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\begin{document} |
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\title{An algorithm for performing Langevin dynamics on rigid bodies of arbitrary shape } |
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|
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\author{Teng Lin, Xiuquan Sun and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: |
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\author{Xiuquan Sun, Teng Lin and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: |
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gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry\\ |
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University of Notre Dame\\ |
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+ \frac{h}{2} {\bf \tau}^b(t + h) . |
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\end{align*} |
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|
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\section{Results and Discussion} |
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|
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The Langevin algorithm described in previous section has been |
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implemented in {\sc oopse}\cite{Meineke2005} and applied to studies |
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of the static and dynamic properties in several systems. |
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|
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\subsection{Temperature Control} |
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|
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As shown in Eq.~\ref{randomForce}, random collisions associated with |
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the solvent's thermal motions is controlled by the external |
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temperature. The capability to maintain the temperature of the whole |
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system was usually used to measure the stability and efficiency of |
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the algorithm. In order to verify the stability of this new |
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algorithm, a series of simulations are performed on system |
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consisiting of 256 SSD water molecules with different viscosities. |
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The initial configuration for the simulations is taken from a 1ns |
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NVT simulation with a cubic box of 19.7166~\AA. All simulation are |
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carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
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with reference temperature at 300~K. The average temperature as a |
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function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
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the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
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1$ poise. The better temperature control at higher viscosity can be |
| 599 |
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explained by the finite size effect and relative slow relaxation |
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rate at lower viscosity regime. |
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\begin{table} |
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\caption{AVERAGE TEMPERATURES FROM LANGEVIN DYNAMICS SIMULATIONS OF |
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SSD WATER MOLECULES WITH REFERENCE TEMPERATURE AT 300~K.} |
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\label{langevin:viscosity} |
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\begin{center} |
| 606 |
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\begin{tabular}{lll} |
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\hline |
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$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
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\hline |
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1 & 300.47 & 10.99 \\ |
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0.1 & 301.19 & 11.136 \\ |
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0.01 & 303.04 & 11.796 \\ |
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< |
\hline |
| 577 |
> |
\section{Results} |
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> |
In order to validate our Langevin integrator for arbitrarily-shaped |
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rigid bodies, we implemented the algorithm in {\sc |
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oopse}\cite{Meineke2005} and compared the results of this algorithm |
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with the known |
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hydrodynamic limiting behavior for a few model systems, and to |
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microcanonical molecular dynamics simulations for some more |
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complicated bodies. The model systems and their analytical behavior |
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> |
(if known) are summarized below. Parameters for the primary particles |
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> |
comprising our model systems are given in table \ref{tab:parameters}, |
| 587 |
> |
and a sketch of the arrangement of these primary particles into the |
| 588 |
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model rigid bodies is shown in figure \ref{fig:models}. In table |
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\ref{tab:parameters}, $d$ and $l$ are the physical dimensions of |
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ellipsoidal (Gay-Berne) particles. For spherical particles, the value |
| 591 |
> |
of the Lennard-Jones $\sigma$ parameter is the particle diameter |
| 592 |
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($d$). Gay-Berne ellipsoids have an energy scaling parameter, |
| 593 |
> |
$\epsilon^s$, which describes the well depth for two identical |
| 594 |
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ellipsoids in a {\it side-by-side} configuration. Additionally, a |
| 595 |
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well depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, |
| 596 |
> |
describes the ratio between the well depths in the {\it end-to-end} |
| 597 |
> |
and side-by-side configurations. For spheres, $\epsilon^r \equiv 1$. |
| 598 |
> |
Moments of inertia are also required to describe the motion of primary |
| 599 |
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particles with orientational degrees of freedom. |
| 600 |
> |
|
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\begin{table*} |
| 602 |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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\caption{Parameters for the primary particles in use by the rigid body |
| 605 |
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models in figure \ref{fig:models}.} |
| 606 |
> |
\begin{tabular}{lrcccccccc} |
| 607 |
> |
\hline |
| 608 |
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& & & & & & & \multicolumn{3}c{$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$)} \\ |
| 609 |
> |
& & $d$ (\AA) & $l$ (\AA) & $\epsilon^s$ (kcal/mol) & $\epsilon^r$ & |
| 610 |
> |
$m$ (amu) & $I_{xx}$ & $I_{yy}$ & $I_{zz}$ \\ \hline |
| 611 |
> |
Sphere & & 6.5 & $= d$ & 0.8 & 1 & 190 & 802.75 & 802.75 & 802.75 \\ |
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> |
Ellipsoid & & 4.6 & 13.8 & 0.8 & 0.2 & 200 & 2105 & 2105 & 421 \\ |
| 613 |
> |
Dumbbell &(2 identical spheres) & 6.5 & $= d$ & 0.8 & 1 & 190 & 802.75 & 802.75 & 802.75 \\ |
| 614 |
> |
Banana &(3 identical ellipsoids)& 4.2 & 11.2 & 0.8 & 0.2 & 240 & 10000 & 10000 & 0 \\ |
| 615 |
> |
Lipid: & Spherical Head & 6.5 & $= d$ & 0.185 & 1 & 196 & & & \\ |
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> |
& Ellipsoidal Tail & 4.6 & 13.8 & 0.8 & 0.2 & 760 & 45000 & 45000 & 9000 \\ |
| 617 |
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Solvent & & 4.7 & $= d$ & 0.8 & 1 & 72.06 & & & \\ |
| 618 |
> |
\hline |
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|
\end{tabular} |
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+ |
\label{tab:parameters} |
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|
\end{center} |
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< |
\end{table} |
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|
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Another set of calculations were performed to study the efficiency of |
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temperature control using different temperature coupling schemes. |
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The starting configuration is cooled to 173~K and evolved using NVE, |
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< |
NVT, and Langevin dynamic with time step of 2 fs. |
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< |
Fig.~\ref{langevin:temperature} shows the heating curve obtained as |
| 623 |
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the systems reach equilibrium. The orange curve in |
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Fig.~\ref{langevin:temperature} represents the simulation using |
| 625 |
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Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
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< |
which gives reasonable tight coupling, while the blue one from |
| 627 |
< |
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
| 628 |
< |
scaling to the desire temperature. When $ \eta = 0$, Langevin dynamics becomes normal |
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< |
NVE (see orange curve in Fig.~\ref{langevin:temperature}) which |
| 630 |
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loses the temperature control ability. |
| 622 |
> |
\end{minipage} |
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> |
\end{table*} |
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|
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|
\begin{figure} |
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|
\centering |
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\includegraphics[width=\linewidth]{temperature.pdf} |
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\caption[Plot of Temperature Fluctuation Versus Time]{Plot of |
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< |
temperature fluctuation versus time.} \label{langevin:temperature} |
| 627 |
> |
\includegraphics[width=3in]{sketch} |
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\caption[Sketch of the model systems]{A sketch of the model systems |
| 629 |
> |
used in evaluating the behavior of the rigid body Langevin |
| 630 |
> |
integrator.} \label{fig:models} |
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|
\end{figure} |
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|
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\subsection{Langevin Dynamics simulation {\it vs} NVE simulations} |
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\subsection{Simulation Methodology} |
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|
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< |
To validate our langevin dynamics simulation. We performed several NVE |
| 636 |
< |
simulations with explicit solvents for different shaped |
| 637 |
< |
molecules. There are one solute molecule and 1929 solvent molecules in |
| 638 |
< |
NVE simulation. The parameters are shown in table |
| 639 |
< |
\ref{tab:parameters}. The force field between spheres is standard |
| 640 |
< |
Lennard-Jones, and ellipsoids interact with other ellipsoids and |
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< |
spheres with generalized Gay-Berne potential. All simulations are |
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< |
carried out at 300 K and 1 Atm. The time step is 25 ns, and a |
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< |
switching function was applied to all potentials to smoothly turn off |
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< |
the interactions between a range of $22$ and $25$ \AA. The switching |
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function was the standard (cubic) function, |
| 635 |
> |
We performed reference microcanonical simulations with explicit |
| 636 |
> |
solvents for each of the different model system. In each case there |
| 637 |
> |
was one solute model and 1929 solvent molecules present in the |
| 638 |
> |
simulation box. All simulations were equilibrated using a |
| 639 |
> |
constant-pressure and temperature integrator with target values of 300 |
| 640 |
> |
K for the temperature and 1 atm for pressure. Following this stage, |
| 641 |
> |
further equilibration and sampling was done in a microcanonical |
| 642 |
> |
ensemble. Since the model bodies are typically quite massive, we were |
| 643 |
> |
able to use a time step of 25 fs. A switching function was applied to |
| 644 |
> |
all potentials to smoothly turn off the interactions between a range |
| 645 |
> |
of $22$ and $25$ \AA. The switching function was the standard (cubic) |
| 646 |
> |
function, |
| 647 |
|
\begin{equation} |
| 648 |
|
s(r) = |
| 649 |
|
\begin{cases} |
| 655 |
|
\end{cases} |
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|
\label{eq:switchingFunc} |
| 657 |
|
\end{equation} |
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< |
We have computed translational diffusion constants for lipid molecules |
| 659 |
< |
from the mean-square displacement, |
| 658 |
> |
To measure shear viscosities from our microcanonical simulations, we |
| 659 |
> |
used the Einstein form of the pressure correlation function,\cite{hess:209} |
| 660 |
|
\begin{equation} |
| 661 |
< |
D = \lim_{t\rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
| 661 |
> |
\eta = \lim_{t->\infty} \frac{V}{2 k_B T} \frac{d}{dt} \langle \left( |
| 662 |
> |
\int_{t_0}^{t_0 + t} P_{xz}(t') dt' \right)^2 \rangle_{t_0}. |
| 663 |
> |
\label{eq:shear} |
| 664 |
|
\end{equation} |
| 665 |
< |
of the solute molecules. Translational diffusion constants for the |
| 669 |
< |
different shaped molecules are shown in table |
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< |
\ref{tab:translation}. We have also computed orientational correlation |
| 671 |
< |
times for different shaped molecules from fits of the second-order |
| 672 |
< |
Legendre polynomial correlation function, |
| 665 |
> |
A similar form exists for the bulk viscosity |
| 666 |
|
\begin{equation} |
| 667 |
< |
C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
| 668 |
< |
\mu}_{i}(0) \right) |
| 667 |
> |
\kappa = \lim_{t->\infty} \frac{V}{2 k_B T} \frac{d}{dt} \langle \left( |
| 668 |
> |
\int_{t_0}^{t_0 + t} |
| 669 |
> |
\left(P\left(t'\right)-\langle P \rangle \right)dt' |
| 670 |
> |
\right)^2 \rangle_{t_0}. |
| 671 |
|
\end{equation} |
| 672 |
< |
the results are shown in table \ref{tab:rotation}. We used einstein |
| 673 |
< |
format of the pressure correlation function, |
| 672 |
> |
Alternatively, the shear viscosity can also be calculated using a |
| 673 |
> |
Green-Kubo formula with the off-diagonal pressure tensor correlation function, |
| 674 |
|
\begin{equation} |
| 675 |
< |
C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
| 676 |
< |
\mu}_{i}(0) \right) |
| 675 |
> |
\eta = \frac{V}{k_B T} \int_0^{\infty} \langle P_{xz}(t_0) P_{xz}(t_0 |
| 676 |
> |
+ t) \rangle_{t_0} dt, |
| 677 |
|
\end{equation} |
| 678 |
< |
to estimate the viscosity of the systems from NVE simulations. The |
| 679 |
< |
viscosity can also be calculated by Green-Kubo pressure correlaton |
| 680 |
< |
function, |
| 681 |
< |
\begin{equation} |
| 682 |
< |
C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
| 683 |
< |
\mu}_{i}(0) \right) |
| 678 |
> |
although this method converges extremely slowly and is not practical |
| 679 |
> |
for obtaining viscosities from molecular dynamics simulations. |
| 680 |
> |
|
| 681 |
> |
The Langevin dynamics for the different model systems were performed |
| 682 |
> |
at the same temperature as the average temperature of the |
| 683 |
> |
microcanonical simulations and with a solvent viscosity taken from |
| 684 |
> |
Eq. (\ref{eq:shear}) applied to these simulations. We used 1024 |
| 685 |
> |
independent solute simulations to obtain statistics on our Langevin |
| 686 |
> |
integrator. |
| 687 |
> |
|
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> |
\subsection{Analysis} |
| 689 |
> |
|
| 690 |
> |
The quantities of interest when comparing the Langevin integrator to |
| 691 |
> |
analytic hydrodynamic equations and to molecular dynamics simulations |
| 692 |
> |
are typically translational diffusion constants and orientational |
| 693 |
> |
relaxation times. Translational diffusion constants for point |
| 694 |
> |
particles are computed easily from the long-time slope of the |
| 695 |
> |
mean-square displacement, |
| 696 |
> |
\begin{equation} |
| 697 |
> |
D = \lim_{t\rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
| 698 |
|
\end{equation} |
| 699 |
< |
However, this method converges slowly, and the statistics are not good |
| 700 |
< |
enough to give us a very accurate value. The langevin dynamics |
| 701 |
< |
simulations for different shaped molecules are performed at the same |
| 702 |
< |
conditions as the NVE simulations with viscosity estimated from NVE |
| 703 |
< |
simulations. To get better statistics, 1024 non-interacting solute |
| 704 |
< |
molecules are put into one simulation box for each langevin |
| 705 |
< |
simulation, this is equal to 1024 simulations for single solute |
| 706 |
< |
systems. The diffusion constants and rotation relaxation times for |
| 699 |
> |
of the solute molecules. For models in which the translational |
| 700 |
> |
diffusion tensor (${\bf D}_{tt}$) has non-degenerate eigenvalues |
| 701 |
> |
(i.e. any non-spherically-symmetric rigid body), it is possible to |
| 702 |
> |
compute the diffusive behavior for motion parallel to each body-fixed |
| 703 |
> |
axis by projecting the displacement of the particle onto the |
| 704 |
> |
body-fixed reference frame at $t=0$. With an isotropic solvent, as we |
| 705 |
> |
have used in this study, there are differences between the three |
| 706 |
> |
diffusion constants, but these must converge to the same value at |
| 707 |
> |
longer times. Translational diffusion constants for the different |
| 708 |
> |
shaped models are shown in table \ref{tab:translation}. |
| 709 |
> |
|
| 710 |
> |
In general, the three eigenvalues ($D_1, D_2, D_3$) of the rotational |
| 711 |
> |
diffusion tensor (${\bf D}_{rr}$) measure the diffusion of an object |
| 712 |
> |
{\it around} a particular body-fixed axis and {\it not} the diffusion |
| 713 |
> |
of a vector pointing along the axis. However, these eigenvalues can |
| 714 |
> |
be combined to find 5 characteristic rotational relaxation |
| 715 |
> |
times,\cite{PhysRev.119.53,Berne90} |
| 716 |
> |
\begin{eqnarray} |
| 717 |
> |
1 / \tau_1 & = & 6 D_r + 2 \Delta \\ |
| 718 |
> |
1 / \tau_2 & = & 6 D_r - 2 \Delta \\ |
| 719 |
> |
1 / \tau_3 & = & 3 (D_r + D_1) \\ |
| 720 |
> |
1 / \tau_4 & = & 3 (D_r + D_2) \\ |
| 721 |
> |
1 / \tau_5 & = & 3 (D_r + D_3) |
| 722 |
> |
\end{eqnarray} |
| 723 |
> |
where |
| 724 |
> |
\begin{equation} |
| 725 |
> |
D_r = \frac{1}{3} \left(D_1 + D_2 + D_3 \right) |
| 726 |
> |
\end{equation} |
| 727 |
> |
and |
| 728 |
> |
\begin{equation} |
| 729 |
> |
\Delta = \left( (D_1 - D_2)^2 + (D_3 - D_1 )(D_3 - D_2)\right)^{1/2} |
| 730 |
> |
\end{equation} |
| 731 |
> |
Each of these characteristic times can be used to predict the decay of |
| 732 |
> |
part of the rotational correlation function when $\ell = 2$, |
| 733 |
> |
\begin{equation} |
| 734 |
> |
C_2(t) = \frac{a^2}{N^2} e^{-t/\tau_1} + \frac{b^2}{N^2} e^{-t/\tau_2}. |
| 735 |
> |
\end{equation} |
| 736 |
> |
This is the same as the $F^2_{0,0}(t)$ correlation function that |
| 737 |
> |
appears in Ref. \citen{Berne90}. The amplitudes of the two decay |
| 738 |
> |
terms are expressed in terms of three dimensionless functions of the |
| 739 |
> |
eigenvalues: $a = \sqrt{3} (D_1 - D_2)$, $b = (2D_3 - D_1 - D_2 + |
| 740 |
> |
2\Delta)$, and $N = 2 \sqrt{\Delta b}$. Similar expressions can be |
| 741 |
> |
obtained for other angular momentum correlation |
| 742 |
> |
functions.\cite{PhysRev.119.53,Berne90} In all of the model systems we |
| 743 |
> |
studied, only one of the amplitudes of the two decay terms was |
| 744 |
> |
non-zero, so it was possible to derive a single relaxation time for |
| 745 |
> |
each of the hydrodynamic tensors. In many cases, these characteristic |
| 746 |
> |
times are averaged and reported in the literature as a single relaxation |
| 747 |
> |
time,\cite{Garcia-de-la-Torre:1997qy} |
| 748 |
> |
\begin{equation} |
| 749 |
> |
1 / \tau_0 = \frac{1}{5} \sum_{i=1}^5 \tau_{i}^{-1}, |
| 750 |
> |
\end{equation} |
| 751 |
> |
although for the cases reported here, this averaging is not necessary |
| 752 |
> |
and only one of the five relaxation times is relevant. |
| 753 |
> |
|
| 754 |
> |
To test the Langevin integrator's behavior for rotational relaxation, |
| 755 |
> |
we have compared the analytical orientational relaxation times (if |
| 756 |
> |
they are known) with the general result from the diffusion tensor and |
| 757 |
> |
with the results from both the explicitly solvated molecular dynamics |
| 758 |
> |
and Langevin simulations. Relaxation times from simulations (both |
| 759 |
> |
microcanonical and Langevin), were computed using Legendre polynomial |
| 760 |
> |
correlation functions for a unit vector (${\bf u}$) fixed along one or |
| 761 |
> |
more of the body-fixed axes of the model. |
| 762 |
> |
\begin{equation} |
| 763 |
> |
C_{\ell}(t) = \langle P_{\ell}\left({\bf u}_{i}(t) \cdot {\bf |
| 764 |
> |
u}_{i}(0) \right) |
| 765 |
> |
\end{equation} |
| 766 |
> |
For simulations in the high-friction limit, orientational correlation |
| 767 |
> |
times can then be obtained from exponential fits of this function, or by |
| 768 |
> |
integrating, |
| 769 |
> |
\begin{equation} |
| 770 |
> |
\tau = \ell (\ell + 1) \int_0^{\infty} C_{\ell}(t) dt. |
| 771 |
> |
\end{equation} |
| 772 |
> |
In lower-friction solvents, the Legendre correlation functions often |
| 773 |
> |
exhibit non-exponential decay, and may not be characterized by a |
| 774 |
> |
single decay constant. |
| 775 |
> |
|
| 776 |
> |
In table \ref{tab:rotation} we show the characteristic rotational |
| 777 |
> |
relaxation times (based on the diffusion tensor) for each of the model |
| 778 |
> |
systems compared with the values obtained via microcanonical and Langevin |
| 779 |
> |
simulations. |
| 780 |
> |
|
| 781 |
> |
\subsection{Spherical particles} |
| 782 |
> |
Our model system for spherical particles was a Lennard-Jones sphere of |
| 783 |
> |
diameter ($\sigma$) 6.5 \AA\ in a sea of smaller spheres ($\sigma$ = |
| 784 |
> |
4.7 \AA). The well depth ($\epsilon$) for both particles was set to |
| 785 |
> |
an arbitrary value of 0.8 kcal/mol. |
| 786 |
> |
|
| 787 |
> |
The Stokes-Einstein behavior of large spherical particles in |
| 788 |
> |
hydrodynamic flows is well known, giving translational friction |
| 789 |
> |
coefficients of $6 \pi \eta R$ (stick boundary conditions) and |
| 790 |
> |
rotational friction coefficients of $8 \pi \eta R^3$. Recently, |
| 791 |
> |
Schmidt and Skinner have computed the behavior of spherical tag |
| 792 |
> |
particles in molecular dynamics simulations, and have shown that {\it |
| 793 |
> |
slip} boundary conditions ($\Xi_{tt} = 4 \pi \eta R$) may be more |
| 794 |
> |
appropriate for molecule-sized spheres embedded in a sea of spherical |
| 795 |
> |
qsolvent particles.\cite{Schmidt:2004fj,Schmidt:2003kx} |
| 796 |
> |
|
| 797 |
> |
Our simulation results show similar behavior to the behavior observed |
| 798 |
> |
by Schmidt and Skinner. The diffusion constant obtained from our |
| 799 |
> |
microcanonical molecular dynamics simulations lies between the slip |
| 800 |
> |
and stick boundary condition results obtained via Stokes-Einstein |
| 801 |
> |
behavior. Since the Langevin integrator assumes Stokes-Einstein stick |
| 802 |
> |
boundary conditions in calculating the drag and random forces for |
| 803 |
> |
spherical particles, our Langevin routine obtains nearly quantitative |
| 804 |
> |
agreement with the hydrodynamic results for spherical particles. One |
| 805 |
> |
avenue for improvement of the method would be to compute elements of |
| 806 |
> |
$\Xi_{tt}$ assuming behavior intermediate between the two boundary |
| 807 |
> |
conditions. |
| 808 |
> |
|
| 809 |
> |
In these simulations, our spherical particles were structureless, so |
| 810 |
> |
there is no way to obtain rotational correlation times for this model |
| 811 |
> |
system. |
| 812 |
> |
|
| 813 |
> |
\subsubsection{Ellipsoids} |
| 814 |
> |
For uniaxial ellipsoids ($a > b = c$) , Perrin's formulae for both |
| 815 |
> |
translational and rotational diffusion of each of the body-fixed axes |
| 816 |
> |
can be combined to give a single translational diffusion |
| 817 |
> |
constant,\cite{Berne90} |
| 818 |
> |
\begin{equation} |
| 819 |
> |
D = \frac{k_B T}{6 \pi \eta a} G(\rho), |
| 820 |
> |
\label{Dperrin} |
| 821 |
> |
\end{equation} |
| 822 |
> |
as well as a single rotational diffusion coefficient, |
| 823 |
> |
\begin{equation} |
| 824 |
> |
\Theta = \frac{3 k_B T}{16 \pi \eta a^3} \left\{ \frac{(2 - \rho^2) |
| 825 |
> |
G(\rho) - 1}{1 - \rho^4} \right\}. |
| 826 |
> |
\label{ThetaPerrin} |
| 827 |
> |
\end{equation} |
| 828 |
> |
In these expressions, $G(\rho)$ is a function of the axial ratio |
| 829 |
> |
($\rho = b / a$), which for prolate ellipsoids, is |
| 830 |
> |
\begin{equation} |
| 831 |
> |
G(\rho) = (1- \rho^2)^{-1/2} \ln \left\{ \frac{1 + (1 - |
| 832 |
> |
\rho^2)^{1/2}}{\rho} \right\} |
| 833 |
> |
\label{GPerrin} |
| 834 |
> |
\end{equation} |
| 835 |
> |
Again, there is some uncertainty about the correct boundary conditions |
| 836 |
> |
to use for molecular-scale ellipsoids in a sea of similarly-sized |
| 837 |
> |
solvent particles. Ravichandran and Bagchi found that {\it slip} |
| 838 |
> |
boundary conditions most closely resembled the simulation |
| 839 |
> |
results,\cite{Ravichandran:1999fk} in agreement with earlier work of |
| 840 |
> |
Tang and Evans.\cite{TANG:1993lr} |
| 841 |
> |
|
| 842 |
> |
Even though there are analytic resistance tensors for ellipsoids, we |
| 843 |
> |
constructed a rough-shell model using 2135 beads (each with a diameter |
| 844 |
> |
of 0.25 \AA) to approximate the shape of the modle ellipsoid. We |
| 845 |
> |
compared the Langevin dynamics from both the simple ellipsoidal |
| 846 |
> |
resistance tensor and the rough shell approximation with |
| 847 |
> |
microcanonical simulations and the predictions of Perrin. As in the |
| 848 |
> |
case of our spherical model system, the Langevin integrator reproduces |
| 849 |
> |
almost exactly the behavior of the Perrin formulae (which is |
| 850 |
> |
unsurprising given that the Perrin formulae were used to derive the |
| 851 |
> |
drag and random forces applied to the ellipsoid). We obtain |
| 852 |
> |
translational diffusion constants and rotational correlation times |
| 853 |
> |
that are within a few percent of the analytic values for both the |
| 854 |
> |
exact treatment of the diffusion tensor as well as the rough-shell |
| 855 |
> |
model for the ellipsoid. |
| 856 |
> |
|
| 857 |
> |
The translational diffusion constants from the microcanonical simulations |
| 858 |
> |
agree well with the predictions of the Perrin model, although the rotational |
| 859 |
> |
correlation times are a factor of 2 shorter than expected from hydrodynamic |
| 860 |
> |
theory. One explanation for the slower rotation |
| 861 |
> |
of explicitly-solvated ellipsoids is the possibility that solute-solvent |
| 862 |
> |
collisions happen at both ends of the solute whenever the principal |
| 863 |
> |
axis of the ellipsoid is turning. In the upper portion of figure |
| 864 |
> |
\ref{fig:explanation} we sketch a physical picture of this explanation. |
| 865 |
> |
Since our Langevin integrator is providing nearly quantitative agreement with |
| 866 |
> |
the Perrin model, it also predicts orientational diffusion for ellipsoids that |
| 867 |
> |
exceed explicitly solvated correlation times by a factor of two. |
| 868 |
> |
|
| 869 |
> |
\begin{figure} |
| 870 |
> |
\centering |
| 871 |
> |
\includegraphics[width=6in]{explanation} |
| 872 |
> |
\caption[Explanations of the differences between orientational correlation times for explicitly-solvated models and hydrodynamics predictions]{Explanations of the differences between orientational correlation times for explicitly-solvated models and hydrodynamic predictions. For the ellipsoids (upper figures), rotation of the principal axis can involve correlated collisions at both sides of the solute. In the rigid dumbbell model (lower figures), the large size of the explicit solvent spheres prevents them from coming in contact with a substantial fraction of the surface area of the dumbbell. Therefore, the explicit solvent only provides drag over a substantially reduced surface area of this model, where the hydrodynamic theories utilize the entire surface area for estimating rotational diffusion. |
| 873 |
> |
} \label{fig:explanation} |
| 874 |
> |
\end{figure} |
| 875 |
> |
|
| 876 |
> |
\subsubsection{Rigid dumbbells} |
| 877 |
> |
Perhaps the only {\it composite} rigid body for which analytic |
| 878 |
> |
expressions for the hydrodynamic tensor are available is the |
| 879 |
> |
two-sphere dumbbell model. This model consists of two non-overlapping |
| 880 |
> |
spheres held by a rigid bond connecting their centers. There are |
| 881 |
> |
competing expressions for the 6x6 resistance tensor for this |
| 882 |
> |
model. Equation (\ref{introEquation:oseenTensor}) above gives the |
| 883 |
> |
original Oseen tensor, while the second order expression introduced by |
| 884 |
> |
Rotne and Prager,\cite{Rotne1969} and improved by Garc\'{i}a de la |
| 885 |
> |
Torre and Bloomfield,\cite{Torre1977} is given above as |
| 886 |
> |
Eq. (\ref{introEquation:RPTensorNonOverlapped}). In our case, we use |
| 887 |
> |
a model dumbbell in which the two spheres are identical Lennard-Jones |
| 888 |
> |
particles ($\sigma$ = 6.5 \AA\ , $\epsilon$ = 0.8 kcal / mol) held at |
| 889 |
> |
a distance of 6.532 \AA. |
| 890 |
> |
|
| 891 |
> |
The theoretical values for the translational diffusion constant of the |
| 892 |
> |
dumbbell are calculated from the work of Stimson and Jeffery, who |
| 893 |
> |
studied the motion of this system in a flow parallel to the |
| 894 |
> |
inter-sphere axis,\cite{Stimson:1926qy} and Davis, who studied the |
| 895 |
> |
motion in a flow {\it perpendicular} to the inter-sphere |
| 896 |
> |
axis.\cite{Davis:1969uq} We know of no analytic solutions for the {\it |
| 897 |
> |
orientational} correlation times for this model system (other than |
| 898 |
> |
those derived from the 6 x 6 tensors mentioned above). |
| 899 |
> |
|
| 900 |
> |
The bead model for this model system comprises the two large spheres |
| 901 |
> |
by themselves, while the rough shell approximation used 3368 separate |
| 902 |
> |
beads (each with a diameter of 0.25 \AA) to approximate the shape of |
| 903 |
> |
the rigid body. The hydrodynamics tensors computed from both the bead |
| 904 |
> |
and rough shell models are remarkably similar. Computing the initial |
| 905 |
> |
hydrodynamic tensor for a rough shell model can be quite expensive (in |
| 906 |
> |
this case it requires inverting a 10104 x 10104 matrix), while the |
| 907 |
> |
bead model is typically easy to compute (in this case requiring |
| 908 |
> |
inversion of a 6 x 6 matrix). |
| 909 |
> |
|
| 910 |
> |
\begin{figure} |
| 911 |
> |
\centering |
| 912 |
> |
\includegraphics[width=3in]{RoughShell} |
| 913 |
> |
\caption[Model rigid bodies and their rough shell approximations]{The |
| 914 |
> |
model rigid bodies (left column) used to test this algorithm and their |
| 915 |
> |
rough-shell approximations (right-column) that were used to compute |
| 916 |
> |
the hydrodynamic tensors. The top two models (ellipsoid and dumbbell) |
| 917 |
> |
have analytic solutions and were used to test the rough shell |
| 918 |
> |
approximation. The lower two models (banana and lipid) were compared |
| 919 |
> |
with explicitly-solvated molecular dynamics simulations. } |
| 920 |
> |
\label{fig:roughShell} |
| 921 |
> |
\end{figure} |
| 922 |
> |
|
| 923 |
> |
|
| 924 |
> |
Once the hydrodynamic tensor has been computed, there is no additional |
| 925 |
> |
penalty for carrying out a Langevin simulation with either of the two |
| 926 |
> |
different hydrodynamics models. Our naive expectation is that since |
| 927 |
> |
the rigid body's surface is roughened under the various shell models, |
| 928 |
> |
the diffusion constants will be even farther from the ``slip'' |
| 929 |
> |
boundary conditions than observed for the bead model (which uses a |
| 930 |
> |
Stokes-Einstein model to arrive at the hydrodynamic tensor). For the |
| 931 |
> |
dumbbell, this prediction is correct although all of the Langevin |
| 932 |
> |
diffusion constants are within 6\% of the diffusion constant predicted |
| 933 |
> |
from the fully solvated system. |
| 934 |
> |
|
| 935 |
> |
For rotational motion, Langevin integration (and the hydrodynamic tensor) |
| 936 |
> |
yields rotational correlation times that are substantially shorter than those |
| 937 |
> |
obtained from explicitly-solvated simulations. It is likely that this is due |
| 938 |
> |
to the large size of the explicit solvent spheres, a feature that prevents |
| 939 |
> |
the solvent from coming in contact with a substantial fraction of the surface |
| 940 |
> |
area of the dumbbell. Therefore, the explicit solvent only provides drag |
| 941 |
> |
over a substantially reduced surface area of this model, while the |
| 942 |
> |
hydrodynamic theories utilize the entire surface area for estimating |
| 943 |
> |
rotational diffusion. A sketch of the free volume available in the explicit |
| 944 |
> |
solvent simulations is shown in figure \ref{fig:explanation}. |
| 945 |
> |
|
| 946 |
> |
\subsubsection{Ellipsoidal-composite banana-shaped molecules} |
| 947 |
> |
Banana-shaped rigid bodies composed of three Gay-Berne ellipsoids |
| 948 |
> |
have been used by Orlandi {\it et al.} to observe |
| 949 |
> |
mesophases in coarse-grained models bent-core liquid crystalline |
| 950 |
> |
molecules.\cite{Orlandi:2006fk} We have used the same overlapping |
| 951 |
> |
ellipsoids as a way to test the behavior of our algorithm for a |
| 952 |
> |
structure of some interest to the materials science community, |
| 953 |
> |
although since we are interested in capturing only the hydrodynamic |
| 954 |
> |
behavior of this model, we have left out the dipolar interactions of the |
| 955 |
> |
original Orlandi model. |
| 956 |
> |
|
| 957 |
> |
A reference system composed of a single banana rigid body embedded in a |
| 958 |
> |
sea of 1929 solvent particles was created and run under standard |
| 959 |
> |
(microcanonical) molecular dynamics. The resulting viscosity of this |
| 960 |
> |
mixture was 0.298 centipoise (as estimated using Eq. (\ref{eq:shear})). |
| 961 |
> |
To calculate the hydrodynamic properties of the banana rigid body model, |
| 962 |
> |
we created a rough shell (see Fig.~\ref{langevin:roughShell}), in which |
| 963 |
> |
the banana is represented as a ``shell'' made of 3321 identical beads |
| 964 |
> |
(0.25 \AA in diameter) distributed on the surface. Applying the |
| 965 |
> |
procedure described in Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 966 |
> |
identified the center of resistance at (0 $\rm{\AA}$, 0.81 $\rm{\AA}$, 0 $\rm{\AA}$), as well as the resistance tensor, |
| 967 |
> |
\[ |
| 968 |
> |
\left( {\begin{array}{*{20}c} |
| 969 |
> |
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
| 970 |
> |
0& 0.9270&-0.007063& 0.08585&0&0\\ |
| 971 |
> |
0&-0.007063&0.7494&0.2057&0&0\\ |
| 972 |
> |
0&0.0858&0.2057& 58.64& 0&0\\0.08585&0&0&0&48.30&3.219&\\0.2057&0&0&0&3.219&10.7373\\\end{array}} \right). |
| 973 |
> |
\] |
| 974 |
> |
where the units for translational, translation-rotation coupling and rotational |
| 975 |
> |
tensors are $\frac{kcal \cdot fs}{mol \cdot \rm{\AA}^2}$, $\frac{kcal \cdot fs}{ |
| 976 |
> |
mol \cdot \rm{\AA} \cdot rad}$ and $\frac{kcal \cdot fs}{mol \cdot rad^2}$ respe |
| 977 |
> |
ctively. |
| 978 |
> |
|
| 979 |
> |
The Langevin rigid-body integrator (and the hydrodynamic diffusion tensor) |
| 980 |
> |
are essentially quantitative for translational diffusion of this model. |
| 981 |
> |
Orientational correlation times under the Langevin rigid-body integrator |
| 982 |
> |
are within 11\% of the values obtained from explicit solvent, but these |
| 983 |
> |
models also exhibit some solvent inaccessible surface area in the |
| 984 |
> |
explicitly-solvated case. |
| 985 |
> |
|
| 986 |
> |
\subsubsection{Composite sphero-ellipsoids} |
| 987 |
> |
Spherical heads perched on the ends of Gay-Berne ellipsoids have been |
| 988 |
> |
used recently as models for lipid molecules.\cite{SunGezelter08,Ayton01} |
| 989 |
> |
|
| 990 |
> |
The diffusion constants and rotation relaxation times for |
| 991 |
|
different shaped molecules are shown in table \ref{tab:translation} |
| 992 |
|
and \ref{tab:rotation} to compare to the results calculated from NVE |
| 993 |
|
simulations. The theoretical values for sphere is calculated from the |
| 994 |
|
Stokes-Einstein law, the theoretical values for ellipsoid is |
| 995 |
< |
calculated from Perrin's fomula, the theoretical values for dumbbell |
| 703 |
< |
is calculated from StinXX and Davis theory. The exact method is |
| 995 |
> |
calculated from Perrin's fomula, The exact method is |
| 996 |
|
applied to the langevin dynamics simulations for sphere and ellipsoid, |
| 997 |
|
the bead model is applied to the simulation for dumbbell molecule, and |
| 998 |
|
the rough shell model is applied to ellipsoid, dumbbell, banana and |
| 1032 |
|
\begin{table*} |
| 1033 |
|
\begin{minipage}{\linewidth} |
| 1034 |
|
\begin{center} |
| 1035 |
< |
\caption{} |
| 1036 |
< |
\begin{tabular}{llccccccc} |
| 1037 |
< |
\hline |
| 1038 |
< |
& & Sphere & Ellipsoid & Dumbbell(2 spheres) & Banana(3 ellpsoids) & |
| 1039 |
< |
Lipid(head) & lipid(tail) & Solvent \\ |
| 1040 |
< |
\hline |
| 1041 |
< |
$d$ (\AA) & & 6.5 & 4.6 & 6.5 & 4.2 & 6.5 & 4.6 & 4.7 \\ |
| 1042 |
< |
$l$ (\AA) & & $= d$ & 13.8 & $=d$ & 11.2 & $=d$ & 13.8 & 4.7 \\ |
| 1043 |
< |
$\epsilon^s$ (kcal/mol) & & 0.8 & 0.8 & 0.8 & 0.8 & 0.185 & 0.8 & 0.8 \\ |
| 1044 |
< |
$\epsilon_r$ (well-depth aspect ratio)& & 1 & 0.2 & 1 & 0.2 & 1 & 0.2 & 1 \\ |
| 753 |
< |
$m$ (amu) & & 190 & 200 & 190 & 240 & 196 & 760 & 72.06 \\ |
| 754 |
< |
%$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$) & & & & \\ |
| 755 |
< |
%\multicolumn{2}c{$I_{xx}$} & 1125 & 45000 & N/A \\ |
| 756 |
< |
%\multicolumn{2}c{$I_{yy}$} & 1125 & 45000 & N/A \\ |
| 757 |
< |
%\multicolumn{2}c{$I_{zz}$} & 0 & 9000 & N/A \\ |
| 758 |
< |
%$\mu$ (Debye) & & varied & 0 & 0 \\ |
| 759 |
< |
\end{tabular} |
| 760 |
< |
\label{tab:parameters} |
| 761 |
< |
\end{center} |
| 762 |
< |
\end{minipage} |
| 763 |
< |
\end{table*} |
| 764 |
< |
|
| 765 |
< |
\begin{table*} |
| 766 |
< |
\begin{minipage}{\linewidth} |
| 767 |
< |
\begin{center} |
| 768 |
< |
\caption{} |
| 769 |
< |
\begin{tabular}{lccccc} |
| 1035 |
> |
\caption{Translational diffusion constants (D) for the model systems |
| 1036 |
> |
calculated using microcanonical simulations (with explicit solvent), |
| 1037 |
> |
theoretical predictions, and Langevin simulations (with implicit solvent). |
| 1038 |
> |
Analytical solutions for the exactly-solved hydrodynamics models are |
| 1039 |
> |
from Refs. \citen{Einstein05} (sphere), \citen{Perrin1934} and \citen{Perrin1936} |
| 1040 |
> |
(ellipsoid), \citen{Stimson:1926qy} and \citen{Davis:1969uq} |
| 1041 |
> |
(dumbbell). The other model systems have no known analytic solution. |
| 1042 |
> |
All diffusion constants are reported in units of $10^{-3}$ cm$^2$ / ps (= |
| 1043 |
> |
$10^{-4}$ \AA$^2$ / fs). } |
| 1044 |
> |
\begin{tabular}{lccccccc} |
| 1045 |
|
\hline |
| 1046 |
< |
& & & & &Translation \\ |
| 1047 |
< |
\hline |
| 1048 |
< |
& NVE & & Theoretical & Langevin & \\ |
| 774 |
< |
\hline |
| 775 |
< |
& $\eta$ & D & D & method & D \\ |
| 1046 |
> |
& \multicolumn{2}c{microcanonical simulation} & & \multicolumn{3}c{Theoretical} & Langevin \\ |
| 1047 |
> |
\cline{2-3} \cline{5-7} |
| 1048 |
> |
model & $\eta$ (centipoise) & D & & Analytical & method & Hydrodynamics & simulation \\ |
| 1049 |
|
\hline |
| 1050 |
< |
sphere & 3.480159e-03 & 1.643135e-04 & 1.942779e-04 & exact & 1.982283e-04 \\ |
| 1051 |
< |
ellipsoid & 2.551262e-03 & 2.437492e-04 & 2.335756e-04 & exact & 2.374905e-04 \\ |
| 1052 |
< |
& 2.551262e-03 & 2.437492e-04 & 2.335756e-04 & rough shell & 2.284088e-04 \\ |
| 1053 |
< |
dumbell & 2.41276e-03 & 2.129432e-04 & 2.090239e-04 & bead model & 2.148098e-04 \\ |
| 1054 |
< |
& 2.41276e-03 & 2.129432e-04 & 2.090239e-04 & rough shell & 2.013219e-04 \\ |
| 1055 |
< |
banana & 2.9846e-03 & 1.527819e-04 & & rough shell & 1.54807e-04 \\ |
| 1056 |
< |
lipid & 3.488661e-03 & 0.9562979e-04 & & rough shell & 1.320987e-04 \\ |
| 1050 |
> |
sphere & 0.261 & ? & & 2.59 & exact & 2.59 & 2.56 \\ |
| 1051 |
> |
ellipsoid & 0.255 & 2.44 & & 2.34 & exact & 2.34 & 2.37 \\ |
| 1052 |
> |
& 0.255 & 2.44 & & 2.34 & rough shell & 2.36 & 2.28 \\ |
| 1053 |
> |
dumbbell & 0.322 & ? & & 1.57 & bead model & 1.57 & 1.57 \\ |
| 1054 |
> |
& 0.322 & ? & & 1.57 & rough shell & ? & ? \\ |
| 1055 |
> |
banana & 0.298 & 1.53 & & & rough shell & 1.56 & 1.55 \\ |
| 1056 |
> |
lipid & 0.349 & 0.96 & & & rough shell & 1.33 & 1.32 \\ |
| 1057 |
|
\end{tabular} |
| 1058 |
|
\label{tab:translation} |
| 1059 |
|
\end{center} |
| 1063 |
|
\begin{table*} |
| 1064 |
|
\begin{minipage}{\linewidth} |
| 1065 |
|
\begin{center} |
| 1066 |
< |
\caption{} |
| 1067 |
< |
\begin{tabular}{lccccc} |
| 1066 |
> |
\caption{Orientational relaxation times ($\tau$) for the model systems using |
| 1067 |
> |
microcanonical simulation (with explicit solvent), theoretical |
| 1068 |
> |
predictions, and Langevin simulations (with implicit solvent). All |
| 1069 |
> |
relaxation times are for the rotational correlation function with |
| 1070 |
> |
$\ell = 2$ and are reported in units of ps. The ellipsoidal model has |
| 1071 |
> |
an exact solution for the orientational correlation time due to |
| 1072 |
> |
Perrin, but the other model systems have no known analytic solution.} |
| 1073 |
> |
\begin{tabular}{lccccccc} |
| 1074 |
|
\hline |
| 1075 |
< |
& & & & &Rotation \\ |
| 1076 |
< |
\hline |
| 1077 |
< |
& NVE & & Theoretical & Langevin & \\ |
| 799 |
< |
\hline |
| 800 |
< |
& $\eta$ & $\tau_0$ & $\tau_0$ & method & $\tau_0$ \\ |
| 1075 |
> |
& \multicolumn{2}c{microcanonical simulation} & & \multicolumn{3}c{Theoretical} & Langevin \\ |
| 1076 |
> |
\cline{2-3} \cline{5-7} |
| 1077 |
> |
model & $\eta$ (centipoise) & $\tau$ & & Perrin & method & Hydrodynamic & simulation \\ |
| 1078 |
|
\hline |
| 1079 |
< |
sphere & 3.480159e-03 & & 1.208178e+04 & exact & 1.20628e+04 \\ |
| 1080 |
< |
ellipsoid & 2.551262e-03 & 4.66806e+04 & 2.198986e+04 & exact & 2.21507e+04 \\ |
| 1081 |
< |
& 2.551262e-03 & 4.66806e+04 & 2.198986e+04 & rough shell & 2.21714e+04 \\ |
| 1082 |
< |
dumbell & 2.41276e-03 & 1.42974e+04 & & bead model & 7.12435e+04 \\ |
| 1083 |
< |
& 2.41276e-03 & 1.42974e+04 & & rough shell & 7.04765e+04 \\ |
| 1084 |
< |
banana & 2.9846e-03 & 6.38323e+04 & & rough shell & 7.0945e+04 \\ |
| 1085 |
< |
lipid & 3.488661e-03 & 7.79595e+04 & & rough shell & 7.78886e+04 \\ |
| 1079 |
> |
sphere & 0.261 & & & 9.06 & exact & 9.06 & 9.11 \\ |
| 1080 |
> |
ellipsoid & 0.255 & 46.7 & & 22.0 & exact & 22.0 & 22.2 \\ |
| 1081 |
> |
& 0.255 & 46.7 & & 22.0 & rough shell & 22.6 & 22.2 \\ |
| 1082 |
> |
dumbbell & 0.322 & 14.0 & & & bead model & 52.3 & 52.8 \\ |
| 1083 |
> |
& 0.322 & 14.0 & & & rough shell & ? & ? \\ |
| 1084 |
> |
banana & 0.298 & 63.8 & & & rough shell & 70.9 & 70.9 \\ |
| 1085 |
> |
lipid & 0.349 & 78.0 & & & rough shell & 76.9 & 77.9 \\ |
| 1086 |
> |
\hline |
| 1087 |
|
\end{tabular} |
| 1088 |
|
\label{tab:rotation} |
| 1089 |
|
\end{center} |
| 1100 |
|
100 ns run. The efficiency of the simulation is increased by one order |
| 1101 |
|
of magnitude. |
| 1102 |
|
|
| 825 |
– |
\subsection{Langevin Dynamics of Banana Shaped Molecules} |
| 826 |
– |
|
| 827 |
– |
In order to verify that Langevin dynamics can mimic the dynamics of |
| 828 |
– |
the systems absent of explicit solvents, we carried out two sets of |
| 829 |
– |
simulations and compare their dynamic properties. |
| 830 |
– |
Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
| 831 |
– |
made of 256 pentane molecules and two banana shaped molecules at |
| 832 |
– |
273~K. It has an equivalent implicit solvent system containing only |
| 833 |
– |
two banana shaped molecules with viscosity of 0.289 center poise. To |
| 834 |
– |
calculate the hydrodynamic properties of the banana shaped molecule, |
| 835 |
– |
we created a rough shell model (see Fig.~\ref{langevin:roughShell}), |
| 836 |
– |
in which the banana shaped molecule is represented as a ``shell'' |
| 837 |
– |
made of 2266 small identical beads with size of 0.3 \AA on the |
| 838 |
– |
surface. Applying the procedure described in |
| 839 |
– |
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 840 |
– |
identified the center of resistance at (0 $\rm{\AA}$, 0.7482 $\rm{\AA}$, |
| 841 |
– |
-0.1988 $\rm{\AA}$), as well as the resistance tensor, |
| 842 |
– |
\[ |
| 843 |
– |
\left( {\begin{array}{*{20}c} |
| 844 |
– |
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
| 845 |
– |
0& 0.9270&-0.007063& 0.08585&0&0\\ |
| 846 |
– |
0&-0.007063&0.7494&0.2057&0&0\\ |
| 847 |
– |
0&0.0858&0.2057& 58.64& 0&0\\ |
| 848 |
– |
0.08585&0&0&0&48.30&3.219&\\ |
| 849 |
– |
0.2057&0&0&0&3.219&10.7373\\ |
| 850 |
– |
\end{array}} \right). |
| 851 |
– |
\] |
| 852 |
– |
where the units for translational, translation-rotation coupling and rotational tensors are $\frac{kcal \cdot fs}{mol \cdot \rm{\AA}^2}$, $\frac{kcal \cdot fs}{mol \cdot \rm{\AA} \cdot rad}$ and $\frac{kcal \cdot fs}{mol \cdot rad^2}$ respectively. |
| 853 |
– |
Curves of the velocity auto-correlation functions in |
| 854 |
– |
Fig.~\ref{langevin:vacf} were shown to match each other very well. |
| 855 |
– |
However, because of the stochastic nature, simulation using Langevin |
| 856 |
– |
dynamics was shown to decay slightly faster than MD. In order to |
| 857 |
– |
study the rotational motion of the molecules, we also calculated the |
| 858 |
– |
auto-correlation function of the principle axis of the second GB |
| 859 |
– |
particle, $u$. The discrepancy shown in Fig.~\ref{langevin:uacf} was |
| 860 |
– |
probably due to the reason that we used the experimental viscosity directly instead of calculating bulk viscosity from simulation. |
| 861 |
– |
|
| 862 |
– |
\begin{figure} |
| 863 |
– |
\centering |
| 864 |
– |
\includegraphics[width=\linewidth]{roughShell.pdf} |
| 865 |
– |
\caption[Rough shell model for banana shaped molecule]{Rough shell |
| 866 |
– |
model for banana shaped molecule.} \label{langevin:roughShell} |
| 867 |
– |
\end{figure} |
| 868 |
– |
|
| 869 |
– |
\begin{figure} |
| 870 |
– |
\centering |
| 871 |
– |
\includegraphics[width=\linewidth]{twoBanana.pdf} |
| 872 |
– |
\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
| 873 |
– |
256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
| 874 |
– |
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
| 875 |
– |
\end{figure} |
| 876 |
– |
|
| 877 |
– |
\begin{figure} |
| 878 |
– |
\centering |
| 879 |
– |
\includegraphics[width=\linewidth]{vacf.pdf} |
| 880 |
– |
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
| 881 |
– |
auto-correlation functions of NVE (explicit solvent) in blue and |
| 882 |
– |
Langevin dynamics (implicit solvent) in red.} \label{langevin:vacf} |
| 883 |
– |
\end{figure} |
| 884 |
– |
|
| 885 |
– |
\begin{figure} |
| 886 |
– |
\centering |
| 887 |
– |
\includegraphics[width=\linewidth]{uacf.pdf} |
| 888 |
– |
\caption[Auto-correlation functions of the principle axis of the |
| 889 |
– |
middle GB particle]{Auto-correlation functions of the principle axis |
| 890 |
– |
of the middle GB particle of NVE (blue) and Langevin dynamics |
| 891 |
– |
(red).} \label{langevin:uacf} |
| 892 |
– |
\end{figure} |
| 893 |
– |
|
| 1103 |
|
\section{Conclusions} |
| 1104 |
|
|
| 1105 |
|
We have presented a new Langevin algorithm by incorporating the |
| 1106 |
|
hydrodynamics properties of arbitrary shaped molecules into an |
| 1107 |
< |
advanced symplectic integration scheme. The temperature control |
| 1108 |
< |
ability of this algorithm was demonstrated by a set of simulations |
| 1109 |
< |
with different viscosities. It was also shown to have significant |
| 1110 |
< |
advantage of producing rapid thermal equilibration over |
| 902 |
< |
Nos\'{e}-Hoover method. Further studies in systems involving banana |
| 903 |
< |
shaped molecules illustrated that the dynamic properties could be |
| 904 |
< |
preserved by using this new algorithm as an implicit solvent model. |
| 1107 |
> |
advanced symplectic integration scheme. Further studies in systems |
| 1108 |
> |
involving banana shaped molecules illustrated that the dynamic |
| 1109 |
> |
properties could be preserved by using this new algorithm as an |
| 1110 |
> |
implicit solvent model. |
| 1111 |
|
|
| 1112 |
|
|
| 1113 |
|
\section{Acknowledgments} |
| 1117 |
|
of Notre Dame. |
| 1118 |
|
\newpage |
| 1119 |
|
|
| 1120 |
< |
\bibliographystyle{jcp2} |
| 1120 |
> |
\bibliographystyle{jcp} |
| 1121 |
|
\bibliography{langevin} |
| 1122 |
|
|
| 1123 |
|
\end{document} |
