| 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 |
| 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} |
| 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} |
| 370 |
|
\end{equation} |
| 371 |
|
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
| 372 |
|
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
| 373 |
< |
$V_i = V_i(v_i,\omega _i)$. The right side of Eq. |
| 373 |
> |
$V_i = V_i(v_i,\omega _i)$. The right side of Eq.~ |
| 374 |
|
(\ref{LDGeneralizedForm}) consists of three generalized forces in |
| 375 |
|
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
| 376 |
|
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
| 404 |
|
\begin{equation} |
| 405 |
|
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
| 406 |
|
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
| 407 |
< |
2k_B T\Xi _R \delta (t - t'). |
| 407 |
> |
2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
| 408 |
|
\end{equation} |
| 409 |
|
|
| 410 |
|
The equation of motion for $v_i$ can be written as |
| 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} |
| 526 |
+ |
|
| 527 |
+ |
As shown in Eq.~\ref{randomForce}, random collisions associated with |
| 528 |
+ |
the solvent's thermal motions is controlled by the external |
| 529 |
+ |
temperature. The capability to maintain the temperature of the whole |
| 530 |
+ |
system was usually used to measure the stability and efficiency of |
| 531 |
+ |
the algorithm. In order to verify the stability of this new |
| 532 |
+ |
algorithm, a series of simulations are performed on system |
| 533 |
+ |
consisiting of 256 SSD water molecules with different viscosities. |
| 534 |
+ |
Initial configuration for the simulations is taken from a 1ns NVT |
| 535 |
+ |
simulation with a cubic box of 19.7166~\AA. All simulation are |
| 536 |
+ |
carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
| 537 |
+ |
with reference temperature at 300~K. Average temperature as a |
| 538 |
+ |
function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
| 539 |
+ |
the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
| 540 |
+ |
1$ poise. The better temperature control at higher viscosity can be |
| 541 |
+ |
explained by the finite size effect and relative slow relaxation |
| 542 |
+ |
rate at lower viscosity regime. |
| 543 |
+ |
\begin{table} |
| 544 |
+ |
\caption{Average temperatures from Langevin dynamics simulations of |
| 545 |
+ |
SSD water molecules with reference temperature at 300~K.} |
| 546 |
+ |
\label{langevin:viscosity} |
| 547 |
+ |
\begin{center} |
| 548 |
+ |
\begin{tabular}{|l|l|l|} |
| 549 |
+ |
\hline |
| 550 |
+ |
$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
| 551 |
+ |
1 & 300.47 & 10.99 \\ |
| 552 |
+ |
0.1 & 301.19 & 11.136 \\ |
| 553 |
+ |
0.01 & 303.04 & 11.796 \\ |
| 554 |
+ |
\hline |
| 555 |
+ |
\end{tabular} |
| 556 |
+ |
\end{center} |
| 557 |
+ |
\end{table} |
| 558 |
+ |
|
| 559 |
+ |
Another set of calculation were performed to study the efficiency of |
| 560 |
+ |
temperature control using different temperature coupling schemes. |
| 561 |
+ |
The starting configuration is cooled to 173~K and evolved using NVE, |
| 562 |
+ |
NVT, and Langevin dynamic with time step of 2 fs. |
| 563 |
+ |
Fig.~\ref{langevin:temperature} shows the heating curve obtained as |
| 564 |
+ |
the systems reach equilibrium. The orange curve in |
| 565 |
+ |
Fig.~\ref{langevin:temperature} represents the simulation using |
| 566 |
+ |
Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
| 567 |
+ |
which gives reasonable tight coupling, while the blue one from |
| 568 |
+ |
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
| 569 |
+ |
scaling to the desire temperature. In extremely lower friction |
| 570 |
+ |
regime (when $ \eta \approx 0$), Langevin dynamics becomes normal |
| 571 |
+ |
NVE (see green curve in Fig.~\ref{langevin:temperature}) which loses |
| 572 |
+ |
the temperature control ability. |
| 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 |
+ |
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 |
+ |
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 |
| 614 |
+ |
\includegraphics[width=\linewidth]{roughShell.eps} |
| 615 |
+ |
\caption[Rough shell model for banana shaped molecule]{Rough shell |
| 616 |
+ |
model for banana shaped molecule.} \label{langevin:roughShell} |
| 617 |
+ |
\end{figure} |
| 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 |
| 634 |
+ |
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
| 635 |
+ |
\end{figure} |
| 636 |
+ |
|
| 637 |
|
\section{Conclusions} |
