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\begin{document} |
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\title{An algorithm for performing Langevin dynamics on rigid bodies of arbitrary shape } |
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\author{Xiuquan Sun, Teng Lin and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: |
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gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle \doublespacing |
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\begin{abstract} |
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\end{abstract} |
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\newpage |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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%applications of langevin dynamics |
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Langevin dynamics, which mimics a simple heat bath with stochastic and |
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dissipative forces, has been applied in a variety of situations as an |
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alternative to molecular dynamics with explicit solvent molecules. |
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The stochastic treatment of the solvent allows the use of simulations |
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with substantially longer time and length scales. In general, the |
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dynamic and structural properties obtained from Langevin simulations |
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agree quite well with similar properties obtained from explicit |
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solvent simulations. |
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Recent examples of the usefulness of Langevin simulations include a |
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study of met-enkephalin in which Langevin simulations predicted |
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dynamical properties that were largely in agreement with explicit |
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solvent simulations.\cite{Shen2002} By applying Langevin dynamics with |
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the UNRES model, Liow and his coworkers suggest that protein folding |
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pathways can be explored within a reasonable amount of |
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time.\cite{Liwo2005} |
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The stochastic nature of Langevin dynamics also enhances the sampling |
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of the system and increases the probability of crossing energy |
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barriers.\cite{Cui2003,Banerjee2004} Combining Langevin dynamics with |
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Kramers' theory, Klimov and Thirumalai identified free-energy |
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barriers by studying the viscosity dependence of the protein folding |
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rates.\cite{Klimov1997} In order to account for solvent induced |
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interactions missing from the implicit solvent model, Kaya |
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incorporated a desolvation free energy barrier into protein |
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folding/unfolding studies and discovered a higher free energy barrier |
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between the native and denatured states.\cite{HuseyinKaya07012005} |
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|
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Because of its stability against noise, Langevin dynamics has also |
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proven useful for studying remagnetization processes in various |
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systems.\cite{Palacios1998,Berkov2002,Denisov2003} [Check: For |
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instance, the oscillation power spectrum of nanoparticles from |
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Langevin dynamics 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}] |
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|
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In typical LD simulations, the friction and random forces on |
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individual atoms are taken from Stokes' law, |
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\begin{eqnarray} |
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m \dot{v}(t) & = & -\nabla U(x) - \xi m v(t) + R(t) \\ |
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\langle R(t) \rangle & = & 0 \\ |
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\langle R(t) R(t') \rangle & = & 2 k_B T \xi m \delta(t - t') |
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\end{eqnarray} |
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where $\xi \approx 6 \pi \eta a$. Here $\eta$ is the viscosity of the |
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implicit solvent, and $a$ is the hydrodynamic radius of the atom. |
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|
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The use of rigid substructures,\cite{Chun:2000fj} |
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coarse-graining,\cite{Ayton01,Golubkov06,Orlandi:2006fk,SunGezelter08} |
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and ellipsoidal representations of protein side chains~\cite{Fogolari:1996lr} |
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has made the use of the Stokes-Einstein approximation problematic. A |
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rigid substructure moves as a single unit with orientational as well |
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as translational degrees of freedom. This requires a more general |
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treatment of the hydrodynamics than the spherical approximation |
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provides. The atoms involved in a rigid or coarse-grained structure |
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should properly have solvent-mediated interactions with each |
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other. The theory of interactions {\it between} bodies moving through |
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a fluid has been developed over the past century and has been applied |
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to simulations of Brownian |
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motion.\cite{FIXMAN:1986lr,Ramachandran1996} |
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|
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In order to account for the diffusion anisotropy of arbitrarily-shaped |
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particles, Fernandes and Garc\'{i}a de la Torre improved the original |
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Brownian 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 rotational evolution scheme consisting of three |
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consecutive rotations.\cite{Fernandes2002} Unfortunately, biases are |
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introduced into the system due to the arbitrary order of applying the |
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noncommuting rotation operators.\cite{Beard2003} Based on the |
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observation the momentum relaxation time is much less than the time |
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step, one may ignore the inertia in Brownian dynamics. However, the |
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assumption of zero average acceleration is not always true for |
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cooperative motion which is common in proteins. An inertial Brownian |
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dynamics (IBD) was proposed to address this issue by adding an |
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inertial correction term.\cite{Beard2000} As a complement to IBD which |
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has a lower bound in time step because of the inertial relaxation |
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time, long-time-step inertial dynamics (LTID) can be used to |
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investigate the inertial behavior of linked polymer segments in a low |
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friction regime.\cite{Beard2000} LTID can also deal with the |
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rotational dynamics for nonskew bodies without translation-rotation |
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coupling by separating the translation and rotation motion and taking |
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advantage of the analytical solution of hydrodynamics |
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properties. However, typical nonskew bodies like cylinders and |
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ellipsoids are inadequate to represent most complex macromolecular |
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assemblies. There is therefore a need for incorporating the |
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hydrodynamics of complex (and potentially skew) rigid bodies in the |
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library of methods available for performing Langevin simulations. |
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\subsection{Rigid Body Dynamics} |
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Rigid bodies are frequently involved in the modeling of large |
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collections of particles that move as a single unit. In molecular |
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simulations, rigid bodies have been used to simplify protein-protein |
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docking,\cite{Gray2003} and lipid bilayer |
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simulations.\cite{SunGezelter08} Many of the water models in common |
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use are also rigid-body |
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models,\cite{Jorgensen83,Berendsen81,Berendsen87} although they are |
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typically evolved using constraints rather than rigid body equations |
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of motion. |
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Euler angles are a natural choice to describe the rotational degrees |
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of freedom. However, due to $\frac{1}{\sin \theta}$ singularities, the |
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numerical integration of corresponding equations of these motion can |
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become inaccurate (and inefficient). Although the use of multiple |
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sets of Euler angles can overcome this problem,\cite{Barojas1973} the |
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computational penalty and the loss of angular momentum conservation |
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remain. A singularity-free representation utilizing quaternions was |
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developed by Evans in 1977.\cite{Evans1977} The Evans quaternion |
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approach uses a nonseparable Hamiltonian, and this has prevented |
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symplectic algorithms from being utilized until very |
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recently.\cite{Miller2002} |
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Another approach is the application of holonomic constraints to the |
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atoms belonging to the rigid body. Each atom moves independently |
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under the normal forces deriving from potential energy and constraints |
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are used to guarantee rigidity. However, due to their iterative |
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nature, the SHAKE and RATTLE algorithms converge very slowly when the |
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number of constraints (and the number of particles that belong to the |
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rigid body) increases.\cite{Ryckaert1977,Andersen1983} |
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In order to develop a stable and efficient integration scheme that |
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preserves most constants of the motion, symplectic propagators are |
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necessary. By introducing a conjugate momentum to the rotation matrix |
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$Q$ and re-formulating Hamilton's equations, a symplectic |
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orientational integrator, RSHAKE,\cite{Kol1997} was proposed to evolve |
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rigid bodies on a constraint manifold by iteratively satisfying the |
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orthogonality constraint $Q^T Q = 1$. An alternative method using the |
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quaternion representation was developed by Omelyan.\cite{Omelyan1998} |
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However, both of these methods are iterative and suffer from some |
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related inefficiencies. A symplectic Lie-Poisson integrator for rigid |
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bodies developed by Dullweber {\it et al.}\cite{Dullweber1997} removes |
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most of the limitations mentioned above and is therefore the basis for |
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our Langevin integrator. |
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The goal of the present work is to develop a Langevin dynamics |
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algorithm for arbitrary-shaped rigid particles by integrating an |
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accurate estimate of the friction tensor from hydrodynamics theory |
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into a symplectic rigid body dynamics propagator. In the sections |
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below, we review some of the theory of hydrodynamic tensors developed |
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primarily for Brownian simulations of multi-particle systems, we then |
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present our integration method for a set of generalized Langevin |
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equations of motion, and we compare the behavior of the new Langevin |
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integrator to dynamical quantities obtained via explicit solvent |
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molecular dynamics. |
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\subsection{\label{introSection:frictionTensor}The Friction Tensor} |
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Theoretically, a complete friction kernel for a solute particle can be |
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determined using the velocity autocorrelation function from a |
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simulation with explicit solvent molecules. However, this approach |
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becomes impractical when the solute becomes complex. Instead, various |
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approaches based on hydrodynamics have been developed to calculate |
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static friction coefficients. In general, the friction tensor $\Xi$ is |
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a $6\times 6$ matrix given by |
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\begin{equation} |
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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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\end{equation} |
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Here, $\Xi^{tt}$ and $\Xi^{rr}$ are $3 \times 3$ translational and |
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rotational resistance (friction) tensors respectively, while |
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$\Xi^{tr}$ is translation-rotation coupling tensor and $\Xi^{rt}$ is |
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rotation-translation coupling tensor. When a particle moves in a |
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fluid, it may experience a friction force ($\mathbf{f}_f$) and torque |
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($\mathbf{\tau}_f$) in opposition to the velocity ($\mathbf{v}$) and |
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body-fixed angular velocity ($\mathbf{\omega}$), |
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\begin{equation} |
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\left( \begin{array}{l} |
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\mathbf{f}_f \\ |
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\mathbf{\tau}_f \\ |
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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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\mathbf{v} \\ |
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\mathbf{\omega} \\ |
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\end{array} \right). |
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\end{equation} |
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\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shapes}} |
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For a spherical body under ``stick'' boundary conditions, the |
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translational and rotational friction tensors can be calculated from |
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Stokes' law,\cite{stokes} |
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\begin{equation} |
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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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\end{equation} |
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and |
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\begin{equation} |
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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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\end{equation} |
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where $\eta$ is the viscosity of the solvent and $R$ is the |
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hydrodynamic radius. |
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Other non-spherical shapes, such as cylinders and ellipsoids, are |
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widely used as references for developing new hydrodynamic theories, |
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because their properties can be calculated exactly. In 1936, Perrin |
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extended Stokes' law to general |
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ellipsoids,\cite{Perrin1934,Perrin1936} described in Cartesian |
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coordinates as |
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\begin{equation} |
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\frac{x^2 }{a^2} + \frac{y^2}{b^2} + \frac{z^2 }{c^2} = 1. |
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\end{equation} |
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Here, the semi-axes are of lengths $a$, $b$, and $c$. Due to the |
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complexity of the elliptic integral, only uniaxial ellipsoids, either |
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prolate ($a \ge b = c$) or oblate ($a < b = c$), were solved |
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exactly. Introducing an elliptic integral parameter $S$ for prolate, |
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\begin{equation} |
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S = \frac{2}{\sqrt{a^2 - b^2}} \ln \frac{a + \sqrt{a^2 - b^2}}{b}, |
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\end{equation} |
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and oblate, |
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\begin{equation} |
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S = \frac{2}{\sqrt {b^2 - a^2 }} \arctan \frac{\sqrt {b^2 - a^2}}{a}, |
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\end{equation} |
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ellipsoids, it is possible to write down exact solutions for the |
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resistance tensors. As is the case for spherical bodies, the translational, |
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\begin{eqnarray} |
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\Xi_a^{tt} & = & 16\pi \eta \frac{a^2 - b^2}{(2a^2 - b^2 )S - 2a}. \\ |
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\Xi_b^{tt} = \Xi_c^{tt} & = & 32\pi \eta \frac{a^2 - b^2 }{(2a^2 - 3b^2 )S + 2a}, |
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\end{eqnarray} |
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and rotational, |
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\begin{eqnarray} |
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\Xi_a^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^2 - b^2 )b^2}{2a - b^2 S}, \\ |
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\Xi_b^{rr} = \Xi_c^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^4 - b^4)}{(2a^2 - b^2 )S - 2a} |
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\end{eqnarray} |
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resistance tensors are diagonal $3 \times 3$ matrices. For both |
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spherical and ellipsoidal particles, the translation-rotation and |
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rotation-translation coupling tensors are zero. |
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\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
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There is no analytical solution for the friction tensor for rigid |
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molecules of arbitrary shape. The ellipsoid of revolution and general |
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triaxial ellipsoid models have been widely used to approximate the |
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hydrodynamic properties of rigid bodies. However, the mapping from all |
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possible ellipsoidal spaces ($r$-space) to all possible combinations |
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of rotational diffusion coefficients ($D$-space) is not |
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unique.\cite{Wegener1979} Additionally, because there is intrinsic |
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coupling between translational and rotational motion of {\it skew} |
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rigid bodies, general ellipsoids are not always suitable for modeling |
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rigid molecules. A number of studies have been devoted to determining |
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the friction tensor for irregular shapes using methods in which the |
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molecule of interest is modeled with a combination of |
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spheres\cite{Carrasco1999} and the hydrodynamic properties of the |
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molecule are then calculated using a set of two-point interaction |
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tensors. We have found the {\it bead} and {\it rough shell} models of |
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Carrasco and Garc\'{i}a de la Torre to be the most useful of these |
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methods,\cite{Carrasco1999} and we review the basic outline of the |
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rough shell approach here. A more thorough explanation can be found |
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in Ref. \citen{Carrasco1999}. |
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|
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Consider a rigid assembly of $N$ small beads moving through a |
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continuous medium. Due to hydrodynamic interactions between the |
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beads, the net velocity of the $i^\mathrm{th}$ bead relative to the |
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medium, ${\bf v}'_i$, is different than its unperturbed velocity ${\bf |
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v}_i$, |
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\begin{equation} |
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{\bf v}'_i = {\bf v}_i - \sum\limits_{j \ne i} {{\bf T}_{ij} {\bf F}_j } |
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\end{equation} |
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gezelter |
3341 |
where ${\bf F}_j$ is the frictional force on the medium due to bead $j$, and |
315 |
|
|
${\bf T}_{ij}$ is the hydrodynamic interaction tensor between the two beads. |
316 |
|
|
The frictional force felt by the $i^\mathrm{th}$ bead is proportional to |
317 |
|
|
its net velocity |
318 |
tim |
2746 |
\begin{equation} |
319 |
gezelter |
3341 |
{\bf F}_i = \zeta_i {\bf v}_i - \zeta _i \sum\limits_{j \ne i} {{\bf T}_{ij} {\bf F}_j }. |
320 |
tim |
2999 |
\label{introEquation:tensorExpression} |
321 |
tim |
2746 |
\end{equation} |
322 |
gezelter |
3341 |
Eq. (\ref{introEquation:tensorExpression}) defines the two-point |
323 |
|
|
hydrodynamic tensor, ${\bf T}_{ij}$. There have been many proposed |
324 |
|
|
solutions to this equation, including the simple solution given by |
325 |
|
|
Oseen and Burgers in 1930 for two beads of identical radius. A second |
326 |
|
|
order expression for beads of different hydrodynamic radii was |
327 |
tim |
2999 |
introduced by Rotne and Prager\cite{Rotne1969} and improved by |
328 |
|
|
Garc\'{i}a de la Torre and Bloomfield,\cite{Torre1977} |
329 |
tim |
2746 |
\begin{equation} |
330 |
gezelter |
3341 |
{\bf T}_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {{\bf I} + |
331 |
|
|
\frac{{{\bf R}_{ij} {\bf R}_{ij}^T }}{{R_{ij}^2 }}} \right) + \frac{{\sigma |
332 |
|
|
_i^2 + \sigma _j^2 }}{{R_{ij}^2 }}\left( {\frac{{\bf I}}{3} - |
333 |
|
|
\frac{{{\bf R}_{ij} {\bf R}_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
334 |
tim |
2999 |
\label{introEquation:RPTensorNonOverlapped} |
335 |
tim |
2746 |
\end{equation} |
336 |
gezelter |
3341 |
Here ${\bf R}_{ij}$ is the distance vector between beads $i$ and $j$. Both |
337 |
|
|
the Oseen-Burgers tensor and |
338 |
|
|
Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption that |
339 |
|
|
the beads do not overlap ($R_{ij} \ge \sigma _i + \sigma _j$). |
340 |
|
|
|
341 |
|
|
To calculate the resistance tensor for a body represented as the union |
342 |
|
|
of many non-overlapping beads, we first pick an arbitrary origin $O$ |
343 |
|
|
and then construct a $3N \times 3N$ supermatrix consisting of $N |
344 |
|
|
\times N$ ${\bf B}_{ij}$ blocks |
345 |
tim |
2746 |
\begin{equation} |
346 |
gezelter |
3341 |
{\bf B} = \left( \begin{array}{*{20}c} |
347 |
|
|
{\bf B}_{11} & \ldots & {\bf B}_{1N} \\ |
348 |
|
|
\vdots & \ddots & \vdots \\ |
349 |
|
|
{\bf B}_{N1} & \cdots & {\bf B}_{NN} |
350 |
|
|
\end{array} \right) |
351 |
tim |
2746 |
\end{equation} |
352 |
gezelter |
3341 |
${\bf B}_{ij}$ is a version of the hydrodynamic tensor which includes the |
353 |
|
|
self-contributions for spheres, |
354 |
tim |
2999 |
\begin{equation} |
355 |
gezelter |
3341 |
{\bf B}_{ij} = \delta _{ij} \frac{{\bf I}}{{6\pi \eta R_{ij}}} + (1 - \delta_{ij} |
356 |
|
|
){\bf T}_{ij} |
357 |
|
|
\end{equation} |
358 |
|
|
where $\delta_{ij}$ is the Kronecker delta function. Inverting the |
359 |
|
|
${\bf B}$ matrix, we obtain |
360 |
|
|
\begin{equation} |
361 |
|
|
{\bf C} = {\bf B}^{ - 1} = \left(\begin{array}{*{20}c} |
362 |
|
|
{\bf C}_{11} & \ldots & {\bf C}_{1N} \\ |
363 |
|
|
\vdots & \ddots & \vdots \\ |
364 |
|
|
{\bf C}_{N1} & \cdots & {\bf C}_{NN} |
365 |
gezelter |
3333 |
\end{array} \right), |
366 |
tim |
2999 |
\end{equation} |
367 |
gezelter |
3341 |
which can be partitioned into $N \times N$ blocks labeled ${\bf C}_{ij}$. |
368 |
|
|
(Each of the ${\bf C}_{ij}$ blocks is a $3 \times 3$ matrix.) Using the |
369 |
|
|
skew matrix, |
370 |
gezelter |
3333 |
\begin{equation} |
371 |
gezelter |
3341 |
{\bf U}_i = \left(\begin{array}{*{20}c} |
372 |
|
|
0 & -z_i & y_i \\ |
373 |
|
|
z_i & 0 & - x_i \\ |
374 |
|
|
-y_i & x_i & 0 |
375 |
|
|
\end{array}\right) |
376 |
|
|
\label{eq:skewMatrix} |
377 |
gezelter |
3333 |
\end{equation} |
378 |
tim |
2999 |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
379 |
gezelter |
3341 |
bead $i$ and origin $O$, the elements of the resistance tensor (at the |
380 |
|
|
arbitrary origin $O$) can be written as |
381 |
tim |
2999 |
\begin{eqnarray} |
382 |
xsun |
3339 |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
383 |
gezelter |
3341 |
\Xi^{tt} & = & \sum\limits_i {\sum\limits_j {{\bf C}_{ij} } } \notag , \\ |
384 |
|
|
\Xi^{tr} = \Xi _{}^{rt} & = & \sum\limits_i {\sum\limits_j {{\bf U}_i {\bf C}_{ij} } } , \\ |
385 |
|
|
\Xi^{rr} & = & -\sum\limits_i \sum\limits_j {\bf U}_i {\bf C}_{ij} {\bf U}_j + 6 \eta V {\bf I}. \notag |
386 |
tim |
2999 |
\end{eqnarray} |
387 |
gezelter |
3341 |
The final term in the expression for $\Xi^{rr}$ is a correction that |
388 |
|
|
accounts for errors in the rotational motion of the bead models. The |
389 |
|
|
additive correction uses the solvent viscosity ($\eta$) as well as the |
390 |
|
|
total volume of the beads that contribute to the hydrodynamic model, |
391 |
gezelter |
3310 |
\begin{equation} |
392 |
|
|
V = \frac{4 \pi}{3} \sum_{i=1}^{N} \sigma_i^3, |
393 |
|
|
\end{equation} |
394 |
|
|
where $\sigma_i$ is the radius of bead $i$. This correction term was |
395 |
|
|
rigorously tested and compared with the analytical results for |
396 |
gezelter |
3341 |
two-sphere and ellipsoidal systems by Garc\'{i}a de la Torre and |
397 |
gezelter |
3310 |
Rodes.\cite{Torre:1983lr} |
398 |
|
|
|
399 |
gezelter |
3341 |
In general, resistance tensors depend on the origin at which they were |
400 |
|
|
computed. However, the proper location for applying the friction |
401 |
|
|
force is the center of resistance, the special point at which the |
402 |
|
|
trace of rotational resistance tensor, $\Xi^{rr}$ reaches a minimum |
403 |
|
|
value. Mathematically, the center of resistance can also be defined |
404 |
|
|
as the unique point for a rigid body at which the translation-rotation |
405 |
|
|
coupling tensors are symmetric, |
406 |
tim |
2999 |
\begin{equation} |
407 |
gezelter |
3341 |
\Xi^{tr} = \left(\Xi^{tr}\right)^T |
408 |
tim |
2999 |
\label{introEquation:definitionCR} |
409 |
|
|
\end{equation} |
410 |
gezelter |
3341 |
From Eq. \ref{introEquation:ResistanceTensorArbitraryOrigin}, we can |
411 |
|
|
easily derive that the {\it translational} resistance tensor is origin |
412 |
|
|
independent, while the rotational resistance tensor and |
413 |
tim |
2999 |
translation-rotation coupling resistance tensor depend on the |
414 |
gezelter |
3341 |
origin. Given the resistance tensor at an arbitrary origin $O$, and a |
415 |
|
|
vector ,${\bf r}_{OP} = (x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we |
416 |
|
|
can obtain the resistance tensor at $P$ by |
417 |
|
|
\begin{eqnarray} |
418 |
|
|
\label{introEquation:resistanceTensorTransformation} |
419 |
|
|
\Xi_P^{tt} & = & \Xi_O^{tt} \notag \\ |
420 |
|
|
\Xi_P^{tr} = \Xi_P^{rt} & = & \Xi_O^{tr} - {\bf U}_{OP} \Xi _O^{tt} \\ |
421 |
|
|
\Xi_P^{rr} & = &\Xi_O^{rr} - {\bf U}_{OP} \Xi_O^{tt} {\bf U}_{OP} |
422 |
|
|
+ \Xi_O^{tr} {\bf U}_{OP} - {\bf U}_{OP} \left( \Xi_O^{tr} |
423 |
|
|
\right)^{^T} \notag |
424 |
|
|
\end{eqnarray} |
425 |
|
|
where ${\bf U}_{OP}$ is the skew matrix (Eq. (\ref{eq:skewMatrix})) |
426 |
|
|
for the vector between the origin $O$ and the point $P$. Using |
427 |
|
|
Eqs.~\ref{introEquation:definitionCR}~and~\ref{introEquation:resistanceTensorTransformation}, |
428 |
|
|
one can locate the position of center of resistance, |
429 |
|
|
\begin{equation*} |
430 |
|
|
\left(\begin{array}{l} |
431 |
|
|
x_{OR} \\ |
432 |
|
|
y_{OR} \\ |
433 |
|
|
z_{OR} |
434 |
|
|
\end{array}\right) = |
435 |
|
|
\left(\begin{array}{*{20}c} |
436 |
|
|
(\Xi_O^{rr})_{yy} + (\Xi_O^{rr})_{zz} & -(\Xi_O^{rr})_{xy} & -(\Xi_O^{rr})_{xz} \\ |
437 |
|
|
-(\Xi_O^{rr})_{xy} & (\Xi_O^{rr})_{zz} + (\Xi_O^{rr})_{xx} & -(\Xi_O^{rr})_{yz} \\ |
438 |
|
|
-(\Xi_O^{rr})_{xz} & -(\Xi_O^{rr})_{yz} & (\Xi_O^{rr})_{xx} + (\Xi_O^{rr})_{yy} \\ |
439 |
|
|
\end{array}\right)^{-1} |
440 |
|
|
\left(\begin{array}{l} |
441 |
|
|
(\Xi_O^{tr})_{yz} - (\Xi_O^{tr})_{zy} \\ |
442 |
|
|
(\Xi_O^{tr})_{zx} - (\Xi_O^{tr})_{xz} \\ |
443 |
|
|
(\Xi_O^{tr})_{xy} - (\Xi_O^{tr})_{yx} |
444 |
|
|
\end{array}\right) |
445 |
|
|
\end{equation*} |
446 |
xsun |
3338 |
where $x_{OR}$, $y_{OR}$, $z_{OR}$ are the components of the vector |
447 |
tim |
2999 |
joining center of resistance $R$ and origin $O$. |
448 |
tim |
2746 |
|
449 |
gezelter |
3341 |
For a general rigid molecular substructure, finding the $6 \times 6$ |
450 |
|
|
resistance tensor can be a computationally demanding task. First, a |
451 |
|
|
lattice of small beads that extends well beyond the boundaries of the |
452 |
|
|
rigid substructure is created. The lattice is typically composed of |
453 |
|
|
0.25 \AA\ beads on a dense FCC lattice. The lattice constant is taken |
454 |
|
|
to be the bead diameter, so that adjacent beads are touching, but do |
455 |
|
|
not overlap. To make a shape corresponding to the rigid structure, |
456 |
|
|
beads that sit on lattice sites that are outside the van der Waals |
457 |
|
|
radii of any atoms comprising the rigid body are excluded from the |
458 |
|
|
calculation. |
459 |
tim |
2746 |
|
460 |
gezelter |
3341 |
For large structures, most of the beads will be deep within the rigid |
461 |
|
|
body and will not contribute to the hydrodynamic tensor. In the {\it |
462 |
|
|
rough shell} approach, beads which have all of their lattice neighbors |
463 |
|
|
inside the structure are considered interior beads, and are removed |
464 |
|
|
from the calculation. After following this procedure, only those |
465 |
|
|
beads in direct contact with the van der Waals surface of the rigid |
466 |
|
|
body are retained. For reasonably large molecular structures, this |
467 |
|
|
truncation can still produce bead assemblies with thousands of |
468 |
|
|
members. |
469 |
|
|
|
470 |
|
|
If all of the atoms comprising the rigid substructure are spherical |
471 |
|
|
and non-overlapping, the tensor in |
472 |
|
|
Eq.~(\ref{introEquation:RPTensorNonOverlapped}) may be used directly |
473 |
|
|
using the atoms themselves as the hydrodynamic beads. This is a |
474 |
|
|
variant of the {\it bead model} approach of Carrasco and Garc\'{i}a de |
475 |
|
|
la Torre.\cite{Carrasco1999} In this case, the size of the ${\bf B}$ matrix |
476 |
|
|
can be quite small, and the calculation of the hydrodynamic tensor is |
477 |
|
|
straightforward. |
478 |
|
|
|
479 |
|
|
In general, the inversion of the ${\bf B}$ matrix is the most |
480 |
|
|
computationally demanding task. This inversion is done only once for |
481 |
|
|
each type of rigid structure. We have been using straightforward |
482 |
|
|
LU-decomposition to solve the linear system and obtain the elements of |
483 |
|
|
${\bf C}$. Once ${\bf C}$ has been obtained, the location of the |
484 |
|
|
center of resistance ($R$) is found and the resistance tensor at this |
485 |
|
|
point is calculated. The $3 \times 1$ vector giving the location of |
486 |
|
|
the rigid body's center of resistance and the $6 \times 6$ resistance |
487 |
|
|
tensor are stored for use in the Langevin dynamics calculation. Note |
488 |
|
|
that these quantities depend on solvent viscosity and temperature and |
489 |
|
|
must be recomputed if different simulation conditions are required. |
490 |
|
|
|
491 |
gezelter |
3310 |
\section{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
492 |
gezelter |
3337 |
|
493 |
tim |
2999 |
Consider the Langevin equations of motion in generalized coordinates |
494 |
tim |
2746 |
\begin{equation} |
495 |
gezelter |
3341 |
{\bf M} \dot{{\bf V}}(t) = {\bf F}_{s}(t) + |
496 |
|
|
{\bf F}_{f}(t) + {\bf F}_{r}(t) |
497 |
tim |
2746 |
\label{LDGeneralizedForm} |
498 |
|
|
\end{equation} |
499 |
gezelter |
3341 |
where ${\bf M}$ is a $6 \times 6$ diagonal mass matrix (which |
500 |
gezelter |
3337 |
includes the mass of the rigid body as well as the moments of inertia |
501 |
gezelter |
3341 |
in the body-fixed frame) and ${\bf V}$ is a generalized velocity, |
502 |
|
|
${\bf V} = |
503 |
|
|
\left\{{\bf v},{\bf \omega}\right\}$. The right side of |
504 |
gezelter |
3333 |
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces: a |
505 |
gezelter |
3341 |
system force (${\bf F}_{s}$), a frictional or dissipative force |
506 |
|
|
(${\bf F}_{f}$) and stochastic force (${\bf F}_{r}$). While the |
507 |
gezelter |
3340 |
evolution of the system in Newtonian mechanics is typically done in |
508 |
gezelter |
3341 |
the lab frame, it is convenient to handle the dynamics of rigid bodies |
509 |
|
|
in body-fixed frames. Thus the friction and random forces on each |
510 |
|
|
substructure are calculated in a body-fixed frame and may converted |
511 |
|
|
back to the lab frame using that substructure's rotation matrix (${\bf |
512 |
|
|
Q}$): |
513 |
gezelter |
3333 |
\begin{equation} |
514 |
gezelter |
3341 |
{\bf F}_{f,r} = |
515 |
gezelter |
3340 |
\left( \begin{array}{c} |
516 |
gezelter |
3341 |
{\bf f}_{f,r} \\ |
517 |
|
|
{\bf \tau}_{f,r} |
518 |
gezelter |
3340 |
\end{array} \right) |
519 |
|
|
= |
520 |
|
|
\left( \begin{array}{c} |
521 |
gezelter |
3341 |
{\bf Q}^{T} {\bf f}^{~b}_{f,r} \\ |
522 |
|
|
{\bf Q}^{T} {\bf \tau}^{~b}_{f,r} |
523 |
gezelter |
3340 |
\end{array} \right) |
524 |
gezelter |
3333 |
\end{equation} |
525 |
gezelter |
3341 |
The body-fixed friction force, ${\bf F}_{f}^{~b}$, is proportional to |
526 |
|
|
the (body-fixed) velocity at the center of resistance |
527 |
|
|
${\bf v}_{R}^{~b}$ and the angular velocity ${\bf \omega}$ |
528 |
tim |
2746 |
\begin{equation} |
529 |
gezelter |
3341 |
{\bf F}_{f}^{~b}(t) = \left( \begin{array}{l} |
530 |
|
|
{\bf f}_{f}^{~b}(t) \\ |
531 |
|
|
{\bf \tau}_{f}^{~b}(t) \\ |
532 |
gezelter |
3333 |
\end{array} \right) = - \left( \begin{array}{*{20}c} |
533 |
gezelter |
3341 |
\Xi_{R}^{tt} & \Xi_{R}^{rt} \\ |
534 |
|
|
\Xi_{R}^{tr} & \Xi_{R}^{rr} \\ |
535 |
gezelter |
3333 |
\end{array} \right)\left( \begin{array}{l} |
536 |
gezelter |
3341 |
{\bf v}_{R}^{~b}(t) \\ |
537 |
|
|
{\bf \omega}(t) \\ |
538 |
tim |
2746 |
\end{array} \right), |
539 |
|
|
\end{equation} |
540 |
gezelter |
3341 |
while the random force, ${\bf F}_{r}$, is a Gaussian stochastic |
541 |
|
|
variable with zero mean and variance, |
542 |
tim |
2746 |
\begin{equation} |
543 |
gezelter |
3341 |
\left\langle {{\bf F}_{r}(t) ({\bf F}_{r}(t'))^T } \right\rangle = |
544 |
|
|
\left\langle {{\bf F}_{r}^{~b} (t) ({\bf F}_{r}^{~b} (t'))^T } \right\rangle = |
545 |
|
|
2 k_B T \Xi_R \delta(t - t'). \label{eq:randomForce} |
546 |
tim |
2746 |
\end{equation} |
547 |
gezelter |
3340 |
$\Xi_R$ is the $6\times6$ resistance tensor at the center of |
548 |
gezelter |
3341 |
resistance. Once this tensor is known for a given rigid body (as |
549 |
|
|
described in the previous section) obtaining a stochastic vector that |
550 |
|
|
has the properties in Eq. (\ref{eq:randomForce}) can be done |
551 |
|
|
efficiently by carrying out a one-time Cholesky decomposition to |
552 |
|
|
obtain the square root matrix of the resistance tensor, |
553 |
|
|
\begin{equation} |
554 |
|
|
\Xi_R = {\bf S} {\bf S}^{T}, |
555 |
|
|
\label{eq:Cholesky} |
556 |
|
|
\end{equation} |
557 |
|
|
where ${\bf S}$ is a lower triangular matrix.\cite{SchlickBook} A |
558 |
|
|
vector with the statistics required for the random force can then be |
559 |
|
|
obtained by multiplying ${\bf S}$ onto a random 6-vector ${\bf Z}$ which |
560 |
|
|
has elements chosen from a Gaussian distribution, such that: |
561 |
gezelter |
3340 |
\begin{equation} |
562 |
gezelter |
3341 |
\langle {\bf Z}_i \rangle = 0, \hspace{1in} \langle {\bf Z}_i \cdot |
563 |
|
|
{\bf Z}_j \rangle = \frac{2 k_B T}{\delta t} \delta_{ij}, |
564 |
gezelter |
3340 |
\end{equation} |
565 |
gezelter |
3341 |
where $\delta t$ is the timestep in use during the simulation. The |
566 |
|
|
random force, ${\bf F}_{r}^{~b} = {\bf S} {\bf Z}$, can be shown to have the |
567 |
|
|
correct properties required by Eq. (\ref{eq:randomForce}). |
568 |
gezelter |
3333 |
|
569 |
gezelter |
3341 |
The equation of motion for the translational velocity of the center of |
570 |
|
|
mass (${\bf v}$) can be written as |
571 |
tim |
2746 |
\begin{equation} |
572 |
gezelter |
3341 |
m \dot{{\bf v}} (t) = {\bf f}_{s}(t) + {\bf f}_{f}(t) + |
573 |
|
|
{\bf f}_{r}(t) |
574 |
tim |
2746 |
\end{equation} |
575 |
gezelter |
3341 |
Since the frictional and random forces are applied at the center of |
576 |
|
|
resistance, which generally does not coincide with the center of mass, |
577 |
|
|
extra torques are exerted at the center of mass. Thus, the net |
578 |
|
|
body-fixed torque at the center of mass, $\tau^{~b}(t)$, |
579 |
|
|
is given by |
580 |
tim |
2746 |
\begin{equation} |
581 |
gezelter |
3341 |
\tau^{~b} \leftarrow \tau_{s}^{~b} + \tau_{f}^{~b} + \tau_{r}^{~b} + {\bf r}_{MR} \times \left( {\bf f}_{f}^{~b} + {\bf f}_{r}^{~b} \right) |
582 |
tim |
2746 |
\end{equation} |
583 |
gezelter |
3341 |
where ${\bf r}_{MR}$ is the vector from the center of mass to the center of |
584 |
|
|
resistance. Instead of integrating the angular velocity in lab-fixed |
585 |
|
|
frame, we consider the equation of motion for the angular momentum |
586 |
|
|
(${\bf j}$) in the body-fixed frame |
587 |
tim |
2746 |
\begin{equation} |
588 |
gezelter |
3341 |
\dot{\bf j}(t) = \tau^{~b}(t) |
589 |
tim |
2746 |
\end{equation} |
590 |
gezelter |
3341 |
Embedding the friction and random forces into the the total force and |
591 |
|
|
torque, one can integrate the Langevin equations of motion for a rigid |
592 |
|
|
body of arbitrary shape in a velocity-Verlet style 2-part algorithm, |
593 |
|
|
where $h = \delta t$: |
594 |
tim |
2746 |
|
595 |
gezelter |
3341 |
{\tt move A:} |
596 |
tim |
2746 |
\begin{align*} |
597 |
tim |
2999 |
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
598 |
|
|
+ \frac{h}{2} \left( {\bf f}(t) / m \right), \\ |
599 |
|
|
% |
600 |
|
|
{\bf r}(t + h) &\leftarrow {\bf r}(t) |
601 |
|
|
+ h {\bf v}\left(t + h / 2 \right), \\ |
602 |
|
|
% |
603 |
|
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
604 |
gezelter |
3341 |
+ \frac{h}{2} {\bf \tau}^{~b}(t), \\ |
605 |
tim |
2999 |
% |
606 |
gezelter |
3341 |
{\bf Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
607 |
tim |
2999 |
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
608 |
tim |
2746 |
\end{align*} |
609 |
gezelter |
3341 |
In this context, $\overleftrightarrow{\mathsf{I}}$ is the diagonal |
610 |
|
|
moment of inertia tensor, and the $\mathrm{rotate}$ function is the |
611 |
|
|
reversible product of the three body-fixed rotations, |
612 |
tim |
2746 |
\begin{equation} |
613 |
|
|
\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
614 |
|
|
\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y |
615 |
|
|
/ 2) \cdot \mathsf{G}_x(a_x /2), |
616 |
|
|
\end{equation} |
617 |
|
|
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
618 |
gezelter |
3341 |
rotates both the rotation matrix ($\mathbf{Q}$) and the body-fixed |
619 |
tim |
2999 |
angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed |
620 |
|
|
axis $\alpha$, |
621 |
tim |
2746 |
\begin{equation} |
622 |
|
|
\mathsf{G}_\alpha( \theta ) = \left\{ |
623 |
|
|
\begin{array}{lcl} |
624 |
gezelter |
3341 |
\mathbf{Q}(t) & \leftarrow & \mathbf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
625 |
tim |
2746 |
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
626 |
|
|
j}(0). |
627 |
|
|
\end{array} |
628 |
|
|
\right. |
629 |
|
|
\end{equation} |
630 |
|
|
$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis |
631 |
|
|
rotation matrix. For example, in the small-angle limit, the |
632 |
|
|
rotation matrix around the body-fixed x-axis can be approximated as |
633 |
|
|
\begin{equation} |
634 |
|
|
\mathsf{R}_x(\theta) \approx \left( |
635 |
|
|
\begin{array}{ccc} |
636 |
|
|
1 & 0 & 0 \\ |
637 |
|
|
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
638 |
|
|
\theta^2 / 4} \\ |
639 |
|
|
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
640 |
|
|
\theta^2 / 4} |
641 |
|
|
\end{array} |
642 |
|
|
\right). |
643 |
|
|
\end{equation} |
644 |
tim |
2999 |
All other rotations follow in a straightforward manner. After the |
645 |
|
|
first part of the propagation, the forces and body-fixed torques are |
646 |
gezelter |
3341 |
calculated at the new positions and orientations. The system forces |
647 |
|
|
and torques are derivatives of the total potential energy function |
648 |
|
|
($U$) with respect to the rigid body positions (${\bf r}$) and the |
649 |
|
|
columns of the transposed rotation matrix ${\bf Q}^T = \left({\bf |
650 |
|
|
u}_x, {\bf u}_y, {\bf u}_z \right)$: |
651 |
tim |
2746 |
|
652 |
gezelter |
3341 |
{\tt Forces:} |
653 |
tim |
2999 |
\begin{align*} |
654 |
gezelter |
3341 |
{\bf f}_{s}(t + h) & \leftarrow |
655 |
|
|
- \left(\frac{\partial U}{\partial {\bf r}}\right)_{{\bf r}(t + h)} \\ |
656 |
tim |
2999 |
% |
657 |
gezelter |
3341 |
{\bf \tau}_{s}(t + h) &\leftarrow {\bf u}(t + h) |
658 |
|
|
\times \frac{\partial U}{\partial {\bf u}} \\ |
659 |
tim |
2999 |
% |
660 |
gezelter |
3341 |
{\bf v}^{b}_{R}(t+h) & \leftarrow \mathbf{Q}(t+h) \cdot \left({\bf v}(t+h) + {\bf \omega}(t+h) \times {\bf r}_{MR} \right) \\ |
661 |
|
|
% |
662 |
|
|
{\bf f}_{R,f}^{b}(t+h) & \leftarrow - \Xi_{R}^{tt} \cdot |
663 |
|
|
{\bf v}^{b}_{R}(t+h) - \Xi_{R}^{rt} \cdot {\bf \omega}(t+h) \\ |
664 |
|
|
% |
665 |
|
|
{\bf \tau}_{R,f}^{b}(t+h) & \leftarrow - \Xi_{R}^{tr} \cdot |
666 |
|
|
{\bf v}^{b}_{R}(t+h) - \Xi_{R}^{rr} \cdot {\bf \omega}(t+h) \\ |
667 |
|
|
% |
668 |
|
|
Z & \leftarrow {\tt GaussianNormal}(2 k_B T / h, 6) \\ |
669 |
|
|
{\bf F}_{R,r}^{b}(t+h) & \leftarrow {\bf S} \cdot Z \\ |
670 |
|
|
% |
671 |
|
|
{\bf f}(t+h) & \leftarrow {\bf f}_{s}(t+h) + \mathbf{Q}^{T}(t+h) |
672 |
|
|
\cdot \left({\bf f}_{R,f}^{~b} + {\bf f}_{R,r}^{~b} \right) \\ |
673 |
|
|
% |
674 |
|
|
\tau(t+h) & \leftarrow \tau_{s}(t+h) + \mathbf{Q}^{T}(t+h) \cdot \left(\tau_{R,f}^{~b} + \tau_{R,r}^{~b} \right) + {\bf r}_{MR} \times \left({\bf f}_{f}(t+h) + {\bf f}_{r}(t+h) \right) \\ |
675 |
|
|
\tau^{~b}(t+h) & \leftarrow \mathbf{Q}(t+h) \cdot \tau(t+h) \\ |
676 |
tim |
2999 |
\end{align*} |
677 |
gezelter |
3341 |
Frictional (and random) forces and torques must be computed at the |
678 |
|
|
center of resistance, so there are additional steps required to find |
679 |
|
|
the body-fixed velocity (${\bf v}_{R}^{~b}$) at this location. Mapping |
680 |
|
|
the frictional and random forces at the center of resistance back to |
681 |
|
|
the center of mass also introduces an additional term in the torque |
682 |
|
|
one obtains at the center of mass. |
683 |
|
|
|
684 |
tim |
2746 |
Once the forces and torques have been obtained at the new time step, |
685 |
|
|
the velocities can be advanced to the same time value. |
686 |
|
|
|
687 |
gezelter |
3341 |
{\tt move B:} |
688 |
tim |
2746 |
\begin{align*} |
689 |
tim |
2999 |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 |
690 |
|
|
\right) |
691 |
|
|
+ \frac{h}{2} \left( {\bf f}(t + h) / m \right), \\ |
692 |
|
|
% |
693 |
|
|
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t + h / 2 |
694 |
|
|
\right) |
695 |
gezelter |
3341 |
+ \frac{h}{2} {\bf \tau}^{~b}(t + h) . |
696 |
tim |
2746 |
\end{align*} |
697 |
|
|
|
698 |
gezelter |
3310 |
\section{Validating the Method\label{sec:validating}} |
699 |
gezelter |
3302 |
In order to validate our Langevin integrator for arbitrarily-shaped |
700 |
gezelter |
3305 |
rigid bodies, we implemented the algorithm in {\sc |
701 |
|
|
oopse}\cite{Meineke2005} and compared the results of this algorithm |
702 |
|
|
with the known |
703 |
gezelter |
3302 |
hydrodynamic limiting behavior for a few model systems, and to |
704 |
|
|
microcanonical molecular dynamics simulations for some more |
705 |
|
|
complicated bodies. The model systems and their analytical behavior |
706 |
|
|
(if known) are summarized below. Parameters for the primary particles |
707 |
|
|
comprising our model systems are given in table \ref{tab:parameters}, |
708 |
|
|
and a sketch of the arrangement of these primary particles into the |
709 |
gezelter |
3305 |
model rigid bodies is shown in figure \ref{fig:models}. In table |
710 |
|
|
\ref{tab:parameters}, $d$ and $l$ are the physical dimensions of |
711 |
|
|
ellipsoidal (Gay-Berne) particles. For spherical particles, the value |
712 |
|
|
of the Lennard-Jones $\sigma$ parameter is the particle diameter |
713 |
|
|
($d$). Gay-Berne ellipsoids have an energy scaling parameter, |
714 |
|
|
$\epsilon^s$, which describes the well depth for two identical |
715 |
|
|
ellipsoids in a {\it side-by-side} configuration. Additionally, a |
716 |
|
|
well depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, |
717 |
|
|
describes the ratio between the well depths in the {\it end-to-end} |
718 |
|
|
and side-by-side configurations. For spheres, $\epsilon^r \equiv 1$. |
719 |
|
|
Moments of inertia are also required to describe the motion of primary |
720 |
|
|
particles with orientational degrees of freedom. |
721 |
gezelter |
3299 |
|
722 |
gezelter |
3302 |
\begin{table*} |
723 |
|
|
\begin{minipage}{\linewidth} |
724 |
|
|
\begin{center} |
725 |
|
|
\caption{Parameters for the primary particles in use by the rigid body |
726 |
|
|
models in figure \ref{fig:models}.} |
727 |
|
|
\begin{tabular}{lrcccccccc} |
728 |
|
|
\hline |
729 |
|
|
& & & & & & & \multicolumn{3}c{$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$)} \\ |
730 |
|
|
& & $d$ (\AA) & $l$ (\AA) & $\epsilon^s$ (kcal/mol) & $\epsilon^r$ & |
731 |
|
|
$m$ (amu) & $I_{xx}$ & $I_{yy}$ & $I_{zz}$ \\ \hline |
732 |
gezelter |
3308 |
Sphere & & 6.5 & $= d$ & 0.8 & 1 & 190 & 802.75 & 802.75 & 802.75 \\ |
733 |
gezelter |
3302 |
Ellipsoid & & 4.6 & 13.8 & 0.8 & 0.2 & 200 & 2105 & 2105 & 421 \\ |
734 |
gezelter |
3308 |
Dumbbell &(2 identical spheres) & 6.5 & $= d$ & 0.8 & 1 & 190 & 802.75 & 802.75 & 802.75 \\ |
735 |
gezelter |
3302 |
Banana &(3 identical ellipsoids)& 4.2 & 11.2 & 0.8 & 0.2 & 240 & 10000 & 10000 & 0 \\ |
736 |
|
|
Lipid: & Spherical Head & 6.5 & $= d$ & 0.185 & 1 & 196 & & & \\ |
737 |
|
|
& Ellipsoidal Tail & 4.6 & 13.8 & 0.8 & 0.2 & 760 & 45000 & 45000 & 9000 \\ |
738 |
|
|
Solvent & & 4.7 & $= d$ & 0.8 & 1 & 72.06 & & & \\ |
739 |
|
|
\hline |
740 |
|
|
\end{tabular} |
741 |
|
|
\label{tab:parameters} |
742 |
|
|
\end{center} |
743 |
|
|
\end{minipage} |
744 |
|
|
\end{table*} |
745 |
|
|
|
746 |
gezelter |
3305 |
\begin{figure} |
747 |
|
|
\centering |
748 |
|
|
\includegraphics[width=3in]{sketch} |
749 |
|
|
\caption[Sketch of the model systems]{A sketch of the model systems |
750 |
|
|
used in evaluating the behavior of the rigid body Langevin |
751 |
|
|
integrator.} \label{fig:models} |
752 |
|
|
\end{figure} |
753 |
|
|
|
754 |
gezelter |
3302 |
\subsection{Simulation Methodology} |
755 |
|
|
We performed reference microcanonical simulations with explicit |
756 |
|
|
solvents for each of the different model system. In each case there |
757 |
|
|
was one solute model and 1929 solvent molecules present in the |
758 |
|
|
simulation box. All simulations were equilibrated using a |
759 |
|
|
constant-pressure and temperature integrator with target values of 300 |
760 |
|
|
K for the temperature and 1 atm for pressure. Following this stage, |
761 |
|
|
further equilibration and sampling was done in a microcanonical |
762 |
gezelter |
3305 |
ensemble. Since the model bodies are typically quite massive, we were |
763 |
gezelter |
3310 |
able to use a time step of 25 fs. |
764 |
|
|
|
765 |
|
|
The model systems studied used both Lennard-Jones spheres as well as |
766 |
|
|
uniaxial Gay-Berne ellipoids. In its original form, the Gay-Berne |
767 |
|
|
potential was a single site model for the interactions of rigid |
768 |
|
|
ellipsoidal molecules.\cite{Gay81} It can be thought of as a |
769 |
|
|
modification of the Gaussian overlap model originally described by |
770 |
|
|
Berne and Pechukas.\cite{Berne72} The potential is constructed in the |
771 |
|
|
familiar form of the Lennard-Jones function using |
772 |
|
|
orientation-dependent $\sigma$ and $\epsilon$ parameters, |
773 |
|
|
\begin{equation*} |
774 |
gezelter |
3341 |
V_{ij}({{\bf \hat u}_i}, {{\bf \hat u}_j}, {{\bf \hat |
775 |
|
|
r}_{ij}}) = 4\epsilon ({{\bf \hat u}_i}, {{\bf \hat u}_j}, |
776 |
|
|
{{\bf \hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({{\bf \hat u |
777 |
gezelter |
3310 |
}_i}, |
778 |
gezelter |
3341 |
{{\bf \hat u}_j}, {{\bf \hat r}_{ij}})+\sigma_0}\right)^{12} |
779 |
|
|
-\left(\frac{\sigma_0}{r_{ij}-\sigma({{\bf \hat u}_i}, {{\bf \hat u}_j}, |
780 |
|
|
{{\bf \hat r}_{ij}})+\sigma_0}\right)^6\right] |
781 |
gezelter |
3310 |
\label{eq:gb} |
782 |
|
|
\end{equation*} |
783 |
|
|
|
784 |
|
|
The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
785 |
|
|
\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
786 |
|
|
\hat{u}}_{j},{\bf \hat{r}}_{ij}))$ parameters |
787 |
|
|
are dependent on the relative orientations of the two ellipsoids (${\bf |
788 |
|
|
\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
789 |
|
|
inter-ellipsoid separation (${\bf \hat{r}}_{ij}$). The shape and |
790 |
|
|
attractiveness of each ellipsoid is governed by a relatively small set |
791 |
|
|
of parameters: $l$ and $d$ describe the length and width of each |
792 |
|
|
uniaxial ellipsoid, while $\epsilon^s$, which describes the well depth |
793 |
|
|
for two identical ellipsoids in a {\it side-by-side} configuration. |
794 |
|
|
Additionally, a well depth aspect ratio, $\epsilon^r = \epsilon^e / |
795 |
|
|
\epsilon^s$, describes the ratio between the well depths in the {\it |
796 |
|
|
end-to-end} and side-by-side configurations. Details of the potential |
797 |
|
|
are given elsewhere,\cite{Luckhurst90,Golubkov06,SunGezelter08} and an |
798 |
|
|
excellent overview of the computational methods that can be used to |
799 |
|
|
efficiently compute forces and torques for this potential can be found |
800 |
|
|
in Ref. \citen{Golubkov06} |
801 |
|
|
|
802 |
|
|
For the interaction between nonequivalent uniaxial ellipsoids (or |
803 |
|
|
between spheres and ellipsoids), the spheres are treated as ellipsoids |
804 |
|
|
with an aspect ratio of 1 ($d = l$) and with an well depth ratio |
805 |
|
|
($\epsilon^r$) of 1 ($\epsilon^e = \epsilon^s$). The form of the |
806 |
|
|
Gay-Berne potential we are using was generalized by Cleaver {\it et |
807 |
|
|
al.} and is appropriate for dissimilar uniaxial |
808 |
|
|
ellipsoids.\cite{Cleaver96} |
809 |
|
|
|
810 |
|
|
A switching function was applied to all potentials to smoothly turn |
811 |
|
|
off the interactions between a range of $22$ and $25$ \AA. The |
812 |
|
|
switching function was the standard (cubic) function, |
813 |
gezelter |
3302 |
\begin{equation} |
814 |
|
|
s(r) = |
815 |
|
|
\begin{cases} |
816 |
|
|
1 & \text{if $r \le r_{\text{sw}}$},\\ |
817 |
|
|
\frac{(r_{\text{cut}} + 2r - 3r_{\text{sw}})(r_{\text{cut}} - r)^2} |
818 |
|
|
{(r_{\text{cut}} - r_{\text{sw}})^3} |
819 |
|
|
& \text{if $r_{\text{sw}} < r \le r_{\text{cut}}$}, \\ |
820 |
|
|
0 & \text{if $r > r_{\text{cut}}$.} |
821 |
|
|
\end{cases} |
822 |
|
|
\label{eq:switchingFunc} |
823 |
|
|
\end{equation} |
824 |
gezelter |
3310 |
|
825 |
gezelter |
3302 |
To measure shear viscosities from our microcanonical simulations, we |
826 |
|
|
used the Einstein form of the pressure correlation function,\cite{hess:209} |
827 |
|
|
\begin{equation} |
828 |
gezelter |
3310 |
\eta = \lim_{t->\infty} \frac{V}{2 k_B T} \frac{d}{dt} \left\langle \left( |
829 |
|
|
\int_{t_0}^{t_0 + t} P_{xz}(t') dt' \right)^2 \right\rangle_{t_0}. |
830 |
gezelter |
3302 |
\label{eq:shear} |
831 |
|
|
\end{equation} |
832 |
|
|
A similar form exists for the bulk viscosity |
833 |
|
|
\begin{equation} |
834 |
gezelter |
3310 |
\kappa = \lim_{t->\infty} \frac{V}{2 k_B T} \frac{d}{dt} \left\langle \left( |
835 |
gezelter |
3302 |
\int_{t_0}^{t_0 + t} |
836 |
gezelter |
3310 |
\left(P\left(t'\right)-\left\langle P \right\rangle \right)dt' |
837 |
|
|
\right)^2 \right\rangle_{t_0}. |
838 |
gezelter |
3302 |
\end{equation} |
839 |
|
|
Alternatively, the shear viscosity can also be calculated using a |
840 |
|
|
Green-Kubo formula with the off-diagonal pressure tensor correlation function, |
841 |
|
|
\begin{equation} |
842 |
gezelter |
3310 |
\eta = \frac{V}{k_B T} \int_0^{\infty} \left\langle P_{xz}(t_0) P_{xz}(t_0 |
843 |
|
|
+ t) \right\rangle_{t_0} dt, |
844 |
gezelter |
3302 |
\end{equation} |
845 |
|
|
although this method converges extremely slowly and is not practical |
846 |
|
|
for obtaining viscosities from molecular dynamics simulations. |
847 |
|
|
|
848 |
|
|
The Langevin dynamics for the different model systems were performed |
849 |
|
|
at the same temperature as the average temperature of the |
850 |
|
|
microcanonical simulations and with a solvent viscosity taken from |
851 |
gezelter |
3305 |
Eq. (\ref{eq:shear}) applied to these simulations. We used 1024 |
852 |
|
|
independent solute simulations to obtain statistics on our Langevin |
853 |
|
|
integrator. |
854 |
gezelter |
3302 |
|
855 |
|
|
\subsection{Analysis} |
856 |
|
|
|
857 |
|
|
The quantities of interest when comparing the Langevin integrator to |
858 |
|
|
analytic hydrodynamic equations and to molecular dynamics simulations |
859 |
|
|
are typically translational diffusion constants and orientational |
860 |
|
|
relaxation times. Translational diffusion constants for point |
861 |
|
|
particles are computed easily from the long-time slope of the |
862 |
|
|
mean-square displacement, |
863 |
|
|
\begin{equation} |
864 |
gezelter |
3310 |
D = \lim_{t\rightarrow \infty} \frac{1}{6 t} \left\langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \right\rangle, |
865 |
gezelter |
3302 |
\end{equation} |
866 |
|
|
of the solute molecules. For models in which the translational |
867 |
gezelter |
3305 |
diffusion tensor (${\bf D}_{tt}$) has non-degenerate eigenvalues |
868 |
|
|
(i.e. any non-spherically-symmetric rigid body), it is possible to |
869 |
|
|
compute the diffusive behavior for motion parallel to each body-fixed |
870 |
|
|
axis by projecting the displacement of the particle onto the |
871 |
|
|
body-fixed reference frame at $t=0$. With an isotropic solvent, as we |
872 |
|
|
have used in this study, there are differences between the three |
873 |
gezelter |
3302 |
diffusion constants, but these must converge to the same value at |
874 |
|
|
longer times. Translational diffusion constants for the different |
875 |
gezelter |
3305 |
shaped models are shown in table \ref{tab:translation}. |
876 |
gezelter |
3302 |
|
877 |
gezelter |
3305 |
In general, the three eigenvalues ($D_1, D_2, D_3$) of the rotational |
878 |
gezelter |
3302 |
diffusion tensor (${\bf D}_{rr}$) measure the diffusion of an object |
879 |
|
|
{\it around} a particular body-fixed axis and {\it not} the diffusion |
880 |
|
|
of a vector pointing along the axis. However, these eigenvalues can |
881 |
|
|
be combined to find 5 characteristic rotational relaxation |
882 |
gezelter |
3305 |
times,\cite{PhysRev.119.53,Berne90} |
883 |
gezelter |
3302 |
\begin{eqnarray} |
884 |
gezelter |
3305 |
1 / \tau_1 & = & 6 D_r + 2 \Delta \\ |
885 |
|
|
1 / \tau_2 & = & 6 D_r - 2 \Delta \\ |
886 |
|
|
1 / \tau_3 & = & 3 (D_r + D_1) \\ |
887 |
|
|
1 / \tau_4 & = & 3 (D_r + D_2) \\ |
888 |
|
|
1 / \tau_5 & = & 3 (D_r + D_3) |
889 |
gezelter |
3302 |
\end{eqnarray} |
890 |
|
|
where |
891 |
|
|
\begin{equation} |
892 |
|
|
D_r = \frac{1}{3} \left(D_1 + D_2 + D_3 \right) |
893 |
|
|
\end{equation} |
894 |
|
|
and |
895 |
|
|
\begin{equation} |
896 |
gezelter |
3305 |
\Delta = \left( (D_1 - D_2)^2 + (D_3 - D_1 )(D_3 - D_2)\right)^{1/2} |
897 |
gezelter |
3302 |
\end{equation} |
898 |
gezelter |
3305 |
Each of these characteristic times can be used to predict the decay of |
899 |
|
|
part of the rotational correlation function when $\ell = 2$, |
900 |
gezelter |
3302 |
\begin{equation} |
901 |
gezelter |
3305 |
C_2(t) = \frac{a^2}{N^2} e^{-t/\tau_1} + \frac{b^2}{N^2} e^{-t/\tau_2}. |
902 |
gezelter |
3302 |
\end{equation} |
903 |
gezelter |
3305 |
This is the same as the $F^2_{0,0}(t)$ correlation function that |
904 |
|
|
appears in Ref. \citen{Berne90}. The amplitudes of the two decay |
905 |
|
|
terms are expressed in terms of three dimensionless functions of the |
906 |
|
|
eigenvalues: $a = \sqrt{3} (D_1 - D_2)$, $b = (2D_3 - D_1 - D_2 + |
907 |
|
|
2\Delta)$, and $N = 2 \sqrt{\Delta b}$. Similar expressions can be |
908 |
|
|
obtained for other angular momentum correlation |
909 |
|
|
functions.\cite{PhysRev.119.53,Berne90} In all of the model systems we |
910 |
|
|
studied, only one of the amplitudes of the two decay terms was |
911 |
|
|
non-zero, so it was possible to derive a single relaxation time for |
912 |
|
|
each of the hydrodynamic tensors. In many cases, these characteristic |
913 |
|
|
times are averaged and reported in the literature as a single relaxation |
914 |
|
|
time,\cite{Garcia-de-la-Torre:1997qy} |
915 |
gezelter |
3302 |
\begin{equation} |
916 |
gezelter |
3305 |
1 / \tau_0 = \frac{1}{5} \sum_{i=1}^5 \tau_{i}^{-1}, |
917 |
|
|
\end{equation} |
918 |
|
|
although for the cases reported here, this averaging is not necessary |
919 |
|
|
and only one of the five relaxation times is relevant. |
920 |
|
|
|
921 |
|
|
To test the Langevin integrator's behavior for rotational relaxation, |
922 |
|
|
we have compared the analytical orientational relaxation times (if |
923 |
|
|
they are known) with the general result from the diffusion tensor and |
924 |
|
|
with the results from both the explicitly solvated molecular dynamics |
925 |
|
|
and Langevin simulations. Relaxation times from simulations (both |
926 |
|
|
microcanonical and Langevin), were computed using Legendre polynomial |
927 |
|
|
correlation functions for a unit vector (${\bf u}$) fixed along one or |
928 |
|
|
more of the body-fixed axes of the model. |
929 |
|
|
\begin{equation} |
930 |
gezelter |
3310 |
C_{\ell}(t) = \left\langle P_{\ell}\left({\bf u}_{i}(t) \cdot {\bf |
931 |
|
|
u}_{i}(0) \right) \right\rangle |
932 |
gezelter |
3302 |
\end{equation} |
933 |
|
|
For simulations in the high-friction limit, orientational correlation |
934 |
|
|
times can then be obtained from exponential fits of this function, or by |
935 |
|
|
integrating, |
936 |
|
|
\begin{equation} |
937 |
gezelter |
3305 |
\tau = \ell (\ell + 1) \int_0^{\infty} C_{\ell}(t) dt. |
938 |
gezelter |
3302 |
\end{equation} |
939 |
gezelter |
3305 |
In lower-friction solvents, the Legendre correlation functions often |
940 |
|
|
exhibit non-exponential decay, and may not be characterized by a |
941 |
|
|
single decay constant. |
942 |
gezelter |
3302 |
|
943 |
|
|
In table \ref{tab:rotation} we show the characteristic rotational |
944 |
|
|
relaxation times (based on the diffusion tensor) for each of the model |
945 |
|
|
systems compared with the values obtained via microcanonical and Langevin |
946 |
|
|
simulations. |
947 |
|
|
|
948 |
gezelter |
3305 |
\subsection{Spherical particles} |
949 |
gezelter |
3299 |
Our model system for spherical particles was a Lennard-Jones sphere of |
950 |
|
|
diameter ($\sigma$) 6.5 \AA\ in a sea of smaller spheres ($\sigma$ = |
951 |
|
|
4.7 \AA). The well depth ($\epsilon$) for both particles was set to |
952 |
gezelter |
3302 |
an arbitrary value of 0.8 kcal/mol. |
953 |
gezelter |
3299 |
|
954 |
|
|
The Stokes-Einstein behavior of large spherical particles in |
955 |
|
|
hydrodynamic flows is well known, giving translational friction |
956 |
|
|
coefficients of $6 \pi \eta R$ (stick boundary conditions) and |
957 |
gezelter |
3302 |
rotational friction coefficients of $8 \pi \eta R^3$. Recently, |
958 |
|
|
Schmidt and Skinner have computed the behavior of spherical tag |
959 |
|
|
particles in molecular dynamics simulations, and have shown that {\it |
960 |
|
|
slip} boundary conditions ($\Xi_{tt} = 4 \pi \eta R$) may be more |
961 |
gezelter |
3299 |
appropriate for molecule-sized spheres embedded in a sea of spherical |
962 |
gezelter |
3310 |
solvent particles.\cite{Schmidt:2004fj,Schmidt:2003kx} |
963 |
gezelter |
3299 |
|
964 |
|
|
Our simulation results show similar behavior to the behavior observed |
965 |
gezelter |
3302 |
by Schmidt and Skinner. The diffusion constant obtained from our |
966 |
gezelter |
3299 |
microcanonical molecular dynamics simulations lies between the slip |
967 |
|
|
and stick boundary condition results obtained via Stokes-Einstein |
968 |
|
|
behavior. Since the Langevin integrator assumes Stokes-Einstein stick |
969 |
|
|
boundary conditions in calculating the drag and random forces for |
970 |
|
|
spherical particles, our Langevin routine obtains nearly quantitative |
971 |
|
|
agreement with the hydrodynamic results for spherical particles. One |
972 |
|
|
avenue for improvement of the method would be to compute elements of |
973 |
|
|
$\Xi_{tt}$ assuming behavior intermediate between the two boundary |
974 |
gezelter |
3302 |
conditions. |
975 |
gezelter |
3299 |
|
976 |
gezelter |
3310 |
In the explicit solvent simulations, both our solute and solvent |
977 |
|
|
particles were structureless, exerting no torques upon each other. |
978 |
|
|
Therefore, there are not rotational correlation times available for |
979 |
|
|
this model system. |
980 |
gezelter |
3299 |
|
981 |
gezelter |
3310 |
\subsection{Ellipsoids} |
982 |
|
|
For uniaxial ellipsoids ($a > b = c$), Perrin's formulae for both |
983 |
gezelter |
3299 |
translational and rotational diffusion of each of the body-fixed axes |
984 |
|
|
can be combined to give a single translational diffusion |
985 |
gezelter |
3302 |
constant,\cite{Berne90} |
986 |
gezelter |
3299 |
\begin{equation} |
987 |
|
|
D = \frac{k_B T}{6 \pi \eta a} G(\rho), |
988 |
|
|
\label{Dperrin} |
989 |
|
|
\end{equation} |
990 |
|
|
as well as a single rotational diffusion coefficient, |
991 |
|
|
\begin{equation} |
992 |
|
|
\Theta = \frac{3 k_B T}{16 \pi \eta a^3} \left\{ \frac{(2 - \rho^2) |
993 |
|
|
G(\rho) - 1}{1 - \rho^4} \right\}. |
994 |
|
|
\label{ThetaPerrin} |
995 |
|
|
\end{equation} |
996 |
|
|
In these expressions, $G(\rho)$ is a function of the axial ratio |
997 |
|
|
($\rho = b / a$), which for prolate ellipsoids, is |
998 |
|
|
\begin{equation} |
999 |
|
|
G(\rho) = (1- \rho^2)^{-1/2} \ln \left\{ \frac{1 + (1 - |
1000 |
|
|
\rho^2)^{1/2}}{\rho} \right\} |
1001 |
|
|
\label{GPerrin} |
1002 |
|
|
\end{equation} |
1003 |
|
|
Again, there is some uncertainty about the correct boundary conditions |
1004 |
|
|
to use for molecular-scale ellipsoids in a sea of similarly-sized |
1005 |
|
|
solvent particles. Ravichandran and Bagchi found that {\it slip} |
1006 |
gezelter |
3302 |
boundary conditions most closely resembled the simulation |
1007 |
|
|
results,\cite{Ravichandran:1999fk} in agreement with earlier work of |
1008 |
|
|
Tang and Evans.\cite{TANG:1993lr} |
1009 |
gezelter |
3299 |
|
1010 |
gezelter |
3305 |
Even though there are analytic resistance tensors for ellipsoids, we |
1011 |
|
|
constructed a rough-shell model using 2135 beads (each with a diameter |
1012 |
gezelter |
3310 |
of 0.25 \AA) to approximate the shape of the model ellipsoid. We |
1013 |
gezelter |
3305 |
compared the Langevin dynamics from both the simple ellipsoidal |
1014 |
|
|
resistance tensor and the rough shell approximation with |
1015 |
|
|
microcanonical simulations and the predictions of Perrin. As in the |
1016 |
|
|
case of our spherical model system, the Langevin integrator reproduces |
1017 |
|
|
almost exactly the behavior of the Perrin formulae (which is |
1018 |
|
|
unsurprising given that the Perrin formulae were used to derive the |
1019 |
gezelter |
3299 |
drag and random forces applied to the ellipsoid). We obtain |
1020 |
|
|
translational diffusion constants and rotational correlation times |
1021 |
|
|
that are within a few percent of the analytic values for both the |
1022 |
|
|
exact treatment of the diffusion tensor as well as the rough-shell |
1023 |
|
|
model for the ellipsoid. |
1024 |
|
|
|
1025 |
gezelter |
3308 |
The translational diffusion constants from the microcanonical simulations |
1026 |
|
|
agree well with the predictions of the Perrin model, although the rotational |
1027 |
|
|
correlation times are a factor of 2 shorter than expected from hydrodynamic |
1028 |
|
|
theory. One explanation for the slower rotation |
1029 |
|
|
of explicitly-solvated ellipsoids is the possibility that solute-solvent |
1030 |
|
|
collisions happen at both ends of the solute whenever the principal |
1031 |
|
|
axis of the ellipsoid is turning. In the upper portion of figure |
1032 |
|
|
\ref{fig:explanation} we sketch a physical picture of this explanation. |
1033 |
|
|
Since our Langevin integrator is providing nearly quantitative agreement with |
1034 |
|
|
the Perrin model, it also predicts orientational diffusion for ellipsoids that |
1035 |
|
|
exceed explicitly solvated correlation times by a factor of two. |
1036 |
gezelter |
3299 |
|
1037 |
gezelter |
3310 |
\subsection{Rigid dumbbells} |
1038 |
gezelter |
3302 |
Perhaps the only {\it composite} rigid body for which analytic |
1039 |
|
|
expressions for the hydrodynamic tensor are available is the |
1040 |
|
|
two-sphere dumbbell model. This model consists of two non-overlapping |
1041 |
|
|
spheres held by a rigid bond connecting their centers. There are |
1042 |
|
|
competing expressions for the 6x6 resistance tensor for this |
1043 |
gezelter |
3341 |
model. The second order expression introduced by Rotne and |
1044 |
|
|
Prager,\cite{Rotne1969} and improved by Garc\'{i}a de la Torre and |
1045 |
|
|
Bloomfield,\cite{Torre1977} is given above as |
1046 |
gezelter |
3299 |
Eq. (\ref{introEquation:RPTensorNonOverlapped}). In our case, we use |
1047 |
|
|
a model dumbbell in which the two spheres are identical Lennard-Jones |
1048 |
|
|
particles ($\sigma$ = 6.5 \AA\ , $\epsilon$ = 0.8 kcal / mol) held at |
1049 |
gezelter |
3302 |
a distance of 6.532 \AA. |
1050 |
gezelter |
3299 |
|
1051 |
|
|
The theoretical values for the translational diffusion constant of the |
1052 |
|
|
dumbbell are calculated from the work of Stimson and Jeffery, who |
1053 |
|
|
studied the motion of this system in a flow parallel to the |
1054 |
gezelter |
3302 |
inter-sphere axis,\cite{Stimson:1926qy} and Davis, who studied the |
1055 |
|
|
motion in a flow {\it perpendicular} to the inter-sphere |
1056 |
|
|
axis.\cite{Davis:1969uq} We know of no analytic solutions for the {\it |
1057 |
|
|
orientational} correlation times for this model system (other than |
1058 |
gezelter |
3341 |
those derived from the 6 x 6 tensor mentioned above). |
1059 |
gezelter |
3299 |
|
1060 |
gezelter |
3305 |
The bead model for this model system comprises the two large spheres |
1061 |
|
|
by themselves, while the rough shell approximation used 3368 separate |
1062 |
|
|
beads (each with a diameter of 0.25 \AA) to approximate the shape of |
1063 |
|
|
the rigid body. The hydrodynamics tensors computed from both the bead |
1064 |
|
|
and rough shell models are remarkably similar. Computing the initial |
1065 |
|
|
hydrodynamic tensor for a rough shell model can be quite expensive (in |
1066 |
|
|
this case it requires inverting a 10104 x 10104 matrix), while the |
1067 |
|
|
bead model is typically easy to compute (in this case requiring |
1068 |
gezelter |
3308 |
inversion of a 6 x 6 matrix). |
1069 |
gezelter |
3305 |
|
1070 |
gezelter |
3308 |
\begin{figure} |
1071 |
|
|
\centering |
1072 |
gezelter |
3310 |
\includegraphics[width=2in]{RoughShell} |
1073 |
gezelter |
3308 |
\caption[Model rigid bodies and their rough shell approximations]{The |
1074 |
|
|
model rigid bodies (left column) used to test this algorithm and their |
1075 |
|
|
rough-shell approximations (right-column) that were used to compute |
1076 |
|
|
the hydrodynamic tensors. The top two models (ellipsoid and dumbbell) |
1077 |
|
|
have analytic solutions and were used to test the rough shell |
1078 |
|
|
approximation. The lower two models (banana and lipid) were compared |
1079 |
|
|
with explicitly-solvated molecular dynamics simulations. } |
1080 |
|
|
\label{fig:roughShell} |
1081 |
|
|
\end{figure} |
1082 |
|
|
|
1083 |
|
|
|
1084 |
gezelter |
3305 |
Once the hydrodynamic tensor has been computed, there is no additional |
1085 |
|
|
penalty for carrying out a Langevin simulation with either of the two |
1086 |
|
|
different hydrodynamics models. Our naive expectation is that since |
1087 |
|
|
the rigid body's surface is roughened under the various shell models, |
1088 |
|
|
the diffusion constants will be even farther from the ``slip'' |
1089 |
|
|
boundary conditions than observed for the bead model (which uses a |
1090 |
|
|
Stokes-Einstein model to arrive at the hydrodynamic tensor). For the |
1091 |
|
|
dumbbell, this prediction is correct although all of the Langevin |
1092 |
|
|
diffusion constants are within 6\% of the diffusion constant predicted |
1093 |
|
|
from the fully solvated system. |
1094 |
|
|
|
1095 |
gezelter |
3308 |
For rotational motion, Langevin integration (and the hydrodynamic tensor) |
1096 |
|
|
yields rotational correlation times that are substantially shorter than those |
1097 |
|
|
obtained from explicitly-solvated simulations. It is likely that this is due |
1098 |
|
|
to the large size of the explicit solvent spheres, a feature that prevents |
1099 |
|
|
the solvent from coming in contact with a substantial fraction of the surface |
1100 |
|
|
area of the dumbbell. Therefore, the explicit solvent only provides drag |
1101 |
|
|
over a substantially reduced surface area of this model, while the |
1102 |
|
|
hydrodynamic theories utilize the entire surface area for estimating |
1103 |
|
|
rotational diffusion. A sketch of the free volume available in the explicit |
1104 |
|
|
solvent simulations is shown in figure \ref{fig:explanation}. |
1105 |
gezelter |
3305 |
|
1106 |
gezelter |
3310 |
|
1107 |
|
|
\begin{figure} |
1108 |
|
|
\centering |
1109 |
|
|
\includegraphics[width=6in]{explanation} |
1110 |
|
|
\caption[Explanations of the differences between orientational |
1111 |
|
|
correlation times for explicitly-solvated models and hydrodynamics |
1112 |
|
|
predictions]{Explanations of the differences between orientational |
1113 |
|
|
correlation times for explicitly-solvated models and hydrodynamic |
1114 |
|
|
predictions. For the ellipsoids (upper figures), rotation of the |
1115 |
|
|
principal axis can involve correlated collisions at both sides of the |
1116 |
|
|
solute. In the rigid dumbbell model (lower figures), the large size |
1117 |
|
|
of the explicit solvent spheres prevents them from coming in contact |
1118 |
|
|
with a substantial fraction of the surface area of the dumbbell. |
1119 |
|
|
Therefore, the explicit solvent only provides drag over a |
1120 |
|
|
substantially reduced surface area of this model, where the |
1121 |
|
|
hydrodynamic theories utilize the entire surface area for estimating |
1122 |
|
|
rotational diffusion. |
1123 |
|
|
} \label{fig:explanation} |
1124 |
|
|
\end{figure} |
1125 |
|
|
|
1126 |
|
|
\subsection{Composite banana-shaped molecules} |
1127 |
|
|
Banana-shaped rigid bodies composed of three Gay-Berne ellipsoids have |
1128 |
|
|
been used by Orlandi {\it et al.} to observe mesophases in |
1129 |
|
|
coarse-grained models for bent-core liquid crystalline |
1130 |
|
|
molecules.\cite{Orlandi:2006fk} We have used the same overlapping |
1131 |
gezelter |
3299 |
ellipsoids as a way to test the behavior of our algorithm for a |
1132 |
|
|
structure of some interest to the materials science community, |
1133 |
|
|
although since we are interested in capturing only the hydrodynamic |
1134 |
gezelter |
3310 |
behavior of this model, we have left out the dipolar interactions of |
1135 |
|
|
the original Orlandi model. |
1136 |
gezelter |
3308 |
|
1137 |
gezelter |
3341 |
A reference system composed of a single banana rigid body embedded in |
1138 |
|
|
a sea of 1929 solvent particles was created and run under standard |
1139 |
|
|
(microcanonical) molecular dynamics. The resulting viscosity of this |
1140 |
|
|
mixture was 0.298 centipoise (as estimated using |
1141 |
|
|
Eq. (\ref{eq:shear})). To calculate the hydrodynamic properties of |
1142 |
|
|
the banana rigid body model, we created a rough shell (see |
1143 |
|
|
Fig.~\ref{fig:roughShell}), in which the banana is represented as a |
1144 |
|
|
``shell'' made of 3321 identical beads (0.25 \AA\ in diameter) |
1145 |
|
|
distributed on the surface. Applying the procedure described in |
1146 |
|
|
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
1147 |
|
|
identified the center of resistance, ${\bf r} = $(0 \AA, 0.81 \AA, 0 |
1148 |
|
|
\AA). |
1149 |
gezelter |
3299 |
|
1150 |
gezelter |
3341 |
The Langevin rigid-body integrator (and the hydrodynamic diffusion |
1151 |
|
|
tensor) are essentially quantitative for translational diffusion of |
1152 |
|
|
this model. Orientational correlation times under the Langevin |
1153 |
|
|
rigid-body integrator are within 11\% of the values obtained from |
1154 |
|
|
explicit solvent, but these models also exhibit some solvent |
1155 |
|
|
inaccessible surface area in the explicitly-solvated case. |
1156 |
gezelter |
3308 |
|
1157 |
gezelter |
3310 |
\subsection{Composite sphero-ellipsoids} |
1158 |
gezelter |
3341 |
|
1159 |
gezelter |
3299 |
Spherical heads perched on the ends of Gay-Berne ellipsoids have been |
1160 |
xsun |
3312 |
used recently as models for lipid |
1161 |
gezelter |
3341 |
molecules.\cite{SunGezelter08,Ayton01} A reference system composed of |
1162 |
|
|
a single lipid rigid body embedded in a sea of 1929 solvent particles |
1163 |
|
|
was created and run under a microcanonical ensemble. The resulting |
1164 |
|
|
viscosity of this mixture was 0.349 centipoise (as estimated using |
1165 |
xsun |
3312 |
Eq. (\ref{eq:shear})). To calculate the hydrodynamic properties of |
1166 |
|
|
the lipid rigid body model, we created a rough shell (see |
1167 |
|
|
Fig.~\ref{fig:roughShell}), in which the lipid is represented as a |
1168 |
|
|
``shell'' made of 3550 identical beads (0.25 \AA\ in diameter) |
1169 |
|
|
distributed on the surface. Applying the procedure described in |
1170 |
|
|
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
1171 |
|
|
identified the center of resistance, ${\bf r} = $(0 \AA, 0 \AA, 1.46 |
1172 |
|
|
\AA). |
1173 |
gezelter |
3310 |
|
1174 |
gezelter |
3341 |
The translational diffusion constants and rotational correlation times |
1175 |
|
|
obtained using the Langevin rigid-body integrator (and the |
1176 |
|
|
hydrodynamic tensor) are essentially quantitative when compared with |
1177 |
|
|
the explicit solvent simulations for this model system. |
1178 |
gezelter |
3315 |
|
1179 |
gezelter |
3310 |
\subsection{Summary} |
1180 |
gezelter |
3341 |
According to our simulations, the Langevin rigid-body integrator we |
1181 |
|
|
have developed is a reliable way to replace explicit solvent |
1182 |
|
|
simulations in cases where the detailed solute-solvent interactions do |
1183 |
|
|
not greatly impact the forces on the solute. In cases where the |
1184 |
|
|
dielectric screening of the solvent, or specific solute-solvent |
1185 |
|
|
interactions become important for structural or dynamic features of |
1186 |
|
|
the solute molecule, this integrator may be less useful. However, for |
1187 |
|
|
the kinds of coarse-grained modeling that have become popular in |
1188 |
|
|
recent years, this integrator may prove itself to be quite valuable. |
1189 |
xsun |
3298 |
|
1190 |
gezelter |
3315 |
\begin{figure} |
1191 |
|
|
\centering |
1192 |
|
|
\includegraphics[width=\linewidth]{graph} |
1193 |
|
|
\caption[Mean squared displacements and orientational |
1194 |
|
|
correlation functions for each of the model rigid bodies.]{The |
1195 |
|
|
mean-squared displacements ($\langle r^2(t) \rangle$) and |
1196 |
|
|
orientational correlation functions ($C_2(t)$) for each of the model |
1197 |
|
|
rigid bodies studied. The circles are the results for microcanonical |
1198 |
|
|
simulations with explicit solvent molecules, while the other data sets |
1199 |
|
|
are results for Langevin dynamics using the different hydrodynamic |
1200 |
|
|
tensor approximations. The Perrin model for the ellipsoids is |
1201 |
|
|
considered the ``exact'' hydrodynamic behavior (this can also be said |
1202 |
|
|
for the translational motion of the dumbbell operating under the bead |
1203 |
|
|
model). In most cases, the various hydrodynamics models reproduce |
1204 |
|
|
each other quantitatively.} |
1205 |
|
|
\label{fig:results} |
1206 |
|
|
\end{figure} |
1207 |
|
|
|
1208 |
xsun |
3298 |
\begin{table*} |
1209 |
|
|
\begin{minipage}{\linewidth} |
1210 |
|
|
\begin{center} |
1211 |
gezelter |
3305 |
\caption{Translational diffusion constants (D) for the model systems |
1212 |
|
|
calculated using microcanonical simulations (with explicit solvent), |
1213 |
|
|
theoretical predictions, and Langevin simulations (with implicit solvent). |
1214 |
|
|
Analytical solutions for the exactly-solved hydrodynamics models are |
1215 |
|
|
from Refs. \citen{Einstein05} (sphere), \citen{Perrin1934} and \citen{Perrin1936} |
1216 |
|
|
(ellipsoid), \citen{Stimson:1926qy} and \citen{Davis:1969uq} |
1217 |
|
|
(dumbbell). The other model systems have no known analytic solution. |
1218 |
|
|
All diffusion constants are reported in units of $10^{-3}$ cm$^2$ / ps (= |
1219 |
|
|
$10^{-4}$ \AA$^2$ / fs). } |
1220 |
|
|
\begin{tabular}{lccccccc} |
1221 |
xsun |
3298 |
\hline |
1222 |
gezelter |
3305 |
& \multicolumn{2}c{microcanonical simulation} & & \multicolumn{3}c{Theoretical} & Langevin \\ |
1223 |
|
|
\cline{2-3} \cline{5-7} |
1224 |
|
|
model & $\eta$ (centipoise) & D & & Analytical & method & Hydrodynamics & simulation \\ |
1225 |
xsun |
3298 |
\hline |
1226 |
xsun |
3312 |
sphere & 0.279 & 3.06 & & 2.42 & exact & 2.42 & 2.33 \\ |
1227 |
gezelter |
3305 |
ellipsoid & 0.255 & 2.44 & & 2.34 & exact & 2.34 & 2.37 \\ |
1228 |
|
|
& 0.255 & 2.44 & & 2.34 & rough shell & 2.36 & 2.28 \\ |
1229 |
xsun |
3312 |
dumbbell & 0.308 & 2.06 & & 1.64 & bead model & 1.65 & 1.62 \\ |
1230 |
|
|
& 0.308 & 2.06 & & 1.64 & rough shell & 1.59 & 1.62 \\ |
1231 |
gezelter |
3305 |
banana & 0.298 & 1.53 & & & rough shell & 1.56 & 1.55 \\ |
1232 |
gezelter |
3341 |
lipid & 0.349 & 1.41 & & & rough shell & 1.33 & 1.32 \\ |
1233 |
xsun |
3298 |
\end{tabular} |
1234 |
|
|
\label{tab:translation} |
1235 |
|
|
\end{center} |
1236 |
|
|
\end{minipage} |
1237 |
|
|
\end{table*} |
1238 |
|
|
|
1239 |
|
|
\begin{table*} |
1240 |
|
|
\begin{minipage}{\linewidth} |
1241 |
|
|
\begin{center} |
1242 |
gezelter |
3305 |
\caption{Orientational relaxation times ($\tau$) for the model systems using |
1243 |
|
|
microcanonical simulation (with explicit solvent), theoretical |
1244 |
|
|
predictions, and Langevin simulations (with implicit solvent). All |
1245 |
|
|
relaxation times are for the rotational correlation function with |
1246 |
|
|
$\ell = 2$ and are reported in units of ps. The ellipsoidal model has |
1247 |
|
|
an exact solution for the orientational correlation time due to |
1248 |
|
|
Perrin, but the other model systems have no known analytic solution.} |
1249 |
|
|
\begin{tabular}{lccccccc} |
1250 |
xsun |
3298 |
\hline |
1251 |
gezelter |
3305 |
& \multicolumn{2}c{microcanonical simulation} & & \multicolumn{3}c{Theoretical} & Langevin \\ |
1252 |
|
|
\cline{2-3} \cline{5-7} |
1253 |
|
|
model & $\eta$ (centipoise) & $\tau$ & & Perrin & method & Hydrodynamic & simulation \\ |
1254 |
xsun |
3298 |
\hline |
1255 |
xsun |
3312 |
sphere & 0.279 & & & 9.69 & exact & 9.69 & 9.64 \\ |
1256 |
gezelter |
3305 |
ellipsoid & 0.255 & 46.7 & & 22.0 & exact & 22.0 & 22.2 \\ |
1257 |
|
|
& 0.255 & 46.7 & & 22.0 & rough shell & 22.6 & 22.2 \\ |
1258 |
xsun |
3312 |
dumbbell & 0.308 & 14.1 & & & bead model & 50.0 & 50.1 \\ |
1259 |
|
|
& 0.308 & 14.1 & & & rough shell & 41.5 & 41.3 \\ |
1260 |
gezelter |
3305 |
banana & 0.298 & 63.8 & & & rough shell & 70.9 & 70.9 \\ |
1261 |
|
|
lipid & 0.349 & 78.0 & & & rough shell & 76.9 & 77.9 \\ |
1262 |
|
|
\hline |
1263 |
xsun |
3298 |
\end{tabular} |
1264 |
|
|
\label{tab:rotation} |
1265 |
|
|
\end{center} |
1266 |
|
|
\end{minipage} |
1267 |
|
|
\end{table*} |
1268 |
|
|
|
1269 |
gezelter |
3310 |
\section{Application: A rigid-body lipid bilayer} |
1270 |
|
|
|
1271 |
|
|
The Langevin dynamics integrator was applied to study the formation of |
1272 |
|
|
corrugated structures emerging from simulations of the coarse grained |
1273 |
|
|
lipid molecular models presented above. The initial configuration is |
1274 |
xsun |
3298 |
taken from our molecular dynamics studies on lipid bilayers with |
1275 |
gezelter |
3310 |
lennard-Jones sphere solvents. The solvent molecules were excluded |
1276 |
|
|
from the system, and the experimental value for the viscosity of water |
1277 |
|
|
at 20C ($\eta = 1.00$ cp) was used to mimic the hydrodynamic effects |
1278 |
|
|
of the solvent. The absence of explicit solvent molecules and the |
1279 |
|
|
stability of the integrator allowed us to take timesteps of 50 fs. A |
1280 |
|
|
total simulation run time of 100 ns was sampled. |
1281 |
|
|
Fig. \ref{fig:bilayer} shows the configuration of the system after 100 |
1282 |
|
|
ns, and the ripple structure remains stable during the entire |
1283 |
|
|
trajectory. Compared with using explicit bead-model solvent |
1284 |
|
|
molecules, the efficiency of the simulation has increased by an order |
1285 |
xsun |
3298 |
of magnitude. |
1286 |
|
|
|
1287 |
gezelter |
3310 |
\begin{figure} |
1288 |
|
|
\centering |
1289 |
|
|
\includegraphics[width=\linewidth]{bilayer} |
1290 |
|
|
\caption[Snapshot of a bilayer of rigid-body models for lipids]{A |
1291 |
|
|
snapshot of a bilayer composed of rigid-body models for lipid |
1292 |
|
|
molecules evolving using the Langevin integrator described in this |
1293 |
|
|
work.} \label{fig:bilayer} |
1294 |
|
|
\end{figure} |
1295 |
|
|
|
1296 |
tim |
2746 |
\section{Conclusions} |
1297 |
|
|
|
1298 |
tim |
2999 |
We have presented a new Langevin algorithm by incorporating the |
1299 |
|
|
hydrodynamics properties of arbitrary shaped molecules into an |
1300 |
gezelter |
3308 |
advanced symplectic integration scheme. Further studies in systems |
1301 |
|
|
involving banana shaped molecules illustrated that the dynamic |
1302 |
|
|
properties could be preserved by using this new algorithm as an |
1303 |
|
|
implicit solvent model. |
1304 |
tim |
2999 |
|
1305 |
|
|
|
1306 |
tim |
2746 |
\section{Acknowledgments} |
1307 |
|
|
Support for this project was provided by the National Science |
1308 |
|
|
Foundation under grant CHE-0134881. T.L. also acknowledges the |
1309 |
|
|
financial support from center of applied mathematics at University |
1310 |
|
|
of Notre Dame. |
1311 |
|
|
\newpage |
1312 |
|
|
|
1313 |
gezelter |
3305 |
\bibliographystyle{jcp} |
1314 |
tim |
2746 |
\bibliography{langevin} |
1315 |
|
|
|
1316 |
|
|
\end{document} |