--- trunk/tengDissertation/Introduction.tex 2006/04/15 01:08:18 2713 +++ trunk/tengDissertation/Introduction.tex 2006/04/17 20:39:26 2716 @@ -1247,8 +1247,14 @@ rigid body \end{equation} \section{\label{introSection:langevinDynamics}Langevin Dynamics} - -\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} +As an alternative to newtonian dynamics, Langevin dynamics, which +mimics a simple heat bath with stochastic and dissipative forces, +has been applied in a variety of studies. This section will review +the theory of Langevin dynamics simulation. A brief derivation of +generalized Langevin Dynamics will be given first. Follow that, we +will discuss the physical meaning of the terms appearing in the +equation as well as the calculation of friction tensor from +hydrodynamics theory. \subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} @@ -1442,16 +1448,224 @@ And since the $q$ coordinates are harmonic oscillators \label{introEquation:secondFluctuationDissipation} \end{equation} -\section{\label{introSection:hydroynamics}Hydrodynamics} - \subsection{\label{introSection:frictionTensor} Friction Tensor} -\subsection{\label{introSection:analyticalApproach}Analytical -Approach} - -\subsection{\label{introSection:approximationApproach}Approximation -Approach} +Theoretically, the friction kernel can be determined using velocity +autocorrelation function. However, this approach become impractical +when the system become more and more complicate. Instead, various +approaches based on hydrodynamics have been developed to calculate +the friction coefficients. The friction effect is isotropic in +Equation, \zeta can be taken as a scalar. In general, friction +tensor \Xi is a $6\times 6$ matrix given by +\[ +\Xi = \left( {\begin{array}{*{20}c} + {\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ + {\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ +\end{array}} \right). +\] +Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction +tensor and rotational friction tensor respectively, while ${\Xi^{tr} +}$ is translation-rotation coupling tensor and $ {\Xi^{rt} }$ is +rotation-translation coupling tensor. -\subsection{\label{introSection:centersRigidBody}Centers of Rigid -Body} +\[ +\left( \begin{array}{l} + F_t \\ + \tau \\ + \end{array} \right) = - \left( {\begin{array}{*{20}c} + {\Xi ^{tt} } & {\Xi ^{rt} } \\ + {\Xi ^{tr} } & {\Xi ^{rr} } \\ +\end{array}} \right)\left( \begin{array}{l} + v \\ + w \\ + \end{array} \right) +\] + +\subsubsection{\label{introSection:analyticalApproach}The Friction Tensor for Regular Shape} +For a spherical particle, the translational and rotational friction +constant can be calculated from Stoke's law, +\[ +\Xi ^{tt} = \left( {\begin{array}{*{20}c} + {6\pi \eta R} & 0 & 0 \\ + 0 & {6\pi \eta R} & 0 \\ + 0 & 0 & {6\pi \eta R} \\ +\end{array}} \right) +\] +and +\[ +\Xi ^{rr} = \left( {\begin{array}{*{20}c} + {8\pi \eta R^3 } & 0 & 0 \\ + 0 & {8\pi \eta R^3 } & 0 \\ + 0 & 0 & {8\pi \eta R^3 } \\ +\end{array}} \right) +\] +where $\eta$ is the viscosity of the solvent and $R$ is the +hydrodynamics radius. -\section{\label{introSection:correlationFunctions}Correlation Functions} +Other non-spherical particles have more complex properties. + +\[ +S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 +} }}{b} +\] + + +\[ +S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } +}}{a} +\] + +\[ +\begin{array}{l} + \Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ + \Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ + \end{array} +\] + +\[ +\begin{array}{l} + \Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ + \Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ + \end{array} +\] + + +\subsubsection{\label{introSection:approximationApproach}The Friction Tensor for Arbitrary Shape} +Unlike spherical and other regular shaped molecules, there is not +analytical solution for friction tensor of any arbitrary shaped +rigid molecules. The ellipsoid of revolution model and general +triaxial ellipsoid model have been used to approximate the +hydrodynamic properties of rigid bodies. However, since the mapping +from all possible ellipsoidal space, $r$-space, to all possible +combination of rotational diffusion coefficients, $D$-space is not +unique\cite{Wegener79} as well as the intrinsic coupling between +translational and rotational motion of rigid body\cite{}, general +ellipsoid is not always suitable for modeling arbitrarily shaped +rigid molecule. A number of studies have been devoted to determine +the friction tensor for irregularly shaped rigid bodies using more +advanced method\cite{} where the molecule of interest was modeled by +combinations of spheres(beads)\cite{} and the hydrodynamics +properties of the molecule can be calculated using the hydrodynamic +interaction tensor. Let us consider a rigid assembly of $N$ beads +immersed in a continuous medium. Due to hydrodynamics interaction, +the ``net'' velocity of $i$th bead, $v'_i$ is different than its +unperturbed velocity $v_i$, +\[ +v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } +\] +where $F_i$ is the frictional force, and $T_{ij}$ is the +hydrodynamic interaction tensor. The friction force of $i$th bead is +proportional to its ``net'' velocity +\begin{equation} +F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. +\label{introEquation:tensorExpression} +\end{equation} +This equation is the basis for deriving the hydrodynamic tensor. In +1930, Oseen and Burgers gave a simple solution to Equation +\ref{introEquation:tensorExpression} +\begin{equation} +T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} +R_{ij}^T }}{{R_{ij}^2 }}} \right). +\label{introEquation:oseenTensor} +\end{equation} +Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. +A second order expression for element of different size was +introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de +la Torre and Bloomfield, +\begin{equation} +T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + +\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma +_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - +\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. +\label{introEquation:RPTensorNonOverlapped} +\end{equation} +Both of the Equation \ref{introEquation:oseenTensor} and Equation +\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} +\ge \sigma _i + \sigma _j$. An alternative expression for +overlapping beads with the same radius, $\sigma$, is given by +\begin{equation} +T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - +\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + +\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] +\label{introEquation:RPTensorOverlapped} +\end{equation} + +%Bead Modeling + +\[ +B = \left( {\begin{array}{*{20}c} + {T_{11} } & \ldots & {T_{1N} } \\ + \vdots & \ddots & \vdots \\ + {T_{N1} } & \cdots & {T_{NN} } \\ +\end{array}} \right) +\] + +\[ +C = B^{ - 1} = \left( {\begin{array}{*{20}c} + {C_{11} } & \ldots & {C_{1N} } \\ + \vdots & \ddots & \vdots \\ + {C_{N1} } & \cdots & {C_{NN} } \\ +\end{array}} \right) +\] + +\begin{equation} +\begin{array}{l} + \Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ + \Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ + \Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ + \end{array} +\end{equation} +where +\[ +U_i = \left( {\begin{array}{*{20}c} + 0 & { - z_i } & {y_i } \\ + {z_i } & 0 & { - x_i } \\ + { - y_i } & {x_i } & 0 \\ +\end{array}} \right) +\] + +\[ +r_{OR} = \left( \begin{array}{l} + x_{OR} \\ + y_{OR} \\ + z_{OR} \\ + \end{array} \right) = \left( {\begin{array}{*{20}c} + {\Xi _{yy}^{rr} + \Xi _{zz}^{rr} } & { - \Xi _{xy}^{rr} } & { - \Xi _{xz}^{rr} } \\ + { - \Xi _{yx}^{rr} } & {\Xi _{zz}^{rr} + \Xi _{xx}^{rr} } & { - \Xi _{yz}^{rr} } \\ + { - \Xi _{zx}^{rr} } & { - \Xi _{yz}^{rr} } & {\Xi _{xx}^{rr} + \Xi _{yy}^{rr} } \\ +\end{array}} \right)^{ - 1} \left( \begin{array}{l} + \Xi _{yz}^{tr} - \Xi _{zy}^{tr} \\ + \Xi _{zx}^{tr} - \Xi _{xz}^{tr} \\ + \Xi _{xy}^{tr} - \Xi _{yx}^{tr} \\ + \end{array} \right) +\] + +\[ +U_{OR} = \left( {\begin{array}{*{20}c} + 0 & { - z_{OR} } & {y_{OR} } \\ + {z_i } & 0 & { - x_{OR} } \\ + { - y_{OR} } & {x_{OR} } & 0 \\ +\end{array}} \right) +\] + +\[ +\begin{array}{l} + \Xi _R^{tt} = \Xi _{}^{tt} \\ + \Xi _R^{tr} = \Xi _R^{rt} = \Xi _{}^{tr} - U_{OR} \Xi _{}^{tt} \\ + \Xi _R^{rr} = \Xi _{}^{rr} - U_{OR} \Xi _{}^{tt} U_{OR} + \Xi _{}^{tr} U_{OR} - U_{OR} \Xi _{}^{tr} ^{^T } \\ + \end{array} +\] + +\[ +D_R = \left( {\begin{array}{*{20}c} + {D_R^{tt} } & {D_R^{rt} } \\ + {D_R^{tr} } & {D_R^{rr} } \\ +\end{array}} \right) = k_b T\left( {\begin{array}{*{20}c} + {\Xi _R^{tt} } & {\Xi _R^{rt} } \\ + {\Xi _R^{tr} } & {\Xi _R^{rr} } \\ +\end{array}} \right)^{ - 1} +\] + + +%Approximation Methods + +%\section{\label{introSection:correlationFunctions}Correlation Functions}