| 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} |
| 580 |
|
|
| 581 |
|
\subsection{Langevin Dynamics of Banana Shaped Molecule} |
| 582 |
|
|
| 583 |
+ |
In order to verify that Langevin dynamics can mimic the dynamics of |
| 584 |
+ |
the systems absent of explicit solvents, we carried out two sets of |
| 585 |
+ |
simulations and compare their dynamic properties. |
| 586 |
|
|
| 587 |
+ |
\subsubsection{Simulations Contain One Banana Shaped Molecule} |
| 588 |
|
|
| 589 |
< |
\begin{figure} |
| 590 |
< |
\centering |
| 591 |
< |
\includegraphics[width=\linewidth]{one_banana.eps} |
| 592 |
< |
\caption[]{.} \label{langevin:banana} |
| 593 |
< |
\end{figure} |
| 589 |
> |
Fig.~\ref{langevin:oneBanana} shows a snapshot of the simulation |
| 590 |
> |
made of 256 pentane molecules and one banana shaped molecule at |
| 591 |
> |
273~K. It has an equivalent implicit solvent model containing only |
| 592 |
> |
one banana shaped molecule with viscosity of 0.289 center poise. To |
| 593 |
> |
calculate the hydrodynamic properties of the banana shaped molecule, |
| 594 |
> |
we create a rough shell model (see Fig.~\ref{langevin:roughShell}), |
| 595 |
> |
in which the banana shaped molecule is represented as a ``shell'' |
| 596 |
> |
made of 2266 small identical beads with size of 0.3 $\AA$ on the |
| 597 |
> |
surface. Applying the procedure described in |
| 598 |
> |
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 599 |
> |
identified the center of resistance at $(0, 0.7482, -0.1988)$, as |
| 600 |
> |
well as the resistance tensor, |
| 601 |
> |
\[ |
| 602 |
> |
\left( {\begin{array}{*{20}c} |
| 603 |
> |
0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\ |
| 604 |
> |
3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\ |
| 605 |
> |
-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\ |
| 606 |
> |
5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\ |
| 607 |
> |
0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\ |
| 608 |
> |
0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\ |
| 609 |
> |
\end{array}} \right). |
| 610 |
> |
\] |
| 611 |
|
|
| 612 |
|
\begin{figure} |
| 613 |
|
\centering |
| 618 |
|
|
| 619 |
|
\begin{figure} |
| 620 |
|
\centering |
| 621 |
+ |
\includegraphics[width=\linewidth]{oneBanana.eps} |
| 622 |
+ |
\caption[Snapshot from Simulation of One Banana Shaped Molecules and |
| 623 |
+ |
256 Pentane Molecules]{Snapshot from simulation of one Banana shaped |
| 624 |
+ |
molecules and 256 pentane molecules.} \label{langevin:oneBanana} |
| 625 |
+ |
\end{figure} |
| 626 |
+ |
|
| 627 |
+ |
\subsubsection{Simulations Contain Two Banana Shaped Molecules} |
| 628 |
+ |
|
| 629 |
+ |
\begin{figure} |
| 630 |
+ |
\centering |
| 631 |
|
\includegraphics[width=\linewidth]{twoBanana.eps} |
| 632 |
|
\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
| 633 |
|
256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
