| 1 |
|
|
| 2 |
< |
\chapter{\label{chapt:methodology}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE} |
| 2 |
> |
\chapter{\label{chapt:langevin}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE} |
| 3 |
|
|
| 4 |
|
\section{Introduction} |
| 5 |
|
|
| 28 |
|
between the native and denatured states. Because of its stability |
| 29 |
|
against noise, Langevin dynamics is very suitable for studying |
| 30 |
|
remagnetization processes in various |
| 31 |
< |
systems\cite{Garcia-Palacios1998,Berkov2002,Denisov2003}. For |
| 32 |
< |
instance, the oscillation power spectrum of nanoparticles from |
| 33 |
< |
Langevin dynamics simulation has the same peak frequencies for |
| 34 |
< |
different wave vectors,which recovers the property of magnetic |
| 35 |
< |
excitations in small finite structures\cite{Berkov2005a}. In an |
| 36 |
< |
attempt to reduce the computational cost of simulation, multiple |
| 37 |
< |
time stepping (MTS) methods have been introduced and have been of |
| 38 |
< |
great interest to macromolecule and protein |
| 39 |
< |
community\cite{Tuckerman1992}. Relying on the observation that |
| 40 |
< |
forces between distant atoms generally demonstrate slower |
| 41 |
< |
fluctuations than forces between close atoms, MTS method are |
| 42 |
< |
generally implemented by evaluating the slowly fluctuating forces |
| 43 |
< |
less frequently than the fast ones. Unfortunately, nonlinear |
| 44 |
< |
instability resulting from increasing timestep in MTS simulation |
| 45 |
< |
have became a critical obstruction preventing the long time |
| 46 |
< |
simulation. Due to the coupling to the heat bath, Langevin dynamics |
| 47 |
< |
has been shown to be able to damp out the resonance artifact more |
| 48 |
< |
efficiently\cite{Sandu1999}. |
| 31 |
> |
systems\cite{Palacios1998,Berkov2002,Denisov2003}. For instance, the |
| 32 |
> |
oscillation power spectrum of nanoparticles from Langevin dynamics |
| 33 |
> |
simulation has the same peak frequencies for different wave |
| 34 |
> |
vectors,which recovers the property of magnetic excitations in small |
| 35 |
> |
finite structures\cite{Berkov2005a}. In an attempt to reduce the |
| 36 |
> |
computational cost of simulation, multiple time stepping (MTS) |
| 37 |
> |
methods have been introduced and have been of great interest to |
| 38 |
> |
macromolecule and protein community\cite{Tuckerman1992}. Relying on |
| 39 |
> |
the observation that forces between distant atoms generally |
| 40 |
> |
demonstrate slower fluctuations than forces between close atoms, MTS |
| 41 |
> |
method are generally implemented by evaluating the slowly |
| 42 |
> |
fluctuating forces less frequently than the fast ones. |
| 43 |
> |
Unfortunately, nonlinear instability resulting from increasing |
| 44 |
> |
timestep in MTS simulation have became a critical obstruction |
| 45 |
> |
preventing the long time simulation. Due to the coupling to the heat |
| 46 |
> |
bath, Langevin dynamics has been shown to be able to damp out the |
| 47 |
> |
resonance artifact more efficiently\cite{Sandu1999}. |
| 48 |
|
|
| 49 |
|
%review langevin/browninan dynamics for arbitrarily shaped rigid body |
| 50 |
|
Combining Langevin or Brownian dynamics with rigid body dynamics, |
| 78 |
|
average acceleration is not always true for cooperative motion which |
| 79 |
|
is common in protein motion. An inertial Brownian dynamics (IBD) was |
| 80 |
|
proposed to address this issue by adding an inertial correction |
| 81 |
< |
term\cite{Beard2001}. As a complement to IBD which has a lower bound |
| 81 |
> |
term\cite{Beard2000}. As a complement to IBD which has a lower bound |
| 82 |
|
in time step because of the inertial relaxation time, long-time-step |
| 83 |
|
inertial dynamics (LTID) can be used to investigate the inertial |
| 84 |
|
behavior of the polymer segments in low friction |
| 85 |
< |
regime\cite{Beard2001}. LTID can also deal with the rotational |
| 85 |
> |
regime\cite{Beard2000}. LTID can also deal with the rotational |
| 86 |
|
dynamics for nonskew bodies without translation-rotation coupling by |
| 87 |
|
separating the translation and rotation motion and taking advantage |
| 88 |
|
of the analytical solution of hydrodynamics properties. However, |
| 97 |
|
estimation of friction tensor from hydrodynamics theory into the |
| 98 |
|
sophisticated rigid body dynamics. |
| 99 |
|
|
| 100 |
< |
\section{Computational methods{\label{methodSec}}} |
| 100 |
> |
\section{Computational Methods{\label{methodSec}}} |
| 101 |
|
|
| 102 |
|
\subsection{\label{introSection:frictionTensor}Friction Tensor} |
| 103 |
< |
Theoretically, the friction kernel can be determined using velocity |
| 104 |
< |
autocorrelation function. However, this approach become impractical |
| 105 |
< |
when the system become more and more complicate. Instead, various |
| 106 |
< |
approaches based on hydrodynamics have been developed to calculate |
| 107 |
< |
the friction coefficients. The friction effect is isotropic in |
| 108 |
< |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
| 110 |
< |
tensor $\Xi$ is a $6\times 6$ matrix given by |
| 103 |
> |
Theoretically, the friction kernel can be determined using the |
| 104 |
> |
velocity autocorrelation function. However, this approach become |
| 105 |
> |
impractical when the system become more and more complicate. |
| 106 |
> |
Instead, various approaches based on hydrodynamics have been |
| 107 |
> |
developed to calculate the friction coefficients. In general, |
| 108 |
> |
friction tensor $\Xi$ is a $6\times 6$ matrix given by |
| 109 |
|
\[ |
| 110 |
|
\Xi = \left( {\begin{array}{*{20}c} |
| 111 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 136 |
|
|
| 137 |
|
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
| 138 |
|
|
| 139 |
< |
For a spherical particle, the translational and rotational friction |
| 140 |
< |
constant can be calculated from Stoke's law, |
| 139 |
> |
For a spherical particle with slip boundary conditions, the |
| 140 |
> |
translational and rotational friction constant can be calculated |
| 141 |
> |
from Stoke's law, |
| 142 |
|
\[ |
| 143 |
|
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 144 |
|
{6\pi \eta R} & 0 & 0 \\ |
| 160 |
|
Other non-spherical shape, such as cylinder and ellipsoid |
| 161 |
|
\textit{etc}, are widely used as reference for developing new |
| 162 |
|
hydrodynamics theory, because their properties can be calculated |
| 163 |
< |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
| 164 |
< |
also called a triaxial ellipsoid, which is given in Cartesian |
| 165 |
< |
coordinates by\cite{Perrin1934, Perrin1936} |
| 163 |
> |
exactly. In 1936, Perrin extended Stokes's law to general |
| 164 |
> |
ellipsoids, also called a triaxial ellipsoid, which is given in |
| 165 |
> |
Cartesian coordinates by\cite{Perrin1934, Perrin1936} |
| 166 |
|
\[ |
| 167 |
|
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 168 |
|
}} = 1 |
| 171 |
|
due to the complexity of the elliptic integral, only the ellipsoid |
| 172 |
|
with the restriction of two axes having to be equal, \textit{i.e.} |
| 173 |
|
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 174 |
< |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
| 174 |
> |
exactly. Introducing an elliptic integral parameter $S$ for prolate |
| 175 |
> |
: |
| 176 |
|
\[ |
| 177 |
|
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 178 |
|
} }}{b}, |
| 179 |
|
\] |
| 180 |
< |
and oblate, |
| 180 |
> |
and oblate : |
| 181 |
|
\[ |
| 182 |
|
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 183 |
|
}}{a} |
| 186 |
|
tensors |
| 187 |
|
\[ |
| 188 |
|
\begin{array}{l} |
| 189 |
< |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 190 |
< |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 191 |
< |
\end{array}, |
| 189 |
> |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 190 |
> |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 191 |
> |
2a}}, |
| 192 |
> |
\end{array} |
| 193 |
|
\] |
| 194 |
|
and |
| 195 |
|
\[ |
| 196 |
|
\begin{array}{l} |
| 197 |
< |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 197 |
> |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
| 198 |
|
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 199 |
|
\end{array}. |
| 200 |
|
\] |
| 201 |
|
|
| 202 |
< |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}} |
| 202 |
> |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
| 203 |
|
|
| 204 |
< |
Unlike spherical and other regular shaped molecules, there is not |
| 205 |
< |
analytical solution for friction tensor of any arbitrary shaped |
| 204 |
> |
Unlike spherical and other simply shaped molecules, there is no |
| 205 |
> |
analytical solution for the friction tensor for arbitrarily shaped |
| 206 |
|
rigid molecules. The ellipsoid of revolution model and general |
| 207 |
|
triaxial ellipsoid model have been used to approximate the |
| 208 |
|
hydrodynamic properties of rigid bodies. However, since the mapping |
| 209 |
< |
from all possible ellipsoidal space, $r$-space, to all possible |
| 210 |
< |
combination of rotational diffusion coefficients, $D$-space is not |
| 209 |
> |
from all possible ellipsoidal spaces, $r$-space, to all possible |
| 210 |
> |
combination of rotational diffusion coefficients, $D$-space, is not |
| 211 |
|
unique\cite{Wegener1979} as well as the intrinsic coupling between |
| 212 |
< |
translational and rotational motion of rigid body, general ellipsoid |
| 213 |
< |
is not always suitable for modeling arbitrarily shaped rigid |
| 214 |
< |
molecule. A number of studies have been devoted to determine the |
| 215 |
< |
friction tensor for irregularly shaped rigid bodies using more |
| 212 |
> |
translational and rotational motion of rigid body, general |
| 213 |
> |
ellipsoids are not always suitable for modeling arbitrarily shaped |
| 214 |
> |
rigid molecule. A number of studies have been devoted to determine |
| 215 |
> |
the friction tensor for irregularly shaped rigid bodies using more |
| 216 |
|
advanced method where the molecule of interest was modeled by |
| 217 |
|
combinations of spheres(beads)\cite{Carrasco1999} and the |
| 218 |
|
hydrodynamics properties of the molecule can be calculated using the |
| 274 |
|
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 275 |
|
)T_{ij} |
| 276 |
|
\] |
| 277 |
< |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
| 278 |
< |
$B$, we obtain |
| 277 |
> |
where $\delta _{ij}$ is Kronecker delta function. Inverting the $B$ |
| 278 |
> |
matrix, we obtain |
| 279 |
|
|
| 280 |
|
\[ |
| 281 |
|
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 307 |
|
|
| 308 |
|
The resistance tensor depends on the origin to which they refer. The |
| 309 |
|
proper location for applying friction force is the center of |
| 310 |
< |
resistance (reaction), at which the trace of rotational resistance |
| 311 |
< |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
| 312 |
< |
resistance is defined as an unique point of the rigid body at which |
| 313 |
< |
the translation-rotation coupling tensor are symmetric, |
| 310 |
> |
resistance (or center of reaction), at which the trace of rotational |
| 311 |
> |
resistance tensor, $ \Xi ^{rr}$ reaches a minimum value. |
| 312 |
> |
Mathematically, the center of resistance is defined as an unique |
| 313 |
> |
point of the rigid body at which the translation-rotation coupling |
| 314 |
> |
tensor are symmetric, |
| 315 |
|
\begin{equation} |
| 316 |
|
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 317 |
|
\label{introEquation:definitionCR} |
| 318 |
|
\end{equation} |
| 319 |
< |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 319 |
> |
From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 320 |
|
we can easily find out that the translational resistance tensor is |
| 321 |
|
origin independent, while the rotational resistance tensor and |
| 322 |
|
translation-rotation coupling resistance tensor depend on the |
| 323 |
< |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 324 |
< |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 323 |
> |
origin. Given the resistance tensor at an arbitrary origin $O$, and |
| 324 |
> |
a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 325 |
|
obtain the resistance tensor at $P$ by |
| 326 |
|
\begin{equation} |
| 327 |
|
\begin{array}{l} |
| 362 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 363 |
|
joining center of resistance $R$ and origin $O$. |
| 364 |
|
|
| 365 |
< |
\subsection{Langevin dynamics for rigid particles of arbitrary shape\label{LDRB}} |
| 365 |
> |
\subsection{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
| 366 |
|
|
| 367 |
|
Consider a Langevin equation of motions in generalized coordinates |
| 368 |
|
\begin{equation} |
| 405 |
|
\begin{equation} |
| 406 |
|
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
| 407 |
|
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
| 408 |
< |
2k_B T\Xi _R \delta (t - t'). |
| 408 |
> |
2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
| 409 |
|
\end{equation} |
| 410 |
|
|
| 411 |
|
The equation of motion for $v_i$ can be written as |
| 517 |
|
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
| 518 |
|
\end{align*} |
| 519 |
|
|
| 520 |
< |
\section{Results and discussion} |
| 520 |
> |
\section{Results and Discussion} |
| 521 |
|
|
| 522 |
< |
The Langevin algorithm described in Sec.~\ref{LDRB} has been |
| 523 |
< |
implemented in {\sc oopse}\cite{Meineke2005} and applied to several |
| 524 |
< |
test systems. |
| 523 |
< |
|
| 524 |
< |
\subsection{Langevin dynamics of} |
| 522 |
> |
The Langevin algorithm described in previous section has been |
| 523 |
> |
implemented in {\sc oopse}\cite{Meineke2005} and applied to the |
| 524 |
> |
studies of kinetics and thermodynamic properties in several systems. |
| 525 |
|
|
| 526 |
+ |
\subsection{Temperature Control} |
| 527 |
+ |
|
| 528 |
+ |
As shown in Eq.~\ref{randomForce}, random collisions associated with |
| 529 |
+ |
the solvent's thermal motions is controlled by the external |
| 530 |
+ |
temperature. The capability to maintain the temperature of the whole |
| 531 |
+ |
system was usually used to measure the stability and efficiency of |
| 532 |
+ |
the algorithm. In order to verify the stability of this new |
| 533 |
+ |
algorithm, a series of simulations are performed on system |
| 534 |
+ |
consisiting of 256 SSD water molecules with different viscosities. |
| 535 |
+ |
Initial configuration for the simulations is taken from a 1ns NVT |
| 536 |
+ |
simulation with a cubic box of 19.7166~\AA. All simulation are |
| 537 |
+ |
carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
| 538 |
+ |
with reference temperature at 300~K. Average temperature as a |
| 539 |
+ |
function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
| 540 |
+ |
the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
| 541 |
+ |
1$ poise. The better temperature control at higher viscosity can be |
| 542 |
+ |
explained by the finite size effect and relative slow relaxation |
| 543 |
+ |
rate at lower viscosity regime. |
| 544 |
+ |
\begin{table} |
| 545 |
+ |
\caption{AVERAGE TEMPERATURES FROM LANGEVIN DYNAMICS SIMULATIONS OF |
| 546 |
+ |
SSD WATER MOLECULES WITH REFERENCE TEMPERATURE AT 300~K.} |
| 547 |
+ |
\label{langevin:viscosity} |
| 548 |
+ |
\begin{center} |
| 549 |
+ |
\begin{tabular}{|l|l|l|} |
| 550 |
+ |
\hline |
| 551 |
+ |
$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
| 552 |
+ |
1 & 300.47 & 10.99 \\ |
| 553 |
+ |
0.1 & 301.19 & 11.136 \\ |
| 554 |
+ |
0.01 & 303.04 & 11.796 \\ |
| 555 |
+ |
\hline |
| 556 |
+ |
\end{tabular} |
| 557 |
+ |
\end{center} |
| 558 |
+ |
\end{table} |
| 559 |
+ |
|
| 560 |
+ |
Another set of calculation were performed to study the efficiency of |
| 561 |
+ |
temperature control using different temperature coupling schemes. |
| 562 |
+ |
The starting configuration is cooled to 173~K and evolved using NVE, |
| 563 |
+ |
NVT, and Langevin dynamic with time step of 2 fs. |
| 564 |
+ |
Fig.~\ref{langevin:temperature} shows the heating curve obtained as |
| 565 |
+ |
the systems reach equilibrium. The orange curve in |
| 566 |
+ |
Fig.~\ref{langevin:temperature} represents the simulation using |
| 567 |
+ |
Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
| 568 |
+ |
which gives reasonable tight coupling, while the blue one from |
| 569 |
+ |
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
| 570 |
+ |
scaling to the desire temperature. In extremely lower friction |
| 571 |
+ |
regime (when $ \eta \approx 0$), Langevin dynamics becomes normal |
| 572 |
+ |
NVE (see green curve in Fig.~\ref{langevin:temperature}) which loses |
| 573 |
+ |
the temperature control ability. |
| 574 |
+ |
|
| 575 |
|
\begin{figure} |
| 576 |
|
\centering |
| 577 |
|
\includegraphics[width=\linewidth]{temperature.eps} |
| 578 |
< |
\caption[]{.} \label{langevin:temperature} |
| 578 |
> |
\caption[Plot of Temperature Fluctuation Versus Time]{Plot of |
| 579 |
> |
temperature fluctuation versus time.} \label{langevin:temperature} |
| 580 |
|
\end{figure} |
| 581 |
|
|
| 582 |
< |
\subsection{LD of banana-shaped molecule} |
| 582 |
> |
\subsection{Langevin Dynamics of Banana Shaped Molecules} |
| 583 |
|
|
| 584 |
+ |
In order to verify that Langevin dynamics can mimic the dynamics of |
| 585 |
+ |
the systems absent of explicit solvents, we carried out two sets of |
| 586 |
+ |
simulations and compare their dynamic properties. |
| 587 |
+ |
Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
| 588 |
+ |
made of 256 pentane molecules and two banana shaped molecules at |
| 589 |
+ |
273~K. It has an equivalent implicit solvent system containing only |
| 590 |
+ |
two banana shaped molecules with viscosity of 0.289 center poise. To |
| 591 |
+ |
calculate the hydrodynamic properties of the banana shaped molecule, |
| 592 |
+ |
we create a rough shell model (see Fig.~\ref{langevin:roughShell}), |
| 593 |
+ |
in which the banana shaped molecule is represented as a ``shell'' |
| 594 |
+ |
made of 2266 small identical beads with size of 0.3 \AA on the |
| 595 |
+ |
surface. Applying the procedure described in |
| 596 |
+ |
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 597 |
+ |
identified the center of resistance at $(0, 0.7482, -0.1988)$, as |
| 598 |
+ |
well as the resistance tensor, |
| 599 |
+ |
\[ |
| 600 |
+ |
\left( {\begin{array}{*{20}c} |
| 601 |
+ |
0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\ |
| 602 |
+ |
3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\ |
| 603 |
+ |
-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\ |
| 604 |
+ |
5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\ |
| 605 |
+ |
0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\ |
| 606 |
+ |
0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\ |
| 607 |
+ |
\end{array}} \right). |
| 608 |
+ |
\] |
| 609 |
+ |
Curves of velocity auto-correlation functions in |
| 610 |
+ |
Fig.~\ref{langevin:vacf} were shown to match each other very well. |
| 611 |
+ |
However, because of the stochastic nature, simulation using Langevin |
| 612 |
+ |
dynamics was shown to decay slightly fast. In order to study the |
| 613 |
+ |
rotational motion of the molecules, we also calculated the auto- |
| 614 |
+ |
correlation function of the principle axis of the second GB |
| 615 |
+ |
particle, $u$. |
| 616 |
+ |
|
| 617 |
|
\begin{figure} |
| 618 |
|
\centering |
| 619 |
< |
\includegraphics[width=\linewidth]{one_banana.eps} |
| 620 |
< |
\caption[]{.} \label{langevin:banana} |
| 619 |
> |
\includegraphics[width=\linewidth]{roughShell.eps} |
| 620 |
> |
\caption[Rough shell model for banana shaped molecule]{Rough shell |
| 621 |
> |
model for banana shaped molecule.} \label{langevin:roughShell} |
| 622 |
|
\end{figure} |
| 623 |
|
|
| 624 |
+ |
\begin{figure} |
| 625 |
+ |
\centering |
| 626 |
+ |
\includegraphics[width=\linewidth]{twoBanana.eps} |
| 627 |
+ |
\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
| 628 |
+ |
256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
| 629 |
+ |
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
| 630 |
+ |
\end{figure} |
| 631 |
+ |
|
| 632 |
+ |
\begin{figure} |
| 633 |
+ |
\centering |
| 634 |
+ |
\includegraphics[width=\linewidth]{vacf.eps} |
| 635 |
+ |
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
| 636 |
+ |
auto-correlation functions in NVE (blue) and Langevin dynamics |
| 637 |
+ |
(red).} \label{langevin:vacf} |
| 638 |
+ |
\end{figure} |
| 639 |
+ |
|
| 640 |
+ |
\begin{figure} |
| 641 |
+ |
\centering |
| 642 |
+ |
\includegraphics[width=\linewidth]{uacf.eps} |
| 643 |
+ |
\caption[Auto-correlation functions of the principle axis of the |
| 644 |
+ |
middle GB particle]{Auto-correlation functions of the principle axis |
| 645 |
+ |
of the middle GB particle in NVE (blue) and Langevin dynamics |
| 646 |
+ |
(red).} \label{langevin:twoBanana} |
| 647 |
+ |
\end{figure} |
| 648 |
+ |
|
| 649 |
|
\section{Conclusions} |
| 650 |
+ |
|
| 651 |
+ |
We have presented a new Langevin algorithm by incorporating the |
| 652 |
+ |
hydrodynamics properties of arbitrary shaped molecules into an |
| 653 |
+ |
advanced symplectic integration scheme. The temperature control |
| 654 |
+ |
ability of this algorithm was demonstrated by a set of simulations |
| 655 |
+ |
with different viscosities. It was also shown to have significant |
| 656 |
+ |
advantage of producing rapid thermal equilibration over |
| 657 |
+ |
Nos\'{e}-Hoover method. Further studies in systems involving banana |
| 658 |
+ |
shaped molecules illustrated that the dynamic properties could be |
| 659 |
+ |
preserved by using this new algorithm as an implicit solvent model. |
