98 |
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estimation of friction tensor from hydrodynamics theory into the |
99 |
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sophisticated rigid body dynamics. |
100 |
|
|
101 |
< |
\section{Method{\label{methodSec}}} |
101 |
> |
\section{Computational methods{\label{methodSec}}} |
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|
|
103 |
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\subsection{\label{introSection:frictionTensor}Friction Tensor} |
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Theoretically, the friction kernel can be determined using velocity |
360 |
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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} |
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> |
\subsection{Langevin dynamics for rigid particles of arbitrary shape\label{LDRB}} |
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|
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Consider a Langevin equation of motions in generalized coordinates |
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\begin{equation} |
369 |
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\end{equation} |
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where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
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and moment of inertial) matrix and $V_i$ is a generalized velocity, |
372 |
< |
$V_i = V_i(v_i,\omega _i)$. The right side of Eq. |
372 |
> |
$V_i = V_i(v_i,\omega _i)$. The right side of Eq.~ |
373 |
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(\ref{LDGeneralizedForm}) consists of three generalized forces in |
374 |
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lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
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$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
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\begin{equation} |
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\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
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\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} |
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\end{equation} |
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|
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The equation of motion for $v_i$ can be written as |
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\end{align*} |
517 |
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|
518 |
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\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 |
+ |
|
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+ |
As shown in Eq.~\ref{randomForce}, random collisions associated with |
527 |
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the solvent's thermal motions is controlled by the external |
528 |
+ |
temperature. The capability to maintain the temperature of the whole |
529 |
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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 & ${\rm{T}}_{{\rm{avg}}}$ & ${\rm{T}}_{{\rm{rms}}}$ \\ |
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]{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} |