| 108 |
|
\label{introEq:SM5} |
| 109 |
|
\end{equation} |
| 110 |
|
Where $k_B$ is the Boltzmann constant. Having defined entropy, one can |
| 111 |
< |
also define the temperature of the system using the relation |
| 111 |
> |
also define the temperature of the system using the Maxwell relation |
| 112 |
|
\begin{equation} |
| 113 |
|
\frac{1}{T} = \biggl ( \frac{\partial S}{\partial E} \biggr )_{N,V} |
| 114 |
|
\label{introEq:SM6} |
| 209 |
|
Where the value of an observable is averaged over the length of time |
| 210 |
|
that the simulation is run. This type of measurement mirrors the |
| 211 |
|
experimental measurement of an observable. In an experiment, the |
| 212 |
< |
instrument analyzing the system must average its observation of the |
| 212 |
> |
instrument analyzing the system must average its observation over the |
| 213 |
|
finite time of the measurement. What is required then, is a principle |
| 214 |
|
to relate the time average to the ensemble average. This is the |
| 215 |
|
ergodic hypothesis. |
| 547 |
|
efficiency of the force evaluation, as particles farther than a |
| 548 |
|
predetermined distance, $r_{\text{cut}}$, are not included in the |
| 549 |
|
calculation.\cite{Frenkel1996} In a simulation with periodic images, |
| 550 |
< |
$r_{\text{cut}}$ has a maximum value of $\text{box}/2$. |
| 551 |
< |
Fig.~\ref{introFig:rMax} illustrates how when using an |
| 552 |
< |
$r_{\text{cut}}$ larger than this value, or in the extreme limit of no |
| 553 |
< |
$r_{\text{cut}}$ at all, the corners of the simulation box are |
| 554 |
< |
unequally weighted due to the lack of particle images in the $x$, $y$, |
| 555 |
< |
or $z$ directions past a distance of $\text{box} / 2$. |
| 550 |
> |
there are two methods to choose from, both with their own cutoff |
| 551 |
> |
limits. In the minimum image convention, $r_{\text{cut}}$ has a |
| 552 |
> |
maximum value of $\text{box}/2$. This is because each atom has only |
| 553 |
> |
one image that is seen by another atom, and further the image used is |
| 554 |
> |
the one that minimizes the distance between the two atoms. A system of |
| 555 |
> |
wrapped images about a central atom therefore has a maximum length |
| 556 |
> |
scale of box on a side (Fig.~\ref{introFig:rMaxMin}). The second |
| 557 |
> |
convention, multiple image convention, has a maximum $r_{\text{cut}}$ |
| 558 |
> |
of box. Here multiple images of each atom are replicated in the |
| 559 |
> |
periodic cells surrounding the central atom, this causes the atom to |
| 560 |
> |
see multiple copies of several atoms. If the cutoff radius is larger |
| 561 |
> |
than box, however, then the atom will see an image of itself, and |
| 562 |
> |
attempt to calculate an unphysical self-self force interaction |
| 563 |
> |
(Fig.~\ref{introFig:rMaxMult}). Due to the increased complexity and |
| 564 |
> |
commputaional ineffeciency of the multiple image method, the minimum |
| 565 |
> |
image method is the periodic method used throughout this research. |
| 566 |
|
|
| 567 |
|
\begin{figure} |
| 568 |
|
\centering |
| 569 |
|
\includegraphics[width=\linewidth]{rCutMaxFig.eps} |
| 570 |
< |
\caption[An explanation of $r_{\text{cut}}$]{The yellow atom has all other images wrapped to itself as the center. If $r_{\text{cut}}=\text{box}/2$, then the distribution is uniform (blue atoms). However, when $r_{\text{cut}}>\text{box}/2$ the corners are disproportionately weighted (green atoms) vs the axial directions (shaded regions).} |
| 571 |
< |
\label{introFig:rMax} |
| 570 |
> |
\caption[An explanation of minimum image convention]{The yellow atom has all other images wrapped to itself as the center. If $r_{\text{cut}}=\text{box}/2$, then the distribution is uniform (blue atoms). However, when $r_{\text{cut}}>\text{box}/2$ the corners are disproportionately weighted (green atoms) vs the axial directions (shaded regions).} |
| 571 |
> |
\label{introFig:rMaxMin} |
| 572 |
|
\end{figure} |
| 573 |
|
|
| 574 |
+ |
\begin{figure} |
| 575 |
+ |
\centering |
| 576 |
+ |
\includegraphics[width=\linewidth]{rCutMaxMultFig.eps} |
| 577 |
+ |
\caption[An explanation of multiple image convention]{The yellow atom is the central wrapping point. The blue atoms are the minimum images of the system about the central atom. The boxes with the green atoms are multiple images of the central box. If $r_{\text{cut}} \geq \{text{box}$ then the central atom sees multiple images of itself (red atom), creating a self-self force evaluation.} |
| 578 |
+ |
\label{introFig:rMaxMult} |
| 579 |
+ |
\end{figure} |
| 580 |
+ |
|
| 581 |
|
With the use of an $r_{\text{cut}}$, however, comes a discontinuity in |
| 582 |
|
the potential energy curve (Fig.~\ref{introFig:shiftPot}). To fix this |
| 583 |
|
discontinuity, one calculates the potential energy at the |
| 908 |
|
|
| 909 |
|
This dissertation is divided as follows:Ch.~\ref{chapt:RSA} |
| 910 |
|
presents the random sequential adsorption simulations of related |
| 911 |
< |
pthalocyanines on a gold (111) surface. Ch.~\ref{chapt:OOPSE} |
| 911 |
> |
pthalocyanines on a gold (111) surface. Ch.~\ref{chapt:oopse} |
| 912 |
|
is about the writing of the molecular dynamics simulation package |
| 913 |
|
{\sc oopse}. Ch.~\ref{chapt:lipid} regards the simulations of |
| 914 |
|
phospholipid bilayers using a mesoscale model. And lastly, |
