| 139 |
|
fully populated. |
| 140 |
|
|
| 141 |
|
To investigate the phase behavior of this model, we have performed a |
| 142 |
< |
series of Metropolis Monte Carlo simulations of moderately-sized (34.3 |
| 143 |
< |
$\sigma$ on a side) patches of membrane hosted on both triangular |
| 144 |
< |
($\gamma = a/b = \sqrt{3}$) and distorted ($\gamma \neq \sqrt{3}$) |
| 145 |
< |
lattices. The linear extent of one edge of the monolayer was $20 a$ |
| 146 |
< |
and the system was kept roughly square. The average distance that |
| 147 |
< |
coplanar dipoles were positioned from their six nearest neighbors was |
| 148 |
< |
1 $\sigma$ (on both triangular and distorted lattices). Typical |
| 149 |
< |
system sizes were 1360 dipoles for the triangular lattices and |
| 142 |
> |
series of Me\-trop\-o\-lis Monte Carlo simulations of moderately-sized |
| 143 |
> |
(34.3 $\sigma$ on a side) patches of membrane hosted on both |
| 144 |
> |
triangular ($\gamma = a/b = \sqrt{3}$) and distorted ($\gamma \neq |
| 145 |
> |
\sqrt{3}$) lattices. The linear extent of one edge of the monolayer |
| 146 |
> |
was $20 a$ and the system was kept roughly square. The average |
| 147 |
> |
distance that coplanar dipoles were positioned from their six nearest |
| 148 |
> |
neighbors was 1 $\sigma$ (on both triangular and distorted lattices). |
| 149 |
> |
Typical system sizes were 1360 dipoles for the triangular lattices and |
| 150 |
|
840-2800 dipoles for the distorted lattices. Two-dimensional periodic |
| 151 |
|
boundary conditions were used, and the cutoff for the dipole-dipole |
| 152 |
|
interaction was set to 4.3 $\sigma$. This cutoff is roughly 2.5 times |
| 159 |
|
methods are computationally expensive,\cite{Spohr97,Yeh99} and are not |
| 160 |
|
necessary in this case. All parameters ($T^{*}$, $\mu^{*}$, and |
| 161 |
|
$\gamma$) were varied systematically to study the effects of these |
| 162 |
< |
parameters on the formation of ripple-like phases. |
| 162 |
> |
parameters on the formation of ripple-like phases. The error bars in |
| 163 |
> |
our results are one $\sigma$ on each side of the average values, where |
| 164 |
> |
$\sigma$ is the standard deviation obtained from repeated observations |
| 165 |
> |
of many configurations. |
| 166 |
|
|
| 167 |
|
\section{Results and Analysis} |
| 168 |
|
\label{mc:sec:results} |
| 196 |
|
|
| 197 |
|
\begin{figure} |
| 198 |
|
\includegraphics[width=\linewidth]{./figures/mcPhase.pdf} |
| 199 |
< |
\caption{\label{mcfig:phase} Top panel: The $P_2$ dipolar order parameter as |
| 200 |
< |
a function of temperature for both triangular ($\gamma = 1.732$) and |
| 201 |
< |
distorted ($\gamma = 1.875$) lattices. Bottom Panel: The phase |
| 202 |
< |
diagram for the dipolar membrane model. The line denotes the division |
| 203 |
< |
between the dipolar ordered (anti-ferroelectric) and disordered phases. |
| 204 |
< |
An enlarged view near the triangular lattice is shown inset.} |
| 199 |
> |
\caption[ The $P_2$ dipolar order parameter as |
| 200 |
> |
a function of temperature and the phase diagram for the dipolar |
| 201 |
> |
membrane model]{\label{mcfig:phase} Top panel: The $P_2$ dipolar order |
| 202 |
> |
parameter as a function of temperature for both triangular ($\gamma = |
| 203 |
> |
1.732$) and distorted ($\gamma = 1.875$) lattices. Bottom Panel: The |
| 204 |
> |
phase diagram for the dipolar membrane model. The line denotes the |
| 205 |
> |
division between the dipolar ordered (anti-ferroelectric) and |
| 206 |
> |
disordered phases. An enlarged view near the triangular lattice is |
| 207 |
> |
shown inset.} |
| 208 |
|
\end{figure} |
| 209 |
|
|
| 210 |
|
There is a clear order-disorder transition in evidence from this data. |
| 211 |
|
Both the triangular and distorted lattices have dipolar-ordered |
| 212 |
< |
low-temperature phases, and orientationally-disordered high |
| 212 |
> |
low-temperature phases, and ori\-en\-ta\-tion\-al\-ly-disordered high |
| 213 |
|
temperature phases. The coexistence temperature for the triangular |
| 214 |
|
lattice is significantly lower than for the distorted lattices, and |
| 215 |
|
the bulk polarization is approximately $0$ for both dipolar ordered |
| 220 |
|
transition is equally smooth at all system sizes that were studied. |
| 221 |
|
Additionally, we have repeated the Monte Carlo simulations over a wide |
| 222 |
|
range of lattice ratios ($\gamma$) to generate a dipolar |
| 223 |
< |
order/disorder phase diagram. The bottom panel in Fig. \ref{mcfig:phase} |
| 224 |
< |
shows that the triangular lattice is a low-temperature cusp in the |
| 225 |
< |
$T^{*}-\gamma$ phase diagram. |
| 223 |
> |
order/disorder phase diagram. The bottom panel in |
| 224 |
> |
Fig. \ref{mcfig:phase} shows that the triangular lattice is a |
| 225 |
> |
low-temperature cusp in the $T^{*}-\gamma$ phase diagram. |
| 226 |
|
|
| 227 |
|
This phase diagram is remarkable in that it shows an |
| 228 |
|
anti-ferroelectric phase near $\gamma=1.732$ where one would expect |
| 242 |
|
|
| 243 |
|
\begin{figure} |
| 244 |
|
\includegraphics[width=\linewidth]{./figures/mcSnapshot.pdf} |
| 245 |
< |
\caption{\label{mcfig:snapshot} Top and Side views of a representative |
| 245 |
> |
\caption[ Top and Side views of a representative |
| 246 |
|
configuration for the dipolar ordered phase supported on the |
| 247 |
< |
triangular lattice. Note the anti-ferroelectric ordering and the long |
| 248 |
< |
wavelength buckling of the membrane. Dipolar ordering has been |
| 249 |
< |
observed in all three equivalent directions on the triangular lattice, |
| 250 |
< |
and the ripple direction is always perpendicular to the director axis |
| 251 |
< |
for the dipoles.} |
| 247 |
> |
triangular lattice]{\label{mcfig:snapshot} Top and Side views of a |
| 248 |
> |
representative configuration for the dipolar ordered phase supported |
| 249 |
> |
on the triangular lattice. Note the anti-ferroelectric ordering and |
| 250 |
> |
the long wavelength buckling of the membrane. Dipolar ordering has |
| 251 |
> |
been observed in all three equivalent directions on the triangular |
| 252 |
> |
lattice, and the ripple direction is always perpendicular to the |
| 253 |
> |
director axis for the dipoles.} |
| 254 |
|
\end{figure} |
| 255 |
|
|
| 256 |
|
Although the snapshot in Fig. \ref{mcfig:snapshot} gives the appearance |
| 280 |
|
|
| 281 |
|
\begin{figure} |
| 282 |
|
\includegraphics[width=\linewidth]{./figures/mcHdc.pdf} |
| 283 |
< |
\caption{\label{mcfig:CrossCorrelation} Contours of the height-dipole |
| 284 |
< |
correlation function as a function of the dot product between the |
| 285 |
< |
dipole ($\hat{\mu}$) and inter-dipole separation vector ($\hat{r}$) |
| 286 |
< |
and the distance ($r$) between the dipoles. Perfect height |
| 287 |
< |
correlation (contours approaching 1) are present in the ordered phase |
| 288 |
< |
when the two dipoles are in the same head-to-tail line. |
| 283 |
> |
\caption[Contours of the height-dipole |
| 284 |
> |
correlation function]{\label{mcfig:CrossCorrelation} Contours of the |
| 285 |
> |
height-dipole correlation function as a function of the dot product |
| 286 |
> |
between the dipole ($\hat{\mu}$) and inter-dipole separation vector |
| 287 |
> |
($\hat{r}$) and the distance ($r$) between the dipoles. Perfect |
| 288 |
> |
height correlation (contours approaching 1) are present in the ordered |
| 289 |
> |
phase when the two dipoles are in the same head-to-tail line. |
| 290 |
|
Anti-correlation (contours below 0) is only seen when the inter-dipole |
| 291 |
|
vector is perpendicular to the dipoles. In the dipole-disordered |
| 292 |
|
portion of the phase diagram, there is only weak correlation in the |
| 378 |
|
|
| 379 |
|
\begin{figure} |
| 380 |
|
\includegraphics[width=\linewidth]{./figures/mcLogFit.pdf} |
| 381 |
< |
\caption{\label{mcfig:fit} Evidence that the observed ripples are {\em |
| 382 |
< |
not} thermal undulations is obtained from the 2-d Fourier transform |
| 383 |
< |
$\langle |h^{*}(\vec{q})|^2 \rangle$ of the height profile ($\langle |
| 384 |
< |
h^{*}(x,y) \rangle$). Rippled samples show low-wavelength peaks that |
| 385 |
< |
are outliers on the Landau free energy fits by an order of magnitude. |
| 386 |
< |
Samples exhibiting only thermal undulations fit Eq. \ref{mceq:fit} |
| 387 |
< |
remarkably well.} |
| 381 |
> |
\caption[Evidence that the observed ripples are {\em not} thermal |
| 382 |
> |
undulations]{\label{mcfig:fit} Evidence that the observed ripples are |
| 383 |
> |
{\em not} thermal undulations is obtained from the 2-d Fourier |
| 384 |
> |
transform $\langle |h^{*}(\vec{q})|^2 \rangle$ of the height profile |
| 385 |
> |
($\langle h^{*}(x,y) \rangle$). Rippled samples show low-wavelength |
| 386 |
> |
peaks that are outliers on the Landau free energy fits by an order of |
| 387 |
> |
magnitude. Samples exhibiting only thermal undulations fit |
| 388 |
> |
Eq. \ref{mceq:fit} remarkably well.} |
| 389 |
|
\end{figure} |
| 390 |
|
|
| 391 |
|
\subsection{Effects of Potential Parameters on Amplitude and Wavelength} |
| 444 |
|
|
| 445 |
|
\begin{figure} |
| 446 |
|
\includegraphics[width=\linewidth]{./figures/mcProperties_sq.pdf} |
| 447 |
< |
\caption{\label{mcfig:Amplitude} a) The amplitude $A^{*}$ of the ripples |
| 448 |
< |
vs. temperature for a triangular lattice. b) The amplitude $A^{*}$ of |
| 449 |
< |
the ripples vs. dipole strength ($\mu^{*}$) for both the triangular |
| 450 |
< |
lattice (circles) and distorted lattice (squares). The reduced |
| 451 |
< |
temperatures were kept fixed at $T^{*} = 94$ for the triangular |
| 452 |
< |
lattice and $T^{*} = 106$ for the distorted lattice (approximately 2/3 |
| 453 |
< |
of the order-disorder transition temperature for each lattice).} |
| 447 |
> |
\caption[ The amplitude $A^{*}$ of the ripples |
| 448 |
> |
vs. temperature and dipole strength |
| 449 |
> |
($\mu^{*}$)]{\label{mcfig:Amplitude} a) The amplitude $A^{*}$ of the |
| 450 |
> |
ripples vs. temperature for a triangular lattice. b) The amplitude |
| 451 |
> |
$A^{*}$ of the ripples vs. dipole strength ($\mu^{*}$) for both the |
| 452 |
> |
triangular lattice (circles) and distorted lattice (squares). The |
| 453 |
> |
reduced temperatures were kept fixed at $T^{*} = 94$ for the |
| 454 |
> |
triangular lattice and $T^{*} = 106$ for the distorted lattice |
| 455 |
> |
(approximately 2/3 of the order-disorder transition temperature for |
| 456 |
> |
each lattice).} |
| 457 |
|
\end{figure} |
| 458 |
|
|
| 459 |
|
The ripples can be made to disappear by increasing the internal |
| 506 |
|
|
| 507 |
|
\begin{figure} |
| 508 |
|
\includegraphics[width=\linewidth]{./figures/mcSystemSize.pdf} |
| 509 |
< |
\caption{\label{mcfig:systemsize} The ripple wavelength (top) and |
| 509 |
> |
\caption[The ripple wavelength and amplitude as a function of system |
| 510 |
> |
size]{\label{mcfig:systemsize} The ripple wavelength (top) and |
| 511 |
|
amplitude (bottom) as a function of system size for a triangular |
| 512 |
|
lattice ($\gamma=1.732$) at $T^{*} = 122$.} |
| 513 |
|
\end{figure} |
