30 |
|
thermodynamic properties of the system are being probed, then chose |
31 |
|
which method best suits that objective. |
32 |
|
|
33 |
< |
\subsection{\label{introSec:statThermo}Statistical Thermodynamics} |
33 |
> |
\subsection{\label{introSec:statThermo}Statistical Mechanics} |
34 |
|
|
35 |
< |
ergodic hypothesis |
35 |
> |
The following section serves as a brief introduction to some of the |
36 |
> |
Statistical Mechanics concepts present in this dissertation. What |
37 |
> |
follows is a brief derivation of Blotzman weighted statistics, and an |
38 |
> |
explanation of how one can use the information to calculate an |
39 |
> |
observable for a system. This section then concludes with a brief |
40 |
> |
discussion of the ergodic hypothesis and its relevance to this |
41 |
> |
research. |
42 |
|
|
43 |
< |
enesemble averages |
43 |
> |
\subsection{\label{introSec:boltzman}Boltzman weighted statistics} |
44 |
> |
|
45 |
> |
Consider a system, $\gamma$, with some total energy,, $E_{\gamma}$. |
46 |
> |
Let $\Omega(E_{gamma})$ represent the number of degenerate ways |
47 |
> |
$\boldsymbol{\Gamma}$, the collection of positions and conjugate |
48 |
> |
momenta of system $\gamma$, can be configured to give |
49 |
> |
$E_{\gamma}$. Further, if $\gamma$ is in contact with a bath system |
50 |
> |
where energy is exchanged between the two systems, $\Omega(E)$, where |
51 |
> |
$E$ is the total energy of both systems, can be represented as |
52 |
> |
\begin{equation} |
53 |
> |
eq here |
54 |
> |
\label{introEq:SM1} |
55 |
> |
\end{equation} |
56 |
> |
Or additively as |
57 |
> |
\begin{equation} |
58 |
> |
eq here |
59 |
> |
\label{introEq:SM2} |
60 |
> |
\end{equation} |
61 |
> |
|
62 |
> |
The solution to Eq.~\ref{introEq:SM2} maximizes the number of |
63 |
> |
degenerative configurations in $E$. \cite{fix} |
64 |
> |
This gives |
65 |
> |
\begin{equation} |
66 |
> |
eq here |
67 |
> |
\label{introEq:SM3} |
68 |
> |
\end{equation} |
69 |
> |
Where $E_{\text{bath}}$ is $E-E_{\gamma}$, and |
70 |
> |
$\frac{partialE_{\text{bath}}}{\partial E_{\gamma}}$ is |
71 |
> |
$-1$. Eq.~\ref{introEq:SM3} becomes |
72 |
> |
\begin{equation} |
73 |
> |
eq here |
74 |
> |
\label{introEq:SM4} |
75 |
> |
\end{equation} |
76 |
> |
|
77 |
> |
At this point, one can draw a relationship between the maximization of |
78 |
> |
degeneracy in Eq.~\ref{introEq:SM3} and the second law of |
79 |
> |
thermodynamics. Namely, that for a closed system, entropy wil |
80 |
> |
increase for an irreversible process.\cite{fix} Here the |
81 |
> |
process is the partitioning of energy among the two systems. This |
82 |
> |
allows the following definition of entropy: |
83 |
> |
\begin{equation} |
84 |
> |
eq here |
85 |
> |
\label{introEq:SM5} |
86 |
> |
\end{equation} |
87 |
> |
Where $k_B$ is the Boltzman constant. Having defined entropy, one can |
88 |
> |
also define the temperature of the system using the relation |
89 |
> |
\begin{equation} |
90 |
> |
eq here |
91 |
> |
\label{introEq:SM6} |
92 |
> |
\end{equation} |
93 |
> |
The temperature in the system $\gamma$ is then |
94 |
> |
\begin{equation} |
95 |
> |
eq here |
96 |
> |
\label{introEq:SM7} |
97 |
> |
\end{equation} |
98 |
> |
Applying this to Eq.~\ref{introEq:SM4} gives the following |
99 |
> |
\begin{equation} |
100 |
> |
eq here |
101 |
> |
\label{introEq:SM8} |
102 |
> |
\end{equation} |
103 |
> |
Showing that the partitioning of energy between the two systems is |
104 |
> |
actually a process of thermal equilibration. \cite{fix} |
105 |
> |
|
106 |
> |
An application of these results is to formulate the form of an |
107 |
> |
expectation value of an observable, $A$, in the canonical ensemble. In |
108 |
> |
the canonical ensemble, the number of particles, $N$, the volume, $V$, |
109 |
> |
and the temperature, $T$, are all held constant while the energy, $E$, |
110 |
> |
is allowed to fluctuate. Returning to the previous example, the bath |
111 |
> |
system is now an infinitly large thermal bath, whose exchange of |
112 |
> |
energy with the system $\gamma$ holds teh temperature constant. The |
113 |
> |
partitioning of energy in the bath system is then related to the total |
114 |
> |
energy of both systems and the fluctuations in $E_{\gamma}}$: |
115 |
> |
\begin{equation} |
116 |
> |
eq here |
117 |
> |
\label{introEq:SM9} |
118 |
> |
\end{equation} |
119 |
> |
As for the expectation value, it can be defined |
120 |
> |
\begin{equation} |
121 |
> |
eq here |
122 |
> |
\label{introEq:SM10} |
123 |
> |
\end{eequation} |
124 |
> |
Where $\int_{\boldsymbol{\Gamma}} d\Boldsymbol{\Gamma}$ denotes an |
125 |
> |
integration over all accessable phase space, $P_{\gamma}$ is the |
126 |
> |
probability of being in a given phase state and |
127 |
> |
$A(\boldsymbol{\Gamma})$ is some observable that is a function of the |
128 |
> |
phase state. |
129 |
> |
|
130 |
> |
Because entropy seeks to maximize the number of degenerate states at a |
131 |
> |
given energy, the probability of being in a particular state in |
132 |
> |
$\gamma$ will be directly proportional to the number of allowable |
133 |
> |
states the coupled system is able to assume. Namely, |
134 |
> |
\begin{equation} |
135 |
> |
eq here |
136 |
> |
\label{introEq:SM11} |
137 |
> |
\end{equation} |
138 |
> |
With $E_{\gamma} \lE$, $\ln \Omega$ can be expanded in a Taylor series: |
139 |
> |
\begin{equation} |
140 |
> |
eq here |
141 |
> |
\label{introEq:SM12} |
142 |
> |
\end{equation} |
143 |
> |
Higher order terms are omitted as $E$ is an infinite thermal |
144 |
> |
bath. Further, using Eq.~\ref{introEq:SM7}, Eq.~\ref{introEq:SM11} can |
145 |
> |
be rewritten: |
146 |
> |
\begin{equation} |
147 |
> |
eq here |
148 |
> |
\label{introEq:SM13} |
149 |
> |
\end{equation} |
150 |
> |
Where $\ln \Omega(E)$ has been factored out of the porpotionality as a |
151 |
> |
constant. Normalizing the probability ($\int_{\boldsymbol{\Gamma}} |
152 |
> |
d\boldsymbol{\Gamma} P_{\gamma} =1$ gives |
153 |
> |
\begin{equation} |
154 |
> |
eq here |
155 |
> |
\label{introEq:SM14} |
156 |
> |
\end{equation} |
157 |
> |
This result is the standard Boltzman statistical distribution. |
158 |
> |
Applying it to Eq.~\ref{introEq:SM10} one can obtain the following relation for ensemble averages: |
159 |
> |
\begin{equation} |
160 |
> |
eq here |
161 |
> |
\label{introEq:SM15} |
162 |
> |
\end{equation} |
163 |
> |
|
164 |
> |
\subsection{\label{introSec:ergodic}The Ergodic Hypothesis} |
165 |
> |
|
166 |
> |
One last important consideration is that of ergodicity. Ergodicity is |
167 |
> |
the assumption that given an infinite amount of time, a system will |
168 |
> |
visit every available point in phase space.\cite{fix} For most |
169 |
> |
systems, this is a valid assumption, except in cases where the system |
170 |
> |
may be trapped in a local feature (\emph{i.~e.~glasses}). When valid, |
171 |
> |
ergodicity allows the unification of a time averaged observation and |
172 |
> |
an ensemble averged one. If an observation is averaged over a |
173 |
> |
sufficiently long time, the system is assumed to visit all |
174 |
> |
appropriately available points in phase space, giving a properly |
175 |
> |
weighted statistical average. This allows the researcher freedom of |
176 |
> |
choice when deciding how best to measure a given observable. When an |
177 |
> |
ensemble averaged approach seems most logical, the Monte Carlo |
178 |
> |
techniques described in Sec.~\ref{introSec:MC} can be utilized. |
179 |
> |
Conversely, if a problem lends itself to a time averaging approach, |
180 |
> |
the Molecular Dynamics techniques in Sec.~\ref{introSec:MD} can be |
181 |
> |
employed. |
182 |
|
|
183 |
|
\subsection{\label{introSec:monteCarlo}Monte Carlo Simulations} |
184 |
|
|
387 |
|
|
388 |
|
The choice of when to use molecular dynamics over Monte Carlo |
389 |
|
techniques, is normally decided by the observables in which the |
390 |
< |
researcher is interested. If the observabvles depend on momenta in |
390 |
> |
researcher is interested. If the observables depend on momenta in |
391 |
|
any fashion, then the only choice is molecular dynamics in some form. |
392 |
|
However, when the observable is dependent only on the configuration, |
393 |
|
then most of the time Monte Carlo techniques will be more efficent. |
450 |
|
|
451 |
|
Another consideration one must resolve, is that in a given simulation |
452 |
|
a disproportionate number of the particles will feel the effects of |
453 |
< |
the surface. \cite{fix} For a cubic system of 1000 particles arranged |
453 |
> |
the surface.\cite{fix} For a cubic system of 1000 particles arranged |
454 |
|
in a $10x10x10$ cube, 488 particles will be exposed to the surface. |
455 |
|
Unless one is simulating an isolated particle group in a vacuum, the |
456 |
|
behavior of the system will be far from the desired bulk |
457 |
|
charecteristics. To offset this, simulations employ the use of |
458 |
< |
periodic boundary images. \cite{fix} |
458 |
> |
periodic boundary images.\cite{fix} |
459 |
|
|
460 |
|
The technique involves the use of an algorithm that replicates the |
461 |
|
simulation box on an infinite lattice in cartesian space. Any given |
473 |
|
cutoff radius be employed. Using a cutoff radius improves the |
474 |
|
efficiency of the force evaluation, as particles farther than a |
475 |
|
predetermined distance, $fix$, are not included in the |
476 |
< |
calculation. \cite{fix} In a simultation with periodic images, $fix$ |
476 |
> |
calculation.\cite{fix} In a simultation with periodic images, $fix$ |
477 |
|
has a maximum value of $fix$. Fig.~\ref{fix} illustrates how using an |
478 |
|
$fix$ larger than this value, or in the extreme limit of no $fix$ at |
479 |
|
all, the corners of the simulation box are unequally weighted due to |
527 |
|
order of $\Delta t^4$. |
528 |
|
|
529 |
|
In practice, however, the simulations in this research were integrated |
530 |
< |
with a velocity reformulation of teh Verlet method. \cite{allen87:csl} |
530 |
> |
with a velocity reformulation of teh Verlet method.\cite{allen87:csl} |
531 |
|
\begin{equation} |
532 |
|
eq here |
533 |
|
\label{introEq:MDvelVerletPos} |
550 |
|
therefore increases, but does not guarantee the ``correctness'' or the |
551 |
|
integrated trajectory. |
552 |
|
|
553 |
< |
It can be shown, \cite{Frenkel1996} that although the Verlet algorithm |
553 |
> |
It can be shown,\cite{Frenkel1996} that although the Verlet algorithm |
554 |
|
does not rigorously preserve the actual Hamiltonian, it does preserve |
555 |
|
a pseudo-Hamiltonian which shadows the real one in phase space. This |
556 |
|
pseudo-Hamiltonian is proveably area-conserving as well as time |
716 |
|
unitary and symplectic, the entire integration scheme is also |
717 |
|
symplectic and time reversible. |
718 |
|
|
719 |
< |
\section{\label{introSec:chapterLayout}Chapter Layout} |
719 |
> |
\section{\label{introSec:layout}Dissertation Layout} |
720 |
|
|
721 |
< |
\subsection{\label{introSec:RSA}Random Sequential Adsorption} |
721 |
> |
This dissertation is divided as follows:Chapt.~\ref{chapt:RSA} |
722 |
> |
presents the random sequential adsorption simulations of related |
723 |
> |
pthalocyanines on a gold (111) surface. Chapt.~\ref{chapt:OOPSE} |
724 |
> |
is about the writing of the molecular dynamics simulation package |
725 |
> |
{\sc oopse}, Chapt.~\ref{chapt:lipid} regards the simulations of |
726 |
> |
phospholipid bilayers using a mesoscale model, and lastly, |
727 |
> |
Chapt.~\ref{chapt:conclusion} concludes this dissertation with a |
728 |
> |
summary of all results. The chapters are arranged in chronological |
729 |
> |
order, and reflect the progression of techniques I employed during my |
730 |
> |
research. |
731 |
|
|
732 |
< |
\subsection{\label{introSec:OOPSE}The OOPSE Simulation Package} |
732 |
> |
The chapter concerning random sequential adsorption |
733 |
> |
simulations is a study in applying the principles of theoretical |
734 |
> |
research in order to obtain a simple model capaable of explaining the |
735 |
> |
results. My advisor, Dr. Gezelter, and I were approached by a |
736 |
> |
colleague, Dr. Lieberman, about possible explanations for partial |
737 |
> |
coverge of a gold surface by a particular compound of hers. We |
738 |
> |
suggested it might be due to the statistical packing fraction of disks |
739 |
> |
on a plane, and set about to simulate this system. As the events in |
740 |
> |
our model were not dynamic in nature, a Monte Carlo method was |
741 |
> |
emplyed. Here, if a molecule landed on the surface without |
742 |
> |
overlapping another, then its landing was accepted. However, if there |
743 |
> |
was overlap, the landing we rejected and a new random landing location |
744 |
> |
was chosen. This defined our acceptance rules and allowed us to |
745 |
> |
construct a Markov chain whose limiting distribution was the surface |
746 |
> |
coverage in which we were interested. |
747 |
|
|
748 |
< |
\subsection{\label{introSec:bilayers}A Mesoscale Model for |
749 |
< |
Phospholipid Bilayers} |
748 |
> |
The following chapter, about the simulation package {\sc oopse}, |
749 |
> |
describes in detail the large body of scientific code that had to be |
750 |
> |
written in order to study phospholipid bilayer. Although there are |
751 |
> |
pre-existing molecular dynamic simulation packages available, none |
752 |
> |
were capable of implementing the models we were developing.{\sc oopse} |
753 |
> |
is a unique package capable of not only integrating the equations of |
754 |
> |
motion in cartesian space, but is also able to integrate the |
755 |
> |
rotational motion of rigid bodies and dipoles. Add to this the |
756 |
> |
ability to perform calculations across parallel processors and a |
757 |
> |
flexible script syntax for creating systems, and {\sc oopse} becomes a |
758 |
> |
very powerful scientific instrument for the exploration of our model. |
759 |
> |
|
760 |
> |
Bringing us to Chapt.~\ref{chapt:lipid}. Using {\sc oopse}, I have been |
761 |
> |
able to parametrize a mesoscale model for phospholipid simulations. |
762 |
> |
This model retains information about solvent ordering about the |
763 |
> |
bilayer, as well as information regarding the interaction of the |
764 |
> |
phospholipid head groups' dipole with each other and the surrounding |
765 |
> |
solvent. These simulations give us insight into the dynamic events |
766 |
> |
that lead to the formation of phospholipid bilayers, as well as |
767 |
> |
provide the foundation for future exploration of bilayer phase |
768 |
> |
behavior with this model. |
769 |
> |
|
770 |
> |
Which leads into the last chapter, where I discuss future directions |
771 |
> |
for both{\sc oopse} and this mesoscale model. Additionally, I will |
772 |
> |
give a summary of results for this dissertation. |
773 |
> |
|
774 |
> |
|