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\chapter{\label{chapt:methodology}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE} |
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\chapter{\label{chapt:langevin}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE} |
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|
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\section{Introduction} |
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|
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between the native and denatured states. Because of its stability |
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against noise, Langevin dynamics is very suitable for studying |
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remagnetization processes in various |
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systems\cite{Garcia-Palacios1998,Berkov2002,Denisov2003}. For |
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instance, the oscillation power spectrum of nanoparticles from |
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Langevin dynamics simulation has the same peak frequencies for |
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different wave vectors,which recovers the property of magnetic |
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excitations in small finite structures\cite{Berkov2005a}. In an |
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attempt to reduce the computational cost of simulation, multiple |
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time stepping (MTS) methods have been introduced and have been of |
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great interest to macromolecule and protein |
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community\cite{Tuckerman1992}. Relying on the observation that |
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forces between distant atoms generally demonstrate slower |
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fluctuations than forces between close atoms, MTS method are |
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generally implemented by evaluating the slowly fluctuating forces |
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less frequently than the fast ones. Unfortunately, nonlinear |
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instability resulting from increasing timestep in MTS simulation |
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have became a critical obstruction preventing the long time |
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simulation. Due to the coupling to the heat bath, Langevin dynamics |
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has been shown to be able to damp out the resonance artifact more |
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efficiently\cite{Sandu1999}. |
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systems\cite{Palacios1998,Berkov2002,Denisov2003}. For instance, the |
32 |
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oscillation power spectrum of nanoparticles from Langevin dynamics |
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simulation has the same peak frequencies for different wave |
34 |
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vectors,which recovers the property of magnetic excitations in small |
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> |
finite structures\cite{Berkov2005a}. In an attempt to reduce the |
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computational cost of simulation, multiple time stepping (MTS) |
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> |
methods have been introduced and have been of great interest to |
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macromolecule and protein community\cite{Tuckerman1992}. Relying on |
39 |
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the observation that forces between distant atoms generally |
40 |
> |
demonstrate slower fluctuations than forces between close atoms, MTS |
41 |
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method are generally implemented by evaluating the slowly |
42 |
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fluctuating forces less frequently than the fast ones. |
43 |
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Unfortunately, nonlinear instability resulting from increasing |
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timestep in MTS simulation have became a critical obstruction |
45 |
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preventing the long time simulation. Due to the coupling to the heat |
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bath, Langevin dynamics has been shown to be able to damp out the |
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resonance artifact more efficiently\cite{Sandu1999}. |
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|
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%review langevin/browninan dynamics for arbitrarily shaped rigid body |
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Combining Langevin or Brownian dynamics with rigid body dynamics, |
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average acceleration is not always true for cooperative motion which |
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is common in protein motion. An inertial Brownian dynamics (IBD) was |
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proposed to address this issue by adding an inertial correction |
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term\cite{Beard2001}. As a complement to IBD which has a lower bound |
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term\cite{Beard2000}. As a complement to IBD which has a lower bound |
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in time step because of the inertial relaxation time, long-time-step |
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inertial dynamics (LTID) can be used to investigate the inertial |
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behavior of the polymer segments in low friction |
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regime\cite{Beard2001}. LTID can also deal with the rotational |
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regime\cite{Beard2000}. LTID can also deal with the rotational |
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dynamics for nonskew bodies without translation-rotation coupling by |
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separating the translation and rotation motion and taking advantage |
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of the analytical solution of hydrodynamics properties. However, |
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estimation of friction tensor from hydrodynamics theory into the |
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sophisticated rigid body dynamics. |
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|
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\section{Computational methods{\label{methodSec}}} |
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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 |
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Theoretically, the friction kernel can be determined using the |
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velocity autocorrelation function. However, this approach become |
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impractical when the system become more and more complicate. |
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Instead, various approaches based on hydrodynamics have been |
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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 |
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translational and rotational friction constant can be calculated |
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from Stoke's law, |
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\[ |
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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} |
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exactly. In 1936, Perrin extended Stokes's law to general |
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ellipsoids, also called a triaxial ellipsoid, which is given in |
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Cartesian coordinates by\cite{Perrin1934, Perrin1936} |
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\[ |
167 |
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\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
168 |
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}} = 1 |
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due to the complexity of the elliptic integral, only the ellipsoid |
172 |
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with the restriction of two axes having to be equal, \textit{i.e.} |
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prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
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exactly. Introducing an elliptic integral parameter $S$ for prolate, |
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exactly. Introducing an elliptic integral parameter $S$ for prolate |
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: |
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\[ |
177 |
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S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
178 |
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} }}{b}, |
179 |
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\] |
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and oblate, |
180 |
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and oblate : |
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\[ |
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S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
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}}{a} |
184 |
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\], |
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}}{a}. |
184 |
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\] |
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one can write down the translational and rotational resistance |
186 |
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tensors |
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\[ |
188 |
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\begin{array}{l} |
189 |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
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\end{array}, |
189 |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
190 |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
191 |
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2a}}, |
192 |
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\end{array} |
193 |
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\] |
194 |
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and |
195 |
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\[ |
196 |
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\begin{array}{l} |
197 |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
197 |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
198 |
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\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
199 |
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\end{array}. |
200 |
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\] |
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|
202 |
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\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}} |
202 |
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\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
203 |
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|
204 |
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Unlike spherical and other regular shaped molecules, there is not |
205 |
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analytical solution for friction tensor of any arbitrary shaped |
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Unlike spherical and other simply shaped molecules, there is no |
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analytical solution for the friction tensor for arbitrarily shaped |
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rigid molecules. The ellipsoid of revolution model and general |
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triaxial ellipsoid model have been used to approximate the |
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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 |
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from all possible ellipsoidal spaces, $r$-space, to all possible |
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combination of rotational diffusion coefficients, $D$-space, is not |
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unique\cite{Wegener1979} as well as the intrinsic coupling between |
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translational and rotational motion of rigid body, general ellipsoid |
213 |
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is not always suitable for modeling arbitrarily shaped rigid |
214 |
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molecule. A number of studies have been devoted to determine the |
215 |
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friction tensor for irregularly shaped rigid bodies using more |
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translational and rotational motion of rigid body, general |
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ellipsoids are not always suitable for modeling arbitrarily shaped |
214 |
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rigid molecule. A number of studies have been devoted to determine |
215 |
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the friction tensor for irregularly shaped rigid bodies using more |
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advanced method where the molecule of interest was modeled by |
217 |
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combinations of spheres(beads)\cite{Carrasco1999} and the |
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hydrodynamics properties of the molecule can be calculated using the |
231 |
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\label{introEquation:tensorExpression} |
232 |
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\end{equation} |
233 |
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This equation is the basis for deriving the hydrodynamic tensor. In |
234 |
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1930, Oseen and Burgers gave a simple solution to Equation |
235 |
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\ref{introEquation:tensorExpression} |
234 |
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1930, Oseen and Burgers gave a simple solution to |
235 |
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Eq.~\ref{introEquation:tensorExpression} |
236 |
|
\begin{equation} |
237 |
|
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
238 |
|
R_{ij}^T }}{{R_{ij}^2 }}} \right). \label{introEquation:oseenTensor} |
248 |
|
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
249 |
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\label{introEquation:RPTensorNonOverlapped} |
250 |
|
\end{equation} |
251 |
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Both of the Equation \ref{introEquation:oseenTensor} and Equation |
252 |
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\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
253 |
< |
\ge \sigma _i + \sigma _j$. An alternative expression for |
251 |
> |
Both of the Eq.~\ref{introEquation:oseenTensor} and |
252 |
> |
Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption |
253 |
> |
$R_{ij} \ge \sigma _i + \sigma _j$. An alternative expression for |
254 |
|
overlapping beads with the same radius, $\sigma$, is given by |
255 |
|
\begin{equation} |
256 |
|
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
258 |
|
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
259 |
|
\label{introEquation:RPTensorOverlapped} |
260 |
|
\end{equation} |
260 |
– |
|
261 |
|
To calculate the resistance tensor at an arbitrary origin $O$, we |
262 |
|
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
263 |
|
$B_{ij}$ blocks |
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 |
278 |
< |
|
276 |
> |
where $\delta _{ij}$ is Kronecker delta function. Inverting the $B$ |
277 |
> |
matrix, we obtain |
278 |
|
\[ |
279 |
|
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
280 |
|
{C_{11} } & \ldots & {C_{1N} } \\ |
281 |
|
\vdots & \ddots & \vdots \\ |
282 |
|
{C_{N1} } & \cdots & {C_{NN} } \\ |
283 |
< |
\end{array}} \right) |
283 |
> |
\end{array}} \right), |
284 |
|
\] |
285 |
< |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
285 |
> |
which can be partitioned into $N \times N$ $3 \times 3$ block |
286 |
|
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
287 |
|
\[ |
288 |
|
U_i = \left( {\begin{array}{*{20}c} |
294 |
|
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
295 |
|
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
296 |
|
arbitrary origin $O$ can be written as |
297 |
< |
\begin{equation} |
298 |
< |
\begin{array}{l} |
300 |
< |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
297 |
> |
\begin{eqnarray} |
298 |
> |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
299 |
|
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
300 |
< |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
303 |
< |
\end{array} |
300 |
> |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
301 |
|
\label{introEquation:ResistanceTensorArbitraryOrigin} |
302 |
< |
\end{equation} |
302 |
> |
\end{eqnarray} |
303 |
|
|
304 |
|
The resistance tensor depends on the origin to which they refer. The |
305 |
|
proper location for applying friction force is the center of |
306 |
< |
resistance (reaction), at which the trace of rotational resistance |
307 |
< |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
308 |
< |
resistance is defined as an unique point of the rigid body at which |
309 |
< |
the translation-rotation coupling tensor are symmetric, |
306 |
> |
resistance (or center of reaction), at which the trace of rotational |
307 |
> |
resistance tensor, $ \Xi ^{rr}$ reaches a minimum value. |
308 |
> |
Mathematically, the center of resistance is defined as an unique |
309 |
> |
point of the rigid body at which the translation-rotation coupling |
310 |
> |
tensor are symmetric, |
311 |
|
\begin{equation} |
312 |
|
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
313 |
|
\label{introEquation:definitionCR} |
314 |
|
\end{equation} |
315 |
< |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
315 |
> |
From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
316 |
|
we can easily find out that the translational resistance tensor is |
317 |
|
origin independent, while the rotational resistance tensor and |
318 |
|
translation-rotation coupling resistance tensor depend on the |
319 |
< |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
320 |
< |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
319 |
> |
origin. Given the resistance tensor at an arbitrary origin $O$, and |
320 |
> |
a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
321 |
|
obtain the resistance tensor at $P$ by |
322 |
|
\begin{equation} |
323 |
|
\begin{array}{l} |
335 |
|
{ - y_{OP} } & {x_{OP} } & 0 \\ |
336 |
|
\end{array}} \right) |
337 |
|
\] |
338 |
< |
Using Equations \ref{introEquation:definitionCR} and |
339 |
< |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
340 |
< |
the position of center of resistance, |
338 |
> |
Using Eq.~\ref{introEquation:definitionCR} and |
339 |
> |
Eq.~\ref{introEquation:resistanceTensorTransformation}, one can |
340 |
> |
locate the position of center of resistance, |
341 |
|
\begin{eqnarray*} |
342 |
|
\left( \begin{array}{l} |
343 |
|
x_{OR} \\ |
354 |
|
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
355 |
|
\end{array} \right) \\ |
356 |
|
\end{eqnarray*} |
359 |
– |
|
357 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
358 |
|
joining center of resistance $R$ and origin $O$. |
359 |
|
|
360 |
< |
\subsection{Langevin dynamics for rigid particles of arbitrary shape\label{LDRB}} |
360 |
> |
\subsection{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
361 |
|
|
362 |
|
Consider a Langevin equation of motions in generalized coordinates |
363 |
|
\begin{equation} |
366 |
|
\end{equation} |
367 |
|
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
368 |
|
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
369 |
< |
$V_i = V_i(v_i,\omega _i)$. The right side of Eq.~ |
370 |
< |
(\ref{LDGeneralizedForm}) consists of three generalized forces in |
369 |
> |
$V_i = V_i(v_i,\omega _i)$. The right side of |
370 |
> |
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces in |
371 |
|
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
372 |
|
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
373 |
|
system in Newtownian mechanics typically refers to lab-fixed frame, |
377 |
|
\[ |
378 |
|
\begin{array}{l} |
379 |
|
F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\ |
380 |
< |
F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\ |
381 |
< |
\end{array}. |
380 |
> |
F_{r,i}^l (t) = A^T F_{r,i}^b (t). \\ |
381 |
> |
\end{array} |
382 |
|
\] |
383 |
|
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
384 |
|
the body-fixed velocity at center of resistance $v_{R,i}^b$ and |
385 |
< |
angular velocity $\omega _i$, |
385 |
> |
angular velocity $\omega _i$ |
386 |
|
\begin{equation} |
387 |
|
F_{r,i}^b (t) = \left( \begin{array}{l} |
388 |
|
f_{r,i}^b (t) \\ |
400 |
|
\begin{equation} |
401 |
|
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
402 |
|
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
403 |
< |
2k_B T\Xi _R \delta (t - t'). |
403 |
> |
2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
404 |
|
\end{equation} |
408 |
– |
|
405 |
|
The equation of motion for $v_i$ can be written as |
406 |
|
\begin{equation} |
407 |
|
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
423 |
|
\dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t) |
424 |
|
+ \tau _{r,i}^b(t) |
425 |
|
\end{equation} |
430 |
– |
|
426 |
|
Embedding the friction terms into force and torque, one can |
427 |
|
integrate the langevin equations of motion for rigid body of |
428 |
|
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
442 |
|
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
443 |
|
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
444 |
|
\end{align*} |
450 |
– |
|
445 |
|
In this context, the $\mathrm{rotate}$ function is the reversible |
446 |
|
product of the three body-fixed rotations, |
447 |
|
\begin{equation} |
476 |
|
\end{array} |
477 |
|
\right). |
478 |
|
\end{equation} |
479 |
< |
All other rotations follow in a straightforward manner. |
479 |
> |
All other rotations follow in a straightforward manner. After the |
480 |
> |
first part of the propagation, the forces and body-fixed torques are |
481 |
> |
calculated at the new positions and orientations |
482 |
|
|
487 |
– |
After the first part of the propagation, the forces and body-fixed |
488 |
– |
torques are calculated at the new positions and orientations |
489 |
– |
|
483 |
|
{\tt doForces:} |
484 |
|
\begin{align*} |
485 |
|
{\bf f}(t + h) &\leftarrow |
491 |
|
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
492 |
|
\cdot {\bf \tau}^s(t + h). |
493 |
|
\end{align*} |
494 |
+ |
Once the forces and torques have been obtained at the new time step, |
495 |
+ |
the velocities can be advanced to the same time value. |
496 |
|
|
502 |
– |
{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
503 |
– |
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
504 |
– |
torques have been obtained at the new time step, the velocities can |
505 |
– |
be advanced to the same time value. |
506 |
– |
|
497 |
|
{\tt moveB:} |
498 |
|
\begin{align*} |
499 |
|
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 |
505 |
|
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
506 |
|
\end{align*} |
507 |
|
|
508 |
< |
\section{Results and discussion} |
508 |
> |
\section{Results and Discussion} |
509 |
|
|
510 |
< |
The Langevin algorithm described in Sec.~\ref{LDRB} has been |
511 |
< |
implemented in {\sc oopse}\cite{Meineke2005} and applied to several |
512 |
< |
test systems. |
510 |
> |
The Langevin algorithm described in previous section has been |
511 |
> |
implemented in {\sc oopse}\cite{Meineke2005} and applied to the |
512 |
> |
studies of kinetics and thermodynamic properties in several systems. |
513 |
|
|
514 |
< |
\subsection{Langevin dynamics of} |
514 |
> |
\subsection{Temperature Control} |
515 |
|
|
516 |
+ |
As shown in Eq.~\ref{randomForce}, random collisions associated with |
517 |
+ |
the solvent's thermal motions is controlled by the external |
518 |
+ |
temperature. The capability to maintain the temperature of the whole |
519 |
+ |
system was usually used to measure the stability and efficiency of |
520 |
+ |
the algorithm. In order to verify the stability of this new |
521 |
+ |
algorithm, a series of simulations are performed on system |
522 |
+ |
consisiting of 256 SSD water molecules with different viscosities. |
523 |
+ |
Initial configuration for the simulations is taken from a 1ns NVT |
524 |
+ |
simulation with a cubic box of 19.7166~\AA. All simulation are |
525 |
+ |
carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
526 |
+ |
with reference temperature at 300~K. Average temperature as a |
527 |
+ |
function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
528 |
+ |
the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
529 |
+ |
1$ poise. The better temperature control at higher viscosity can be |
530 |
+ |
explained by the finite size effect and relative slow relaxation |
531 |
+ |
rate at lower viscosity regime. |
532 |
+ |
\begin{table} |
533 |
+ |
\caption{AVERAGE TEMPERATURES FROM LANGEVIN DYNAMICS SIMULATIONS OF |
534 |
+ |
SSD WATER MOLECULES WITH REFERENCE TEMPERATURE AT 300~K.} |
535 |
+ |
\label{langevin:viscosity} |
536 |
+ |
\begin{center} |
537 |
+ |
\begin{tabular}{lll} |
538 |
+ |
\hline |
539 |
+ |
$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
540 |
+ |
\hline |
541 |
+ |
1 & 300.47 & 10.99 \\ |
542 |
+ |
0.1 & 301.19 & 11.136 \\ |
543 |
+ |
0.01 & 303.04 & 11.796 \\ |
544 |
+ |
\hline |
545 |
+ |
\end{tabular} |
546 |
+ |
\end{center} |
547 |
+ |
\end{table} |
548 |
+ |
|
549 |
+ |
Another set of calculation were performed to study the efficiency of |
550 |
+ |
temperature control using different temperature coupling schemes. |
551 |
+ |
The starting configuration is cooled to 173~K and evolved using NVE, |
552 |
+ |
NVT, and Langevin dynamic with time step of 2 fs. |
553 |
+ |
Fig.~\ref{langevin:temperature} shows the heating curve obtained as |
554 |
+ |
the systems reach equilibrium. The orange curve in |
555 |
+ |
Fig.~\ref{langevin:temperature} represents the simulation using |
556 |
+ |
Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
557 |
+ |
which gives reasonable tight coupling, while the blue one from |
558 |
+ |
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
559 |
+ |
scaling to the desire temperature. In extremely lower friction |
560 |
+ |
regime (when $ \eta \approx 0$), Langevin dynamics becomes normal |
561 |
+ |
NVE (see green curve in Fig.~\ref{langevin:temperature}) which loses |
562 |
+ |
the temperature control ability. |
563 |
+ |
|
564 |
|
\begin{figure} |
565 |
|
\centering |
566 |
|
\includegraphics[width=\linewidth]{temperature.eps} |
567 |
< |
\caption[]{.} \label{langevin:temperature} |
567 |
> |
\caption[Plot of Temperature Fluctuation Versus Time]{Plot of |
568 |
> |
temperature fluctuation versus time.} \label{langevin:temperature} |
569 |
|
\end{figure} |
570 |
|
|
571 |
< |
\subsection{LD of banana-shaped molecule} |
571 |
> |
\subsection{Langevin Dynamics of Banana Shaped Molecules} |
572 |
|
|
573 |
+ |
In order to verify that Langevin dynamics can mimic the dynamics of |
574 |
+ |
the systems absent of explicit solvents, we carried out two sets of |
575 |
+ |
simulations and compare their dynamic properties. |
576 |
+ |
Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
577 |
+ |
made of 256 pentane molecules and two banana shaped molecules at |
578 |
+ |
273~K. It has an equivalent implicit solvent system containing only |
579 |
+ |
two banana shaped molecules with viscosity of 0.289 center poise. To |
580 |
+ |
calculate the hydrodynamic properties of the banana shaped molecule, |
581 |
+ |
we create a rough shell model (see Fig.~\ref{langevin:roughShell}), |
582 |
+ |
in which the banana shaped molecule is represented as a ``shell'' |
583 |
+ |
made of 2266 small identical beads with size of 0.3 \AA on the |
584 |
+ |
surface. Applying the procedure described in |
585 |
+ |
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
586 |
+ |
identified the center of resistance at $(0, 0.7482, -0.1988)$, as |
587 |
+ |
well as the resistance tensor, |
588 |
+ |
\[ |
589 |
+ |
\left( {\begin{array}{*{20}c} |
590 |
+ |
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
591 |
+ |
0& 0.9270&-0.007063& 0.08585&0&0\\ |
592 |
+ |
0&0.007063&0.7494&0.2057&0&0\\ |
593 |
+ |
0&0.0858&0.2057& 58.64& 0&-8.5736\\ |
594 |
+ |
0.08585&0&0&0&48.30&3.219&\\ |
595 |
+ |
0.2057&0&0&0&3.219&10.7373\\ |
596 |
+ |
\end{array}} \right). |
597 |
+ |
\] |
598 |
+ |
%\[ |
599 |
+ |
%\left( {\begin{array}{*{20}c} |
600 |
+ |
%0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\ |
601 |
+ |
%3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\ |
602 |
+ |
%-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\ |
603 |
+ |
%5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\ |
604 |
+ |
%0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\ |
605 |
+ |
%0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\ |
606 |
+ |
%\end{array}} \right). |
607 |
+ |
%\] |
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]{roughShell.eps} |
627 |
< |
\caption[Rough Shell]{Rough Shell.} \label{langevin:roughShell} |
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]{twoBanana.eps} |
643 |
< |
\caption[Two Banana Shaped Molecules]{Two Banana Shaped Molecules.} |
644 |
< |
\label{langevin:twoBanana} |
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. |