--- trunk/mattDisertation/Introduction.tex	2004/02/02 21:56:16	1003
+++ trunk/mattDisertation/Introduction.tex	2004/02/03 17:41:56	1008
@@ -6,15 +6,15 @@ Carlo. Molecular Dynamic simulations integrate the equ
 The techniques used in the course of this research fall under the two
 main classes of molecular simulation: Molecular Dynamics and Monte
 Carlo. Molecular Dynamic simulations integrate the equations of motion
-for a given system of particles, allowing the researher to gain
+for a given system of particles, allowing the researcher to gain
 insight into the time dependent evolution of a system. Diffusion
 phenomena are readily studied with this simulation technique, making
 Molecular Dynamics the main simulation technique used in this
 research. Other aspects of the research fall under the Monte Carlo
 class of simulations. In Monte Carlo, the configuration space
-available to the collection of particles is sampled stochastichally,
+available to the collection of particles is sampled stochastically,
 or randomly. Each configuration is chosen with a given probability
-based on the Maxwell Boltzman distribution. These types of simulations
+based on the Maxwell Boltzmann distribution. These types of simulations
 are best used to probe properties of a system that are only dependent
 only on the state of the system. Structural information about a system
 is most readily obtained through these types of methods.
@@ -31,13 +31,13 @@ Statistical Mechanics concepts present in this dissert
 
 The following section serves as a brief introduction to some of the
 Statistical Mechanics concepts present in this dissertation.  What
-follows is a brief derivation of Blotzman weighted statistics, and an
+follows is a brief derivation of Boltzmann weighted statistics, and an
 explanation of how one can use the information to calculate an
 observable for a system.  This section then concludes with a brief
 discussion of the ergodic hypothesis and its relevance to this
 research.
 
-\subsection{\label{introSec:boltzman}Boltzman weighted statistics}
+\subsection{\label{introSec:boltzman}Boltzmann weighted statistics}
 
 Consider a system, $\gamma$, with some total energy,, $E_{\gamma}$.
 Let $\Omega(E_{\gamma})$ represent the number of degenerate ways
@@ -86,7 +86,7 @@ S = k_B \ln \Omega(E)
 S = k_B \ln \Omega(E)
 \label{introEq:SM5}
 \end{equation}
-Where $k_B$ is the Boltzman constant.  Having defined entropy, one can
+Where $k_B$ is the Boltzmann constant.  Having defined entropy, one can
 also define the temperature of the system using the relation
 \begin{equation}
 \frac{1}{T} = \biggl ( \frac{\partial S}{\partial E} \biggr )_{N,V}
@@ -111,7 +111,7 @@ is allowed to fluctuate. Returning to the previous exa
 the canonical ensemble, the number of particles, $N$, the volume, $V$,
 and the temperature, $T$, are all held constant while the energy, $E$,
 is allowed to fluctuate. Returning to the previous example, the bath
-system is now an infinitly large thermal bath, whose exchange of
+system is now an infinitely large thermal bath, whose exchange of
 energy with the system $\gamma$ holds the temperature constant.  The
 partitioning of energy in the bath system is then related to the total
 energy of both systems and the fluctuations in $E_{\gamma}$:
@@ -127,7 +127,7 @@ Where $\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{
 \label{introEq:SM10}
 \end{equation}
 Where $\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma}$ denotes
-an integration over all accessable phase space, $P_{\gamma}$ is the
+an integration over all accessible phase space, $P_{\gamma}$ is the
 probability of being in a given phase state and
 $A(\boldsymbol{\Gamma})$ is some observable that is a function of the
 phase state.
@@ -156,7 +156,7 @@ P_{\gamma} \propto e^{-\beta E_{\gamma}}
 P_{\gamma} \propto e^{-\beta E_{\gamma}}
 \label{introEq:SM13}
 \end{equation}
-Where $\ln \Omega(E)$ has been factored out of the porpotionality as a
+Where $\ln \Omega(E)$ has been factored out of the proportionality as a
 constant.  Normalizing the probability ($\int\limits_{\boldsymbol{\Gamma}}
 d\boldsymbol{\Gamma} P_{\gamma} = 1$) gives 
 \begin{equation}
@@ -164,7 +164,7 @@ P_{\gamma} = \frac{e^{-\beta E_{\gamma}}}
 {\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} e^{-\beta E_{\gamma}}}
 \label{introEq:SM14}
 \end{equation}
-This result is the standard Boltzman statistical distribution.
+This result is the standard Boltzmann statistical distribution.
 Applying it to Eq.~\ref{introEq:SM10} one can obtain the following relation for ensemble averages:
 \begin{equation}
 \langle A \rangle = 
@@ -182,7 +182,7 @@ ergodicity allows the unification of a time averaged o
 systems, this is a valid assumption, except in cases where the system
 may be trapped in a local feature (\emph{e.g.}~glasses). When valid,
 ergodicity allows the unification of a time averaged observation and
-an ensemble averged one. If an observation is averaged over a
+an ensemble averaged one. If an observation is averaged over a
 sufficiently long time, the system is assumed to visit all
 appropriately available points in phase space, giving a properly
 weighted statistical average. This allows the researcher freedom of
@@ -217,7 +217,7 @@ the value of $f(x)$ at each point. The accumulated ave
 $[a,b]$. The calculation of the integral could then be solved by
 randomly choosing points along the interval $[a,b]$ and calculating
 the value of $f(x)$ at each point. The accumulated average would then
-approach $I$ in the limit where the number of trials is infintely
+approach $I$ in the limit where the number of trials is infinitely
 large.
 
 However, in Statistical Mechanics, one is typically interested in
@@ -235,7 +235,7 @@ brute force method to this system would yield highly i
 momentum. Therefore the momenta contribution of the integral can be
 factored out, leaving the configurational integral. Application of the
 brute force method to this system would yield highly inefficient
-results. Due to the Boltzman weighting of this integral, most random
+results. Due to the Boltzmann weighting of this integral, most random
 configurations will have a near zero contribution to the ensemble
 average. This is where importance sampling comes into
 play.\cite{allen87:csl}
@@ -263,7 +263,7 @@ Looking at Eq.~\ref{eq:mcEnsAvg}, and realizing
 	{\int d^N \mathbf{r}~e^{-\beta V(\mathbf{r}^N)}}
 \label{introEq:MCboltzman}
 \end{equation}
-Where $\rho_{kT}$ is the boltzman distribution.  The ensemble average
+Where $\rho_{kT}$ is the Boltzmann distribution.  The ensemble average
 can be rewritten as
 \begin{equation}
 \langle A \rangle = \int d^N \mathbf{r}~A(\mathbf{r}^N) 
@@ -297,8 +297,8 @@ conditions:\cite{leach01:mm}
 \item The outcome of each trial depends only on the outcome of the previous trial.
 \item Each trial belongs to a finite set of outcomes called the state space.
 \end{enumerate}
-If given two configuartions, $\mathbf{r}^N_m$ and $\mathbf{r}^N_n$,
-$\rho_m$ and $\rho_n$ are the probablilities of being in state
+If given two configurations, $\mathbf{r}^N_m$ and $\mathbf{r}^N_n$,
+$\rho_m$ and $\rho_n$ are the probabilities of being in state
 $\mathbf{r}^N_m$ and $\mathbf{r}^N_n$ respectively.  Further, the two
 states are linked by a transition probability, $\pi_{mn}$, which is the
 probability of going from state $m$ to state $n$.
@@ -343,7 +343,7 @@ Eq.~\ref{introEq:MCmarkovEquil} is solved such that
 
 In the Metropolis method\cite{metropolis:1953}
 Eq.~\ref{introEq:MCmarkovEquil} is solved such that
-$\boldsymbol{\rho}_{\text{limit}}$ matches the Boltzman distribution
+$\boldsymbol{\rho}_{\text{limit}}$ matches the Boltzmann distribution
 of states.  The method accomplishes this by imposing the strong
 condition of microscopic reversibility on the equilibrium
 distribution.  Meaning, that at equilibrium the probability of going
@@ -352,7 +352,7 @@ from $m$ to $n$ is the same as going from $n$ to $m$.
 \rho_m\pi_{mn} = \rho_n\pi_{nm}
 \label{introEq:MCmicroReverse}
 \end{equation}
-Further, $\boldsymbol{\alpha}$ is chosen to be a symetric matrix in
+Further, $\boldsymbol{\alpha}$ is chosen to be a symmetric matrix in
 the Metropolis method.  Using Eq.~\ref{introEq:MCpi},
 Eq.~\ref{introEq:MCmicroReverse} becomes
 \begin{equation}
@@ -360,7 +360,7 @@ Eq.~\ref{introEq:MCmicroReverse} becomes
 	\frac{\rho_n}{\rho_m}
 \label{introEq:MCmicro2}
 \end{equation}
-For a Boltxman limiting distribution,
+For a Boltzmann limiting distribution,
 \begin{equation}
 \frac{\rho_n}{\rho_m} = e^{-\beta[\mathcal{U}(n) - \mathcal{U}(m)]}
 	= e^{-\beta \Delta \mathcal{U}}
@@ -380,16 +380,16 @@ Metropolis method proceeds as follows
 Metropolis method proceeds as follows
 \begin{enumerate}
 \item Generate an initial configuration $\mathbf{r}^N$ which has some finite probability in $\rho_{kT}$.
-\item Modify $\mathbf{r}^N$, to generate configuratioon $\mathbf{r^{\prime}}^N$.
+\item Modify $\mathbf{r}^N$, to generate configuration $\mathbf{r^{\prime}}^N$.
 \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 $e^{-\beta \Delta \mathcal{U}}$.
-\item Accumulate the average for the configurational observable of intereest.
+\item Accumulate the average for the configurational observable of interest.
 \item Repeat from step 2 until the average converges.
 \end{enumerate}
 One important note is that the average is accumulated whether the move
 is accepted or not, this ensures proper weighting of the average.
 Using Eq.~\ref{introEq:Importance4} it becomes clear that the
 accumulated averages are the ensemble averages, as this method ensures
-that the limiting distribution is the Boltzman distribution.
+that the limiting distribution is the Boltzmann distribution.
 
 \section{\label{introSec:MD}Molecular Dynamics Simulations}
 
@@ -409,7 +409,7 @@ However, when the observable is dependent only on the 
 researcher is interested.  If the observables depend on momenta in
 any fashion, then the only choice is molecular dynamics in some form.
 However, when the observable is dependent only on the configuration,
-then most of the time Monte Carlo techniques will be more efficent.
+then most of the time Monte Carlo techniques will be more efficient.
 
 The focus of research in the second half of this dissertation is
 centered around the dynamic properties of phospholipid bilayers,
@@ -432,24 +432,24 @@ positions were selected that in some cases dispersed t
 Ch.~\ref{chapt:lipid} deals with the formation and equilibrium dynamics of
 phospholipid membranes.  Therefore in these simulations initial
 positions were selected that in some cases dispersed the lipids in
-water, and in other cases structured the lipids into preformed
+water, and in other cases structured the lipids into performed
 bilayers.  Important considerations at this stage of the simulation are:
 \begin{itemize}
 \item There are no major overlaps of molecular or atomic orbitals
-\item Velocities are chosen in such a way as to not gie the system a non=zero total momentum or angular momentum.
-\item It is also sometimes desireable to select the velocities to correctly sample the target temperature.
+\item Velocities are chosen in such a way as to not give the system a non=zero total momentum or angular momentum.
+\item It is also sometimes desirable to select the velocities to correctly sample the target temperature.
 \end{itemize}
 
 The first point is important due to the amount of potential energy
 generated by having two particles too close together.  If overlap
 occurs, the first evaluation of forces will return numbers so large as
-to render the numerical integration of teh motion meaningless.  The
+to render the numerical integration of the motion meaningless.  The
 second consideration keeps the system from drifting or rotating as a
 whole.  This arises from the fact that most simulations are of systems
 in equilibrium in the absence of outside forces.  Therefore any net
 movement would be unphysical and an artifact of the simulation method
 used.  The final point addresses the selection of the magnitude of the
-initial velocities.  For many simulations it is convienient to use
+initial velocities.  For many simulations it is convenient to use
 this opportunity to scale the amount of kinetic energy to reflect the
 desired thermal distribution of the system.  However, it must be noted
 that most systems will require further velocity rescaling after the
@@ -474,11 +474,11 @@ group in a vacuum, the behavior of the system will be 
 arranged in a $10 \times 10 \times 10$ cube, 488 particles will be
 exposed to the surface.  Unless one is simulating an isolated particle
 group in a vacuum, the behavior of the system will be far from the
-desired bulk charecteristics.  To offset this, simulations employ the
+desired bulk characteristics.  To offset this, simulations employ the
 use of periodic boundary images.\cite{born:1912}
 
 The technique involves the use of an algorithm that replicates the
-simulation box on an infinite lattice in cartesian space.  Any given
+simulation box on an infinite lattice in Cartesian space.  Any given
 particle leaving the simulation box on one side will have an image of
 itself enter on the opposite side (see Fig.~\ref{introFig:pbc}).  In
 addition, this sets that any given particle pair has an image, real or
@@ -489,7 +489,7 @@ Sec.\ref{oopseSec:pbc}.
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{pbcFig.eps}
-\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.}
+\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.}
 \label{introFig:pbc}
 \end{figure}
 
@@ -498,28 +498,36 @@ predetermined distance, $r_{\text{cut}}$, are not incl
 cutoff radius be employed.  Using a cutoff radius improves the
 efficiency of the force evaluation, as particles farther than a
 predetermined distance, $r_{\text{cut}}$, are not included in the
-calculation.\cite{Frenkel1996} In a simultation with periodic images,
+calculation.\cite{Frenkel1996} In a simulation with periodic images,
 $r_{\text{cut}}$ has a maximum value of $\text{box}/2$.
 Fig.~\ref{introFig:rMax} illustrates how when using an
 $r_{\text{cut}}$ larger than this value, or in the extreme limit of no
 $r_{\text{cut}}$ at all, the corners of the simulation box are
 unequally weighted due to the lack of particle images in the $x$, $y$,
-or $z$ directions past a disance of $\text{box} / 2$.
+or $z$ directions past a distance of $\text{box} / 2$.
 
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{rCutMaxFig.eps}
-\caption
+\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).}
 \label{introFig:rMax}
 \end{figure}
 
-With the use of an $fix$, however, comes a discontinuity in the
-potential energy curve (Fig.~\ref{fix}). To fix this discontinuity,
-one calculates the potential energy at the $r_{\text{cut}}$, and add
-that value to the potential.  This causes the function to go smoothly
-to zero at the cutoff radius.  This ensures conservation of energy
-when integrating the Newtonian equations of motion.
+With the use of an $r_{\text{cut}}$, however, comes a discontinuity in
+the potential energy curve (Fig.~\ref{introFig:shiftPot}). To fix this
+discontinuity, one calculates the potential energy at the
+$r_{\text{cut}}$, and adds that value to the potential, causing
+the function to go smoothly to zero at the cutoff radius.  This
+shifted potential ensures conservation of energy when integrating the
+Newtonian equations of motion.
 
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{shiftedPot.eps}
+\caption[Shifting the Lennard-Jones Potential]{The Lennard-Jones potential (blue line) is shifted (red line) to remove the discontinuity at $r_{\text{cut}}$.}
+\label{introFig:shiftPot}
+\end{figure}
+
 The second main simplification used in this research is the Verlet
 neighbor list. \cite{allen87:csl} In the Verlet method, one generates
 a list of all neighbor atoms, $j$, surrounding atom $i$ within some
@@ -534,15 +542,19 @@ A starting point for the discussion of molecular dynam
 \subsection{\label{introSec:mdIntegrate} Integration of the equations of motion}
 
 A starting point for the discussion of molecular dynamics integrators
-is the Verlet algorithm. \cite{Frenkel1996} It begins with a Taylor
+is the Verlet algorithm.\cite{Frenkel1996} It begins with a Taylor
 expansion of position in time:
 \begin{equation}
-eq here
+q(t+\Delta t)= q(t) + v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 +
+	\frac{\Delta t^3}{3!}\frac{\partial q(t)}{\partial t} +
+	\mathcal{O}(\Delta t^4) 
 \label{introEq:verletForward}
 \end{equation}
 As well as,
 \begin{equation}
-eq here
+q(t-\Delta t)= q(t) - v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 -
+	\frac{\Delta t^3}{3!}\frac{\partial q(t)}{\partial t} +
+	\mathcal{O}(\Delta t^4) 
 \label{introEq:verletBack}
 \end{equation}
 Adding together Eq.~\ref{introEq:verletForward} and
@@ -560,7 +572,7 @@ In practice, however, the simulations in this research
 order of $\Delta t^4$.
 
 In practice, however, the simulations in this research were integrated
-with a velocity reformulation of teh Verlet method.\cite{allen87:csl}
+with a velocity reformulation of the Verlet method.\cite{allen87:csl}
 \begin{equation}
 eq here
 \label{introEq:MDvelVerletPos}
@@ -575,7 +587,7 @@ importance, as it is a measure of how closely one is f
 very little long term drift in energy conservation.  Energy
 conservation in a molecular dynamics simulation is of extreme
 importance, as it is a measure of how closely one is following the
-``true'' trajectory wtih the finite integration scheme.  An exact
+``true'' trajectory with the finite integration scheme.  An exact
 solution to the integration will conserve area in phase space, as well
 as be reversible in time, that is, the trajectory integrated forward
 or backwards will exactly match itself.  Having a finite algorithm
@@ -586,7 +598,7 @@ a pseudo-Hamiltonian which shadows the real one in pha
 It can be shown,\cite{Frenkel1996} that although the Verlet algorithm
 does not rigorously preserve the actual Hamiltonian, it does preserve
 a pseudo-Hamiltonian which shadows the real one in phase space.  This
-pseudo-Hamiltonian is proveably area-conserving as well as time
+pseudo-Hamiltonian is provably area-conserving as well as time
 reversible.  The fact that it shadows the true Hamiltonian in phase
 space is acceptable in actual simulations as one is interested in the
 ensemble average of the observable being measured.  From the ergodic
@@ -597,7 +609,7 @@ parameters are needed to describe the motions.  The si
 \subsection{\label{introSec:MDfurther}Further Considerations}
 In the simulations presented in this research, a few additional
 parameters are needed to describe the motions.  The simulations
-involving water and phospholipids in Chapt.~\ref{chaptLipids} are
+involving water and phospholipids in Ch.~\ref{chaptLipids} are
 required to integrate the equations of motions for dipoles on atoms.
 This involves an additional three parameters be specified for each
 dipole atom: $\phi$, $\theta$, and $\psi$.  These three angles are
@@ -619,7 +631,7 @@ Where $\omega^s_i$ is the angular velocity in the lab 
 \label{introEq:MDeuleeerPsi}
 \end{equation}
 Where $\omega^s_i$ is the angular velocity in the lab space frame
-along cartesian coordinate $i$.  However, a difficulty arises when
+along Cartesian coordinate $i$.  However, a difficulty arises when
 attempting to integrate Eq.~\ref{introEq:MDeulerPhi} and
 Eq.~\ref{introEq:MDeulerPsi}. The $\frac{1}{\sin \theta}$ present in
 both equations means there is a non-physical instability present when
@@ -649,7 +661,7 @@ degree of freedom, and $f_j$ is the force on that degr
 \end{equation}
 Here, $r_j$ and $p_j$ are the position and conjugate momenta of a
 degree of freedom, and $f_j$ is the force on that degree of freedom.
-$\Gamma$ is defined as the set of all positions nad conjugate momenta,
+$\Gamma$ is defined as the set of all positions and conjugate momenta,
 $\{r_j,p_j\}$, and the propagator, $U(t)$, is defined
 \begin {equation}
 eq here
@@ -705,7 +717,7 @@ comprised of unitary propagators, it is symplectic, an
 This is the velocity Verlet formulation presented in
 Sec.~\ref{introSec:MDintegrate}.  Because this integration scheme is
 comprised of unitary propagators, it is symplectic, and therefore area
-preserving in phase space.  From the preceeding fatorization, one can
+preserving in phase space.  From the preceding factorization, one can
 see that the integration of the equations of motion would follow:
 \begin{enumerate}
 \item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position.
@@ -729,15 +741,15 @@ eq here
 eq here
 \label{introEq:SR1}
 \end{equation}
-Where $\boldsymbol{\tau}(\mathbf{A})$ are the tourques of the system
+Where $\boldsymbol{\tau}(\mathbf{A})$ are the torques of the system
 due to the configuration, and $\boldsymbol{/pi}$ are the conjugate
 angular momenta of the system. The propagator, $G(\Delta t)$, becomes
 \begin{equation}
 eq here
 \label{introEq:SR2}
 \end{equation}
-Propagation fo the linear and angular momenta follows as in the Verlet
-scheme.  The propagation of positions also follows the verlet scheme
+Propagation of the linear and angular momenta follows as in the Verlet
+scheme.  The propagation of positions also follows the Verlet scheme
 with the addition of a further symplectic splitting of the rotation
 matrix propagation, $\mathcal{G}_{\text{rot}}(\Delta t)$.
 \begin{equation}
@@ -753,25 +765,25 @@ presents the random sequential adsorption simulations 
 
 This dissertation is divided as follows:Chapt.~\ref{chapt:RSA}
 presents the random sequential adsorption simulations of related
-pthalocyanines on a gold (111) surface. Chapt.~\ref{chapt:OOPSE}
+pthalocyanines on a gold (111) surface. Ch.~\ref{chapt:OOPSE}
 is about the writing of the molecular dynamics simulation package
-{\sc oopse}, Chapt.~\ref{chapt:lipid} regards the simulations of
+{\sc oopse}, Ch.~\ref{chapt:lipid} regards the simulations of
 phospholipid bilayers using a mesoscale model, and lastly,
-Chapt.~\ref{chapt:conclusion} concludes this dissertation with a
+Ch.~\ref{chapt:conclusion} concludes this dissertation with a
 summary of all results. The chapters are arranged in chronological
 order, and reflect the progression of techniques I employed during my
 research.  
 
 The chapter concerning random sequential adsorption
 simulations is a study in applying the principles of theoretical
-research in order to obtain a simple model capaable of explaining the
+research in order to obtain a simple model capable of explaining the
 results.  My advisor, Dr. Gezelter, and I were approached by a
 colleague, Dr. Lieberman, about possible explanations for partial
-coverge of a gold surface by a particular compound of hers. We
+coverage of a gold surface by a particular compound of hers. We
 suggested it might be due to the statistical packing fraction of disks
 on a plane, and set about to simulate this system.  As the events in
 our model were not dynamic in nature, a Monte Carlo method was
-emplyed.  Here, if a molecule landed on the surface without
+employed.  Here, if a molecule landed on the surface without
 overlapping another, then its landing was accepted.  However, if there
 was overlap, the landing we rejected and a new random landing location
 was chosen.  This defined our acceptance rules and allowed us to
@@ -784,14 +796,14 @@ is a unique package capable of not only integrating th
 pre-existing molecular dynamic simulation packages available, none
 were capable of implementing the models we were developing.{\sc oopse}
 is a unique package capable of not only integrating the equations of
-motion in cartesian space, but is also able to integrate the
+motion in Cartesian space, but is also able to integrate the
 rotational motion of rigid bodies and dipoles.  Add to this the
 ability to perform calculations across parallel processors and a
 flexible script syntax for creating systems, and {\sc oopse} becomes a
 very powerful scientific instrument for the exploration of our model.
 
-Bringing us to Chapt.~\ref{chapt:lipid}. Using {\sc oopse}, I have been
-able to parametrize a mesoscale model for phospholipid simulations.
+Bringing us to Ch.~\ref{chapt:lipid}. Using {\sc oopse}, I have been
+able to parameterize a mesoscale model for phospholipid simulations.
 This model retains information about solvent ordering about the
 bilayer, as well as information regarding the interaction of the
 phospholipid head groups' dipole with each other and the surrounding
