| 22 |
|
translational freedom along one coordinate (out of the plane of the |
| 23 |
|
membrane). There is an additional harmonic surface tension which |
| 24 |
|
binds each of the dipoles to the six nearest neighbors on either |
| 25 |
< |
hexagonal or distorted-hexagonal lattices. The translational freedom |
| 26 |
< |
of the dipoles allows hexagonal lattices to find states that break out |
| 25 |
> |
triangular or distorted lattices. The translational freedom |
| 26 |
> |
of the dipoles allows triangular lattices to find states that break out |
| 27 |
|
of the normal orientational disorder of frustrated configurations and |
| 28 |
|
which are stabilized by long-range antiferroelectric ordering. In |
| 29 |
|
order to break out of the frustrated states, the dipolar membranes |
| 30 |
|
form corrugated or ``rippled'' phases that make the lattices |
| 31 |
< |
effectively non-hexagonal. We observe three common features of the |
| 31 |
> |
effectively non-triangular. We observe three common features of the |
| 32 |
|
corrugated dipolar membranes: 1) the corrugated phases develop easily |
| 33 |
< |
when hosted on hexagonal lattices, 2) the wave vectors for the surface |
| 33 |
> |
when hosted on triangular lattices, 2) the wave vectors for the surface |
| 34 |
|
ripples are always found to be perpendicular to the dipole director |
| 35 |
< |
axis, and 3) on hexagonal lattices, the dipole director axis is found |
| 35 |
> |
axis, and 3) on triangular lattices, the dipole director axis is found |
| 36 |
|
to be parallel to any of the three equivalent lattice directions. |
| 37 |
|
\end{abstract} |
| 38 |
|
|
| 87 |
|
et al.} used atomic force microscopy (AFM) to observe ripple phase |
| 88 |
|
morphology in bilayers supported on mica.~\cite{Kaasgaard03} The |
| 89 |
|
experimental results provide strong support for a 2-dimensional |
| 90 |
< |
hexagonal packing lattice of the lipid molecules within the ripple |
| 90 |
> |
triangular packing lattice of the lipid molecules within the ripple |
| 91 |
|
phase. This is a notable change from the observed lipid packing |
| 92 |
|
within the gel phase.~\cite{Cevc87} |
| 93 |
|
|
| 131 |
|
Kubica has suggested that a lattice model of polar head groups could |
| 132 |
|
be valuable in trying to understand bilayer phase |
| 133 |
|
formation.~\cite{Kubica02} Bannerjee used Monte Carlo simulations of |
| 134 |
< |
lamellar stacks of hexagonal lattices to show that large headgroups |
| 134 |
> |
lamellar stacks of triangular lattices to show that large headgroups |
| 135 |
|
and molecular tilt with respect to the membrane normal vector can |
| 136 |
|
cause bulk rippling.~\cite{Bannerjee02} |
| 137 |
|
|
| 166 |
|
At the other extreme in density from the traditional simulations of |
| 167 |
|
dipolar fluids is the behavior of dipoles locked on regular lattices. |
| 168 |
|
Ferroelectric states (with long-range dipolar order) can be observed |
| 169 |
< |
in dipolar systems with non-hexagonal packings. However, {\em |
| 170 |
< |
hexagonally}-packed 2-D dipolar systems are inherently frustrated and |
| 169 |
> |
in dipolar systems with non-triangular packings. However, {\em |
| 170 |
> |
triangularly}-packed 2-D dipolar systems are inherently frustrated and |
| 171 |
|
one would expect a dipolar-disordered phase to be the lowest free |
| 172 |
|
energy configuration. Therefore, it would seem unlikely that a |
| 173 |
|
frustrated lattice in a dipolar-disordered state could exhibit the |
| 193 |
|
of a two-dimensional dipolar medium. Since molecules in the ripple |
| 194 |
|
phase have limited translational freedom, we have chosen a lattice to |
| 195 |
|
support the dipoles in the x-y plane. The lattice may be either |
| 196 |
< |
hexagonal (lattice constants $a/b = \sqrt{3}$) or non-hexagonal. |
| 196 |
> |
triangular (lattice constants $a/b = \sqrt{3}$) or distorted. |
| 197 |
|
However, each dipole has 3 degrees of freedom. They may move freely |
| 198 |
|
{\em out} of the x-y plane (along the $z$ axis), and they have |
| 199 |
|
complete orientational freedom ($0 <= \theta <= \pi$, $0 <= \phi < 2 |
| 231 |
|
|
| 232 |
|
To investigate the phase behavior of this model, we have performed a |
| 233 |
|
series of Metropolis Monte Carlo simulations of moderately-sized (34.3 |
| 234 |
< |
$\sigma$ on a side) patches of membrane hosted on both hexagonal |
| 235 |
< |
($\gamma = a/b = \sqrt{3}$) and non-hexagonal ($\gamma \neq \sqrt{3}$) |
| 234 |
> |
$\sigma$ on a side) patches of membrane hosted on both triangular |
| 235 |
> |
($\gamma = a/b = \sqrt{3}$) and distorted ($\gamma \neq \sqrt{3}$) |
| 236 |
|
lattices. The linear extent of one edge of the monolayer was $20 a$ |
| 237 |
|
and the system was kept roughly square. The average distance that |
| 238 |
|
coplanar dipoles were positioned from their six nearest neighbors was |
| 239 |
< |
1 $\sigma$ (on both hexagonal and non-hexagonal lattices). Typical |
| 240 |
< |
system sizes were 1360 dipoles for the hexagonal lattices and 840-2800 |
| 241 |
< |
dipoles for the non-hexagonal lattices. Periodic boundary conditions |
| 242 |
< |
were used, and the cutoff for the dipole-dipole interaction was set to |
| 243 |
< |
4.3 $\sigma$. All parameters ($T^{*}$, $\mu^{*}$, and $\gamma$) were |
| 244 |
< |
varied systematically to study the effects of these parameters on the |
| 245 |
< |
formation of ripple-like phases. |
| 239 |
> |
1 $\sigma$ (on both triangular and distorted lattices). Typical |
| 240 |
> |
system sizes were 1360 dipoles for the triangular lattices and |
| 241 |
> |
840-2800 dipoles for the distorted lattices. Two-dimensional periodic |
| 242 |
> |
boundary conditions were used, and the cutoff for the dipole-dipole |
| 243 |
> |
interaction was set to 4.3 $\sigma$. Since dipole-dipole interactions |
| 244 |
> |
decay rapidly with distance, and since the intrinsic three-dimensional |
| 245 |
> |
periodicity of the Ewald sum can give artifacts in 2-d systems, we |
| 246 |
> |
have chosen not to use it in these calculations. Although the Ewald |
| 247 |
> |
sum has been reformulated to handle 2-D |
| 248 |
> |
systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89} these methods |
| 249 |
> |
are computationally expensive,\cite{Spohr97,Yeh99} and are not |
| 250 |
> |
necessary in this case. All parameters ($T^{*}$, $\mu^{*}$, and |
| 251 |
> |
$\gamma$) were varied systematically to study the effects of these |
| 252 |
> |
parameters on the formation of ripple-like phases. |
| 253 |
|
|
| 254 |
|
\section{Results and Analysis} |
| 255 |
|
\label{sec:results} |
| 278 |
|
the director axis, which can be used to determine a privileged dipolar |
| 279 |
|
axis for dipole-ordered systems. The top panel in Fig. \ref{phase} |
| 280 |
|
shows the values of $P_2$ as a function of temperature for both |
| 281 |
< |
hexagonal ($\gamma = 1.732$) and non-hexagonal ($\gamma=1.875$) |
| 281 |
> |
triangular ($\gamma = 1.732$) and distorted ($\gamma=1.875$) |
| 282 |
|
lattices. |
| 283 |
|
|
| 284 |
|
\begin{figure}[ht] |
| 285 |
|
\centering |
| 286 |
|
\caption{Top panel: The $P_2$ dipolar order parameter as a function of |
| 287 |
< |
temperature for both hexagonal ($\gamma = 1.732$) and non-hexagonal |
| 287 |
> |
temperature for both triangular ($\gamma = 1.732$) and distorted |
| 288 |
|
($\gamma = 1.875$) lattices. Bottom Panel: The phase diagram for the |
| 289 |
|
dipolar membrane model. The line denotes the division between the |
| 290 |
|
dipolar ordered (antiferroelectric) and disordered phases. An |
| 291 |
< |
enlarged view near the hexagonal lattice is shown inset.} |
| 291 |
> |
enlarged view near the triangular lattice is shown inset.} |
| 292 |
|
\includegraphics[width=\linewidth]{phase.pdf} |
| 293 |
|
\label{phase} |
| 294 |
|
\end{figure} |
| 295 |
|
|
| 296 |
|
There is a clear order-disorder transition in evidence from this data. |
| 297 |
< |
Both the hexagonal and non-hexagonal lattices have dipolar-ordered |
| 297 |
> |
Both the triangular and distorted lattices have dipolar-ordered |
| 298 |
|
low-temperature phases, and orientationally-disordered high |
| 299 |
< |
temperature phases. The coexistence temperature for the hexagonal |
| 300 |
< |
lattice is significantly lower than for the non-hexagonal lattices, |
| 301 |
< |
and the bulk polarization is approximately $0$ for both dipolar |
| 302 |
< |
ordered and disordered phases. This gives strong evidence that the |
| 303 |
< |
dipolar ordered phase is antiferroelectric. We have repeated the |
| 304 |
< |
Monte Carlo simulations over a wide range of lattice ratios ($\gamma$) |
| 305 |
< |
to generate a dipolar order/disorder phase diagram. The bottom panel |
| 306 |
< |
in Fig. \ref{phase} shows that the hexagonal lattice is a |
| 307 |
< |
low-temperature cusp in the $T^{*}-\gamma$ phase diagram. |
| 299 |
> |
temperature phases. The coexistence temperature for the triangular |
| 300 |
> |
lattice is significantly lower than for the distorted lattices, and |
| 301 |
> |
the bulk polarization is approximately $0$ for both dipolar ordered |
| 302 |
> |
and disordered phases. This gives strong evidence that the dipolar |
| 303 |
> |
ordered phase is antiferroelectric. We have verified that this |
| 304 |
> |
dipolar ordering transition is not a function of system size by |
| 305 |
> |
performing identical calculations with systems twice as large. The |
| 306 |
> |
transition is equally smooth at all system sizes that were studied. |
| 307 |
> |
Additionally, we have repeated the Monte Carlo simulations over a wide |
| 308 |
> |
range of lattice ratios ($\gamma$) to generate a dipolar |
| 309 |
> |
order/disorder phase diagram. The bottom panel in Fig. \ref{phase} |
| 310 |
> |
shows that the triangular lattice is a low-temperature cusp in the |
| 311 |
> |
$T^{*}-\gamma$ phase diagram. |
| 312 |
|
|
| 313 |
|
This phase diagram is remarkable in that it shows an antiferroelectric |
| 314 |
|
phase near $\gamma=1.732$ where one would expect lattice frustration |
| 316 |
|
the configurations in this phase show clearly that the system has |
| 317 |
|
accomplished dipolar orderering by forming large ripple-like |
| 318 |
|
structures. We have observed antiferroelectric ordering in all three |
| 319 |
< |
of the equivalent directions on the hexagonal lattice, and the dipoles |
| 319 |
> |
of the equivalent directions on the triangular lattice, and the dipoles |
| 320 |
|
have been observed to organize perpendicular to the membrane normal |
| 321 |
|
(in the plane of the membrane). It is particularly interesting to |
| 322 |
|
note that the ripple-like structures have also been observed to |
| 328 |
|
\begin{figure}[ht] |
| 329 |
|
\centering |
| 330 |
|
\caption{Top and Side views of a representative configuration for the |
| 331 |
< |
dipolar ordered phase supported on the hexagonal lattice. Note the |
| 331 |
> |
dipolar ordered phase supported on the triangular lattice. Note the |
| 332 |
|
antiferroelectric ordering and the long wavelength buckling of the |
| 333 |
|
membrane. Dipolar ordering has been observed in all three equivalent |
| 334 |
< |
directions on the hexagonal lattice, and the ripple direction is |
| 334 |
> |
directions on the triangular lattice, and the ripple direction is |
| 335 |
|
always perpendicular to the director axis for the dipoles.} |
| 336 |
|
\includegraphics[width=5.5in]{snapshot.pdf} |
| 337 |
|
\label{fig:snapshot} |
| 338 |
|
\end{figure} |
| 339 |
|
|
| 340 |
+ |
Although the snapshot in Fig. \ref{fig:snapshot} gives the appearance |
| 341 |
+ |
of three-row stair-like structures, these appear to be transient. On |
| 342 |
+ |
average, the corrugation of the membrane is a relatively smooth, |
| 343 |
+ |
long-wavelength phenomenon, with occasional steep drops between |
| 344 |
+ |
adjacent lines of anti-aligned dipoles. |
| 345 |
+ |
|
| 346 |
+ |
The height-dipole correlation function ($C(r, \cos \theta)$) makes the |
| 347 |
+ |
connection between dipolar ordering and the wave vector of the ripple |
| 348 |
+ |
even more explicit. $C(r, \cos \theta)$ is an angle-dependent pair |
| 349 |
+ |
distribution function. The angle ($\theta$) is defined by the |
| 350 |
+ |
intermolecular vector $\vec{r}_{ij}$ and dipolar-axis of atom $i$, |
| 351 |
+ |
\begin{equation} |
| 352 |
+ |
C(r, \cos \theta) = \langle \sum_{i} |
| 353 |
+ |
\sum_{j} h_i \cdot h_j \delta(r - r_{ij}) \delta(\cos \theta_{ij} - \cos \theta)\rangle / \langle h^2 \rangle |
| 354 |
+ |
\end{equation} |
| 355 |
+ |
where $\cos \theta_{ij} = \hat{\mu}_{i} \cdot \hat{r}_{ij}$ and |
| 356 |
+ |
$\hat{r}_{ij} = \vec{r}_{ij} / r_{ij}$. Fig. \ref{fig:CrossCorrelation} |
| 357 |
+ |
shows contours of this correlation function for both anti-ferroelectric, rippled |
| 358 |
+ |
membranes as well as for the dipole-disordered portion of the phase diagram. |
| 359 |
+ |
|
| 360 |
+ |
\begin{figure}[ht] |
| 361 |
+ |
\centering |
| 362 |
+ |
\caption{Contours of the height-dipole correlation function as a function |
| 363 |
+ |
of the dot product between the dipole ($\hat{\mu}$) and inter-dipole |
| 364 |
+ |
separation vector ($\hat{r}$) and the distance ($r$) between the dipoles. |
| 365 |
+ |
Perfect height correlation (contours approaching 1) are present in the |
| 366 |
+ |
ordered phase when the two dipoles are in the same head-to-tail line. |
| 367 |
+ |
Anti-correlation (contours below 0) is only seen when the inter-dipole |
| 368 |
+ |
vector is perpendicular to the dipoles. In the dipole-disordered portion |
| 369 |
+ |
of the phase diagram, there is only weak correlation in the dipole direction |
| 370 |
+ |
and this correlation decays rapidly to zero for intermolecular vectors that are |
| 371 |
+ |
not dipole-aligned.} |
| 372 |
+ |
\includegraphics[width=\linewidth]{height-dipole-correlation.pdf} |
| 373 |
+ |
\label{fig:CrossCorrelation} |
| 374 |
+ |
\end{figure} |
| 375 |
+ |
|
| 376 |
|
\subsection{Discriminating Ripples from Thermal Undulations} |
| 377 |
|
|
| 378 |
|
In order to be sure that the structures we have observed are actually |
| 383 |
|
h(\vec{r}) e^{-i \vec{q}\cdot\vec{r}} |
| 384 |
|
\end{equation} |
| 385 |
|
where $h(\vec{r})$ is the height of the membrane at location $\vec{r} |
| 386 |
< |
= (x,y)$.~\cite{Safran94} In simple (and more complicated) elastic |
| 387 |
< |
continuum models, Brannigan {\it et al.} have shown that in the $NVT$ |
| 388 |
< |
ensemble, the absolute value of the undulation spectrum can be |
| 342 |
< |
written, |
| 386 |
> |
= (x,y)$.~\cite{Safran94,Seifert97} In simple (and more complicated) |
| 387 |
> |
elastic continuum models, it can shown that in the $NVT$ ensemble, the |
| 388 |
> |
absolute value of the undulation spectrum can be written, |
| 389 |
|
\begin{equation} |
| 390 |
|
\langle | h(q)|^2 \rangle_{NVT} = \frac{k_B T}{k_c |\vec{q}|^4 + |
| 391 |
|
\tilde{\gamma}|\vec{q}|^2}, |
| 392 |
|
\label{eq:fit} |
| 393 |
|
\end{equation} |
| 394 |
|
where $k_c$ is the bending modulus for the membrane, and |
| 395 |
< |
$\tilde{\gamma}$ is the mechanical surface |
| 396 |
< |
tension.~\cite{Brannigan04b} |
| 395 |
> |
$\tilde{\gamma}$ is the mechanical surface tension.~\cite{Safran94} |
| 396 |
> |
The systems studied in this paper have essentially zero bending moduli |
| 397 |
> |
($k_c$) and relatively large mechanical surface tensions |
| 398 |
> |
($\tilde{\gamma}$), so a much simpler form can be written, |
| 399 |
> |
\begin{equation} |
| 400 |
> |
\langle | h(q)|^2 \rangle_{NVT} = \frac{k_B T}{\tilde{\gamma}|\vec{q}|^2}, |
| 401 |
> |
\label{eq:fit2} |
| 402 |
> |
\end{equation} |
| 403 |
|
|
| 404 |
|
The undulation spectrum is computed by superimposing a rectangular |
| 405 |
|
grid on top of the membrane, and by assigning height ($h(\vec{r})$) |
| 407 |
|
given $\vec{r}+d\vec{r}$ grid area. Empty grid pixels are assigned |
| 408 |
|
height values by interpolation from the nearest neighbor pixels. A |
| 409 |
|
standard 2-d Fourier transform is then used to obtain $\langle | |
| 410 |
< |
h(q)|^2 \rangle$. |
| 410 |
> |
h(q)|^2 \rangle$. Alternatively, since the dipoles sit on a Bravais |
| 411 |
> |
lattice, one could use the heights of the lattice points themselves as |
| 412 |
> |
the grid for the Fourier transform (without interpolating to a square |
| 413 |
> |
grid). However, if lateral translational freedom is added to this |
| 414 |
> |
model, an interpolated method for computing undulation spectra will be |
| 415 |
> |
required. |
| 416 |
|
|
| 417 |
< |
The systems studied in this paper have relatively small bending moduli |
| 418 |
< |
($k_c$) and relatively large mechanical surface tensions |
| 362 |
< |
($\tilde{\gamma}$). In practice, the best fits to our undulation |
| 363 |
< |
spectra are obtained by approximating the value of $k_c$ to 0. In |
| 417 |
> |
As mentioned above, the best fits to our undulation spectra are |
| 418 |
> |
obtained by approximating the value of $k_c$ to 0. In |
| 419 |
|
Fig. \ref{fig:fit} we show typical undulation spectra for two |
| 420 |
|
different regions of the phase diagram along with their fits from the |
| 421 |
< |
Landau free energy approach (Eq. \ref{eq:fit}). In the |
| 421 |
> |
Landau free energy approach (Eq. \ref{eq:fit2}). In the |
| 422 |
|
high-temperature disordered phase, the Landau fits can be nearly |
| 423 |
< |
perfect, and from these fits we can estimate the bending modulus and |
| 424 |
< |
the mechanical surface tension. |
| 423 |
> |
perfect, and from these fits we can estimate the tension in the |
| 424 |
> |
surface. |
| 425 |
|
|
| 426 |
< |
For the dipolar-ordered hexagonal lattice near the coexistence |
| 426 |
> |
For the dipolar-ordered triangular lattice near the coexistence |
| 427 |
|
temperature, however, we observe long wavelength undulations that are |
| 428 |
|
far outliers to the fits. That is, the Landau free energy fits are |
| 429 |
< |
well within error bars for all other points, but can be off by {\em |
| 430 |
< |
orders of magnitude} for a few low frequency components. |
| 429 |
> |
well within error bars for most of the other points, but can be off by |
| 430 |
> |
{\em orders of magnitude} for a few low frequency components. |
| 431 |
|
|
| 432 |
|
We interpret these outliers as evidence that these low frequency modes |
| 433 |
|
are {\em non-thermal undulations}. We take this as evidence that we |
| 441 |
|
\rangle$). Rippled samples show low-wavelength peaks that are |
| 442 |
|
outliers on the Landau free energy fits. Samples exhibiting only |
| 443 |
|
thermal undulations fit Eq. \ref{eq:fit} remarkably well.} |
| 444 |
< |
\includegraphics[width=5.5in]{fit.pdf} |
| 444 |
> |
\includegraphics[width=5.5in]{logFit.pdf} |
| 445 |
|
\label{fig:fit} |
| 446 |
|
\end{figure} |
| 447 |
|
|
| 475 |
|
estimated by measuring peak-to-trough distances in |
| 476 |
|
$h(q_{\mathrm{rip}})$ itself. |
| 477 |
|
|
| 423 |
– |
\begin{figure}[ht] |
| 424 |
– |
\centering |
| 425 |
– |
\caption{Contours of the height-dipole correlation function as a function |
| 426 |
– |
of the dot product between the dipole ($\hat{\mu}$) and inter-dipole |
| 427 |
– |
separation vector ($\hat{r}$) and the distance ($r$) between the dipoles. |
| 428 |
– |
Perfect height correlation (contours approaching 1) are present in the |
| 429 |
– |
ordered phase when the two dipoles are in the same head-to-tail line. |
| 430 |
– |
Anti-correlation (contours below 0) is only seen when the inter-dipole |
| 431 |
– |
vector is perpendicular to the dipoles. } |
| 432 |
– |
\includegraphics[width=\linewidth]{height-dipole-correlation.pdf} |
| 433 |
– |
\label{fig:CrossCorrelation} |
| 434 |
– |
\end{figure} |
| 435 |
– |
|
| 478 |
|
A second, more accurate, and simpler method for estimating ripple |
| 479 |
|
shape is to extract the wavelength and height information directly |
| 480 |
|
from the largest non-thermal peak in the undulation spectrum. For |
| 481 |
|
large-amplitude ripples, the two methods give similar results. The |
| 482 |
|
one-dimensional projection method is more prone to noise (particularly |
| 483 |
< |
in the amplitude estimates for the non-hexagonal lattices). We report |
| 483 |
> |
in the amplitude estimates for the distorted lattices). We report |
| 484 |
|
amplitudes and wavelengths taken directly from the undulation spectrum |
| 485 |
|
below. |
| 486 |
|
|
| 487 |
< |
In the hexagonal lattice ($\gamma = \sqrt{3}$), the rippling is |
| 487 |
> |
In the triangular lattice ($\gamma = \sqrt{3}$), the rippling is |
| 488 |
|
observed for temperatures ($T^{*}$) from $61-122$. The wavelength of |
| 489 |
|
the ripples is remarkably stable at 21.4~$\sigma$ for all but the |
| 490 |
|
temperatures closest to the order-disorder transition. At $T^{*} = |
| 502 |
|
\begin{figure}[ht] |
| 503 |
|
\centering |
| 504 |
|
\caption{a) The amplitude $A^{*}$ of the ripples vs. temperature for a |
| 505 |
< |
hexagonal lattice. b) The amplitude $A^{*}$ of the ripples vs. dipole |
| 506 |
< |
strength ($\mu^{*}$) for both the hexagonal lattice (circles) and |
| 507 |
< |
non-hexagonal lattice (squares). The reduced temperatures were kept |
| 508 |
< |
fixed at $T^{*} = 94$ for the hexagonal lattice and $T^{*} = 106$ for |
| 509 |
< |
the non-hexagonal lattice (approximately 2/3 of the order-disorder |
| 505 |
> |
triangular lattice. b) The amplitude $A^{*}$ of the ripples vs. dipole |
| 506 |
> |
strength ($\mu^{*}$) for both the triangular lattice (circles) and |
| 507 |
> |
distorted lattice (squares). The reduced temperatures were kept |
| 508 |
> |
fixed at $T^{*} = 94$ for the triangular lattice and $T^{*} = 106$ for |
| 509 |
> |
the distorted lattice (approximately 2/3 of the order-disorder |
| 510 |
|
transition temperature for each lattice).} |
| 511 |
|
\includegraphics[width=\linewidth]{properties_sq.pdf} |
| 512 |
|
\label{fig:Amplitude} |
| 523 |
|
of ripple amplitude on the dipolar strength in |
| 524 |
|
Fig. \ref{fig:Amplitude}. |
| 525 |
|
|
| 526 |
< |
\subsection{Non-hexagonal lattices} |
| 526 |
> |
\subsection{Distorted lattices} |
| 527 |
|
|
| 528 |
|
We have also investigated the effect of the lattice geometry by |
| 529 |
|
changing the ratio of lattice constants ($\gamma$) while keeping the |
| 530 |
|
average nearest-neighbor spacing constant. The antiferroelectric state |
| 531 |
|
is accessible for all $\gamma$ values we have used, although the |
| 532 |
< |
distorted hexagonal lattices prefer a particular director axis due to |
| 532 |
> |
distorted triangular lattices prefer a particular director axis due to |
| 533 |
|
the anisotropy of the lattice. |
| 534 |
|
|
| 535 |
< |
Our observation of rippling behavior was not limited to the hexagonal |
| 536 |
< |
lattices. In non-hexagonal lattices the antiferroelectric phase can |
| 535 |
> |
Our observation of rippling behavior was not limited to the triangular |
| 536 |
> |
lattices. In distorted lattices the antiferroelectric phase can |
| 537 |
|
develop nearly instantaneously in the Monte Carlo simulations, and |
| 538 |
|
these dipolar-ordered phases tend to be remarkably flat. Whenever |
| 539 |
< |
rippling has been observed in these non-hexagonal lattices |
| 539 |
> |
rippling has been observed in these distorted lattices |
| 540 |
|
(e.g. $\gamma = 1.875$), we see relatively short ripple wavelengths |
| 541 |
|
(14 $\sigma$) and amplitudes of 2.4~$\sigma$. These ripples are |
| 542 |
|
weakly dependent on dipolar strength (see Fig. \ref{fig:Amplitude}), |
| 548 |
|
\gamma < 1.875$. Outside this range, the order-disorder transition in |
| 549 |
|
the dipoles remains, but the ordered dipolar phase has only thermal |
| 550 |
|
undulations. This is one of our strongest pieces of evidence that |
| 551 |
< |
rippling is a symmetry-breaking phenomenon for hexagonal and |
| 552 |
< |
nearly-hexagonal lattices. |
| 551 |
> |
rippling is a symmetry-breaking phenomenon for triangular and |
| 552 |
> |
nearly-triangular lattices. |
| 553 |
|
|
| 554 |
|
\subsection{Effects of System Size} |
| 555 |
|
To evaluate the effect of finite system size, we have performed a |
| 556 |
< |
series of simulations on the hexagonal lattice at a reduced |
| 556 |
> |
series of simulations on the triangular lattice at a reduced |
| 557 |
|
temperature of 122, which is just below the order-disorder transition |
| 558 |
|
temperature ($T^{*} = 139$). These conditions are in the |
| 559 |
|
dipole-ordered and rippled portion of the phase diagram. These are |
| 563 |
|
\begin{figure}[ht] |
| 564 |
|
\centering |
| 565 |
|
\caption{The ripple wavelength (top) and amplitude (bottom) as a |
| 566 |
< |
function of system size for a hexagonal lattice ($\gamma=1.732$) at $T^{*} = |
| 566 |
> |
function of system size for a triangular lattice ($\gamma=1.732$) at $T^{*} = |
| 567 |
|
122$.} |
| 568 |
|
\includegraphics[width=\linewidth]{SystemSize.pdf} |
| 569 |
|
\label{fig:systemsize} |
| 596 |
|
stable long-wavelength non-thermal surface corrugations. The best |
| 597 |
|
explanation for this behavior is that the ability of the dipoles to |
| 598 |
|
translate out of the plane of the membrane is enough to break the |
| 599 |
< |
symmetry of the hexagonal lattice and allow the energetic benefit from |
| 599 |
> |
symmetry of the triangular lattice and allow the energetic benefit from |
| 600 |
|
the formation of a bulk antiferroelectric phase. Were the weak |
| 601 |
|
surface tension absent from our model, it would be possible for the |
| 602 |
|
entire lattice to ``tilt'' using $z$-translation. Tilting the lattice |
| 603 |
< |
in this way would yield an effectively non-hexagonal lattice which |
| 603 |
> |
in this way would yield an effectively non-triangular lattice which |
| 604 |
|
would avoid dipolar frustration altogether. With the surface tension |
| 605 |
|
in place, bulk tilt causes a large strain, and the simplest way to |
| 606 |
|
release this strain is along line defects. Line defects will result |
| 616 |
|
relative to the surface tension can cause the corrugated phase to |
| 617 |
|
disappear. |
| 618 |
|
|
| 619 |
< |
The packing of the dipoles into a nearly-hexagonal lattice is clearly |
| 619 |
> |
The packing of the dipoles into a nearly-triangular lattice is clearly |
| 620 |
|
an important piece of the puzzle. The dipolar head groups of lipid |
| 621 |
|
molecules are sterically (as well as electrostatically) anisotropic, |
| 622 |
< |
and would not be able to pack hexagonally without the steric |
| 623 |
< |
interference of adjacent molecular bodies. Since we only see rippled |
| 624 |
< |
phases in the neighborhood of $\gamma=\sqrt{3}$, this implies that |
| 625 |
< |
there is a role played by the lipid chains in the organization of the |
| 626 |
< |
hexagonally ordered phases which support ripples in realistic lipid |
| 627 |
< |
bilayers. |
| 622 |
> |
and would not be able to pack in triangular arrangements without the |
| 623 |
> |
steric interference of adjacent molecular bodies. Since we only see |
| 624 |
> |
rippled phases in the neighborhood of $\gamma=\sqrt{3}$, this implies |
| 625 |
> |
that there is a role played by the lipid chains in the organization of |
| 626 |
> |
the triangularly ordered phases which support ripples in realistic |
| 627 |
> |
lipid bilayers. |
| 628 |
|
|
| 629 |
|
The most important prediction we can make using the results from this |
| 630 |
|
simple model is that if dipolar ordering is driving the surface |
| 637 |
|
|
| 638 |
|
Our other observation about the ripple and dipolar directionality is |
| 639 |
|
that the dipole director axis can be found to be parallel to any of |
| 640 |
< |
the three equivalent lattice vectors in the hexagonal lattice. |
| 640 |
> |
the three equivalent lattice vectors in the triangular lattice. |
| 641 |
|
Defects in the ordering of the dipoles can cause the dipole director |
| 642 |
|
(and consequently the surface corrugation) of small regions to be |
| 643 |
|
rotated relative to each other by 120$^{\circ}$. This is a similar |
