| 98 |
|
estimation of friction tensor from hydrodynamics theory into the |
| 99 |
|
sophisticated rigid body dynamics. |
| 100 |
|
|
| 101 |
< |
\section{Method{\label{methodSec}}} |
| 101 |
> |
\section{Computational methods{\label{methodSec}}} |
| 102 |
|
|
| 103 |
|
\subsection{\label{introSection:frictionTensor}Friction Tensor} |
| 104 |
|
Theoretically, the friction kernel can be determined using velocity |
| 360 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 361 |
|
joining center of resistance $R$ and origin $O$. |
| 362 |
|
|
| 363 |
< |
\subsection{Langevin dynamics for rigid particles of arbitrary shape} |
| 363 |
> |
\subsection{Langevin dynamics for rigid particles of arbitrary shape\label{LDRB}} |
| 364 |
|
|
| 365 |
|
Consider a Langevin equation of motions in generalized coordinates |
| 366 |
|
\begin{equation} |
| 403 |
|
\begin{equation} |
| 404 |
|
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
| 405 |
|
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
| 406 |
< |
2k_B T\Xi _R \delta (t - t'). |
| 406 |
> |
2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
| 407 |
|
\end{equation} |
| 408 |
|
|
| 409 |
|
The equation of motion for $v_i$ can be written as |
| 516 |
|
\end{align*} |
| 517 |
|
|
| 518 |
|
\section{Results and discussion} |
| 519 |
+ |
|
| 520 |
+ |
The Langevin algorithm described in previous section has been |
| 521 |
+ |
implemented in {\sc oopse}\cite{Meineke2005} and applied to the |
| 522 |
+ |
studies of kinetics and thermodynamic properties in several systems. |
| 523 |
+ |
|
| 524 |
+ |
\subsection{Temperature control} |
| 525 |
+ |
|
| 526 |
+ |
As shown in Eq.~\ref{randomForce}, random collisions associated with |
| 527 |
+ |
the solvent's thermal motions is controlled by the external |
| 528 |
+ |
temperature. The capability to maintain the temperature of the whole |
| 529 |
+ |
system was usually used to measure the stability and efficiency of |
| 530 |
+ |
the algorithm. In order to verify the stability of this new |
| 531 |
+ |
algorithm, a series of simulations are performed on system |
| 532 |
+ |
consisiting of 256 SSD water molecules with different viscosities. |
| 533 |
+ |
Initial configuration for the simulations is taken from a 1ns NVT |
| 534 |
+ |
simulation with a cubic box of 19.7166~\AA. All simulation are |
| 535 |
+ |
carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
| 536 |
+ |
with reference temperature at 300~K. Average temperature as a |
| 537 |
+ |
function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
| 538 |
+ |
the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
| 539 |
+ |
1$ poise. The better temperature control at higher viscosity can be |
| 540 |
+ |
explained by the finite size effect and relative slow relaxation |
| 541 |
+ |
rate at lower viscosity regime. |
| 542 |
+ |
\begin{table} |
| 543 |
+ |
\caption{Average temperatures from Langevin dynamics simulations of |
| 544 |
+ |
SSD water molecules with reference temperature at 300~K.} |
| 545 |
+ |
\label{langevin:viscosity} |
| 546 |
+ |
\begin{center} |
| 547 |
+ |
\begin{tabular}{|l|l|l|} |
| 548 |
+ |
\hline |
| 549 |
+ |
$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
| 550 |
+ |
1 & 300.47 & 10.99 \\ |
| 551 |
+ |
0.1 & 301.19 & 11.136 \\ |
| 552 |
+ |
0.01 & 303.04 & 11.796 \\ |
| 553 |
+ |
\hline |
| 554 |
+ |
\end{tabular} |
| 555 |
+ |
\end{center} |
| 556 |
+ |
\end{table} |
| 557 |
|
|
| 558 |
+ |
Another set of calculation were performed to study the efficiency of |
| 559 |
+ |
temperature control using different temperature coupling schemes. |
| 560 |
+ |
The starting configuration is cooled to 173~K and evolved using NVE, |
| 561 |
+ |
NVT, and Langevin dynamic with time step of 2 fs. |
| 562 |
+ |
Fig.~\ref{langevin:temperature} shows the heating curve obtained as |
| 563 |
+ |
the systems reach equilibrium. The orange curve in |
| 564 |
+ |
Fig.~\ref{langevin:temperature} represents the simulation using |
| 565 |
+ |
Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
| 566 |
+ |
which gives reasonable tight coupling, while the blue one from |
| 567 |
+ |
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
| 568 |
+ |
scaling to the desire temperature. In extremely lower friction |
| 569 |
+ |
regime (when $ \eta \approx 0$), Langevin dynamics becomes normal |
| 570 |
+ |
NVE (see green curve in Fig.~\ref{langevin:temperature}) which loses |
| 571 |
+ |
the temperature control ability. |
| 572 |
+ |
|
| 573 |
+ |
|
| 574 |
|
\begin{figure} |
| 575 |
|
\centering |
| 576 |
+ |
\includegraphics[width=\linewidth]{temperature.eps} |
| 577 |
+ |
\caption[Plot of Temperature Fluctuation Versus Time]{Plot of |
| 578 |
+ |
temperature fluctuation versus time.} \label{langevin:temperature} |
| 579 |
+ |
\end{figure} |
| 580 |
+ |
|
| 581 |
+ |
\subsection{Langevin dynamics of banana-shaped molecule} |
| 582 |
+ |
|
| 583 |
+ |
\begin{figure} |
| 584 |
+ |
\centering |
| 585 |
|
\includegraphics[width=\linewidth]{one_banana.eps} |
| 586 |
|
\caption[]{.} \label{langevin:banana} |
| 587 |
|
\end{figure} |
| 588 |
|
|
| 589 |
|
\begin{figure} |
| 590 |
|
\centering |
| 591 |
< |
\includegraphics[width=\linewidth]{temperature.eps} |
| 592 |
< |
\caption[]{.} \label{langevin:temperature} |
| 591 |
> |
\includegraphics[width=\linewidth]{roughShell.eps} |
| 592 |
> |
\caption[Rough Shell]{Rough Shell.} \label{langevin:roughShell} |
| 593 |
|
\end{figure} |
| 594 |
|
|
| 595 |
+ |
\begin{figure} |
| 596 |
+ |
\centering |
| 597 |
+ |
\includegraphics[width=\linewidth]{twoBanana.eps} |
| 598 |
+ |
\caption[Two Banana Shaped Molecules]{Two Banana Shaped Molecules.} |
| 599 |
+ |
\label{langevin:twoBanana} |
| 600 |
+ |
\end{figure} |
| 601 |
+ |
|
| 602 |
|
\section{Conclusions} |
