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\section{Computational Methods{\label{methodSec}}} |
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|
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\subsection{\label{introSection:frictionTensor}Friction Tensor} |
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Theoretically, the friction kernel can be determined using velocity |
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autocorrelation function. However, this approach become impractical |
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when the system become more and more complicate. Instead, various |
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< |
approaches based on hydrodynamics have been developed to calculate |
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< |
the friction coefficients. The friction effect is isotropic in |
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< |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
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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. |
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> |
Instead, various approaches based on hydrodynamics have been |
| 108 |
> |
developed to calculate the friction coefficients. In general, |
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> |
friction tensor $\Xi$ is a $6\times 6$ matrix given by |
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|
\[ |
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\Xi = \left( {\begin{array}{*{20}c} |
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{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
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|
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\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
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|
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For a spherical particle, the translational and rotational friction |
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< |
constant can be calculated from Stoke's law, |
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> |
For a spherical particle with slip boundary conditions, the |
| 141 |
> |
translational and rotational friction constant can be calculated |
| 142 |
> |
from Stoke's law, |
| 143 |
|
\[ |
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\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
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{6\pi \eta R} & 0 & 0 \\ |
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Other non-spherical shape, such as cylinder and ellipsoid |
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\textit{etc}, are widely used as reference for developing new |
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hydrodynamics theory, because their properties can be calculated |
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exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
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also called a triaxial ellipsoid, which is given in Cartesian |
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coordinates by\cite{Perrin1934, Perrin1936} |
| 164 |
> |
exactly. In 1936, Perrin extended Stokes's law to general |
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> |
ellipsoids, also called a triaxial ellipsoid, which is given in |
| 166 |
> |
Cartesian coordinates by\cite{Perrin1934, Perrin1936} |
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|
\[ |
| 168 |
|
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 169 |
|
}} = 1 |
| 198 |
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\end{array}. |
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|
\] |
| 200 |
|
|
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< |
\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 |
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< |
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 |
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|
triaxial ellipsoid model have been used to approximate the |
| 207 |
|
hydrodynamic properties of rigid bodies. However, since the mapping |
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< |
from all possible ellipsoidal space, $r$-space, to all possible |
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< |
combination of rotational diffusion coefficients, $D$-space is not |
| 208 |
> |
from all possible ellipsoidal spaces, $r$-space, to all possible |
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> |
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 |
< |
|
| 583 |
> |
In order to compare the |
| 584 |
|
|
| 585 |
|
\begin{figure} |
| 586 |
|
\centering |
