| 98 |
|
estimation of friction tensor from hydrodynamics theory into the |
| 99 |
|
sophisticated rigid body dynamics. |
| 100 |
|
|
| 101 |
< |
\section{Computational methods{\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 |
| 105 |
< |
autocorrelation function. However, this approach become impractical |
| 106 |
< |
when the system become more and more complicate. Instead, various |
| 107 |
< |
approaches based on hydrodynamics have been developed to calculate |
| 108 |
< |
the friction coefficients. The friction effect is isotropic in |
| 109 |
< |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
| 110 |
< |
tensor $\Xi$ is a $6\times 6$ matrix given by |
| 104 |
> |
Theoretically, the friction kernel can be determined using the |
| 105 |
> |
velocity autocorrelation function. However, this approach become |
| 106 |
> |
impractical when the system become more and more complicate. |
| 107 |
> |
Instead, various approaches based on hydrodynamics have been |
| 108 |
> |
developed to calculate the friction coefficients. In general, |
| 109 |
> |
friction tensor $\Xi$ is a $6\times 6$ matrix given by |
| 110 |
|
\[ |
| 111 |
|
\Xi = \left( {\begin{array}{*{20}c} |
| 112 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 137 |
|
|
| 138 |
|
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
| 139 |
|
|
| 140 |
< |
For a spherical particle, the translational and rotational friction |
| 141 |
< |
constant can be calculated from Stoke's law, |
| 140 |
> |
For a spherical particle with slip boundary conditions, the |
| 141 |
> |
translational and rotational friction constant can be calculated |
| 142 |
> |
from Stoke's law, |
| 143 |
|
\[ |
| 144 |
|
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 145 |
|
{6\pi \eta R} & 0 & 0 \\ |
| 161 |
|
Other non-spherical shape, such as cylinder and ellipsoid |
| 162 |
|
\textit{etc}, are widely used as reference for developing new |
| 163 |
|
hydrodynamics theory, because their properties can be calculated |
| 164 |
< |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
| 165 |
< |
also called a triaxial ellipsoid, which is given in Cartesian |
| 166 |
< |
coordinates by\cite{Perrin1934, Perrin1936} |
| 164 |
> |
exactly. In 1936, Perrin extended Stokes's law to general |
| 165 |
> |
ellipsoids, also called a triaxial ellipsoid, which is given in |
| 166 |
> |
Cartesian coordinates by\cite{Perrin1934, Perrin1936} |
| 167 |
|
\[ |
| 168 |
|
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 169 |
|
}} = 1 |
| 198 |
|
\end{array}. |
| 199 |
|
\] |
| 200 |
|
|
| 201 |
< |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}} |
| 201 |
> |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
| 202 |
|
|
| 203 |
< |
Unlike spherical and other regular shaped molecules, there is not |
| 204 |
< |
analytical solution for friction tensor of any arbitrary shaped |
| 203 |
> |
Unlike spherical and other simply shaped molecules, there is no |
| 204 |
> |
analytical solution for the friction tensor for arbitrarily shaped |
| 205 |
|
rigid molecules. The ellipsoid of revolution model and general |
| 206 |
|
triaxial ellipsoid model have been used to approximate the |
| 207 |
|
hydrodynamic properties of rigid bodies. However, since the mapping |
| 208 |
< |
from all possible ellipsoidal space, $r$-space, to all possible |
| 209 |
< |
combination of rotational diffusion coefficients, $D$-space is not |
| 208 |
> |
from all possible ellipsoidal spaces, $r$-space, to all possible |
| 209 |
> |
combination of rotational diffusion coefficients, $D$-space, is not |
| 210 |
|
unique\cite{Wegener1979} as well as the intrinsic coupling between |
| 211 |
< |
translational and rotational motion of rigid body, general ellipsoid |
| 212 |
< |
is not always suitable for modeling arbitrarily shaped rigid |
| 213 |
< |
molecule. A number of studies have been devoted to determine the |
| 214 |
< |
friction tensor for irregularly shaped rigid bodies using more |
| 211 |
> |
translational and rotational motion of rigid body, general |
| 212 |
> |
ellipsoids are not always suitable for modeling arbitrarily shaped |
| 213 |
> |
rigid molecule. A number of studies have been devoted to determine |
| 214 |
> |
the friction tensor for irregularly shaped rigid bodies using more |
| 215 |
|
advanced method where the molecule of interest was modeled by |
| 216 |
|
combinations of spheres(beads)\cite{Carrasco1999} and the |
| 217 |
|
hydrodynamics properties of the molecule can be calculated using the |
| 273 |
|
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 274 |
|
)T_{ij} |
| 275 |
|
\] |
| 276 |
< |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
| 277 |
< |
$B$, we obtain |
| 276 |
> |
where $\delta _{ij}$ is Kronecker delta function. Inverting the $B$ |
| 277 |
> |
matrix, we obtain |
| 278 |
|
|
| 279 |
|
\[ |
| 280 |
|
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 306 |
|
|
| 307 |
|
The resistance tensor depends on the origin to which they refer. The |
| 308 |
|
proper location for applying friction force is the center of |
| 309 |
< |
resistance (reaction), at which the trace of rotational resistance |
| 310 |
< |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
| 311 |
< |
resistance is defined as an unique point of the rigid body at which |
| 312 |
< |
the translation-rotation coupling tensor are symmetric, |
| 309 |
> |
resistance (or center of reaction), at which the trace of rotational |
| 310 |
> |
resistance tensor, $ \Xi ^{rr}$ reaches a minimum value. |
| 311 |
> |
Mathematically, the center of resistance is defined as an unique |
| 312 |
> |
point of the rigid body at which the translation-rotation coupling |
| 313 |
> |
tensor are symmetric, |
| 314 |
|
\begin{equation} |
| 315 |
|
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 316 |
|
\label{introEquation:definitionCR} |
| 317 |
|
\end{equation} |
| 318 |
< |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 318 |
> |
From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 319 |
|
we can easily find out that the translational resistance tensor is |
| 320 |
|
origin independent, while the rotational resistance tensor and |
| 321 |
|
translation-rotation coupling resistance tensor depend on the |
| 322 |
< |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 323 |
< |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 322 |
> |
origin. Given the resistance tensor at an arbitrary origin $O$, and |
| 323 |
> |
a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 324 |
|
obtain the resistance tensor at $P$ by |
| 325 |
|
\begin{equation} |
| 326 |
|
\begin{array}{l} |
| 361 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 362 |
|
joining center of resistance $R$ and origin $O$. |
| 363 |
|
|
| 364 |
< |
\subsection{Langevin dynamics for rigid particles of arbitrary shape\label{LDRB}} |
| 364 |
> |
\subsection{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
| 365 |
|
|
| 366 |
|
Consider a Langevin equation of motions in generalized coordinates |
| 367 |
|
\begin{equation} |
| 516 |
|
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
| 517 |
|
\end{align*} |
| 518 |
|
|
| 519 |
< |
\section{Results and discussion} |
| 519 |
> |
\section{Results and Discussion} |
| 520 |
|
|
| 521 |
|
The Langevin algorithm described in previous section has been |
| 522 |
|
implemented in {\sc oopse}\cite{Meineke2005} and applied to the |
| 523 |
|
studies of kinetics and thermodynamic properties in several systems. |
| 524 |
|
|
| 525 |
< |
\subsection{Temperature control} |
| 525 |
> |
\subsection{Temperature Control} |
| 526 |
|
|
| 527 |
|
As shown in Eq.~\ref{randomForce}, random collisions associated with |
| 528 |
|
the solvent's thermal motions is controlled by the external |
| 571 |
|
NVE (see green curve in Fig.~\ref{langevin:temperature}) which loses |
| 572 |
|
the temperature control ability. |
| 573 |
|
|
| 573 |
– |
|
| 574 |
|
\begin{figure} |
| 575 |
|
\centering |
| 576 |
|
\includegraphics[width=\linewidth]{temperature.eps} |
| 578 |
|
temperature fluctuation versus time.} \label{langevin:temperature} |
| 579 |
|
\end{figure} |
| 580 |
|
|
| 581 |
< |
\subsection{Langevin dynamics of banana-shaped molecule} |
| 581 |
> |
\subsection{Langevin Dynamics of Banana Shaped Molecule} |
| 582 |
|
|
| 583 |
+ |
In order to compare the |
| 584 |
+ |
|
| 585 |
|
\begin{figure} |
| 586 |
|
\centering |
| 587 |
|
\includegraphics[width=\linewidth]{one_banana.eps} |
| 591 |
|
\begin{figure} |
| 592 |
|
\centering |
| 593 |
|
\includegraphics[width=\linewidth]{roughShell.eps} |
| 594 |
< |
\caption[Rough Shell]{Rough Shell.} \label{langevin:roughShell} |
| 594 |
> |
\caption[Rough shell model for banana shaped molecule]{Rough shell |
| 595 |
> |
model for banana shaped molecule.} \label{langevin:roughShell} |
| 596 |
|
\end{figure} |
| 597 |
|
|
| 598 |
|
\begin{figure} |
| 599 |
|
\centering |
| 600 |
|
\includegraphics[width=\linewidth]{twoBanana.eps} |
| 601 |
< |
\caption[Two Banana Shaped Molecules]{Two Banana Shaped Molecules.} |
| 602 |
< |
\label{langevin:twoBanana} |
| 601 |
> |
\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
| 602 |
> |
256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
| 603 |
> |
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
| 604 |
|
\end{figure} |
| 605 |
|
|
| 606 |
|
\section{Conclusions} |
