| 366 |
|
the set of resistors which spans the gold/thiolate interface, thiolate |
| 367 |
|
chains, and thiolate/solvent interface simplifies to |
| 368 |
|
\begin{equation} |
| 369 |
< |
\frac{T_{n}-T_{1}}{J_z}, |
| 369 |
> |
R_{K} = \frac{T_{n}-T_{1}}{J_z}, |
| 370 |
|
\label{eq:finalG} |
| 371 |
|
\end{equation} |
| 372 |
|
or the temperature difference between the gold side of the |
| 533 |
|
% \label{eq:biexponential} |
| 534 |
|
% \end{equation} |
| 535 |
|
% to determine short and long residence timescales and the relative populations of solvent molecules that can escape rapidly. |
| 536 |
< |
We define the mobility of solvent molecules at the interface as |
| 536 |
> |
We define the escape rate for trapped solvent molecules at the interface as |
| 537 |
|
\begin{equation} |
| 538 |
< |
P_{mobile} = \frac{1}{T}\int_{0}^{T} 1 - C(t) dt, |
| 538 |
> |
k_{escape} = \left( \int_0^T C(t) dt \right)^{-1} |
| 539 |
|
\label{eq:mobility} |
| 540 |
|
\end{equation} |
| 541 |
< |
where T is the length of the simulation. If all solvent molecule originally entangled in the thiolate chains leave the interfacial layer during time T, $P_{mobile} = 1$. Conversely, if the solvent molecules remain completely entrenched within the chains, $P_{mobile} = 0$. |
| 542 |
< |
In figure \ref{figure:res} we show that interfacial solvent mobility decreases as the percentage of long thiolate chains increases. |
| 541 |
> |
where T is the length of the simulation. This is a direct measure of |
| 542 |
> |
the rate at which solvent molecules entangled in the thiolate layer |
| 543 |
> |
can escape into the bulk. As $k_{escape} \rightarrow \infty$, the |
| 544 |
> |
solvent has become permanently trapped in the thiolate layer. In |
| 545 |
> |
figure \ref{figure:res} we show that interfacial solvent mobility |
| 546 |
> |
decreases as the percentage of long thiolate chains increases. |
| 547 |
|
|
| 548 |
|
|
| 549 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 579 |
|
molecular alignment at any time $t$. The overlap between the director |
| 580 |
|
axes of the thiolates and the entrapped solvent is time-averaged, |
| 581 |
|
\begin{equation} |
| 582 |
< |
\left \langle \mathbf{d}_{thiolates} \left( t \right) \cdot |
| 583 |
< |
\mathbf{d}_{solvent} \left( t \right) \right \rangle, |
| 582 |
> |
\bar{d} = \langle \mathbf{d}_{thiolates} \left( t \right) \cdot |
| 583 |
> |
\mathbf{d}_{solvent} \left( t \right) \rangle_t |
| 584 |
|
\label{eq:orientation} |
| 585 |
|
\end{equation} |
| 586 |
< |
and reported in table \ref{table:ordering}. |
| 586 |
> |
and reported in figure \ref{fig:Gstack}. |
| 587 |
|
|
| 588 |
|
Once the solvent molecules have picked up thermal energy from the |
| 589 |
|
thiolates, they carry heat away from the gold as they diffuse back |
