| 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 |
< |
|
| 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 |
|
\[ |
| 1473 |
|
\left( \begin{array}{l} |
| 1474 |
< |
F_t \\ |
| 1475 |
< |
\tau \\ |
| 1474 |
> |
F_R \\ |
| 1475 |
> |
\tau _R \\ |
| 1476 |
|
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 1477 |
|
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 1478 |
|
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 1481 |
|
w \\ |
| 1482 |
|
\end{array} \right) |
| 1483 |
|
\] |
| 1484 |
+ |
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 1485 |
+ |
toque. |
| 1486 |
|
|
| 1487 |
< |
\subsubsection{\label{introSection:analyticalApproach}The Friction Tensor for Regular Shape} |
| 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 |
|
\[ |
| 1506 |
|
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 1507 |
|
hydrodynamics radius. |
| 1508 |
|
|
| 1509 |
< |
Other non-spherical particles have more complex properties. |
| 1510 |
< |
|
| 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 |
+ |
\[ |
| 1525 |
|
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 1526 |
< |
} }}{b} |
| 1526 |
> |
} }}{b}, |
| 1527 |
|
\] |
| 1528 |
< |
|
| 1511 |
< |
|
| 1528 |
> |
and oblate, |
| 1529 |
|
\[ |
| 1530 |
|
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 1531 |
|
}}{a} |
| 1532 |
< |
\] |
| 1533 |
< |
|
| 1532 |
> |
\], |
| 1533 |
> |
one can write down the translational and rotational resistance |
| 1534 |
> |
tensors |
| 1535 |
|
\[ |
| 1536 |
|
\begin{array}{l} |
| 1537 |
|
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1538 |
|
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 1539 |
< |
\end{array} |
| 1539 |
> |
\end{array}, |
| 1540 |
|
\] |
| 1541 |
< |
|
| 1541 |
> |
and |
| 1542 |
|
\[ |
| 1543 |
|
\begin{array}{l} |
| 1544 |
|
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 1545 |
|
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1546 |
< |
\end{array} |
| 1546 |
> |
\end{array}. |
| 1547 |
|
\] |
| 1548 |
|
|
| 1549 |
+ |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} |
| 1550 |
|
|
| 1532 |
– |
\subsubsection{\label{introSection:approximationApproach}The Friction Tensor for Arbitrary Shape} |
| 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 |
| 1607 |
|
\label{introEquation:RPTensorOverlapped} |
| 1608 |
|
\end{equation} |
| 1609 |
|
|
| 1610 |
< |
%Bead Modeling |
| 1611 |
< |
|
| 1612 |
< |
\[ |
| 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 |
< |
{T_{11} } & \ldots & {T_{1N} } \\ |
| 1615 |
> |
{B_{11} } & \ldots & {B_{1N} } \\ |
| 1616 |
|
\vdots & \ddots & \vdots \\ |
| 1617 |
< |
{T_{N1} } & \cdots & {T_{NN} } \\ |
| 1618 |
< |
\end{array}} \right) |
| 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 |
|
\[ |
| 1629 |
|
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 1632 |
|
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 1633 |
|
\end{array}} \right) |
| 1634 |
|
\] |
| 1635 |
< |
|
| 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 |
|
\[ |
| 1683 |
< |
U_i = \left( {\begin{array}{*{20}c} |
| 1684 |
< |
0 & { - z_i } & {y_i } \\ |
| 1685 |
< |
{z_i } & 0 & { - x_i } \\ |
| 1686 |
< |
{ - y_i } & {x_i } & 0 \\ |
| 1683 |
> |
U_{OP} = \left( {\begin{array}{*{20}c} |
| 1684 |
> |
0 & { - z_{OP} } & {y_{OP} } \\ |
| 1685 |
> |
{z_i } & 0 & { - x_{OP} } \\ |
| 1686 |
> |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 1687 |
|
\end{array}} \right) |
| 1688 |
|
\] |
| 1689 |
< |
|
| 1689 |
> |
Using Equations \ref{introEquation:definitionCR} and |
| 1690 |
> |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
| 1691 |
> |
the position of center of resistance, |
| 1692 |
|
\[ |
| 1693 |
< |
r_{OR} = \left( \begin{array}{l} |
| 1693 |
> |
\left( \begin{array}{l} |
| 1694 |
|
x_{OR} \\ |
| 1695 |
|
y_{OR} \\ |
| 1696 |
|
z_{OR} \\ |
| 1697 |
|
\end{array} \right) = \left( {\begin{array}{*{20}c} |
| 1698 |
< |
{\Xi _{yy}^{rr} + \Xi _{zz}^{rr} } & { - \Xi _{xy}^{rr} } & { - \Xi _{xz}^{rr} } \\ |
| 1699 |
< |
{ - \Xi _{yx}^{rr} } & {\Xi _{zz}^{rr} + \Xi _{xx}^{rr} } & { - \Xi _{yz}^{rr} } \\ |
| 1700 |
< |
{ - \Xi _{zx}^{rr} } & { - \Xi _{yz}^{rr} } & {\Xi _{xx}^{rr} + \Xi _{yy}^{rr} } \\ |
| 1698 |
> |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
| 1699 |
> |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
| 1700 |
> |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
| 1701 |
|
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
| 1702 |
< |
\Xi _{yz}^{tr} - \Xi _{zy}^{tr} \\ |
| 1703 |
< |
\Xi _{zx}^{tr} - \Xi _{xz}^{tr} \\ |
| 1704 |
< |
\Xi _{xy}^{tr} - \Xi _{yx}^{tr} \\ |
| 1705 |
< |
\end{array} \right) |
| 1702 |
> |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
| 1703 |
> |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
| 1704 |
> |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
| 1705 |
> |
\end{array} \right). |
| 1706 |
|
\] |
| 1707 |
+ |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 1708 |
+ |
joining center of resistance $R$ and origin $O$. |
| 1709 |
|
|
| 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 |
– |
|
| 1710 |
|
%\section{\label{introSection:correlationFunctions}Correlation Functions} |
