
< |
\[ |
| 92 |
< |
\begin{array}{l} |
| 93 |
< |
w(r_{ij} ,\Omega _i ,\Omega _j ) = \sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij} \\ |
| 94 |
< |
w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij} - 0.6)^2 (\cos \theta _{ij} + 0.8)^2 - w_0 \\ |
| 95 |
< |
\end{array} |
| 96 |
< |
\] |
| 97 |
< |
Although the dipole-dipole and sticky interactions are more |
| 90 |
> |
interaction respectively: |
| 91 |
> |
\begin{eqnarray} |
| 92 |
> |
w(r_{ij} ,\Omega _i ,\Omega _j )& = &\sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij}, \\ |
| 93 |
> |
w'(r_{ij} ,\Omega _i ,\Omega _j )& = &(\cos \theta _{ij} - 0.6)^2 (\cos \theta _{ij} + 0.8)^2 - |
| 94 |
> |
w_0. |
| 95 |
> |
\end{eqnarray} |
| 96 |
> |
Here $\theta _{ij}$ and $\phi _{ij}$ are the spherical polar angles |
| 97 |
> |
representing relative orientations between molecule $i$ and molecule |
| 98 |
> |
$j$. Although the dipole-dipole and sticky interactions are more |
| 99 |
|
mathematically complicated than Coulomb interactions, the number of |
| 100 |
|
pair interactions is reduced dramatically both because the model |
| 101 |
|
only contains a single-point as well as "short range" nature of the |
