| 172 |
|
one can write down the translational and rotational resistance |
| 173 |
|
tensors |
| 174 |
|
\begin{eqnarray*} |
| 175 |
< |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 176 |
< |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 175 |
> |
\Xi _a^{tt} & = & 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 176 |
> |
\Xi _b^{tt} & = & \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 177 |
|
2a}}, |
| 178 |
|
\end{eqnarray*} |
| 179 |
|
and |
| 180 |
|
\begin{eqnarray*} |
| 181 |
< |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
| 182 |
< |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}}. |
| 181 |
> |
\Xi _a^{rr} & = & \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
| 182 |
> |
\Xi _b^{rr} & = & \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}}. |
| 183 |
|
\end{eqnarray*} |
| 184 |
|
|
| 185 |
|
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
| 565 |
|
made of 2266 small identical beads with size of 0.3 \AA on the |
| 566 |
|
surface. Applying the procedure described in |
| 567 |
|
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 568 |
< |
identified the center of resistance at $(0\AA, 0.7482\AA, |
| 569 |
< |
-0.1988\AA)$, as well as the resistance tensor, |
| 568 |
> |
identified the center of resistance at $(0 \AA, 0.7482 \AA, |
| 569 |
> |
-0.1988 \AA)$, as well as the resistance tensor, |
| 570 |
|
\[ |
| 571 |
|
\left( {\begin{array}{*{20}c} |
| 572 |
|
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
| 573 |
|
0& 0.9270&-0.007063& 0.08585&0&0\\ |
| 574 |
< |
0&0.007063&0.7494&0.2057&0&0\\ |
| 575 |
< |
0&0.0858&0.2057& 58.64& 0&-8.5736\\ |
| 574 |
> |
0&-0.007063&0.7494&0.2057&0&0\\ |
| 575 |
> |
0&0.0858&0.2057& 58.64& 0&0\\ |
| 576 |
|
0.08585&0&0&0&48.30&3.219&\\ |
| 577 |
|
0.2057&0&0&0&3.219&10.7373\\ |
| 578 |
|
\end{array}} \right). |
| 593 |
|
However, because of the stochastic nature, simulation using Langevin |
| 594 |
|
dynamics was shown to decay slightly faster than MD. In order to |
| 595 |
|
study the rotational motion of the molecules, we also calculated the |
| 596 |
< |
auto- correlation function of the principle axis of the second GB |
| 596 |
> |
auto-correlation function of the principle axis of the second GB |
| 597 |
|
particle, $u$. The discrepancy shown in Fig.~\ref{langevin:uacf} was |
| 598 |
|
probably due to the reason that the viscosity using in the |
| 599 |
|
simulations only partially preserved the dynamics of the system. |
| 617 |
|
\centering |
| 618 |
|
\includegraphics[width=\linewidth]{vacf.eps} |
| 619 |
|
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
| 620 |
< |
auto-correlation functions of NVE (explicit solvent) in blue) and |
| 620 |
> |
auto-correlation functions of NVE (explicit solvent) in blue and |
| 621 |
|
Langevin dynamics (implicit solvent) in red.} \label{langevin:vacf} |
| 622 |
|
\end{figure} |
| 623 |
|
|
