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\chapter{\label{chapt:methodology}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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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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|
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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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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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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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%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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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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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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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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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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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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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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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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\section{Method{\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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\[ |
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\Xi = \left( {\begin{array}{*{20}c} |
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{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
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{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
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\end{array}} \right). |
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\] |
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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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\[ |
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\left( \begin{array}{l} |
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F_R \\ |
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\tau _R \\ |
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\end{array} \right) = - \left( {\begin{array}{*{20}c} |
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{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
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{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
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\end{array}} \right)\left( \begin{array}{l} |
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v \\ |
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w \\ |
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\end{array} \right) |
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\] |
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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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\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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\[ |
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\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
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{6\pi \eta R} & 0 & 0 \\ |
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0 & {6\pi \eta R} & 0 \\ |
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0 & 0 & {6\pi \eta R} \\ |
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\end{array}} \right) |
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\] |
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and |
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\[ |
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\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
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{8\pi \eta R^3 } & 0 & 0 \\ |
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0 & {8\pi \eta R^3 } & 0 \\ |
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0 & 0 & {8\pi \eta R^3 } \\ |
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\end{array}} \right) |
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\] |
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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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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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\[ |
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\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
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}} = 1 |
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\] |
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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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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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\[ |
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S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
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} }}{b}, |
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\] |
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and oblate, |
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\[ |
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S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
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}}{a} |
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\], |
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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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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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\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}} |
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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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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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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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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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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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\[ |
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v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
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\] |
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where $F_i$ is the frictional force, and $T_{ij}$ is the |
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hydrodynamic interaction tensor. The friction force of $i$th bead is |
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proportional to its ``net'' velocity |
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\begin{equation} |
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F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
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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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\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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\end{equation} |
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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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\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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_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
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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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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} }}{\sigma }} \right)I + |
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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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\begin{equation} |
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B = \left( {\begin{array}{*{20}c} |
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{B_{11} } & \ldots & {B_{1N} } \\ |
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\vdots & \ddots & \vdots \\ |
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{B_{N1} } & \cdots & {B_{NN} } \\ |
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\end{array}} \right), |
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\end{equation} |
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where $B_{ij}$ is given by |
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\[ |
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B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
327 |
|
|
)T_{ij} |
328 |
|
|
\] |
329 |
|
|
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
330 |
|
|
$B$, we obtain |
331 |
|
|
|
332 |
|
|
\[ |
333 |
|
|
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
334 |
|
|
{C_{11} } & \ldots & {C_{1N} } \\ |
335 |
|
|
\vdots & \ddots & \vdots \\ |
336 |
|
|
{C_{N1} } & \cdots & {C_{NN} } \\ |
337 |
|
|
\end{array}} \right) |
338 |
|
|
\] |
339 |
|
|
, which can be partitioned into $N \times N$ $3 \times 3$ block |
340 |
|
|
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
341 |
|
|
\[ |
342 |
|
|
U_i = \left( {\begin{array}{*{20}c} |
343 |
|
|
0 & { - z_i } & {y_i } \\ |
344 |
|
|
{z_i } & 0 & { - x_i } \\ |
345 |
|
|
{ - y_i } & {x_i } & 0 \\ |
346 |
|
|
\end{array}} \right) |
347 |
|
|
\] |
348 |
|
|
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
349 |
|
|
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
350 |
|
|
arbitrary origin $O$ can be written as |
351 |
|
|
\begin{equation} |
352 |
|
|
\begin{array}{l} |
353 |
|
|
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
354 |
|
|
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
355 |
|
|
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
356 |
|
|
\end{array} |
357 |
|
|
\label{introEquation:ResistanceTensorArbitraryOrigin} |
358 |
|
|
\end{equation} |
359 |
|
|
|
360 |
|
|
The resistance tensor depends on the origin to which they refer. The |
361 |
|
|
proper location for applying friction force is the center of |
362 |
|
|
resistance (reaction), at which the trace of rotational resistance |
363 |
|
|
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
364 |
|
|
resistance is defined as an unique point of the rigid body at which |
365 |
|
|
the translation-rotation coupling tensor are symmetric, |
366 |
|
|
\begin{equation} |
367 |
|
|
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
368 |
|
|
\label{introEquation:definitionCR} |
369 |
|
|
\end{equation} |
370 |
|
|
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
371 |
|
|
we can easily find out that the translational resistance tensor is |
372 |
|
|
origin independent, while the rotational resistance tensor and |
373 |
|
|
translation-rotation coupling resistance tensor depend on the |
374 |
|
|
origin. Given resistance tensor at an arbitrary origin $O$, and a |
375 |
|
|
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
376 |
|
|
obtain the resistance tensor at $P$ by |
377 |
|
|
\begin{equation} |
378 |
|
|
\begin{array}{l} |
379 |
|
|
\Xi _P^{tt} = \Xi _O^{tt} \\ |
380 |
|
|
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
381 |
|
|
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{{tr} ^{^T }} \\ |
382 |
|
|
\end{array} |
383 |
|
|
\label{introEquation:resistanceTensorTransformation} |
384 |
|
|
\end{equation} |
385 |
|
|
where |
386 |
|
|
\[ |
387 |
|
|
U_{OP} = \left( {\begin{array}{*{20}c} |
388 |
|
|
0 & { - z_{OP} } & {y_{OP} } \\ |
389 |
|
|
{z_i } & 0 & { - x_{OP} } \\ |
390 |
|
|
{ - y_{OP} } & {x_{OP} } & 0 \\ |
391 |
|
|
\end{array}} \right) |
392 |
|
|
\] |
393 |
|
|
Using Equations \ref{introEquation:definitionCR} and |
394 |
|
|
\ref{introEquation:resistanceTensorTransformation}, one can locate |
395 |
|
|
the position of center of resistance, |
396 |
|
|
\begin{eqnarray*} |
397 |
|
|
\left( \begin{array}{l} |
398 |
|
|
x_{OR} \\ |
399 |
|
|
y_{OR} \\ |
400 |
|
|
z_{OR} \\ |
401 |
|
|
\end{array} \right) & = &\left( {\begin{array}{*{20}c} |
402 |
|
|
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
403 |
|
|
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
404 |
|
|
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
405 |
|
|
\end{array}} \right)^{ - 1} \\ |
406 |
|
|
& & \left( \begin{array}{l} |
407 |
|
|
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
408 |
|
|
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
409 |
|
|
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
410 |
|
|
\end{array} \right) \\ |
411 |
|
|
\end{eqnarray*} |
412 |
|
|
|
413 |
|
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
414 |
|
|
joining center of resistance $R$ and origin $O$. |
415 |
|
|
|
416 |
|
|
\subsection{Langevin dynamics for rigid particles of arbitrary shape} |
417 |
|
|
|
418 |
|
|
Consider a Langevin equation of motions in generalized coordinates |
419 |
|
|
\begin{equation} |
420 |
|
|
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
421 |
|
|
\label{LDGeneralizedForm} |
422 |
|
|
\end{equation} |
423 |
|
|
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
424 |
|
|
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
425 |
|
|
$V_i = V_i(v_i,\omega _i)$. The right side of Eq. |
426 |
|
|
(\ref{LDGeneralizedForm}) consists of three generalized forces in |
427 |
|
|
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
428 |
|
|
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
429 |
|
|
system in Newtownian mechanics typically refers to lab-fixed frame, |
430 |
|
|
it is also convenient to handle the rotation of rigid body in |
431 |
|
|
body-fixed frame. Thus the friction and random forces are calculated |
432 |
|
|
in body-fixed frame and converted back to lab-fixed frame by: |
433 |
|
|
\[ |
434 |
|
|
\begin{array}{l} |
435 |
|
|
F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\ |
436 |
|
|
F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\ |
437 |
|
|
\end{array}. |
438 |
|
|
\] |
439 |
|
|
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
440 |
|
|
the body-fixed velocity at center of resistance $v_{R,i}^b$ and |
441 |
|
|
angular velocity $\omega _i$, |
442 |
|
|
\begin{equation} |
443 |
|
|
F_{r,i}^b (t) = \left( \begin{array}{l} |
444 |
|
|
f_{r,i}^b (t) \\ |
445 |
|
|
\tau _{r,i}^b (t) \\ |
446 |
|
|
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
447 |
|
|
{\Xi _{R,t} } & {\Xi _{R,c}^T } \\ |
448 |
|
|
{\Xi _{R,c} } & {\Xi _{R,r} } \\ |
449 |
|
|
\end{array}} \right)\left( \begin{array}{l} |
450 |
|
|
v_{R,i}^b (t) \\ |
451 |
|
|
\omega _i (t) \\ |
452 |
|
|
\end{array} \right), |
453 |
|
|
\end{equation} |
454 |
|
|
while the random force $F_{r,i}^l$ is a Gaussian stochastic variable |
455 |
|
|
with zero mean and variance |
456 |
|
|
\begin{equation} |
457 |
|
|
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
458 |
|
|
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
459 |
|
|
2k_B T\Xi _R \delta (t - t'). |
460 |
|
|
\end{equation} |
461 |
|
|
|
462 |
|
|
The equation of motion for $v_i$ can be written as |
463 |
|
|
\begin{equation} |
464 |
|
|
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
465 |
|
|
f_{r,i}^l (t) |
466 |
|
|
\end{equation} |
467 |
|
|
Since the frictional force is applied at the center of resistance |
468 |
|
|
which generally does not coincide with the center of mass, an extra |
469 |
|
|
torque is exerted at the center of mass. Thus, the net body-fixed |
470 |
|
|
frictional torque at the center of mass, $\tau _{n,i}^b (t)$, is |
471 |
|
|
given by |
472 |
|
|
\begin{equation} |
473 |
|
|
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b |
474 |
|
|
\end{equation} |
475 |
|
|
where $r_{MR}$ is the vector from the center of mass to the center |
476 |
|
|
of the resistance. Instead of integrating angular velocity in |
477 |
|
|
lab-fixed frame, we consider the equation of motion of angular |
478 |
|
|
momentum in body-fixed frame |
479 |
|
|
\begin{equation} |
480 |
|
|
\dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t) |
481 |
|
|
+ \tau _{r,i}^b(t) |
482 |
|
|
\end{equation} |
483 |
|
|
|
484 |
|
|
Embedding the friction terms into force and torque, one can |
485 |
|
|
integrate the langevin equations of motion for rigid body of |
486 |
|
|
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
487 |
|
|
$h= \delta t$: |
488 |
|
|
|
489 |
|
|
{\tt moveA:} |
490 |
|
|
\begin{align*} |
491 |
|
|
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
492 |
|
|
+ \frac{h}{2} \left( {\bf f}(t) / m \right), \\ |
493 |
|
|
% |
494 |
|
|
{\bf r}(t + h) &\leftarrow {\bf r}(t) |
495 |
|
|
+ h {\bf v}\left(t + h / 2 \right), \\ |
496 |
|
|
% |
497 |
|
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
498 |
|
|
+ \frac{h}{2} {\bf \tau}^b(t), \\ |
499 |
|
|
% |
500 |
|
|
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
501 |
|
|
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
502 |
|
|
\end{align*} |
503 |
|
|
|
504 |
|
|
In this context, the $\mathrm{rotate}$ function is the reversible |
505 |
|
|
product of the three body-fixed rotations, |
506 |
|
|
\begin{equation} |
507 |
|
|
\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
508 |
|
|
\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y |
509 |
|
|
/ 2) \cdot \mathsf{G}_x(a_x /2), |
510 |
|
|
\end{equation} |
511 |
|
|
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
512 |
|
|
rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed |
513 |
|
|
angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed |
514 |
|
|
axis $\alpha$, |
515 |
|
|
\begin{equation} |
516 |
|
|
\mathsf{G}_\alpha( \theta ) = \left\{ |
517 |
|
|
\begin{array}{lcl} |
518 |
|
|
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
519 |
|
|
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
520 |
|
|
j}(0). |
521 |
|
|
\end{array} |
522 |
|
|
\right. |
523 |
|
|
\end{equation} |
524 |
|
|
$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis |
525 |
|
|
rotation matrix. For example, in the small-angle limit, the |
526 |
|
|
rotation matrix around the body-fixed x-axis can be approximated as |
527 |
|
|
\begin{equation} |
528 |
|
|
\mathsf{R}_x(\theta) \approx \left( |
529 |
|
|
\begin{array}{ccc} |
530 |
|
|
1 & 0 & 0 \\ |
531 |
|
|
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
532 |
|
|
\theta^2 / 4} \\ |
533 |
|
|
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
534 |
|
|
\theta^2 / 4} |
535 |
|
|
\end{array} |
536 |
|
|
\right). |
537 |
|
|
\end{equation} |
538 |
|
|
All other rotations follow in a straightforward manner. |
539 |
|
|
|
540 |
|
|
After the first part of the propagation, the forces and body-fixed |
541 |
|
|
torques are calculated at the new positions and orientations |
542 |
|
|
|
543 |
|
|
{\tt doForces:} |
544 |
|
|
\begin{align*} |
545 |
|
|
{\bf f}(t + h) &\leftarrow |
546 |
|
|
- \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\ |
547 |
|
|
% |
548 |
|
|
{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
549 |
|
|
\times \frac{\partial V}{\partial {\bf u}}, \\ |
550 |
|
|
% |
551 |
|
|
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
552 |
|
|
\cdot {\bf \tau}^s(t + h). |
553 |
|
|
\end{align*} |
554 |
|
|
|
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 |
|
|
|
560 |
|
|
{\tt moveB:} |
561 |
|
|
\begin{align*} |
562 |
|
|
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 |
563 |
|
|
\right) |
564 |
|
|
+ \frac{h}{2} \left( {\bf f}(t + h) / m \right), \\ |
565 |
|
|
% |
566 |
|
|
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t + h / 2 |
567 |
|
|
\right) |
568 |
|
|
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
569 |
|
|
\end{align*} |
570 |
|
|
|
571 |
|
|
\section{Results and discussion} |
572 |
|
|
|
573 |
|
|
\section{Conclusions} |