1247 |
|
\end{equation} |
1248 |
|
|
1249 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1250 |
< |
|
1251 |
< |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
1250 |
> |
As an alternative to newtonian dynamics, Langevin dynamics, which |
1251 |
> |
mimics a simple heat bath with stochastic and dissipative forces, |
1252 |
> |
has been applied in a variety of studies. This section will review |
1253 |
> |
the theory of Langevin dynamics simulation. A brief derivation of |
1254 |
> |
generalized Langevin Dynamics will be given first. Follow that, we |
1255 |
> |
will discuss the physical meaning of the terms appearing in the |
1256 |
> |
equation as well as the calculation of friction tensor from |
1257 |
> |
hydrodynamics theory. |
1258 |
|
|
1259 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
1260 |
|
|
1448 |
|
\label{introEquation:secondFluctuationDissipation} |
1449 |
|
\end{equation} |
1450 |
|
|
1445 |
– |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
1446 |
– |
|
1451 |
|
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
1452 |
< |
\subsection{\label{introSection:analyticalApproach}Analytical |
1453 |
< |
Approach} |
1454 |
< |
|
1455 |
< |
\subsection{\label{introSection:approximationApproach}Approximation |
1456 |
< |
Approach} |
1452 |
> |
Theoretically, the friction kernel can be determined using velocity |
1453 |
> |
autocorrelation function. However, this approach become impractical |
1454 |
> |
when the system become more and more complicate. Instead, various |
1455 |
> |
approaches based on hydrodynamics have been developed to calculate |
1456 |
> |
the friction coefficients. The friction effect is isotropic in |
1457 |
> |
Equation, \zeta can be taken as a scalar. In general, friction |
1458 |
> |
tensor \Xi is a $6\times 6$ matrix given by |
1459 |
> |
\[ |
1460 |
> |
\Xi = \left( {\begin{array}{*{20}c} |
1461 |
> |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
1462 |
> |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
1463 |
> |
\end{array}} \right). |
1464 |
> |
\] |
1465 |
> |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
1466 |
> |
tensor and rotational friction tensor respectively, while ${\Xi^{tr} |
1467 |
> |
}$ is translation-rotation coupling tensor and $ {\Xi^{rt} }$ is |
1468 |
> |
rotation-translation coupling tensor. |
1469 |
|
|
1470 |
< |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
1471 |
< |
Body} |
1470 |
> |
\[ |
1471 |
> |
\left( \begin{array}{l} |
1472 |
> |
F_t \\ |
1473 |
> |
\tau \\ |
1474 |
> |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
1475 |
> |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
1476 |
> |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
1477 |
> |
\end{array}} \right)\left( \begin{array}{l} |
1478 |
> |
v \\ |
1479 |
> |
w \\ |
1480 |
> |
\end{array} \right) |
1481 |
> |
\] |
1482 |
> |
|
1483 |
> |
\subsubsection{\label{introSection:analyticalApproach}The Friction Tensor for Regular Shape} |
1484 |
> |
For a spherical particle, the translational and rotational friction |
1485 |
> |
constant can be calculated from Stoke's law, |
1486 |
> |
\[ |
1487 |
> |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
1488 |
> |
{6\pi \eta R} & 0 & 0 \\ |
1489 |
> |
0 & {6\pi \eta R} & 0 \\ |
1490 |
> |
0 & 0 & {6\pi \eta R} \\ |
1491 |
> |
\end{array}} \right) |
1492 |
> |
\] |
1493 |
> |
and |
1494 |
> |
\[ |
1495 |
> |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
1496 |
> |
{8\pi \eta R^3 } & 0 & 0 \\ |
1497 |
> |
0 & {8\pi \eta R^3 } & 0 \\ |
1498 |
> |
0 & 0 & {8\pi \eta R^3 } \\ |
1499 |
> |
\end{array}} \right) |
1500 |
> |
\] |
1501 |
> |
where $\eta$ is the viscosity of the solvent and $R$ is the |
1502 |
> |
hydrodynamics radius. |
1503 |
|
|
1504 |
< |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
1504 |
> |
Other non-spherical particles have more complex properties. |
1505 |
> |
|
1506 |
> |
\[ |
1507 |
> |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
1508 |
> |
} }}{b} |
1509 |
> |
\] |
1510 |
> |
|
1511 |
> |
|
1512 |
> |
\[ |
1513 |
> |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
1514 |
> |
}}{a} |
1515 |
> |
\] |
1516 |
> |
|
1517 |
> |
\[ |
1518 |
> |
\begin{array}{l} |
1519 |
> |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
1520 |
> |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
1521 |
> |
\end{array} |
1522 |
> |
\] |
1523 |
> |
|
1524 |
> |
\[ |
1525 |
> |
\begin{array}{l} |
1526 |
> |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
1527 |
> |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
1528 |
> |
\end{array} |
1529 |
> |
\] |
1530 |
> |
|
1531 |
> |
|
1532 |
> |
\subsubsection{\label{introSection:approximationApproach}The Friction Tensor for Arbitrary Shape} |
1533 |
> |
Unlike spherical and other regular shaped molecules, there is not |
1534 |
> |
analytical solution for friction tensor of any arbitrary shaped |
1535 |
> |
rigid molecules. The ellipsoid of revolution model and general |
1536 |
> |
triaxial ellipsoid model have been used to approximate the |
1537 |
> |
hydrodynamic properties of rigid bodies. However, since the mapping |
1538 |
> |
from all possible ellipsoidal space, $r$-space, to all possible |
1539 |
> |
combination of rotational diffusion coefficients, $D$-space is not |
1540 |
> |
unique\cite{Wegener79} as well as the intrinsic coupling between |
1541 |
> |
translational and rotational motion of rigid body\cite{}, general |
1542 |
> |
ellipsoid is not always suitable for modeling arbitrarily shaped |
1543 |
> |
rigid molecule. A number of studies have been devoted to determine |
1544 |
> |
the friction tensor for irregularly shaped rigid bodies using more |
1545 |
> |
advanced method\cite{} where the molecule of interest was modeled by |
1546 |
> |
combinations of spheres(beads)\cite{} and the hydrodynamics |
1547 |
> |
properties of the molecule can be calculated using the hydrodynamic |
1548 |
> |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
1549 |
> |
immersed in a continuous medium. Due to hydrodynamics interaction, |
1550 |
> |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
1551 |
> |
unperturbed velocity $v_i$, |
1552 |
> |
\[ |
1553 |
> |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
1554 |
> |
\] |
1555 |
> |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
1556 |
> |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
1557 |
> |
proportional to its ``net'' velocity |
1558 |
> |
\begin{equation} |
1559 |
> |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
1560 |
> |
\label{introEquation:tensorExpression} |
1561 |
> |
\end{equation} |
1562 |
> |
This equation is the basis for deriving the hydrodynamic tensor. In |
1563 |
> |
1930, Oseen and Burgers gave a simple solution to Equation |
1564 |
> |
\ref{introEquation:tensorExpression} |
1565 |
> |
\begin{equation} |
1566 |
> |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
1567 |
> |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
1568 |
> |
\label{introEquation:oseenTensor} |
1569 |
> |
\end{equation} |
1570 |
> |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
1571 |
> |
A second order expression for element of different size was |
1572 |
> |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
1573 |
> |
la Torre and Bloomfield, |
1574 |
> |
\begin{equation} |
1575 |
> |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
1576 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
1577 |
> |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
1578 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
1579 |
> |
\label{introEquation:RPTensorNonOverlapped} |
1580 |
> |
\end{equation} |
1581 |
> |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
1582 |
> |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
1583 |
> |
\ge \sigma _i + \sigma _j$. An alternative expression for |
1584 |
> |
overlapping beads with the same radius, $\sigma$, is given by |
1585 |
> |
\begin{equation} |
1586 |
> |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
1587 |
> |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
1588 |
> |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
1589 |
> |
\label{introEquation:RPTensorOverlapped} |
1590 |
> |
\end{equation} |
1591 |
> |
|
1592 |
> |
%Bead Modeling |
1593 |
> |
|
1594 |
> |
\[ |
1595 |
> |
B = \left( {\begin{array}{*{20}c} |
1596 |
> |
{T_{11} } & \ldots & {T_{1N} } \\ |
1597 |
> |
\vdots & \ddots & \vdots \\ |
1598 |
> |
{T_{N1} } & \cdots & {T_{NN} } \\ |
1599 |
> |
\end{array}} \right) |
1600 |
> |
\] |
1601 |
> |
|
1602 |
> |
\[ |
1603 |
> |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
1604 |
> |
{C_{11} } & \ldots & {C_{1N} } \\ |
1605 |
> |
\vdots & \ddots & \vdots \\ |
1606 |
> |
{C_{N1} } & \cdots & {C_{NN} } \\ |
1607 |
> |
\end{array}} \right) |
1608 |
> |
\] |
1609 |
> |
|
1610 |
> |
\begin{equation} |
1611 |
> |
\begin{array}{l} |
1612 |
> |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
1613 |
> |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
1614 |
> |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
1615 |
> |
\end{array} |
1616 |
> |
\end{equation} |
1617 |
> |
where |
1618 |
> |
\[ |
1619 |
> |
U_i = \left( {\begin{array}{*{20}c} |
1620 |
> |
0 & { - z_i } & {y_i } \\ |
1621 |
> |
{z_i } & 0 & { - x_i } \\ |
1622 |
> |
{ - y_i } & {x_i } & 0 \\ |
1623 |
> |
\end{array}} \right) |
1624 |
> |
\] |
1625 |
> |
|
1626 |
> |
\[ |
1627 |
> |
r_{OR} = \left( \begin{array}{l} |
1628 |
> |
x_{OR} \\ |
1629 |
> |
y_{OR} \\ |
1630 |
> |
z_{OR} \\ |
1631 |
> |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
1632 |
> |
{\Xi _{yy}^{rr} + \Xi _{zz}^{rr} } & { - \Xi _{xy}^{rr} } & { - \Xi _{xz}^{rr} } \\ |
1633 |
> |
{ - \Xi _{yx}^{rr} } & {\Xi _{zz}^{rr} + \Xi _{xx}^{rr} } & { - \Xi _{yz}^{rr} } \\ |
1634 |
> |
{ - \Xi _{zx}^{rr} } & { - \Xi _{yz}^{rr} } & {\Xi _{xx}^{rr} + \Xi _{yy}^{rr} } \\ |
1635 |
> |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
1636 |
> |
\Xi _{yz}^{tr} - \Xi _{zy}^{tr} \\ |
1637 |
> |
\Xi _{zx}^{tr} - \Xi _{xz}^{tr} \\ |
1638 |
> |
\Xi _{xy}^{tr} - \Xi _{yx}^{tr} \\ |
1639 |
> |
\end{array} \right) |
1640 |
> |
\] |
1641 |
> |
|
1642 |
> |
\[ |
1643 |
> |
U_{OR} = \left( {\begin{array}{*{20}c} |
1644 |
> |
0 & { - z_{OR} } & {y_{OR} } \\ |
1645 |
> |
{z_i } & 0 & { - x_{OR} } \\ |
1646 |
> |
{ - y_{OR} } & {x_{OR} } & 0 \\ |
1647 |
> |
\end{array}} \right) |
1648 |
> |
\] |
1649 |
> |
|
1650 |
> |
\[ |
1651 |
> |
\begin{array}{l} |
1652 |
> |
\Xi _R^{tt} = \Xi _{}^{tt} \\ |
1653 |
> |
\Xi _R^{tr} = \Xi _R^{rt} = \Xi _{}^{tr} - U_{OR} \Xi _{}^{tt} \\ |
1654 |
> |
\Xi _R^{rr} = \Xi _{}^{rr} - U_{OR} \Xi _{}^{tt} U_{OR} + \Xi _{}^{tr} U_{OR} - U_{OR} \Xi _{}^{tr} ^{^T } \\ |
1655 |
> |
\end{array} |
1656 |
> |
\] |
1657 |
> |
|
1658 |
> |
\[ |
1659 |
> |
D_R = \left( {\begin{array}{*{20}c} |
1660 |
> |
{D_R^{tt} } & {D_R^{rt} } \\ |
1661 |
> |
{D_R^{tr} } & {D_R^{rr} } \\ |
1662 |
> |
\end{array}} \right) = k_b T\left( {\begin{array}{*{20}c} |
1663 |
> |
{\Xi _R^{tt} } & {\Xi _R^{rt} } \\ |
1664 |
> |
{\Xi _R^{tr} } & {\Xi _R^{rr} } \\ |
1665 |
> |
\end{array}} \right)^{ - 1} |
1666 |
> |
\] |
1667 |
> |
|
1668 |
> |
|
1669 |
> |
%Approximation Methods |
1670 |
> |
|
1671 |
> |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |