| 6 |
|
The techniques used in the course of this research fall under the two |
| 7 |
|
main classes of molecular simulation: Molecular Dynamics and Monte |
| 8 |
|
Carlo. Molecular Dynamic simulations integrate the equations of motion |
| 9 |
< |
for a given system of particles, allowing the researher to gain |
| 9 |
> |
for a given system of particles, allowing the researcher to gain |
| 10 |
|
insight into the time dependent evolution of a system. Diffusion |
| 11 |
|
phenomena are readily studied with this simulation technique, making |
| 12 |
|
Molecular Dynamics the main simulation technique used in this |
| 13 |
|
research. Other aspects of the research fall under the Monte Carlo |
| 14 |
|
class of simulations. In Monte Carlo, the configuration space |
| 15 |
< |
available to the collection of particles is sampled stochastichally, |
| 15 |
> |
available to the collection of particles is sampled stochastically, |
| 16 |
|
or randomly. Each configuration is chosen with a given probability |
| 17 |
< |
based on the Maxwell Boltzman distribution. These types of simulations |
| 17 |
> |
based on the Maxwell Boltzmann distribution. These types of simulations |
| 18 |
|
are best used to probe properties of a system that are only dependent |
| 19 |
|
only on the state of the system. Structural information about a system |
| 20 |
|
is most readily obtained through these types of methods. |
| 31 |
|
|
| 32 |
|
The following section serves as a brief introduction to some of the |
| 33 |
|
Statistical Mechanics concepts present in this dissertation. What |
| 34 |
< |
follows is a brief derivation of Blotzman weighted statistics, and an |
| 34 |
> |
follows is a brief derivation of Boltzmann weighted statistics, and an |
| 35 |
|
explanation of how one can use the information to calculate an |
| 36 |
|
observable for a system. This section then concludes with a brief |
| 37 |
|
discussion of the ergodic hypothesis and its relevance to this |
| 38 |
|
research. |
| 39 |
|
|
| 40 |
< |
\subsection{\label{introSec:boltzman}Boltzman weighted statistics} |
| 40 |
> |
\subsection{\label{introSec:boltzman}Boltzmann weighted statistics} |
| 41 |
|
|
| 42 |
|
Consider a system, $\gamma$, with some total energy,, $E_{\gamma}$. |
| 43 |
|
Let $\Omega(E_{\gamma})$ represent the number of degenerate ways |
| 86 |
|
S = k_B \ln \Omega(E) |
| 87 |
|
\label{introEq:SM5} |
| 88 |
|
\end{equation} |
| 89 |
< |
Where $k_B$ is the Boltzman constant. Having defined entropy, one can |
| 89 |
> |
Where $k_B$ is the Boltzmann constant. Having defined entropy, one can |
| 90 |
|
also define the temperature of the system using the relation |
| 91 |
|
\begin{equation} |
| 92 |
|
\frac{1}{T} = \biggl ( \frac{\partial S}{\partial E} \biggr )_{N,V} |
| 111 |
|
the canonical ensemble, the number of particles, $N$, the volume, $V$, |
| 112 |
|
and the temperature, $T$, are all held constant while the energy, $E$, |
| 113 |
|
is allowed to fluctuate. Returning to the previous example, the bath |
| 114 |
< |
system is now an infinitly large thermal bath, whose exchange of |
| 114 |
> |
system is now an infinitely large thermal bath, whose exchange of |
| 115 |
|
energy with the system $\gamma$ holds the temperature constant. The |
| 116 |
|
partitioning of energy in the bath system is then related to the total |
| 117 |
|
energy of both systems and the fluctuations in $E_{\gamma}$: |
| 127 |
|
\label{introEq:SM10} |
| 128 |
|
\end{equation} |
| 129 |
|
Where $\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma}$ denotes |
| 130 |
< |
an integration over all accessable phase space, $P_{\gamma}$ is the |
| 130 |
> |
an integration over all accessible phase space, $P_{\gamma}$ is the |
| 131 |
|
probability of being in a given phase state and |
| 132 |
|
$A(\boldsymbol{\Gamma})$ is some observable that is a function of the |
| 133 |
|
phase state. |
| 156 |
|
P_{\gamma} \propto e^{-\beta E_{\gamma}} |
| 157 |
|
\label{introEq:SM13} |
| 158 |
|
\end{equation} |
| 159 |
< |
Where $\ln \Omega(E)$ has been factored out of the porpotionality as a |
| 159 |
> |
Where $\ln \Omega(E)$ has been factored out of the proportionality as a |
| 160 |
|
constant. Normalizing the probability ($\int\limits_{\boldsymbol{\Gamma}} |
| 161 |
|
d\boldsymbol{\Gamma} P_{\gamma} = 1$) gives |
| 162 |
|
\begin{equation} |
| 164 |
|
{\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} e^{-\beta E_{\gamma}}} |
| 165 |
|
\label{introEq:SM14} |
| 166 |
|
\end{equation} |
| 167 |
< |
This result is the standard Boltzman statistical distribution. |
| 167 |
> |
This result is the standard Boltzmann statistical distribution. |
| 168 |
|
Applying it to Eq.~\ref{introEq:SM10} one can obtain the following relation for ensemble averages: |
| 169 |
|
\begin{equation} |
| 170 |
|
\langle A \rangle = |
| 182 |
|
systems, this is a valid assumption, except in cases where the system |
| 183 |
|
may be trapped in a local feature (\emph{e.g.}~glasses). When valid, |
| 184 |
|
ergodicity allows the unification of a time averaged observation and |
| 185 |
< |
an ensemble averged one. If an observation is averaged over a |
| 185 |
> |
an ensemble averaged one. If an observation is averaged over a |
| 186 |
|
sufficiently long time, the system is assumed to visit all |
| 187 |
|
appropriately available points in phase space, giving a properly |
| 188 |
|
weighted statistical average. This allows the researcher freedom of |
| 217 |
|
$[a,b]$. The calculation of the integral could then be solved by |
| 218 |
|
randomly choosing points along the interval $[a,b]$ and calculating |
| 219 |
|
the value of $f(x)$ at each point. The accumulated average would then |
| 220 |
< |
approach $I$ in the limit where the number of trials is infintely |
| 220 |
> |
approach $I$ in the limit where the number of trials is infinitely |
| 221 |
|
large. |
| 222 |
|
|
| 223 |
|
However, in Statistical Mechanics, one is typically interested in |
| 235 |
|
momentum. Therefore the momenta contribution of the integral can be |
| 236 |
|
factored out, leaving the configurational integral. Application of the |
| 237 |
|
brute force method to this system would yield highly inefficient |
| 238 |
< |
results. Due to the Boltzman weighting of this integral, most random |
| 238 |
> |
results. Due to the Boltzmann weighting of this integral, most random |
| 239 |
|
configurations will have a near zero contribution to the ensemble |
| 240 |
|
average. This is where importance sampling comes into |
| 241 |
|
play.\cite{allen87:csl} |
| 263 |
|
{\int d^N \mathbf{r}~e^{-\beta V(\mathbf{r}^N)}} |
| 264 |
|
\label{introEq:MCboltzman} |
| 265 |
|
\end{equation} |
| 266 |
< |
Where $\rho_{kT}$ is the boltzman distribution. The ensemble average |
| 266 |
> |
Where $\rho_{kT}$ is the Boltzmann distribution. The ensemble average |
| 267 |
|
can be rewritten as |
| 268 |
|
\begin{equation} |
| 269 |
|
\langle A \rangle = \int d^N \mathbf{r}~A(\mathbf{r}^N) |
| 285 |
|
\end{equation} |
| 286 |
|
The difficulty is selecting points $\mathbf{r}^N$ such that they are |
| 287 |
|
sampled from the distribution $\rho_{kT}(\mathbf{r}^N)$. A solution |
| 288 |
< |
was proposed by Metropolis et al.\cite{metropolis:1953} which involved |
| 288 |
> |
was proposed by Metropolis \emph{et al}.\cite{metropolis:1953} which involved |
| 289 |
|
the use of a Markov chain whose limiting distribution was |
| 290 |
|
$\rho_{kT}(\mathbf{r}^N)$. |
| 291 |
|
|
| 297 |
|
\item The outcome of each trial depends only on the outcome of the previous trial. |
| 298 |
|
\item Each trial belongs to a finite set of outcomes called the state space. |
| 299 |
|
\end{enumerate} |
| 300 |
< |
If given two configuartions, $\mathbf{r}^N_m$ and $\mathbf{r}^N_n$, |
| 301 |
< |
$\rho_m$ and $\rho_n$ are the probablilities of being in state |
| 300 |
> |
If given two configurations, $\mathbf{r}^N_m$ and $\mathbf{r}^N_n$, |
| 301 |
> |
$\rho_m$ and $\rho_n$ are the probabilities of being in state |
| 302 |
|
$\mathbf{r}^N_m$ and $\mathbf{r}^N_n$ respectively. Further, the two |
| 303 |
|
states are linked by a transition probability, $\pi_{mn}$, which is the |
| 304 |
|
probability of going from state $m$ to state $n$. |
| 343 |
|
|
| 344 |
|
In the Metropolis method\cite{metropolis:1953} |
| 345 |
|
Eq.~\ref{introEq:MCmarkovEquil} is solved such that |
| 346 |
< |
$\boldsymbol{\rho}_{\text{limit}}$ matches the Boltzman distribution |
| 346 |
> |
$\boldsymbol{\rho}_{\text{limit}}$ matches the Boltzmann distribution |
| 347 |
|
of states. The method accomplishes this by imposing the strong |
| 348 |
|
condition of microscopic reversibility on the equilibrium |
| 349 |
|
distribution. Meaning, that at equilibrium the probability of going |
| 352 |
|
\rho_m\pi_{mn} = \rho_n\pi_{nm} |
| 353 |
|
\label{introEq:MCmicroReverse} |
| 354 |
|
\end{equation} |
| 355 |
< |
Further, $\boldsymbol{\alpha}$ is chosen to be a symetric matrix in |
| 355 |
> |
Further, $\boldsymbol{\alpha}$ is chosen to be a symmetric matrix in |
| 356 |
|
the Metropolis method. Using Eq.~\ref{introEq:MCpi}, |
| 357 |
|
Eq.~\ref{introEq:MCmicroReverse} becomes |
| 358 |
|
\begin{equation} |
| 360 |
|
\frac{\rho_n}{\rho_m} |
| 361 |
|
\label{introEq:MCmicro2} |
| 362 |
|
\end{equation} |
| 363 |
< |
For a Boltxman limiting distribution, |
| 363 |
> |
For a Boltzmann limiting distribution, |
| 364 |
|
\begin{equation} |
| 365 |
|
\frac{\rho_n}{\rho_m} = e^{-\beta[\mathcal{U}(n) - \mathcal{U}(m)]} |
| 366 |
|
= e^{-\beta \Delta \mathcal{U}} |
| 380 |
|
Metropolis method proceeds as follows |
| 381 |
|
\begin{enumerate} |
| 382 |
|
\item Generate an initial configuration $\mathbf{r}^N$ which has some finite probability in $\rho_{kT}$. |
| 383 |
< |
\item Modify $\mathbf{r}^N$, to generate configuratioon $\mathbf{r^{\prime}}^N$. |
| 383 |
> |
\item Modify $\mathbf{r}^N$, to generate configuration $\mathbf{r^{\prime}}^N$. |
| 384 |
|
\item If the new configuration lowers the energy of the system, accept the move with unity ($\mathbf{r}^N$ becomes $\mathbf{r^{\prime}}^N$). Otherwise accept with probability $e^{-\beta \Delta \mathcal{U}}$. |
| 385 |
< |
\item Accumulate the average for the configurational observable of intereest. |
| 385 |
> |
\item Accumulate the average for the configurational observable of interest. |
| 386 |
|
\item Repeat from step 2 until the average converges. |
| 387 |
|
\end{enumerate} |
| 388 |
|
One important note is that the average is accumulated whether the move |
| 389 |
|
is accepted or not, this ensures proper weighting of the average. |
| 390 |
|
Using Eq.~\ref{introEq:Importance4} it becomes clear that the |
| 391 |
|
accumulated averages are the ensemble averages, as this method ensures |
| 392 |
< |
that the limiting distribution is the Boltzman distribution. |
| 392 |
> |
that the limiting distribution is the Boltzmann distribution. |
| 393 |
|
|
| 394 |
|
\section{\label{introSec:MD}Molecular Dynamics Simulations} |
| 395 |
|
|
| 409 |
|
researcher is interested. If the observables depend on momenta in |
| 410 |
|
any fashion, then the only choice is molecular dynamics in some form. |
| 411 |
|
However, when the observable is dependent only on the configuration, |
| 412 |
< |
then most of the time Monte Carlo techniques will be more efficent. |
| 412 |
> |
then most of the time Monte Carlo techniques will be more efficient. |
| 413 |
|
|
| 414 |
|
The focus of research in the second half of this dissertation is |
| 415 |
|
centered around the dynamic properties of phospholipid bilayers, |
| 432 |
|
Ch.~\ref{chapt:lipid} deals with the formation and equilibrium dynamics of |
| 433 |
|
phospholipid membranes. Therefore in these simulations initial |
| 434 |
|
positions were selected that in some cases dispersed the lipids in |
| 435 |
< |
water, and in other cases structured the lipids into preformed |
| 435 |
> |
water, and in other cases structured the lipids into performed |
| 436 |
|
bilayers. Important considerations at this stage of the simulation are: |
| 437 |
|
\begin{itemize} |
| 438 |
|
\item There are no major overlaps of molecular or atomic orbitals |
| 439 |
< |
\item Velocities are chosen in such a way as to not gie the system a non=zero total momentum or angular momentum. |
| 440 |
< |
\item It is also sometimes desireable to select the velocities to correctly sample the target temperature. |
| 439 |
> |
\item Velocities are chosen in such a way as to not give the system a non=zero total momentum or angular momentum. |
| 440 |
> |
\item It is also sometimes desirable to select the velocities to correctly sample the target temperature. |
| 441 |
|
\end{itemize} |
| 442 |
|
|
| 443 |
|
The first point is important due to the amount of potential energy |
| 444 |
|
generated by having two particles too close together. If overlap |
| 445 |
|
occurs, the first evaluation of forces will return numbers so large as |
| 446 |
< |
to render the numerical integration of teh motion meaningless. The |
| 446 |
> |
to render the numerical integration of the motion meaningless. The |
| 447 |
|
second consideration keeps the system from drifting or rotating as a |
| 448 |
|
whole. This arises from the fact that most simulations are of systems |
| 449 |
|
in equilibrium in the absence of outside forces. Therefore any net |
| 450 |
|
movement would be unphysical and an artifact of the simulation method |
| 451 |
|
used. The final point addresses the selection of the magnitude of the |
| 452 |
< |
initial velocities. For many simulations it is convienient to use |
| 452 |
> |
initial velocities. For many simulations it is convenient to use |
| 453 |
|
this opportunity to scale the amount of kinetic energy to reflect the |
| 454 |
|
desired thermal distribution of the system. However, it must be noted |
| 455 |
|
that most systems will require further velocity rescaling after the |
| 474 |
|
arranged in a $10 \times 10 \times 10$ cube, 488 particles will be |
| 475 |
|
exposed to the surface. Unless one is simulating an isolated particle |
| 476 |
|
group in a vacuum, the behavior of the system will be far from the |
| 477 |
< |
desired bulk charecteristics. To offset this, simulations employ the |
| 477 |
> |
desired bulk characteristics. To offset this, simulations employ the |
| 478 |
|
use of periodic boundary images.\cite{born:1912} |
| 479 |
|
|
| 480 |
|
The technique involves the use of an algorithm that replicates the |
| 481 |
< |
simulation box on an infinite lattice in cartesian space. Any given |
| 481 |
> |
simulation box on an infinite lattice in Cartesian space. Any given |
| 482 |
|
particle leaving the simulation box on one side will have an image of |
| 483 |
|
itself enter on the opposite side (see Fig.~\ref{introFig:pbc}). In |
| 484 |
< |
addition, this sets that any given particle pair has an image, real or |
| 485 |
< |
periodic, within $fix$ of each other. A discussion of the method used |
| 486 |
< |
to calculate the periodic image can be found in |
| 484 |
> |
addition, this sets that any two particles have an image, real or |
| 485 |
> |
periodic, within $\text{box}/2$ of each other. A discussion of the |
| 486 |
> |
method used to calculate the periodic image can be found in |
| 487 |
|
Sec.\ref{oopseSec:pbc}. |
| 488 |
|
|
| 489 |
|
\begin{figure} |
| 490 |
|
\centering |
| 491 |
|
\includegraphics[width=\linewidth]{pbcFig.eps} |
| 492 |
< |
\caption[An illustration of periodic boundry conditions]{A 2-D illustration of periodic boundry conditions. As one particle leaves the right of the simulation box, an image of it enters the left.} |
| 492 |
> |
\caption[An illustration of periodic boundary conditions]{A 2-D illustration of periodic boundary conditions. As one particle leaves the right of the simulation box, an image of it enters the left.} |
| 493 |
|
\label{introFig:pbc} |
| 494 |
|
\end{figure} |
| 495 |
|
|
| 498 |
|
cutoff radius be employed. Using a cutoff radius improves the |
| 499 |
|
efficiency of the force evaluation, as particles farther than a |
| 500 |
|
predetermined distance, $r_{\text{cut}}$, are not included in the |
| 501 |
< |
calculation.\cite{Frenkel1996} In a simultation with periodic images, |
| 501 |
> |
calculation.\cite{Frenkel1996} In a simulation with periodic images, |
| 502 |
|
$r_{\text{cut}}$ has a maximum value of $\text{box}/2$. |
| 503 |
|
Fig.~\ref{introFig:rMax} illustrates how when using an |
| 504 |
|
$r_{\text{cut}}$ larger than this value, or in the extreme limit of no |
| 505 |
|
$r_{\text{cut}}$ at all, the corners of the simulation box are |
| 506 |
|
unequally weighted due to the lack of particle images in the $x$, $y$, |
| 507 |
< |
or $z$ directions past a disance of $\text{box} / 2$. |
| 507 |
> |
or $z$ directions past a distance of $\text{box} / 2$. |
| 508 |
|
|
| 509 |
|
\begin{figure} |
| 510 |
|
\centering |
| 524 |
|
\begin{figure} |
| 525 |
|
\centering |
| 526 |
|
\includegraphics[width=\linewidth]{shiftedPot.eps} |
| 527 |
< |
\caption[Shifting the Lennard-Jones Potential]{The Lennard-Jones potential is shifted to remove the discontiuity at $r_{\text{cut}}$.} |
| 527 |
> |
\caption[Shifting the Lennard-Jones Potential]{The Lennard-Jones potential (blue line) is shifted (red line) to remove the discontinuity at $r_{\text{cut}}$.} |
| 528 |
|
\label{introFig:shiftPot} |
| 529 |
|
\end{figure} |
| 530 |
|
|
| 542 |
|
\subsection{\label{introSec:mdIntegrate} Integration of the equations of motion} |
| 543 |
|
|
| 544 |
|
A starting point for the discussion of molecular dynamics integrators |
| 545 |
< |
is the Verlet algorithm. \cite{Frenkel1996} It begins with a Taylor |
| 545 |
> |
is the Verlet algorithm.\cite{Frenkel1996} It begins with a Taylor |
| 546 |
|
expansion of position in time: |
| 547 |
|
\begin{equation} |
| 548 |
< |
eq here |
| 548 |
> |
q(t+\Delta t)= q(t) + v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 + |
| 549 |
> |
\frac{\Delta t^3}{3!}\frac{\partial q(t)}{\partial t} + |
| 550 |
> |
\mathcal{O}(\Delta t^4) |
| 551 |
|
\label{introEq:verletForward} |
| 552 |
|
\end{equation} |
| 553 |
|
As well as, |
| 554 |
|
\begin{equation} |
| 555 |
< |
eq here |
| 555 |
> |
q(t-\Delta t)= q(t) - v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 - |
| 556 |
> |
\frac{\Delta t^3}{3!}\frac{\partial q(t)}{\partial t} + |
| 557 |
> |
\mathcal{O}(\Delta t^4) |
| 558 |
|
\label{introEq:verletBack} |
| 559 |
|
\end{equation} |
| 560 |
< |
Adding together Eq.~\ref{introEq:verletForward} and |
| 560 |
> |
Where $m$ is the mass of the particle, $q(t)$ is the position at time |
| 561 |
> |
$t$, $v(t)$ the velocity, and $F(t)$ the force acting on the |
| 562 |
> |
particle. Adding together Eq.~\ref{introEq:verletForward} and |
| 563 |
|
Eq.~\ref{introEq:verletBack} results in, |
| 564 |
|
\begin{equation} |
| 565 |
< |
eq here |
| 565 |
> |
q(t+\Delta t)+q(t-\Delta t) = |
| 566 |
> |
2q(t) + \frac{F(t)}{m}\Delta t^2 + \mathcal{O}(\Delta t^4) |
| 567 |
|
\label{introEq:verletSum} |
| 568 |
|
\end{equation} |
| 569 |
|
Or equivalently, |
| 570 |
|
\begin{equation} |
| 571 |
< |
eq here |
| 571 |
> |
q(t+\Delta t) = |
| 572 |
> |
2q(t) - q(t-\Delta t) + \frac{F(t)}{m}\Delta t^2 + |
| 573 |
> |
\mathcal{O}(\Delta t^4) |
| 574 |
|
\label{introEq:verletFinal} |
| 575 |
|
\end{equation} |
| 576 |
|
Which contains an error in the estimate of the new positions on the |
| 577 |
|
order of $\Delta t^4$. |
| 578 |
|
|
| 579 |
|
In practice, however, the simulations in this research were integrated |
| 580 |
< |
with a velocity reformulation of teh Verlet method.\cite{allen87:csl} |
| 580 |
> |
with a velocity reformulation of the Verlet method.\cite{allen87:csl} |
| 581 |
|
\begin{equation} |
| 582 |
< |
eq here |
| 582 |
> |
q(t+\Delta t)= q(t) + v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 |
| 583 |
|
\label{introEq:MDvelVerletPos} |
| 584 |
|
\end{equation} |
| 585 |
|
\begin{equation} |
| 586 |
< |
eq here |
| 586 |
> |
v(t+\Delta t) = v(t) + \frac{\Delta t}{2m}[F(t) + F(t+\Delta t)] |
| 587 |
|
\label{introEq:MDvelVerletVel} |
| 588 |
|
\end{equation} |
| 589 |
|
The original Verlet algorithm can be regained by substituting the |
| 592 |
|
very little long term drift in energy conservation. Energy |
| 593 |
|
conservation in a molecular dynamics simulation is of extreme |
| 594 |
|
importance, as it is a measure of how closely one is following the |
| 595 |
< |
``true'' trajectory wtih the finite integration scheme. An exact |
| 595 |
> |
``true'' trajectory with the finite integration scheme. An exact |
| 596 |
|
solution to the integration will conserve area in phase space, as well |
| 597 |
|
as be reversible in time, that is, the trajectory integrated forward |
| 598 |
|
or backwards will exactly match itself. Having a finite algorithm |
| 603 |
|
It can be shown,\cite{Frenkel1996} that although the Verlet algorithm |
| 604 |
|
does not rigorously preserve the actual Hamiltonian, it does preserve |
| 605 |
|
a pseudo-Hamiltonian which shadows the real one in phase space. This |
| 606 |
< |
pseudo-Hamiltonian is proveably area-conserving as well as time |
| 606 |
> |
pseudo-Hamiltonian is provably area-conserving as well as time |
| 607 |
|
reversible. The fact that it shadows the true Hamiltonian in phase |
| 608 |
|
space is acceptable in actual simulations as one is interested in the |
| 609 |
|
ensemble average of the observable being measured. From the ergodic |
| 614 |
|
\subsection{\label{introSec:MDfurther}Further Considerations} |
| 615 |
|
In the simulations presented in this research, a few additional |
| 616 |
|
parameters are needed to describe the motions. The simulations |
| 617 |
< |
involving water and phospholipids in Chapt.~\ref{chaptLipids} are |
| 617 |
> |
involving water and phospholipids in Ch.~\ref{chaptLipids} are |
| 618 |
|
required to integrate the equations of motions for dipoles on atoms. |
| 619 |
|
This involves an additional three parameters be specified for each |
| 620 |
|
dipole atom: $\phi$, $\theta$, and $\psi$. These three angles are |
| 636 |
|
\label{introEq:MDeuleeerPsi} |
| 637 |
|
\end{equation} |
| 638 |
|
Where $\omega^s_i$ is the angular velocity in the lab space frame |
| 639 |
< |
along cartesian coordinate $i$. However, a difficulty arises when |
| 639 |
> |
along Cartesian coordinate $i$. However, a difficulty arises when |
| 640 |
|
attempting to integrate Eq.~\ref{introEq:MDeulerPhi} and |
| 641 |
|
Eq.~\ref{introEq:MDeulerPsi}. The $\frac{1}{\sin \theta}$ present in |
| 642 |
|
both equations means there is a non-physical instability present when |
| 666 |
|
\end{equation} |
| 667 |
|
Here, $r_j$ and $p_j$ are the position and conjugate momenta of a |
| 668 |
|
degree of freedom, and $f_j$ is the force on that degree of freedom. |
| 669 |
< |
$\Gamma$ is defined as the set of all positions nad conjugate momenta, |
| 669 |
> |
$\Gamma$ is defined as the set of all positions and conjugate momenta, |
| 670 |
|
$\{r_j,p_j\}$, and the propagator, $U(t)$, is defined |
| 671 |
|
\begin {equation} |
| 672 |
|
eq here |
| 722 |
|
This is the velocity Verlet formulation presented in |
| 723 |
|
Sec.~\ref{introSec:MDintegrate}. Because this integration scheme is |
| 724 |
|
comprised of unitary propagators, it is symplectic, and therefore area |
| 725 |
< |
preserving in phase space. From the preceeding fatorization, one can |
| 725 |
> |
preserving in phase space. From the preceding factorization, one can |
| 726 |
|
see that the integration of the equations of motion would follow: |
| 727 |
|
\begin{enumerate} |
| 728 |
|
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
| 746 |
|
eq here |
| 747 |
|
\label{introEq:SR1} |
| 748 |
|
\end{equation} |
| 749 |
< |
Where $\boldsymbol{\tau}(\mathbf{A})$ are the tourques of the system |
| 749 |
> |
Where $\boldsymbol{\tau}(\mathbf{A})$ are the torques of the system |
| 750 |
|
due to the configuration, and $\boldsymbol{/pi}$ are the conjugate |
| 751 |
|
angular momenta of the system. The propagator, $G(\Delta t)$, becomes |
| 752 |
|
\begin{equation} |
| 753 |
|
eq here |
| 754 |
|
\label{introEq:SR2} |
| 755 |
|
\end{equation} |
| 756 |
< |
Propagation fo the linear and angular momenta follows as in the Verlet |
| 757 |
< |
scheme. The propagation of positions also follows the verlet scheme |
| 756 |
> |
Propagation of the linear and angular momenta follows as in the Verlet |
| 757 |
> |
scheme. The propagation of positions also follows the Verlet scheme |
| 758 |
|
with the addition of a further symplectic splitting of the rotation |
| 759 |
|
matrix propagation, $\mathcal{G}_{\text{rot}}(\Delta t)$. |
| 760 |
|
\begin{equation} |
| 770 |
|
|
| 771 |
|
This dissertation is divided as follows:Chapt.~\ref{chapt:RSA} |
| 772 |
|
presents the random sequential adsorption simulations of related |
| 773 |
< |
pthalocyanines on a gold (111) surface. Chapt.~\ref{chapt:OOPSE} |
| 773 |
> |
pthalocyanines on a gold (111) surface. Ch.~\ref{chapt:OOPSE} |
| 774 |
|
is about the writing of the molecular dynamics simulation package |
| 775 |
< |
{\sc oopse}, Chapt.~\ref{chapt:lipid} regards the simulations of |
| 775 |
> |
{\sc oopse}, Ch.~\ref{chapt:lipid} regards the simulations of |
| 776 |
|
phospholipid bilayers using a mesoscale model, and lastly, |
| 777 |
< |
Chapt.~\ref{chapt:conclusion} concludes this dissertation with a |
| 777 |
> |
Ch.~\ref{chapt:conclusion} concludes this dissertation with a |
| 778 |
|
summary of all results. The chapters are arranged in chronological |
| 779 |
|
order, and reflect the progression of techniques I employed during my |
| 780 |
|
research. |
| 781 |
|
|
| 782 |
|
The chapter concerning random sequential adsorption |
| 783 |
|
simulations is a study in applying the principles of theoretical |
| 784 |
< |
research in order to obtain a simple model capaable of explaining the |
| 784 |
> |
research in order to obtain a simple model capable of explaining the |
| 785 |
|
results. My advisor, Dr. Gezelter, and I were approached by a |
| 786 |
|
colleague, Dr. Lieberman, about possible explanations for partial |
| 787 |
< |
coverge of a gold surface by a particular compound of hers. We |
| 787 |
> |
coverage of a gold surface by a particular compound of hers. We |
| 788 |
|
suggested it might be due to the statistical packing fraction of disks |
| 789 |
|
on a plane, and set about to simulate this system. As the events in |
| 790 |
|
our model were not dynamic in nature, a Monte Carlo method was |
| 791 |
< |
emplyed. Here, if a molecule landed on the surface without |
| 791 |
> |
employed. Here, if a molecule landed on the surface without |
| 792 |
|
overlapping another, then its landing was accepted. However, if there |
| 793 |
|
was overlap, the landing we rejected and a new random landing location |
| 794 |
|
was chosen. This defined our acceptance rules and allowed us to |
| 801 |
|
pre-existing molecular dynamic simulation packages available, none |
| 802 |
|
were capable of implementing the models we were developing.{\sc oopse} |
| 803 |
|
is a unique package capable of not only integrating the equations of |
| 804 |
< |
motion in cartesian space, but is also able to integrate the |
| 804 |
> |
motion in Cartesian space, but is also able to integrate the |
| 805 |
|
rotational motion of rigid bodies and dipoles. Add to this the |
| 806 |
|
ability to perform calculations across parallel processors and a |
| 807 |
|
flexible script syntax for creating systems, and {\sc oopse} becomes a |
| 808 |
|
very powerful scientific instrument for the exploration of our model. |
| 809 |
|
|
| 810 |
< |
Bringing us to Chapt.~\ref{chapt:lipid}. Using {\sc oopse}, I have been |
| 811 |
< |
able to parametrize a mesoscale model for phospholipid simulations. |
| 810 |
> |
Bringing us to Ch.~\ref{chapt:lipid}. Using {\sc oopse}, I have been |
| 811 |
> |
able to parameterize a mesoscale model for phospholipid simulations. |
| 812 |
|
This model retains information about solvent ordering about the |
| 813 |
|
bilayer, as well as information regarding the interaction of the |
| 814 |
|
phospholipid head groups' dipole with each other and the surrounding |
