1 |
mmeineke |
58 |
\documentclass[11pt]{article} |
2 |
|
|
\usepackage{epsf} |
3 |
|
|
\usepackage[ref]{overcite} |
4 |
|
|
\usepackage{setspace} |
5 |
|
|
\usepackage{tabularx} |
6 |
|
|
\pagestyle{plain} |
7 |
|
|
\pagenumbering{arabic} |
8 |
|
|
\oddsidemargin 0.0cm \evensidemargin 0.0cm |
9 |
|
|
\topmargin -21pt \headsep 10pt |
10 |
|
|
\textheight 9.0in \textwidth 6.5in |
11 |
|
|
\brokenpenalty=10000 |
12 |
|
|
\renewcommand{\baselinestretch}{1.2} |
13 |
|
|
\renewcommand\citemid{\ } % no comma in optional reference note |
14 |
|
|
|
15 |
|
|
|
16 |
|
|
\begin{document} |
17 |
|
|
|
18 |
|
|
\title{Elucidation of Structural Changes in Osmotically Swollen Polymer Hydrogels} |
19 |
|
|
|
20 |
|
|
\author{Matthew A. Meineke\\ |
21 |
|
|
Department of Chemistry and Biochemistry\\ |
22 |
|
|
University of Notre Dame\\ |
23 |
|
|
Notre Dame, Indiana 46556} |
24 |
|
|
|
25 |
|
|
\date{\today} |
26 |
|
|
\maketitle |
27 |
|
|
|
28 |
|
|
\section{Background} |
29 |
|
|
|
30 |
|
|
In the past twenty years, research on polymer gel networks has |
31 |
|
|
revealed interesting information about their osmotically swollen |
32 |
|
|
states and their physiological\cite{Tasaki92,Tasaki94} and mechanical |
33 |
|
|
properties. Of particular interest is the observance of a reversible |
34 |
|
|
gel collapse in polyelectrolyte\cite{Horkay2000} and non-ionic |
35 |
|
|
gels.\cite{Tanaka78} The gel collapse has been linked to many |
36 |
|
|
different environment variables including temperature, solvent |
37 |
|
|
polarity, and in the case of polyelectrolyte gels, salt composition. |
38 |
|
|
|
39 |
|
|
\begin{figure} |
40 |
|
|
\begin{center} |
41 |
|
|
\epsfxsize=3in |
42 |
|
|
\epsfbox{all_poly.eps} |
43 |
|
|
\end{center} |
44 |
|
|
\caption{Shown here, are the three polymers discussed within this proposal.} |
45 |
|
|
\label{fig:poly} |
46 |
|
|
\end{figure} |
47 |
|
|
|
48 |
|
|
Shibayama \emph{et al.}\cite{Shibayama92} have investigated a weakly |
49 |
|
|
charged polyelectrolyte gel with small angle neutron scattering in the |
50 |
|
|
phase region near the volume change. The gel composition was a |
51 |
|
|
copolymer of N-isopropylacrylamide-\emph{co}-acrylic acid (NIPA/AAc) |
52 |
|
|
(structures shown in figure \ref{fig:poly}). In their study the |
53 |
|
|
collapse was studied by varying either the temperature or the |
54 |
|
|
solvent's acetone:water ratio. In their work they found evidence of |
55 |
|
|
microphase separation in the collapsed gel with phase regions on the |
56 |
|
|
order of $300 \mbox{\AA}$. This would imply that there are regions of |
57 |
|
|
aggregates formed from the combination of electrostatic repulsions of |
58 |
|
|
AAc segments and the attractive interactions of the hydrophobic NIPA |
59 |
|
|
segments. However they were unable to reconcile the length scale of |
60 |
|
|
the microphase separation with the theoretical composition of the |
61 |
|
|
polymer strands. |
62 |
|
|
|
63 |
|
|
In a system investigated by Horkay \emph{et al.},\cite{Horkay2000} the |
64 |
|
|
gel studied was a neutralized polyelectrolyte network, sodium |
65 |
|
|
polyacrylate, shown in figure \ref{fig:poly}. The system was studied |
66 |
|
|
near the volume collapse by varying the monovalent:divalent salt |
67 |
|
|
cation ratio in solution. Measurements of the shear modulus and the |
68 |
|
|
osmotic pressure were compared against the volume fraction of the |
69 |
|
|
gel. It was found that the shear modulus was unaffected by the increase in |
70 |
|
|
the divalent cation concentration near the volume change. This implies |
71 |
|
|
that the cross-linking density is unaffected by the increase in the |
72 |
|
|
divalent:monovalent cation ratio in the gel, suggesting the method of |
73 |
|
|
collapse is based only on an increase in the interaction between |
74 |
|
|
polymer strands facilitated by the divalent cations. What remains to |
75 |
|
|
be known, however, is the exact structural nature of these |
76 |
|
|
interactions. |
77 |
|
|
|
78 |
|
|
\section{Research Goals} |
79 |
|
|
|
80 |
|
|
It is the goal of this research to elucidate the nature of the gel |
81 |
|
|
collapse in the two systems investigated by Shibayama \emph{et al.} and |
82 |
|
|
Horkay \emph{et al.} Through Gibbs Monte Carlo techniques, this |
83 |
|
|
research will simulate the two systems at state points surrounding |
84 |
|
|
their respective gel collapses. Information gained from the |
85 |
|
|
equilibrium configurations will be used to give structural |
86 |
|
|
insight into the nature of the sudden volume changes in the gels. |
87 |
|
|
|
88 |
|
|
The first system to be investigated will be the NIPA/AAc system of |
89 |
|
|
Shibayama \emph{et al.} Their system is ideal to begin with because it |
90 |
|
|
will allow the development of Gibbs techniques without the difficulty |
91 |
|
|
of inserting and deleting charged particles into the gel system (here |
92 |
|
|
the solvent is simply a water-acetone mixture). This system will allow |
93 |
|
|
the investigation to answer such questions as the following: How |
94 |
|
|
atomistic does the model have to be to correctly predict the |
95 |
|
|
microphase separation? Does the distribution of AAc monomers affect |
96 |
|
|
the size of the cavities formed? If so, in what way? How do aggregates |
97 |
|
|
form? Do the gel strands collapse laterally together, or longitudinally |
98 |
|
|
along the polymerization axis? |
99 |
|
|
|
100 |
|
|
After simulating the NIPA/AAc system, the investigation will turn to |
101 |
|
|
the system of Horkay \emph{et al.} Here the system is a neutralized |
102 |
|
|
sodium polyacrylate gel. The first question to answer is then, will |
103 |
|
|
the atomistic detail from the first simulation be sufficient for the |
104 |
|
|
more highly charged gel, or will the system require more detail in the |
105 |
|
|
model. Upon completion of the simulation the following questions will |
106 |
|
|
be addressed: Is a microphase separation observed in the |
107 |
|
|
polyelectrolyte gel collapse? If so, what is the distribution and size |
108 |
|
|
of these cavities? Is the collapse longitudinal or lateral with |
109 |
|
|
respect to the gel structure? How would the system be affected by the |
110 |
|
|
introduction of a trivalent cation? Would the system collapse in the |
111 |
|
|
same way or would a different type of polymer strand interaction be |
112 |
|
|
observed. |
113 |
|
|
|
114 |
|
|
Having simulated both systems, it will then be of interest to compare |
115 |
|
|
the results of the two systems. Here, such questions as the following |
116 |
|
|
can be investigated: Do both systems show similar structural features |
117 |
|
|
when collapsed? Is there a simple model to account for and predict |
118 |
|
|
those features in one or both systems? |
119 |
|
|
|
120 |
|
|
\section{Methodology} |
121 |
|
|
|
122 |
|
|
In the research, I will be employing Gibbs ensemble Monte Carlo |
123 |
|
|
techniques.\cite{Panag2000,Frenkel_Smit} The Gibbs ensemble allows |
124 |
|
|
the simulation of two phases in equilibrium, by simulating two |
125 |
|
|
separate systems, one for each phase, and linking the two systems |
126 |
|
|
through particle and volume exchanges. The total system is then |
127 |
|
|
considered to be at constant $NVT$ where $T$ is the constant |
128 |
|
|
temperature of the system, $N$ is the total number of particles, and |
129 |
|
|
$V$ is the total volume. Each particle and volume exchange must then |
130 |
|
|
be constrained to $N$ and $V$: $N = N_{I} + N_{II}$ and $V = V_{I} + |
131 |
|
|
V_{II}$. The whole system then has the following partition function: |
132 |
|
|
|
133 |
|
|
\begin{equation} |
134 |
|
|
Q_{NVT} = \frac{1}{\Lambda^{3N}N!} \sum_{N_{I}=0}^{N} \left( |
135 |
|
|
\begin{array}{c} |
136 |
|
|
N \\ |
137 |
|
|
N_{I} |
138 |
|
|
\end{array} |
139 |
|
|
\right) \int_{0}^{V} dV_{I}\,V_{I}^{N_{I}}V_{II}^{N_{II}} |
140 |
|
|
\int d\xi_{I}^{N_{I}} \mbox{exp} \left[ -\beta U_{I}(N_{I}) \right] |
141 |
|
|
\int d\xi_{II}^{N_{II}} \mbox{exp} \left[ -\beta U_{II}(N_{II}) \right] |
142 |
|
|
\label{eq:partition_function} |
143 |
|
|
\end{equation} |
144 |
|
|
|
145 |
|
|
Where $\Lambda$ is the thermal de Broglie wavelength, $\xi$ is the |
146 |
|
|
scaled coordinates of the particles in the two regions, $\beta = |
147 |
|
|
\frac{1}{k_{B}T}$, here $k_{B}$ is Boltzman's constant, and $U(N)$ is |
148 |
|
|
the total intermolecular potential for the interactions of $N$ |
149 |
|
|
particles. An ensemble with this given partition function will have the |
150 |
|
|
following probability density: |
151 |
|
|
|
152 |
|
|
\begin{equation} |
153 |
|
|
\rho(N_{I},V_{I}; N, V, T) \propto |
154 |
|
|
\frac{N!}{N_{I}! N_{II}!} |
155 |
|
|
\mbox{exp} |
156 |
|
|
\left[ |
157 |
|
|
N_{I} \ln V_{I} + N_{II} \ln V_{II} |
158 |
|
|
- \beta U_{I}(N_{I}) - \beta U_{II}(N_{II}) |
159 |
|
|
\right] |
160 |
|
|
\label{eq:probability_density} |
161 |
|
|
\end{equation} |
162 |
|
|
|
163 |
|
|
With these two equations, one can then specify the probability of |
164 |
|
|
accepting or rejecting any of the Monte Carlo moves in the |
165 |
|
|
simulation. For Gibbs ensemble Monte Carlo, there are three types of |
166 |
|
|
moves at each trial step. The first is a random translational move |
167 |
|
|
within one of the simulation boxes. This has the following acceptance |
168 |
|
|
probability: |
169 |
|
|
|
170 |
|
|
\begin{equation} |
171 |
|
|
\mathcal{P}_{\mbox{translation}} = |
172 |
|
|
\mbox{min} |
173 |
|
|
\left[ 1, |
174 |
|
|
\mbox{exp} \left( -\beta \Delta U \right) |
175 |
|
|
\right] |
176 |
|
|
\label{eq:translate_accept} |
177 |
|
|
\end{equation} |
178 |
|
|
|
179 |
|
|
Where $\Delta U$ is the change in the total energy of the box due to |
180 |
|
|
the configurational change. The second type of move is a volume |
181 |
|
|
change, where box~I is expanded or contracted by $\Delta V$ and box~II |
182 |
|
|
is contracted or expanded by the same amount to satisfy the |
183 |
|
|
constraint: $V = V_{I} + V_{II}$. this move has the following |
184 |
|
|
acceptance probability: |
185 |
|
|
|
186 |
|
|
\begin{equation} |
187 |
|
|
\mathcal{P}_{\mbox{volume}} = |
188 |
|
|
\mbox{min} |
189 |
|
|
\left[ 1, |
190 |
|
|
\mbox{exp} \left( |
191 |
|
|
-\beta \Delta U_{I} - \beta \Delta U_{I} |
192 |
|
|
+ N_{I} \ln \frac{V_{I} + \Delta V}{V_{I}} |
193 |
|
|
+ N_{II} \ln \frac{V_{II} - \Delta V}{V_{II}} |
194 |
|
|
\right) |
195 |
|
|
\right] |
196 |
|
|
\label{eq:volume_accept} |
197 |
|
|
\end{equation} |
198 |
|
|
|
199 |
|
|
The third possible move is the exchange of particle between boxes. The |
200 |
|
|
following acceptance probability is for the transfer of a particle |
201 |
|
|
from box~II to box~I. |
202 |
|
|
|
203 |
|
|
\begin{equation} |
204 |
|
|
\mathcal{P}_{\mbox{transfer}} = |
205 |
|
|
\mbox{min} \left[ 1, |
206 |
|
|
\frac{N_{II} V_{I}}{(N_{I} + 1) V_{II}} |
207 |
|
|
\mbox{exp} \left( |
208 |
|
|
-\beta \Delta U_{I} - \beta \Delta U_{II} |
209 |
|
|
\right) |
210 |
|
|
\right] |
211 |
|
|
\label{eq:transfer_accept} |
212 |
|
|
\end{equation} |
213 |
|
|
|
214 |
|
|
For a multicomponent mixture, equation \ref{eq:transfer_accept} is |
215 |
|
|
changed by replacing $N_{I}$ and $N_{II}$ with $N_{I,j}$ and $N_{II,j}$ |
216 |
|
|
respectively. Where $j$, is the species of the particle being |
217 |
|
|
transfered. |
218 |
|
|
|
219 |
|
|
In addition to the standard Gibbs ensemble trial moves, I will also |
220 |
|
|
make use of a hybrid Monte Carlo\cite{Frenkel_Smit,Mehlig92} move for |
221 |
|
|
the box containing the polymer gel network. This will allow me to run |
222 |
|
|
short molecular dynamic integration paths in order to relax the |
223 |
|
|
polymer structure. In the hybrid Monte Carlo trial move, the system is |
224 |
|
|
given a Boltzmann weighted distribution of velocities and allowed to |
225 |
|
|
evolve through the integration of the equations of motion for a given |
226 |
|
|
number of time steps. The final configuration is then accepted or |
227 |
|
|
rejected with the following probability: |
228 |
|
|
|
229 |
|
|
\begin{equation} |
230 |
|
|
\mathcal{P}_{\mbox{dynamics}} = |
231 |
|
|
\mbox{min} \left[ 1, |
232 |
|
|
\mbox{exp} \left( -\beta \Delta \mathcal{H} \right) |
233 |
|
|
\right] |
234 |
|
|
\label{eq:hybrid_accept} |
235 |
|
|
\end{equation} |
236 |
|
|
Where $\Delta \mathcal{H}$ is the change in the total Hamiltonian of |
237 |
|
|
the system. |
238 |
|
|
|
239 |
|
|
These four moves will form the basis of the simulation, however, there |
240 |
|
|
are still some considerations with which the final form of the |
241 |
|
|
simulation will need to deal. The main point needing development, will |
242 |
|
|
be a method of setting the chemical potential in the solvent reservoir |
243 |
|
|
box. Because of the particle transfer move in the Gibbs ensemble, the |
244 |
|
|
chemical potentials of both boxes will fluctuate as particles are |
245 |
|
|
exchanged between the boxes. The two boxes will then |
246 |
|
|
settle to the same chemical potential once equilibrium is |
247 |
|
|
reached. This poses a problem for the solvent reservoir box, as it |
248 |
|
|
will not truly be a solvent reservoir if it's chemical potential |
249 |
|
|
fluctuates. Meaning, if the solvent concentration in the box |
250 |
|
|
fluctuates, then it is not truly representative of the bulk solvent, |
251 |
|
|
which experimentally would stay at constant concentration. Therefore, a |
252 |
|
|
method will need to be developed that will allow me to regulate the |
253 |
|
|
chemical potential in the solvent box. |
254 |
|
|
|
255 |
|
|
One possible method, is that of parallel tempering.\cite{Yamamoto2000} |
256 |
|
|
Normally, parallel tempering is used as a method to more fully sample |
257 |
|
|
the phase space of a simulation by simulating parallel systems at |
258 |
|
|
higher temperatures, and at intervals attempting to swap one of the |
259 |
|
|
configurations of the high temperature runs with that of the low |
260 |
|
|
temperature simulation you are interested in. I propose to modify this |
261 |
|
|
method slightly, and instead of simulating other systems at higher |
262 |
|
|
temperatures, run simulations of duplicates of the solvent boxes at a |
263 |
|
|
set chemical potential, and periodically swap their configurations |
264 |
|
|
with that of the solvent reservoir that is equilibrating with the gel |
265 |
|
|
box. The acceptance probability for such an exchange will then need to |
266 |
|
|
be determined before this method could be implemented within the |
267 |
|
|
simulation. |
268 |
|
|
|
269 |
|
|
A second problem arises for the simulation of the ionic salts in |
270 |
|
|
solution. Here the particle insertion and deletion methods run the |
271 |
|
|
possibility of being rejected at every attempt. The reason for this is |
272 |
|
|
due to the immense change in configurational energy associated with |
273 |
|
|
the removal or insertion of a charged species into a solution. One |
274 |
|
|
possible method around this is a method similar to that of Lyubarsev |
275 |
|
|
\emph{et al.}\cite{Lyubarsev98} Whereby particle insertions are |
276 |
|
|
attempted through the slow growth of the particle into the system. In |
277 |
|
|
this case, the simulation would allow the slow growth of a charge onto |
278 |
|
|
an inserted ``neutral'' ion. Therefore allowing time for the solvent |
279 |
|
|
to rearrange itself about the new charge and increase the likelihood |
280 |
|
|
of the insertion being accepted. |
281 |
|
|
|
282 |
|
|
\section{Proposal Summary} |
283 |
|
|
|
284 |
|
|
Through the use of standard Monte Carlo techniques I propose to |
285 |
|
|
simulate first the system of Shibayama \emph{et al.}, and then second |
286 |
|
|
that of Horkay \emph{et al.} In the first simulation I plan to work |
287 |
|
|
out many of the details concerning the methodology and the atomistic |
288 |
|
|
detail of the model employed. In the second simulation, I will use the |
289 |
|
|
model and methodology from the previous simulation and add to it the |
290 |
|
|
capability of simulating ions within the solution. Results from both |
291 |
|
|
systems will be used to determine the micro-structural details of the |
292 |
|
|
reversible gel collapse observed in both polymer gels. |
293 |
|
|
|
294 |
|
|
\bibliographystyle{achemso} |
295 |
|
|
\bibliography{original_proposal} \end{document} |