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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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\section{Introduction} |
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%applications of langevin dynamics |
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As an excellent alternative to newtonian dynamics, Langevin |
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dynamics, which mimics a simple heat bath with stochastic and |
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dissipative forces, has been applied in a variety of studies. The |
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stochastic treatment of the solvent enables us to carry out |
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substantially longer time simulation. Implicit solvent Langevin |
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dynamics simulation of met-enkephalin not only outperforms explicit |
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solvent simulation on computation efficiency, but also agrees very |
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well with explicit solvent simulation on dynamics |
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properties\cite{Shen2002}. Recently, applying Langevin dynamics with |
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UNRES model, Liow and his coworkers suggest that protein folding |
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pathways can be possibly exploited within a reasonable amount of |
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time\cite{Liwo2005}. The stochastic nature of the Langevin dynamics |
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also enhances the sampling of the system and increases the |
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probability of crossing energy barrier\cite{Banerjee2004, Cui2003}. |
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Combining Langevin dynamics with Kramers's theory, Klimov and |
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Thirumalai identified the free-energy barrier by studying the |
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viscosity dependence of the protein folding rates\cite{Klimov1997}. |
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In order to account for solvent induced interactions missing from |
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implicit solvent model, Kaya incorporated desolvation free energy |
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barrier into implicit coarse-grained solvent model in protein |
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folding/unfolding study and discovered a higher free energy barrier |
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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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As alternative to Newtonian dynamics, Langevin dynamics, which |
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mimics a simple heat bath with stochastic and dissipative forces, |
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has been applied in a variety of studies. The stochastic treatment |
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of the solvent enables us to carry out substantially longer time |
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simulations. Implicit solvent Langevin dynamics simulations of |
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met-enkephalin not only outperform explicit solvent simulations for |
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computational efficiency, but also agrees very well with explicit |
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solvent simulations for dynamical properties.\cite{Shen2002} |
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Recently, applying Langevin dynamics with the UNRES model, Liow and |
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his coworkers suggest that protein folding pathways can be possibly |
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explored within a reasonable amount of time.\cite{Liwo2005} The |
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stochastic nature of the Langevin dynamics also enhances the |
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sampling of the system and increases the probability of crossing |
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energy barriers.\cite{Banerjee2004, Cui2003} Combining Langevin |
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dynamics with Kramers's theory, Klimov and Thirumalai identified |
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free-energy barriers by studying the viscosity dependence of the |
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protein folding rates.\cite{Klimov1997} In order to account for |
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solvent induced interactions missing from implicit solvent model, |
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Kaya incorporated desolvation free energy barrier into implicit |
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coarse-grained solvent model in protein folding/unfolding studies |
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and discovered a higher free energy barrier between the native and |
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denatured states. Because of its stability against noise, Langevin |
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dynamics is very suitable for studying remagnetization processes in |
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various systems.\cite{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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different wave vectors, which recovers the property of magnetic |
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excitations in small finite structures.\cite{Berkov2005a} |
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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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one can study the slow processes in biomolecular systems. Modeling |
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the DNA as a chain of rigid spheres beads, which subject to harmonic |
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potentials as well as excluded volume potentials, Mielke and his |
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coworkers discover rapid superhelical stress generations from the |
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stochastic simulation of twin supercoiling DNA with response to |
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induced torques\cite{Mielke2004}. Membrane fusion is another key |
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biological process which controls a variety of physiological |
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functions, such as release of neurotransmitters \textit{etc}. A |
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typical fusion event happens on the time scale of millisecond, which |
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is impracticable to study using all atomistic model with newtonian |
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mechanics. With the help of coarse-grained rigid body model and |
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stochastic dynamics, the fusion pathways were exploited by many |
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researchers\cite{Noguchi2001,Noguchi2002,Shillcock2005}. Due to the |
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difficulty of numerical integration of anisotropy rotation, most of |
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the rigid body models are simply modeled by sphere, cylinder, |
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ellipsoid or other regular shapes in stochastic simulations. In an |
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effort to account for the diffusion anisotropy of the arbitrary |
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one can study slow processes in biomolecular systems. Modeling DNA |
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as a chain of rigid beads, which are subject to harmonic potentials |
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as well as excluded volume potentials, Mielke and his coworkers |
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discovered rapid superhelical stress generations from the stochastic |
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simulation of twin supercoiling DNA with response to induced |
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torques.\cite{Mielke2004} Membrane fusion is another key biological |
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process which controls a variety of physiological functions, such as |
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release of neurotransmitters \textit{etc}. A typical fusion event |
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happens on the time scale of a millisecond, which is impractical to |
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study using atomistic models with newtonian mechanics. With the help |
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of coarse-grained rigid body model and stochastic dynamics, the |
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fusion pathways were explored by many |
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researchers.\cite{Noguchi2001,Noguchi2002,Shillcock2005} Due to the |
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difficulty of numerical integration of anisotropic rotation, most of |
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the rigid body models are simply modeled using spheres, cylinders, |
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ellipsoids or other regular shapes in stochastic simulations. In an |
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effort to account for the diffusion anisotropy of arbitrary |
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particles, Fernandes and de la Torre improved the original Brownian |
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dynamics simulation algorithm\cite{Ermak1978,Allison1991} by |
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incorporating a generalized $6\times6$ diffusion tensor and |
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introducing a simple rotation evolution scheme consisting of three |
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consecutive rotations\cite{Fernandes2002}. Unfortunately, unexpected |
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error and bias are introduced into the system due to the arbitrary |
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order of applying the noncommuting rotation |
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operators\cite{Beard2003}. Based on the observation the momentum |
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consecutive rotations.\cite{Fernandes2002} Unfortunately, unexpected |
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errors and biases are introduced into the system due to the |
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arbitrary order of applying the noncommuting rotation |
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operators.\cite{Beard2003} Based on the observation the momentum |
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relaxation time is much less than the time step, one may ignore the |
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inertia in Brownian dynamics. However, assumption of the zero |
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inertia in Brownian dynamics. However, the assumption of zero |
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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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typical nonskew bodies like cylinder and ellipsoid are inadequate to |
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represent most complex macromolecule assemblies. These intricate |
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typical nonskew bodies like cylinders and ellipsoids are inadequate |
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to represent most complex macromolecule assemblies. These intricate |
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molecules have been represented by a set of beads and their |
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hydrodynamics properties can be calculated using variant |
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hydrodynamic interaction tensors. |
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hydrodynamic properties can be calculated using variants on the |
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standard hydrodynamic interaction tensors. |
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The goal of the present work is to develop a Langevin dynamics |
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algorithm for arbitrary rigid particles by integrating the accurate |
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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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algorithm for arbitrary-shaped rigid particles by integrating the |
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accurate estimation of friction tensor from hydrodynamics theory |
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into the sophisticated rigid body dynamics algorithms. |
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\section{Computational methods{\label{methodSec}}} |
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\section{Computational Methods{\label{methodSec}}} |
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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 becomes |
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impractical when the system becomes more and more complicated. |
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Instead, various approaches based on hydrodynamics have been |
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developed to calculate the friction coefficients. In general, the |
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friction tensor $\Xi$ is a $6\times 6$ matrix given by |
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\[ |
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\Xi = \left( {\begin{array}{*{20}c} |
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{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
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{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
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\end{array}} \right). |
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\] |
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Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
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tensor and rotational resistance (friction) tensor respectively, |
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while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
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{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
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particle moves in a fluid, it may experience friction force or |
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torque along the opposite direction of the velocity or angular |
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velocity, |
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Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are $3 \times 3$ |
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translational friction tensor and rotational resistance (friction) |
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tensor respectively, while ${\Xi^{tr} }$ is translation-rotation |
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coupling tensor and $ {\Xi^{rt} }$ is rotation-translation coupling |
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tensor. When a particle moves in a fluid, it may experience friction |
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force or torque along the opposite direction of the velocity or |
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angular velocity, |
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\[ |
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\left( \begin{array}{l} |
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F_R \\ |
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\end{array} \right) |
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\] |
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where $F_r$ is the friction force and $\tau _R$ is the friction |
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toque. |
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torque. |
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|
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\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
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\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shapes}} |
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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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\[ |
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\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
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{6\pi \eta R} & 0 & 0 \\ |
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\end{array}} \right) |
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\] |
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where $\eta$ is the viscosity of the solvent and $R$ is the |
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hydrodynamics radius. |
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hydrodynamic radius. |
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|
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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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Other non-spherical shapes, such as cylinders and ellipsoids, are |
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widely used as references for developing new hydrodynamics theory, |
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because their properties can be calculated exactly. In 1936, Perrin |
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extended Stokes's law to general ellipsoids, also called a triaxial |
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ellipsoid, which is given in Cartesian coordinates |
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by\cite{Perrin1934, Perrin1936} |
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\[ |
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\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
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}} = 1 |
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\] |
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where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
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due to the complexity of the elliptic integral, only the ellipsoid |
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with the restriction of two axes having to be equal, \textit{i.e.} |
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with the restriction of two axes being 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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ellipsoids : |
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\[ |
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S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
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} }}{b}, |
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\] |
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and oblate, |
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and oblate ellipsoids: |
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\[ |
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S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
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}}{a} |
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\], |
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}}{a}, |
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\] |
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one can write down the translational and rotational resistance |
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tensors |
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\[ |
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\begin{array}{l} |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
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\end{array}, |
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\] |
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\begin{eqnarray*} |
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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 + |
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2a}}, |
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\end{eqnarray*} |
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and |
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\[ |
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\begin{array}{l} |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
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\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
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\end{array}. |
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\] |
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\begin{eqnarray*} |
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\Xi _a^{rr} & = & \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
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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}}. |
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\end{eqnarray*} |
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|
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\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}} |
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\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
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|
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Unlike spherical and other regular shaped molecules, there is not |
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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 |
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is not always suitable for modeling arbitrarily shaped rigid |
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molecule. A number of studies have been devoted to determine the |
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friction tensor for irregularly shaped rigid bodies using more |
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advanced method where the molecule of interest was modeled by |
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combinations of spheres(beads)\cite{Carrasco1999} and the |
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translational and rotational motion of rigid bodies, general |
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ellipsoids are not always suitable for modeling arbitrarily shaped |
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rigid molecules. A number of studies have been devoted to |
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determining the friction tensor for irregularly shaped rigid bodies |
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using more advanced methods where the molecule of interest was |
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modeled by a combinations of spheres\cite{Carrasco1999} and the |
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hydrodynamics properties of the molecule can be calculated using the |
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hydrodynamic interaction tensor. Let us consider a rigid assembly of |
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$N$ beads immersed in a continuous medium. Due to hydrodynamics |
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$N$ beads immersed in a continuous medium. Due to hydrodynamic |
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interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
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than its unperturbed velocity $v_i$, |
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\[ |
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\label{introEquation:tensorExpression} |
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\end{equation} |
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This equation is the basis for deriving the hydrodynamic tensor. In |
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1930, Oseen and Burgers gave a simple solution to Equation |
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\ref{introEquation:tensorExpression} |
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1930, Oseen and Burgers gave a simple solution to |
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Eq.~\ref{introEquation:tensorExpression} |
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\begin{equation} |
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T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
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R_{ij}^T }}{{R_{ij}^2 }}} \right). \label{introEquation:oseenTensor} |
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Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
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A second order expression for element of different size was |
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introduced by Rotne and Prager\cite{Rotne1969} and improved by |
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Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977}, |
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Garc\'{i}a de la Torre and Bloomfield,\cite{Torre1977} |
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\begin{equation} |
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T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
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\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
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\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 232 |
|
\label{introEquation:RPTensorNonOverlapped} |
| 233 |
|
\end{equation} |
| 234 |
< |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
| 235 |
< |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
| 236 |
< |
\ge \sigma _i + \sigma _j$. An alternative expression for |
| 234 |
> |
Both of the Eq.~\ref{introEquation:oseenTensor} and |
| 235 |
> |
Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption |
| 236 |
> |
$R_{ij} \ge \sigma _i + \sigma _j$. An alternative expression for |
| 237 |
|
overlapping beads with the same radius, $\sigma$, is given by |
| 238 |
|
\begin{equation} |
| 239 |
|
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 241 |
|
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 242 |
|
\label{introEquation:RPTensorOverlapped} |
| 243 |
|
\end{equation} |
| 260 |
– |
|
| 244 |
|
To calculate the resistance tensor at an arbitrary origin $O$, we |
| 245 |
|
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
| 246 |
|
$B_{ij}$ blocks |
| 256 |
|
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 257 |
|
)T_{ij} |
| 258 |
|
\] |
| 259 |
< |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
| 260 |
< |
$B$, we obtain |
| 278 |
< |
|
| 259 |
> |
where $\delta _{ij}$ is the Kronecker delta function. Inverting the |
| 260 |
> |
$B$ matrix, we obtain |
| 261 |
|
\[ |
| 262 |
|
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 263 |
|
{C_{11} } & \ldots & {C_{1N} } \\ |
| 264 |
|
\vdots & \ddots & \vdots \\ |
| 265 |
|
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 266 |
< |
\end{array}} \right) |
| 266 |
> |
\end{array}} \right), |
| 267 |
|
\] |
| 268 |
< |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
| 269 |
< |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
| 268 |
> |
which can be partitioned into $N \times N$ $3 \times 3$ block |
| 269 |
> |
$C_{ij}$. With the help of $C_{ij}$ and the skew matrix $U_i$ |
| 270 |
|
\[ |
| 271 |
|
U_i = \left( {\begin{array}{*{20}c} |
| 272 |
|
0 & { - z_i } & {y_i } \\ |
| 275 |
|
\end{array}} \right) |
| 276 |
|
\] |
| 277 |
|
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 278 |
< |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
| 278 |
> |
bead $i$ and origin $O$, the elements of resistance tensor at |
| 279 |
|
arbitrary origin $O$ can be written as |
| 280 |
< |
\begin{equation} |
| 281 |
< |
\begin{array}{l} |
| 282 |
< |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
| 283 |
< |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 302 |
< |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
| 303 |
< |
\end{array} |
| 280 |
> |
\begin{eqnarray} |
| 281 |
> |
\Xi _{}^{tt} & = & \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
| 282 |
> |
\Xi _{}^{tr} & = & \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 283 |
> |
\Xi _{}^{rr} & = & - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
| 284 |
|
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 285 |
< |
\end{equation} |
| 306 |
< |
|
| 285 |
> |
\end{eqnarray} |
| 286 |
|
The resistance tensor depends on the origin to which they refer. The |
| 287 |
< |
proper location for applying friction force is the center of |
| 288 |
< |
resistance (reaction), at which the trace of rotational resistance |
| 289 |
< |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
| 290 |
< |
resistance is defined as an unique point of the rigid body at which |
| 291 |
< |
the translation-rotation coupling tensor are symmetric, |
| 287 |
> |
proper location for applying the friction force is the center of |
| 288 |
> |
resistance (or center of reaction), at which the trace of rotational |
| 289 |
> |
resistance tensor, $ \Xi ^{rr}$ reaches a minimum value. |
| 290 |
> |
Mathematically, the center of resistance is defined as an unique |
| 291 |
> |
point of the rigid body at which the translation-rotation coupling |
| 292 |
> |
tensors are symmetric, |
| 293 |
|
\begin{equation} |
| 294 |
|
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 295 |
|
\label{introEquation:definitionCR} |
| 296 |
|
\end{equation} |
| 297 |
< |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 298 |
< |
we can easily find out that the translational resistance tensor is |
| 297 |
> |
From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 298 |
> |
we can easily derive that the translational resistance tensor is |
| 299 |
|
origin independent, while the rotational resistance tensor and |
| 300 |
|
translation-rotation coupling resistance tensor depend on the |
| 301 |
< |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 302 |
< |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 301 |
> |
origin. Given the resistance tensor at an arbitrary origin $O$, and |
| 302 |
> |
a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 303 |
|
obtain the resistance tensor at $P$ by |
| 304 |
|
\begin{equation} |
| 305 |
|
\begin{array}{l} |
| 317 |
|
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 318 |
|
\end{array}} \right) |
| 319 |
|
\] |
| 320 |
< |
Using Equations \ref{introEquation:definitionCR} and |
| 321 |
< |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
| 322 |
< |
the position of center of resistance, |
| 320 |
> |
Using Eq.~\ref{introEquation:definitionCR} and |
| 321 |
> |
Eq.~\ref{introEquation:resistanceTensorTransformation}, one can |
| 322 |
> |
locate the position of center of resistance, |
| 323 |
|
\begin{eqnarray*} |
| 324 |
|
\left( \begin{array}{l} |
| 325 |
|
x_{OR} \\ |
| 336 |
|
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
| 337 |
|
\end{array} \right) \\ |
| 338 |
|
\end{eqnarray*} |
| 359 |
– |
|
| 339 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 340 |
|
joining center of resistance $R$ and origin $O$. |
| 341 |
|
|
| 342 |
< |
\subsection{Langevin dynamics for rigid particles of arbitrary shape\label{LDRB}} |
| 342 |
> |
\subsection{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
| 343 |
|
|
| 344 |
< |
Consider a Langevin equation of motions in generalized coordinates |
| 344 |
> |
Consider the Langevin equations of motion in generalized coordinates |
| 345 |
|
\begin{equation} |
| 346 |
|
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
| 347 |
|
\label{LDGeneralizedForm} |
| 348 |
|
\end{equation} |
| 349 |
|
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
| 350 |
|
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
| 351 |
< |
$V_i = V_i(v_i,\omega _i)$. The right side of Eq.~ |
| 352 |
< |
(\ref{LDGeneralizedForm}) consists of three generalized forces in |
| 351 |
> |
$V_i = V_i(v_i,\omega _i)$. The right side of |
| 352 |
> |
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces in |
| 353 |
|
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
| 354 |
|
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
| 355 |
|
system in Newtownian mechanics typically refers to lab-fixed frame, |
| 358 |
|
in body-fixed frame and converted back to lab-fixed frame by: |
| 359 |
|
\[ |
| 360 |
|
\begin{array}{l} |
| 361 |
< |
F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\ |
| 362 |
< |
F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\ |
| 363 |
< |
\end{array}. |
| 361 |
> |
F_{f,i}^l (t) = Q^T F_{f,i}^b (t), \\ |
| 362 |
> |
F_{r,i}^l (t) = Q^T F_{r,i}^b (t). \\ |
| 363 |
> |
\end{array} |
| 364 |
|
\] |
| 365 |
|
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
| 366 |
|
the body-fixed velocity at center of resistance $v_{R,i}^b$ and |
| 367 |
< |
angular velocity $\omega _i$, |
| 367 |
> |
angular velocity $\omega _i$ |
| 368 |
|
\begin{equation} |
| 369 |
|
F_{r,i}^b (t) = \left( \begin{array}{l} |
| 370 |
|
f_{r,i}^b (t) \\ |
| 382 |
|
\begin{equation} |
| 383 |
|
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
| 384 |
|
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
| 385 |
< |
2k_B T\Xi _R \delta (t - t'). |
| 385 |
> |
2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
| 386 |
|
\end{equation} |
| 408 |
– |
|
| 387 |
|
The equation of motion for $v_i$ can be written as |
| 388 |
|
\begin{equation} |
| 389 |
|
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
| 398 |
|
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b |
| 399 |
|
\end{equation} |
| 400 |
|
where $r_{MR}$ is the vector from the center of mass to the center |
| 401 |
< |
of the resistance. Instead of integrating angular velocity in |
| 402 |
< |
lab-fixed frame, we consider the equation of motion of angular |
| 403 |
< |
momentum in body-fixed frame |
| 401 |
> |
of the resistance. Instead of integrating the angular velocity in |
| 402 |
> |
lab-fixed frame, we consider the equation of angular momentum in |
| 403 |
> |
body-fixed frame |
| 404 |
|
\begin{equation} |
| 405 |
|
\dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t) |
| 406 |
|
+ \tau _{r,i}^b(t) |
| 407 |
|
\end{equation} |
| 430 |
– |
|
| 408 |
|
Embedding the friction terms into force and torque, one can |
| 409 |
|
integrate the langevin equations of motion for rigid body of |
| 410 |
|
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
| 421 |
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 422 |
|
+ \frac{h}{2} {\bf \tau}^b(t), \\ |
| 423 |
|
% |
| 424 |
< |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 424 |
> |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 425 |
|
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
| 426 |
|
\end{align*} |
| 450 |
– |
|
| 427 |
|
In this context, the $\mathrm{rotate}$ function is the reversible |
| 428 |
|
product of the three body-fixed rotations, |
| 429 |
|
\begin{equation} |
| 432 |
|
/ 2) \cdot \mathsf{G}_x(a_x /2), |
| 433 |
|
\end{equation} |
| 434 |
|
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
| 435 |
< |
rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed |
| 435 |
> |
rotates both the rotation matrix ($\mathsf{Q}$) and the body-fixed |
| 436 |
|
angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed |
| 437 |
|
axis $\alpha$, |
| 438 |
|
\begin{equation} |
| 439 |
|
\mathsf{G}_\alpha( \theta ) = \left\{ |
| 440 |
|
\begin{array}{lcl} |
| 441 |
< |
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 441 |
> |
\mathsf{Q}(t) & \leftarrow & \mathsf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 442 |
|
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
| 443 |
|
j}(0). |
| 444 |
|
\end{array} |
| 458 |
|
\end{array} |
| 459 |
|
\right). |
| 460 |
|
\end{equation} |
| 461 |
< |
All other rotations follow in a straightforward manner. |
| 461 |
> |
All other rotations follow in a straightforward manner. After the |
| 462 |
> |
first part of the propagation, the forces and body-fixed torques are |
| 463 |
> |
calculated at the new positions and orientations |
| 464 |
|
|
| 487 |
– |
After the first part of the propagation, the forces and body-fixed |
| 488 |
– |
torques are calculated at the new positions and orientations |
| 489 |
– |
|
| 465 |
|
{\tt doForces:} |
| 466 |
|
\begin{align*} |
| 467 |
|
{\bf f}(t + h) &\leftarrow |
| 470 |
|
{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
| 471 |
|
\times \frac{\partial V}{\partial {\bf u}}, \\ |
| 472 |
|
% |
| 473 |
< |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
| 473 |
> |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{Q}(t + h) |
| 474 |
|
\cdot {\bf \tau}^s(t + h). |
| 475 |
|
\end{align*} |
| 476 |
+ |
Once the forces and torques have been obtained at the new time step, |
| 477 |
+ |
the velocities can be advanced to the same time value. |
| 478 |
|
|
| 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 |
– |
|
| 479 |
|
{\tt moveB:} |
| 480 |
|
\begin{align*} |
| 481 |
|
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 |
| 487 |
|
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
| 488 |
|
\end{align*} |
| 489 |
|
|
| 490 |
< |
\section{Results and discussion} |
| 490 |
> |
\section{Results and Discussion} |
| 491 |
|
|
| 492 |
< |
The Langevin algorithm described in Sec.~\ref{LDRB} has been |
| 493 |
< |
implemented in {\sc oopse}\cite{Meineke2005} and applied to several |
| 494 |
< |
test systems. |
| 492 |
> |
The Langevin algorithm described in previous section has been |
| 493 |
> |
implemented in {\sc oopse}\cite{Meineke2005} and applied to studies |
| 494 |
> |
of the static and dynamic properties in several systems. |
| 495 |
|
|
| 496 |
< |
\subsection{Langevin dynamics of} |
| 496 |
> |
\subsection{Temperature Control} |
| 497 |
|
|
| 498 |
+ |
As shown in Eq.~\ref{randomForce}, random collisions associated with |
| 499 |
+ |
the solvent's thermal motions is controlled by the external |
| 500 |
+ |
temperature. The capability to maintain the temperature of the whole |
| 501 |
+ |
system was usually used to measure the stability and efficiency of |
| 502 |
+ |
the algorithm. In order to verify the stability of this new |
| 503 |
+ |
algorithm, a series of simulations are performed on system |
| 504 |
+ |
consisiting of 256 SSD water molecules with different viscosities. |
| 505 |
+ |
The initial configuration for the simulations is taken from a 1ns |
| 506 |
+ |
NVT simulation with a cubic box of 19.7166~\AA. All simulation are |
| 507 |
+ |
carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
| 508 |
+ |
with reference temperature at 300~K. The average temperature as a |
| 509 |
+ |
function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
| 510 |
+ |
the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
| 511 |
+ |
1$ poise. The better temperature control at higher viscosity can be |
| 512 |
+ |
explained by the finite size effect and relative slow relaxation |
| 513 |
+ |
rate at lower viscosity regime. |
| 514 |
+ |
\begin{table} |
| 515 |
+ |
\caption{AVERAGE TEMPERATURES FROM LANGEVIN DYNAMICS SIMULATIONS OF |
| 516 |
+ |
SSD WATER MOLECULES WITH REFERENCE TEMPERATURE AT 300~K.} |
| 517 |
+ |
\label{langevin:viscosity} |
| 518 |
+ |
\begin{center} |
| 519 |
+ |
\begin{tabular}{lll} |
| 520 |
+ |
\hline |
| 521 |
+ |
$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
| 522 |
+ |
\hline |
| 523 |
+ |
1 & 300.47 & 10.99 \\ |
| 524 |
+ |
0.1 & 301.19 & 11.136 \\ |
| 525 |
+ |
0.01 & 303.04 & 11.796 \\ |
| 526 |
+ |
\hline |
| 527 |
+ |
\end{tabular} |
| 528 |
+ |
\end{center} |
| 529 |
+ |
\end{table} |
| 530 |
+ |
|
| 531 |
+ |
Another set of calculations were performed to study the efficiency of |
| 532 |
+ |
temperature control using different temperature coupling schemes. |
| 533 |
+ |
The starting configuration is cooled to 173~K and evolved using NVE, |
| 534 |
+ |
NVT, and Langevin dynamic with time step of 2 fs. |
| 535 |
+ |
Fig.~\ref{langevin:temperature} shows the heating curve obtained as |
| 536 |
+ |
the systems reach equilibrium. The orange curve in |
| 537 |
+ |
Fig.~\ref{langevin:temperature} represents the simulation using |
| 538 |
+ |
Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
| 539 |
+ |
which gives reasonable tight coupling, while the blue one from |
| 540 |
+ |
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
| 541 |
+ |
scaling to the desire temperature. When $ \eta = 0$, Langevin dynamics becomes normal |
| 542 |
+ |
NVE (see orange curve in Fig.~\ref{langevin:temperature}) which |
| 543 |
+ |
loses the temperature control ability. |
| 544 |
+ |
|
| 545 |
|
\begin{figure} |
| 546 |
|
\centering |
| 547 |
|
\includegraphics[width=\linewidth]{temperature.eps} |
| 548 |
< |
\caption[]{.} \label{langevin:temperature} |
| 548 |
> |
\caption[Plot of Temperature Fluctuation Versus Time]{Plot of |
| 549 |
> |
temperature fluctuation versus time.} \label{langevin:temperature} |
| 550 |
|
\end{figure} |
| 551 |
|
|
| 552 |
< |
\subsection{LD of banana-shaped molecule} |
| 552 |
> |
\subsection{Langevin Dynamics of Banana Shaped Molecules} |
| 553 |
|
|
| 554 |
+ |
In order to verify that Langevin dynamics can mimic the dynamics of |
| 555 |
+ |
the systems absent of explicit solvents, we carried out two sets of |
| 556 |
+ |
simulations and compare their dynamic properties. |
| 557 |
+ |
Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
| 558 |
+ |
made of 256 pentane molecules and two banana shaped molecules at |
| 559 |
+ |
273~K. It has an equivalent implicit solvent system containing only |
| 560 |
+ |
two banana shaped molecules with viscosity of 0.289 center poise. To |
| 561 |
+ |
calculate the hydrodynamic properties of the banana shaped molecule, |
| 562 |
+ |
we created a rough shell model (see Fig.~\ref{langevin:roughShell}), |
| 563 |
+ |
in which the banana shaped molecule is represented as a ``shell'' |
| 564 |
+ |
made of 2266 small identical beads with size of 0.3 \AA on the |
| 565 |
+ |
surface. Applying the procedure described in |
| 566 |
+ |
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 567 |
+ |
identified the center of resistance at (0 $\rm{\AA}$, 0.7482 $\rm{\AA}$, |
| 568 |
+ |
-0.1988 $\rm{\AA}$), as well as the resistance tensor, |
| 569 |
+ |
\[ |
| 570 |
+ |
\left( {\begin{array}{*{20}c} |
| 571 |
+ |
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
| 572 |
+ |
0& 0.9270&-0.007063& 0.08585&0&0\\ |
| 573 |
+ |
0&-0.007063&0.7494&0.2057&0&0\\ |
| 574 |
+ |
0&0.0858&0.2057& 58.64& 0&0\\ |
| 575 |
+ |
0.08585&0&0&0&48.30&3.219&\\ |
| 576 |
+ |
0.2057&0&0&0&3.219&10.7373\\ |
| 577 |
+ |
\end{array}} \right). |
| 578 |
+ |
\] |
| 579 |
+ |
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. |
| 580 |
+ |
Curves of the velocity auto-correlation functions in |
| 581 |
+ |
Fig.~\ref{langevin:vacf} were shown to match each other very well. |
| 582 |
+ |
However, because of the stochastic nature, simulation using Langevin |
| 583 |
+ |
dynamics was shown to decay slightly faster than MD. In order to |
| 584 |
+ |
study the rotational motion of the molecules, we also calculated the |
| 585 |
+ |
auto-correlation function of the principle axis of the second GB |
| 586 |
+ |
particle, $u$. The discrepancy shown in Fig.~\ref{langevin:uacf} was |
| 587 |
+ |
probably due to the reason that we used the experimental viscosity directly instead of calculating bulk viscosity from simulation. |
| 588 |
+ |
|
| 589 |
|
\begin{figure} |
| 590 |
|
\centering |
| 591 |
< |
\includegraphics[width=\linewidth]{one_banana.eps} |
| 592 |
< |
\caption[]{.} \label{langevin:banana} |
| 591 |
> |
\includegraphics[width=\linewidth]{roughShell.eps} |
| 592 |
> |
\caption[Rough shell model for banana shaped molecule]{Rough shell |
| 593 |
> |
model for banana shaped molecule.} \label{langevin:roughShell} |
| 594 |
|
\end{figure} |
| 595 |
|
|
| 596 |
+ |
\begin{figure} |
| 597 |
+ |
\centering |
| 598 |
+ |
\includegraphics[width=\linewidth]{twoBanana.eps} |
| 599 |
+ |
\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
| 600 |
+ |
256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
| 601 |
+ |
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
| 602 |
+ |
\end{figure} |
| 603 |
+ |
|
| 604 |
+ |
\begin{figure} |
| 605 |
+ |
\centering |
| 606 |
+ |
\includegraphics[width=\linewidth]{vacf.eps} |
| 607 |
+ |
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
| 608 |
+ |
auto-correlation functions of NVE (explicit solvent) in blue and |
| 609 |
+ |
Langevin dynamics (implicit solvent) in red.} \label{langevin:vacf} |
| 610 |
+ |
\end{figure} |
| 611 |
+ |
|
| 612 |
+ |
\begin{figure} |
| 613 |
+ |
\centering |
| 614 |
+ |
\includegraphics[width=\linewidth]{uacf.eps} |
| 615 |
+ |
\caption[Auto-correlation functions of the principle axis of the |
| 616 |
+ |
middle GB particle]{Auto-correlation functions of the principle axis |
| 617 |
+ |
of the middle GB particle of NVE (blue) and Langevin dynamics |
| 618 |
+ |
(red).} \label{langevin:uacf} |
| 619 |
+ |
\end{figure} |
| 620 |
+ |
|
| 621 |
|
\section{Conclusions} |
| 622 |
+ |
|
| 623 |
+ |
We have presented a new Langevin algorithm by incorporating the |
| 624 |
+ |
hydrodynamics properties of arbitrary shaped molecules into an |
| 625 |
+ |
advanced symplectic integration scheme. The temperature control |
| 626 |
+ |
ability of this algorithm was demonstrated by a set of simulations |
| 627 |
+ |
with different viscosities. It was also shown to have significant |
| 628 |
+ |
advantage of producing rapid thermal equilibration over |
| 629 |
+ |
Nos\'{e}-Hoover method. Further studies in systems involving banana |
| 630 |
+ |
shaped molecules illustrated that the dynamic properties could be |
| 631 |
+ |
preserved by using this new algorithm as an implicit solvent model. |
