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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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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 |
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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 |
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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 |
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the observation that forces between distant atoms generally |
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demonstrate slower fluctuations than forces between close atoms, MTS |
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method are generally implemented by evaluating the slowly |
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fluctuating forces less frequently than the fast ones. |
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Unfortunately, nonlinear instability resulting from increasing |
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timestep in MTS simulation have became a critical obstruction |
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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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%review rigid body dynamics |
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Rigid bodies are frequently involved in the modeling of different |
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areas, from engineering, physics, to chemistry. For example, |
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missiles and vehicle are usually modeled by rigid bodies. The |
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movement of the objects in 3D gaming engine or other physics |
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simulator is governed by the rigid body dynamics. In molecular |
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simulation, rigid body is used to simplify the model in |
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protein-protein docking study{\cite{Gray2003}}. |
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|
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It is very important to develop stable and efficient methods to |
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integrate the equations of motion of orientational degrees of |
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freedom. Euler angles are the nature choice to describe the |
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rotational degrees of freedom. However, due to its singularity, the |
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numerical integration of corresponding equations of motion is very |
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inefficient and inaccurate. Although an alternative integrator using |
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different sets of Euler angles can overcome this difficulty\cite{}, |
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the computational penalty and the lost of angular momentum |
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conservation still remain. In 1977, a singularity free |
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representation utilizing quaternions was developed by |
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Evans\cite{Evans1977}. Unfortunately, this approach suffer from the |
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nonseparable Hamiltonian resulted from quaternion representation, |
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which prevents the symplectic algorithm to be utilized. Another |
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different approach is to apply holonomic constraints to the atoms |
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belonging to the rigid body\cite{}. Each atom moves independently |
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under the normal forces deriving from potential energy and |
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constraint forces which are used to guarantee the rigidness. |
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However, due to their iterative nature, SHAKE and Rattle algorithm |
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converge very slowly when the number of constraint increases. |
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|
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The break through in geometric literature suggests that, in order to |
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develop a long-term integration scheme, one should preserve the |
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geometric structure of the flow. Matubayasi and Nakahara developed a |
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time-reversible integrator for rigid bodies in quaternion |
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representation. Although it is not symplectic, this integrator still |
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demonstrates a better long-time energy conservation than traditional |
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methods because of the time-reversible nature. Extending |
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Trotter-Suzuki to general system with a flat phase space, Miller and |
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his colleagues devised an novel symplectic, time-reversible and |
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volume-preserving integrator in quaternion representation. However, |
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all of the integrators in quaternion representation suffer from the |
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computational penalty of constructing a rotation matrix from |
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quaternions to evolve coordinates and velocities at every time step. |
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An alternative integration scheme utilizing rotation matrix directly |
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is RSHAKE , in which a conjugate momentum to rotation matrix is |
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introduced to re-formulate the Hamiltonian's equation and the |
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Hamiltonian is evolved in a constraint manifold by iteratively |
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satisfying the orthogonality constraint. However, RSHAKE is |
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inefficient because of the iterative procedure. An extremely |
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efficient integration scheme in rotation matrix representation, |
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which also preserves the same structural properties of the |
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Hamiltonian flow as Miller's integrator, is proposed by Dullweber, |
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Leimkuhler and McLachlan (DLM). |
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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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one can study the slow processes in biomolecular systems. Modeling |
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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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\section{Method{\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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\subsection{\label{introSection:frictionTensor}Friction Tensor} |
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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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\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
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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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\[ |
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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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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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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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\[ |
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S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
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} }}{b}, |
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\] |
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and oblate, |
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and oblate : |
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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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\[ |
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\begin{array}{l} |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
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\end{array}, |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
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2a}}, |
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\end{array} |
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\] |
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and |
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\[ |
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\begin{array}{l} |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
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\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
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\end{array}. |
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\] |
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|
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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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translational and rotational motion of rigid body, general |
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ellipsoids are not always suitable for modeling arbitrarily shaped |
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rigid molecule. A number of studies have been devoted to determine |
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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 |
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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 |
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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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\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
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\label{introEquation:RPTensorNonOverlapped} |
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\end{equation} |
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Both of the Equation \ref{introEquation:oseenTensor} and Equation |
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\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
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\ge \sigma _i + \sigma _j$. An alternative expression for |
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Both of the Eq.~\ref{introEquation:oseenTensor} and |
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Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption |
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$R_{ij} \ge \sigma _i + \sigma _j$. An alternative expression for |
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overlapping beads with the same radius, $\sigma$, is given by |
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\begin{equation} |
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T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
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\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
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\label{introEquation:RPTensorOverlapped} |
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\end{equation} |
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To calculate the resistance tensor at an arbitrary origin $O$, we |
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construct a $3N \times 3N$ matrix consisting of $N \times N$ |
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$B_{ij}$ blocks |
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B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
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)T_{ij} |
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\] |
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where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
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$B$, we obtain |
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|
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where $\delta _{ij}$ is Kronecker delta function. Inverting the $B$ |
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matrix, we obtain |
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\[ |
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C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
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{C_{11} } & \ldots & {C_{1N} } \\ |
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\vdots & \ddots & \vdots \\ |
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{C_{N1} } & \cdots & {C_{NN} } \\ |
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\end{array}} \right) |
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\end{array}} \right), |
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\] |
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, which can be partitioned into $N \times N$ $3 \times 3$ block |
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which can be partitioned into $N \times N$ $3 \times 3$ block |
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$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
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\[ |
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U_i = \left( {\begin{array}{*{20}c} |
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where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
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bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
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arbitrary origin $O$ can be written as |
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\begin{equation} |
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\begin{array}{l} |
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\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
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\begin{eqnarray} |
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\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
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\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
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\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
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\end{array} |
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\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
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\label{introEquation:ResistanceTensorArbitraryOrigin} |
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\end{equation} |
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\end{eqnarray} |
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|
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The resistance tensor depends on the origin to which they refer. The |
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proper location for applying friction force is the center of |
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resistance (reaction), at which the trace of rotational resistance |
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tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
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resistance is defined as an unique point of the rigid body at which |
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the translation-rotation coupling tensor are symmetric, |
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resistance (or center of reaction), at which the trace of rotational |
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resistance tensor, $ \Xi ^{rr}$ reaches a minimum value. |
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Mathematically, the center of resistance is defined as an unique |
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point of the rigid body at which the translation-rotation coupling |
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tensor are symmetric, |
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\begin{equation} |
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\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
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\label{introEquation:definitionCR} |
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\end{equation} |
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Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
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From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
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we can easily find out that the translational resistance tensor is |
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origin independent, while the rotational resistance tensor and |
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translation-rotation coupling resistance tensor depend on the |
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origin. Given resistance tensor at an arbitrary origin $O$, and a |
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vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
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origin. Given the resistance tensor at an arbitrary origin $O$, and |
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a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
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obtain the resistance tensor at $P$ by |
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\begin{equation} |
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\begin{array}{l} |
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{ - y_{OP} } & {x_{OP} } & 0 \\ |
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\end{array}} \right) |
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\] |
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Using Equations \ref{introEquation:definitionCR} and |
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\ref{introEquation:resistanceTensorTransformation}, one can locate |
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the position of center of resistance, |
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Using Eq.~\ref{introEquation:definitionCR} and |
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Eq.~\ref{introEquation:resistanceTensorTransformation}, one can |
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locate the position of center of resistance, |
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\begin{eqnarray*} |
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\left( \begin{array}{l} |
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x_{OR} \\ |
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(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
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\end{array} \right) \\ |
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\end{eqnarray*} |
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|
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where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
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joining center of resistance $R$ and origin $O$. |
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|
| 360 |
< |
\subsection{Langevin dynamics for rigid particles of arbitrary shape} |
| 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} |
| 461 |
– |
|
| 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} |
| 483 |
– |
|
| 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*} |
| 503 |
– |
|
| 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 |
|
|
| 540 |
– |
After the first part of the propagation, the forces and body-fixed |
| 541 |
– |
torques are calculated at the new positions and orientations |
| 542 |
– |
|
| 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 |
|
|
| 555 |
– |
{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
| 556 |
– |
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
| 557 |
– |
torques have been obtained at the new time step, the velocities can |
| 558 |
– |
be advanced to the same time value. |
| 559 |
– |
|
| 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 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{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[Plot of Temperature Fluctuation Versus Time]{Plot of |
| 568 |
+ |
temperature fluctuation versus time.} \label{langevin:temperature} |
| 569 |
+ |
\end{figure} |
| 570 |
+ |
|
| 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]{roughShell.eps} |
| 620 |
+ |
\caption[Rough shell model for banana shaped molecule]{Rough shell |
| 621 |
+ |
model for banana shaped molecule.} \label{langevin:roughShell} |
| 622 |
+ |
\end{figure} |
| 623 |
+ |
|
| 624 |
+ |
\begin{figure} |
| 625 |
+ |
\centering |
| 626 |
+ |
\includegraphics[width=\linewidth]{twoBanana.eps} |
| 627 |
+ |
\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
| 628 |
+ |
256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
| 629 |
+ |
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
| 630 |
+ |
\end{figure} |
| 631 |
+ |
|
| 632 |
+ |
\begin{figure} |
| 633 |
+ |
\centering |
| 634 |
+ |
\includegraphics[width=\linewidth]{vacf.eps} |
| 635 |
+ |
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
| 636 |
+ |
auto-correlation functions in NVE (blue) and Langevin dynamics |
| 637 |
+ |
(red).} \label{langevin:vacf} |
| 638 |
+ |
\end{figure} |
| 639 |
+ |
|
| 640 |
+ |
\begin{figure} |
| 641 |
+ |
\centering |
| 642 |
+ |
\includegraphics[width=\linewidth]{uacf.eps} |
| 643 |
+ |
\caption[Auto-correlation functions of the principle axis of the |
| 644 |
+ |
middle GB particle]{Auto-correlation functions of the principle axis |
| 645 |
+ |
of the middle GB particle in NVE (blue) and Langevin dynamics |
| 646 |
+ |
(red).} \label{langevin:twoBanana} |
| 647 |
+ |
\end{figure} |
| 648 |
+ |
|
| 649 |
|
\section{Conclusions} |
| 650 |
+ |
|
| 651 |
+ |
We have presented a new Langevin algorithm by incorporating the |
| 652 |
+ |
hydrodynamics properties of arbitrary shaped molecules into an |
| 653 |
+ |
advanced symplectic integration scheme. The temperature control |
| 654 |
+ |
ability of this algorithm was demonstrated by a set of simulations |
| 655 |
+ |
with different viscosities. It was also shown to have significant |
| 656 |
+ |
advantage of producing rapid thermal equilibration over |
| 657 |
+ |
Nos\'{e}-Hoover method. Further studies in systems involving banana |
| 658 |
+ |
shaped molecules illustrated that the dynamic properties could be |
| 659 |
+ |
preserved by using this new algorithm as an implicit solvent model. |
