| 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 resistance (friction) tensor respectively, |
| 1467 |
> |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
| 1468 |
> |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
| 1469 |
> |
particle moves in a fluid, it may experience friction force or |
| 1470 |
> |
torque along the opposite direction of the velocity or angular |
| 1471 |
> |
velocity, |
| 1472 |
> |
\[ |
| 1473 |
> |
\left( \begin{array}{l} |
| 1474 |
> |
F_R \\ |
| 1475 |
> |
\tau _R \\ |
| 1476 |
> |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 1477 |
> |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 1478 |
> |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 1479 |
> |
\end{array}} \right)\left( \begin{array}{l} |
| 1480 |
> |
v \\ |
| 1481 |
> |
w \\ |
| 1482 |
> |
\end{array} \right) |
| 1483 |
> |
\] |
| 1484 |
> |
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 1485 |
> |
toque. |
| 1486 |
|
|
| 1487 |
< |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
| 1488 |
< |
Body} |
| 1487 |
> |
\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} |
| 1488 |
> |
|
| 1489 |
> |
For a spherical particle, the translational and rotational friction |
| 1490 |
> |
constant can be calculated from Stoke's law, |
| 1491 |
> |
\[ |
| 1492 |
> |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 1493 |
> |
{6\pi \eta R} & 0 & 0 \\ |
| 1494 |
> |
0 & {6\pi \eta R} & 0 \\ |
| 1495 |
> |
0 & 0 & {6\pi \eta R} \\ |
| 1496 |
> |
\end{array}} \right) |
| 1497 |
> |
\] |
| 1498 |
> |
and |
| 1499 |
> |
\[ |
| 1500 |
> |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 1501 |
> |
{8\pi \eta R^3 } & 0 & 0 \\ |
| 1502 |
> |
0 & {8\pi \eta R^3 } & 0 \\ |
| 1503 |
> |
0 & 0 & {8\pi \eta R^3 } \\ |
| 1504 |
> |
\end{array}} \right) |
| 1505 |
> |
\] |
| 1506 |
> |
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 1507 |
> |
hydrodynamics radius. |
| 1508 |
|
|
| 1509 |
< |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
| 1509 |
> |
Other non-spherical shape, such as cylinder and ellipsoid |
| 1510 |
> |
\textit{etc}, are widely used as reference for developing new |
| 1511 |
> |
hydrodynamics theory, because their properties can be calculated |
| 1512 |
> |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
| 1513 |
> |
also called a triaxial ellipsoid, which is given in Cartesian |
| 1514 |
> |
coordinates by |
| 1515 |
> |
\[ |
| 1516 |
> |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 1517 |
> |
}} = 1 |
| 1518 |
> |
\] |
| 1519 |
> |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 1520 |
> |
due to the complexity of the elliptic integral, only the ellipsoid |
| 1521 |
> |
with the restriction of two axes having to be equal, \textit{i.e.} |
| 1522 |
> |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 1523 |
> |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
| 1524 |
> |
\[ |
| 1525 |
> |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 1526 |
> |
} }}{b}, |
| 1527 |
> |
\] |
| 1528 |
> |
and oblate, |
| 1529 |
> |
\[ |
| 1530 |
> |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 1531 |
> |
}}{a} |
| 1532 |
> |
\], |
| 1533 |
> |
one can write down the translational and rotational resistance |
| 1534 |
> |
tensors |
| 1535 |
> |
\[ |
| 1536 |
> |
\begin{array}{l} |
| 1537 |
> |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1538 |
> |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 1539 |
> |
\end{array}, |
| 1540 |
> |
\] |
| 1541 |
> |
and |
| 1542 |
> |
\[ |
| 1543 |
> |
\begin{array}{l} |
| 1544 |
> |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 1545 |
> |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1546 |
> |
\end{array}. |
| 1547 |
> |
\] |
| 1548 |
> |
|
| 1549 |
> |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} |
| 1550 |
> |
|
| 1551 |
> |
Unlike spherical and other regular shaped molecules, there is not |
| 1552 |
> |
analytical solution for friction tensor of any arbitrary shaped |
| 1553 |
> |
rigid molecules. The ellipsoid of revolution model and general |
| 1554 |
> |
triaxial ellipsoid model have been used to approximate the |
| 1555 |
> |
hydrodynamic properties of rigid bodies. However, since the mapping |
| 1556 |
> |
from all possible ellipsoidal space, $r$-space, to all possible |
| 1557 |
> |
combination of rotational diffusion coefficients, $D$-space is not |
| 1558 |
> |
unique\cite{Wegener79} as well as the intrinsic coupling between |
| 1559 |
> |
translational and rotational motion of rigid body\cite{}, general |
| 1560 |
> |
ellipsoid is not always suitable for modeling arbitrarily shaped |
| 1561 |
> |
rigid molecule. A number of studies have been devoted to determine |
| 1562 |
> |
the friction tensor for irregularly shaped rigid bodies using more |
| 1563 |
> |
advanced method\cite{} where the molecule of interest was modeled by |
| 1564 |
> |
combinations of spheres(beads)\cite{} and the hydrodynamics |
| 1565 |
> |
properties of the molecule can be calculated using the hydrodynamic |
| 1566 |
> |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
| 1567 |
> |
immersed in a continuous medium. Due to hydrodynamics interaction, |
| 1568 |
> |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
| 1569 |
> |
unperturbed velocity $v_i$, |
| 1570 |
> |
\[ |
| 1571 |
> |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
| 1572 |
> |
\] |
| 1573 |
> |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
| 1574 |
> |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
| 1575 |
> |
proportional to its ``net'' velocity |
| 1576 |
> |
\begin{equation} |
| 1577 |
> |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
| 1578 |
> |
\label{introEquation:tensorExpression} |
| 1579 |
> |
\end{equation} |
| 1580 |
> |
This equation is the basis for deriving the hydrodynamic tensor. In |
| 1581 |
> |
1930, Oseen and Burgers gave a simple solution to Equation |
| 1582 |
> |
\ref{introEquation:tensorExpression} |
| 1583 |
> |
\begin{equation} |
| 1584 |
> |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
| 1585 |
> |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
| 1586 |
> |
\label{introEquation:oseenTensor} |
| 1587 |
> |
\end{equation} |
| 1588 |
> |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
| 1589 |
> |
A second order expression for element of different size was |
| 1590 |
> |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
| 1591 |
> |
la Torre and Bloomfield, |
| 1592 |
> |
\begin{equation} |
| 1593 |
> |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
| 1594 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
| 1595 |
> |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
| 1596 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 1597 |
> |
\label{introEquation:RPTensorNonOverlapped} |
| 1598 |
> |
\end{equation} |
| 1599 |
> |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
| 1600 |
> |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
| 1601 |
> |
\ge \sigma _i + \sigma _j$. An alternative expression for |
| 1602 |
> |
overlapping beads with the same radius, $\sigma$, is given by |
| 1603 |
> |
\begin{equation} |
| 1604 |
> |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 1605 |
> |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
| 1606 |
> |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 1607 |
> |
\label{introEquation:RPTensorOverlapped} |
| 1608 |
> |
\end{equation} |
| 1609 |
> |
|
| 1610 |
> |
To calculate the resistance tensor at an arbitrary origin $O$, we |
| 1611 |
> |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
| 1612 |
> |
$B_{ij}$ blocks |
| 1613 |
> |
\begin{equation} |
| 1614 |
> |
B = \left( {\begin{array}{*{20}c} |
| 1615 |
> |
{B_{11} } & \ldots & {B_{1N} } \\ |
| 1616 |
> |
\vdots & \ddots & \vdots \\ |
| 1617 |
> |
{B_{N1} } & \cdots & {B_{NN} } \\ |
| 1618 |
> |
\end{array}} \right), |
| 1619 |
> |
\end{equation} |
| 1620 |
> |
where $B_{ij}$ is given by |
| 1621 |
> |
\[ |
| 1622 |
> |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 1623 |
> |
)T_{ij} |
| 1624 |
> |
\] |
| 1625 |
> |
where \delta _{ij} is Kronecker delta function. Inverting matrix |
| 1626 |
> |
$B$, we obtain |
| 1627 |
> |
|
| 1628 |
> |
\[ |
| 1629 |
> |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 1630 |
> |
{C_{11} } & \ldots & {C_{1N} } \\ |
| 1631 |
> |
\vdots & \ddots & \vdots \\ |
| 1632 |
> |
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 1633 |
> |
\end{array}} \right) |
| 1634 |
> |
\] |
| 1635 |
> |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
| 1636 |
> |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
| 1637 |
> |
\[ |
| 1638 |
> |
U_i = \left( {\begin{array}{*{20}c} |
| 1639 |
> |
0 & { - z_i } & {y_i } \\ |
| 1640 |
> |
{z_i } & 0 & { - x_i } \\ |
| 1641 |
> |
{ - y_i } & {x_i } & 0 \\ |
| 1642 |
> |
\end{array}} \right) |
| 1643 |
> |
\] |
| 1644 |
> |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 1645 |
> |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
| 1646 |
> |
arbitrary origin $O$ can be written as |
| 1647 |
> |
\begin{equation} |
| 1648 |
> |
\begin{array}{l} |
| 1649 |
> |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
| 1650 |
> |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 1651 |
> |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
| 1652 |
> |
\end{array} |
| 1653 |
> |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 1654 |
> |
\end{equation} |
| 1655 |
> |
|
| 1656 |
> |
The resistance tensor depends on the origin to which they refer. The |
| 1657 |
> |
proper location for applying friction force is the center of |
| 1658 |
> |
resistance (reaction), at which the trace of rotational resistance |
| 1659 |
> |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
| 1660 |
> |
resistance is defined as an unique point of the rigid body at which |
| 1661 |
> |
the translation-rotation coupling tensor are symmetric, |
| 1662 |
> |
\begin{equation} |
| 1663 |
> |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 1664 |
> |
\label{introEquation:definitionCR} |
| 1665 |
> |
\end{equation} |
| 1666 |
> |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 1667 |
> |
we can easily find out that the translational resistance tensor is |
| 1668 |
> |
origin independent, while the rotational resistance tensor and |
| 1669 |
> |
translation-rotation coupling resistance tensor do depend on the |
| 1670 |
> |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 1671 |
> |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 1672 |
> |
obtain the resistance tensor at $P$ by |
| 1673 |
> |
\begin{equation} |
| 1674 |
> |
\begin{array}{l} |
| 1675 |
> |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
| 1676 |
> |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
| 1677 |
> |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{tr} ^{^T } \\ |
| 1678 |
> |
\end{array} |
| 1679 |
> |
\label{introEquation:resistanceTensorTransformation} |
| 1680 |
> |
\end{equation} |
| 1681 |
> |
where |
| 1682 |
> |
\[ |
| 1683 |
> |
U_{OP} = \left( {\begin{array}{*{20}c} |
| 1684 |
> |
0 & { - z_{OP} } & {y_{OP} } \\ |
| 1685 |
> |
{z_i } & 0 & { - x_{OP} } \\ |
| 1686 |
> |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 1687 |
> |
\end{array}} \right) |
| 1688 |
> |
\] |
| 1689 |
> |
Using Equations \ref{introEquation:definitionCR} and |
| 1690 |
> |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
| 1691 |
> |
the position of center of resistance, |
| 1692 |
> |
\[ |
| 1693 |
> |
\left( \begin{array}{l} |
| 1694 |
> |
x_{OR} \\ |
| 1695 |
> |
y_{OR} \\ |
| 1696 |
> |
z_{OR} \\ |
| 1697 |
> |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
| 1698 |
> |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
| 1699 |
> |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
| 1700 |
> |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
| 1701 |
> |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
| 1702 |
> |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
| 1703 |
> |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
| 1704 |
> |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
| 1705 |
> |
\end{array} \right). |
| 1706 |
> |
\] |
| 1707 |
> |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 1708 |
> |
joining center of resistance $R$ and origin $O$. |
| 1709 |
> |
|
| 1710 |
> |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |
