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\begin{document} |
26 |
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distorted lattices. The translational freedom of the dipoles allows |
27 |
|
triangular lattices to find states that break out of the normal |
28 |
|
orientational disorder of frustrated configurations and which are |
29 |
< |
stabilized by long-range antiferroelectric ordering. In order to |
29 |
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stabilized by long-range anti-ferroelectric ordering. In order to |
30 |
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break out of the frustrated states, the dipolar membranes form |
31 |
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corrugated or ``rippled'' phases that make the lattices effectively |
32 |
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non-triangular. We observe three common features of the corrugated |
64 |
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|
65 |
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Buckling behavior in liquid crystalline and biological membranes is a |
66 |
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well-known phenomenon. Relatively pure phosphatidylcholine (PC) |
67 |
< |
bilayers are known to form a corrugated or ``rippled'' phase |
68 |
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($P_{\beta'}$) which appears as an intermediate phase between the gel |
69 |
< |
($L_\beta$) and fluid ($L_{\alpha}$) phases. The $P_{\beta'}$ phase |
70 |
< |
has attracted substantial experimental interest over the past 30 |
71 |
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years. Most structural information of the ripple phase has been |
72 |
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obtained by the X-ray diffraction~\cite{Sun96,Katsaras00} and |
73 |
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freeze-fracture electron microscopy (FFEM).~\cite{Copeland80,Meyer96} |
74 |
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Recently, Kaasgaard {\it et al.} used atomic force microscopy (AFM) to |
75 |
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observe ripple phase morphology in bilayers supported on |
76 |
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mica.~\cite{Kaasgaard03} The experimental results provide strong |
77 |
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support for a 2-dimensional triangular packing lattice of the lipid |
78 |
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molecules within the ripple phase. This is a notable change from the |
79 |
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observed lipid packing within the gel phase.~\cite{Cevc87} There have |
80 |
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been a number of theoretical |
67 |
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bilayers form a corrugated or ``rippled'' phase ($P_{\beta'}$) which |
68 |
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appears as an intermediate phase between the gel ($L_\beta$) and fluid |
69 |
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($L_{\alpha}$) phases. The $P_{\beta'}$ phase has attracted |
70 |
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substantial experimental interest over the past 30 years. Most |
71 |
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structural information of the ripple phase has been obtained by the |
72 |
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X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron |
73 |
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microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it |
74 |
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et al.} used atomic force microscopy (AFM) to observe ripple phase |
75 |
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morphology in bilayers supported on mica.~\cite{Kaasgaard03} The |
76 |
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experimental results provide strong support for a 2-dimensional |
77 |
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triangular packing lattice of the lipid molecules within the ripple |
78 |
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phase. This is a notable change from the observed lipid packing |
79 |
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within the gel phase.~\cite{Cevc87} There have been a number of |
80 |
> |
theoretical |
81 |
|
approaches~\cite{Marder84,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02} |
82 |
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(and some heroic |
83 |
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simulations~\cite{Ayton02,Jiang04,Brannigan04a,deVries05,deJoannis06}) |
90 |
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detailed enough to rule in favor of (or against) any of these |
91 |
|
explanations for the $P_{\beta'}$ phase. |
92 |
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|
93 |
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Another interesting properties of elastic membranes containing |
94 |
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electrostatic dipoles is the phenomenon of flexoelectricity,\cite{} |
95 |
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which is the ability of mechanical deformations of the membrane to |
96 |
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result in electrostatic organization of the membrane. This phenomenon |
97 |
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is a curvature-induced membrane polarization which can lead to |
98 |
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potential differences across a membrane. Reverse flexoelectric |
99 |
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behavior (in which applied alternating currents affect membrane |
100 |
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curvature) has also been observed. Explanations of the details of |
101 |
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these effects have typically utilized membrane polarization parallel |
102 |
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to the membrane normal.\cite{} |
93 |
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Membranes containing electrostatic dipoles can also exhibit the |
94 |
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flexoelectric effect,\cite{Todorova2004,Harden2006,Petrov2006} which |
95 |
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is the ability of mechanical deformations to result in electrostatic |
96 |
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organization of the membrane. This phenomenon is a curvature-induced |
97 |
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membrane polarization which can lead to potential differences across a |
98 |
> |
membrane. Reverse flexoelectric behavior (in which applied currents |
99 |
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effect membrane curvature) has also been observed. Explanations of |
100 |
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the details of these effects have typically utilized membrane |
101 |
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polarization perpendicular to the face of the |
102 |
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membrane,\cite{Petrov2006} and the effect has been observed in both |
103 |
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biological,\cite{Raphael2000} bent-core liquid |
104 |
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crystalline,\cite{Harden2006} and polymer-dispersed liquid crystalline |
105 |
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membranes.\cite{Todorova2004} |
106 |
|
|
107 |
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The problem with using atomistic and even coarse-grained approaches to |
108 |
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study membrane buckling phenomena is that only a relatively small |
143 |
|
The point of developing this model was to arrive at the simplest |
144 |
|
possible theoretical model which could exhibit spontaneous corrugation |
145 |
|
of a two-dimensional dipolar medium. Since molecules in polymerized |
146 |
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membranes and in in the $P_{\beta'}$ ripple phase have limited |
146 |
> |
membranes and in the $P_{\beta'}$ ripple phase have limited |
147 |
|
translational freedom, we have chosen a lattice to support the dipoles |
148 |
|
in the x-y plane. The lattice may be either triangular (lattice |
149 |
|
constants $a/b = |
166 |
|
\label{eq:pot} |
167 |
|
\end{eqnarray} |
168 |
|
|
166 |
– |
|
169 |
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In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing |
170 |
|
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
171 |
|
pointing along the inter-dipole vector $\mathbf{r}_{ij}$. The entire |
182 |
|
reduced distances, ($r^{*} = r / \sigma$), temperatures ($T^{*} = 2 |
183 |
|
k_B T / k_r \sigma^2$), densities ($\rho^{*} = N \sigma^2 / L_x L_y$), |
184 |
|
and dipole moments ($\mu^{*} = \mu / \sqrt{4 \pi \epsilon_0 \sigma^5 |
185 |
< |
k_r / 2}$). |
185 |
> |
k_r / 2}$). It should be noted that the density ($\rho^{*}$) depends |
186 |
> |
only on the mean particle spacing in the $x-y$ plane; the lattice is |
187 |
> |
fully populated. |
188 |
|
|
189 |
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To investigate the phase behavior of this model, we have performed a |
190 |
|
series of Metropolis Monte Carlo simulations of moderately-sized (34.3 |
197 |
|
system sizes were 1360 dipoles for the triangular lattices and |
198 |
|
840-2800 dipoles for the distorted lattices. Two-dimensional periodic |
199 |
|
boundary conditions were used, and the cutoff for the dipole-dipole |
200 |
< |
interaction was set to 4.3 $\sigma$. Since dipole-dipole interactions |
201 |
< |
decay rapidly with distance, and since the intrinsic three-dimensional |
202 |
< |
periodicity of the Ewald sum can give artifacts in 2-d systems, we |
203 |
< |
have chosen not to use it in these calculations. Although the Ewald |
204 |
< |
sum has been reformulated to handle 2-D |
205 |
< |
systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89} these methods |
206 |
< |
are computationally expensive,\cite{Spohr97,Yeh99} and are not |
200 |
> |
interaction was set to 4.3 $\sigma$. This cutoff is roughly 2.5 times |
201 |
> |
the typical real-space electrostatic cutoff for molecular systems. |
202 |
> |
Since dipole-dipole interactions decay rapidly with distance, and |
203 |
> |
since the intrinsic three-dimensional periodicity of the Ewald sum can |
204 |
> |
give artifacts in 2-d systems, we have chosen not to use it in these |
205 |
> |
calculations. Although the Ewald sum has been reformulated to handle |
206 |
> |
2-D systems,\cite{Parry75,Parry76,Heyes77,deLeeuw79,Rhee89} these |
207 |
> |
methods are computationally expensive,\cite{Spohr97,Yeh99} and are not |
208 |
|
necessary in this case. All parameters ($T^{*}$, $\mu^{*}$, and |
209 |
|
$\gamma$) were varied systematically to study the effects of these |
210 |
|
parameters on the formation of ripple-like phases. |
230 |
|
for dipole $i$. $P_2$ will be $1.0$ for a perfectly-ordered system |
231 |
|
and near $0$ for a randomized system. Note that this order parameter |
232 |
|
is {\em not} equal to the polarization of the system. For example, |
233 |
< |
the polarization of the perfect antiferroelectric system is $0$, but |
234 |
< |
$P_2$ for an antiferroelectric system is $1$. The eigenvector of |
233 |
> |
the polarization of the perfect anti-ferroelectric system is $0$, but |
234 |
> |
$P_2$ for an anti-ferroelectric system is $1$. The eigenvector of |
235 |
|
$\mathsf{S}$ corresponding to the largest eigenvalue is familiar as |
236 |
|
the director axis, which can be used to determine a privileged dipolar |
237 |
|
axis for dipole-ordered systems. The top panel in Fig. \ref{phase} |
245 |
|
a function of temperature for both triangular ($\gamma = 1.732$) and |
246 |
|
distorted ($\gamma = 1.875$) lattices. Bottom Panel: The phase |
247 |
|
diagram for the dipolar membrane model. The line denotes the division |
248 |
< |
between the dipolar ordered (antiferroelectric) and disordered phases. |
248 |
> |
between the dipolar ordered (anti-ferroelectric) and disordered phases. |
249 |
|
An enlarged view near the triangular lattice is shown inset.} |
250 |
|
\end{figure} |
251 |
|
|
256 |
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lattice is significantly lower than for the distorted lattices, and |
257 |
|
the bulk polarization is approximately $0$ for both dipolar ordered |
258 |
|
and disordered phases. This gives strong evidence that the dipolar |
259 |
< |
ordered phase is antiferroelectric. We have verified that this |
259 |
> |
ordered phase is anti-ferroelectric. We have verified that this |
260 |
|
dipolar ordering transition is not a function of system size by |
261 |
|
performing identical calculations with systems twice as large. The |
262 |
|
transition is equally smooth at all system sizes that were studied. |
266 |
|
shows that the triangular lattice is a low-temperature cusp in the |
267 |
|
$T^{*}-\gamma$ phase diagram. |
268 |
|
|
269 |
< |
This phase diagram is remarkable in that it shows an antiferroelectric |
270 |
< |
phase near $\gamma=1.732$ where one would expect lattice frustration |
271 |
< |
to result in disordered phases at all temperatures. Observations of |
272 |
< |
the configurations in this phase show clearly that the system has |
273 |
< |
accomplished dipolar orderering by forming large ripple-like |
274 |
< |
structures. We have observed antiferroelectric ordering in all three |
275 |
< |
of the equivalent directions on the triangular lattice, and the dipoles |
276 |
< |
have been observed to organize perpendicular to the membrane normal |
277 |
< |
(in the plane of the membrane). It is particularly interesting to |
278 |
< |
note that the ripple-like structures have also been observed to |
279 |
< |
propagate in the three equivalent directions on the lattice, but the |
280 |
< |
{\em direction of ripple propagation is always perpendicular to the |
281 |
< |
dipole director axis}. A snapshot of a typical antiferroelectric |
282 |
< |
rippled structure is shown in Fig. \ref{fig:snapshot}. |
269 |
> |
This phase diagram is remarkable in that it shows an |
270 |
> |
anti-ferroelectric phase near $\gamma=1.732$ where one would expect |
271 |
> |
lattice frustration to result in disordered phases at all |
272 |
> |
temperatures. Observations of the configurations in this phase show |
273 |
> |
clearly that the system has accomplished dipolar ordering by forming |
274 |
> |
large ripple-like structures. We have observed anti-ferroelectric |
275 |
> |
ordering in all three of the equivalent directions on the triangular |
276 |
> |
lattice, and the dipoles have been observed to organize perpendicular |
277 |
> |
to the membrane normal (in the plane of the membrane). It is |
278 |
> |
particularly interesting to note that the ripple-like structures have |
279 |
> |
also been observed to propagate in the three equivalent directions on |
280 |
> |
the lattice, but the {\em direction of ripple propagation is always |
281 |
> |
perpendicular to the dipole director axis}. A snapshot of a typical |
282 |
> |
anti-ferroelectric rippled structure is shown in |
283 |
> |
Fig. \ref{fig:snapshot}. |
284 |
|
|
285 |
|
\begin{figure} |
286 |
|
\includegraphics[width=\linewidth]{snapshot} |
287 |
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\caption{\label{fig:snapshot} Top and Side views of a representative |
288 |
|
configuration for the dipolar ordered phase supported on the |
289 |
< |
triangular lattice. Note the antiferroelectric ordering and the long |
289 |
> |
triangular lattice. Note the anti-ferroelectric ordering and the long |
290 |
|
wavelength buckling of the membrane. Dipolar ordering has been |
291 |
|
observed in all three equivalent directions on the triangular lattice, |
292 |
|
and the ripple direction is always perpendicular to the director axis |
299 |
|
long-wavelength phenomenon, with occasional steep drops between |
300 |
|
adjacent lines of anti-aligned dipoles. |
301 |
|
|
302 |
< |
The height-dipole correlation function ($C(r, \cos \theta)$) makes the |
303 |
< |
connection between dipolar ordering and the wave vector of the ripple |
304 |
< |
even more explicit. $C(r, \cos \theta)$ is an angle-dependent pair |
305 |
< |
distribution function. The angle ($\theta$) is defined by the |
306 |
< |
intermolecular vector $\vec{r}_{ij}$ and direction of dipole $i$, |
302 |
> |
The height-dipole correlation function ($C_{\textrm{hd}}(r, \cos |
303 |
> |
\theta)$) makes the connection between dipolar ordering and the wave |
304 |
> |
vector of the ripple even more explicit. $C_{\textrm{hd}}(r, \cos |
305 |
> |
\theta)$ is an angle-dependent pair distribution function. The angle |
306 |
> |
($\theta$) is the angle between the intermolecular vector |
307 |
> |
$\vec{r}_{ij}$ and direction of dipole $i$, |
308 |
|
\begin{equation} |
309 |
< |
C(r, \cos \theta) = \frac{\langle \sum_{i} |
310 |
< |
\sum_{j} h_i \cdot h_j \delta(r - r_{ij}) \delta(\cos \theta_{ij} - |
309 |
> |
C_{\textrm{hd}} = \frac{\langle \frac{1}{n(r)} \sum_{i}\sum_{j>i} |
310 |
> |
h_i \cdot h_j \delta(r - r_{ij}) \delta(\cos \theta_{ij} - |
311 |
|
\cos \theta)\rangle} {\langle h^2 \rangle} |
312 |
|
\end{equation} |
313 |
|
where $\cos \theta_{ij} = \hat{\mu}_{i} \cdot \hat{r}_{ij}$ and |
314 |
< |
$\hat{r}_{ij} = \vec{r}_{ij} / r_{ij}$. Fig. \ref{fig:CrossCorrelation} |
315 |
< |
shows contours of this correlation function for both anti-ferroelectric, rippled |
316 |
< |
membranes as well as for the dipole-disordered portion of the phase diagram. |
314 |
> |
$\hat{r}_{ij} = \vec{r}_{ij} / r_{ij}$. $n(r)$ is the number of |
315 |
> |
dipoles found in a cylindrical shell between $r$ and $r+\delta r$ of |
316 |
> |
the central particle. Fig. \ref{fig:CrossCorrelation} shows contours |
317 |
> |
of this correlation function for both anti-ferroelectric, rippled |
318 |
> |
membranes as well as for the dipole-disordered portion of the phase |
319 |
> |
diagram. |
320 |
|
|
321 |
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\begin{figure} |
322 |
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\includegraphics[width=\linewidth]{hdc} |
333 |
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intermolecular vectors that are not dipole-aligned.} |
334 |
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\end{figure} |
335 |
|
|
336 |
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The height-dipole correlation function gives a map of how the topology |
337 |
+ |
of the membrane surface varies with angular deviation around a given |
338 |
+ |
dipole. The upper panel of Fig. \ref{fig:CrossCorrelation} shows that |
339 |
+ |
in the anti-ferroelectric phase, the dipole heights are strongly |
340 |
+ |
correlated for dipoles in head-to-tail arrangements, and this |
341 |
+ |
correlation persists for very long distances (up to 15 $\sigma$). For |
342 |
+ |
portions of the membrane located perpendicular to a given dipole, the |
343 |
+ |
membrane height becomes anti-correlated at distances of 10 $\sigma$. |
344 |
+ |
The correlation function is relatively smooth; there are no steep |
345 |
+ |
jumps or steps, so the stair-like structures in |
346 |
+ |
Fig. \ref{fig:snapshot} are indeed transient and disappear when |
347 |
+ |
averaged over many configurations. In the dipole-disordered phase, |
348 |
+ |
the height-dipole correlation function is relatively flat (and hovers |
349 |
+ |
near zero). The only significant height correlations are for axial |
350 |
+ |
dipoles at very short distances ($r \approx |
351 |
+ |
\sigma$). |
352 |
+ |
|
353 |
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\subsection{Discriminating Ripples from Thermal Undulations} |
354 |
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|
355 |
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In order to be sure that the structures we have observed are actually |
418 |
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\begin{figure} |
419 |
|
\includegraphics[width=\linewidth]{logFit} |
420 |
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\caption{\label{fig:fit} Evidence that the observed ripples are {\em |
421 |
< |
not} thermal undulations is obtained from the 2-d fourier transform |
421 |
> |
not} thermal undulations is obtained from the 2-d Fourier transform |
422 |
|
$\langle |h^{*}(\vec{q})|^2 \rangle$ of the height profile ($\langle |
423 |
|
h^{*}(x,y) \rangle$). Rippled samples show low-wavelength peaks that |
424 |
|
are outliers on the Landau free energy fits by an order of magnitude. |
452 |
|
axis by projecting heights of the dipoles to obtain a one-dimensional |
453 |
|
height profile, $h(q_{\mathrm{rip}})$. Ripple wavelengths can be |
454 |
|
estimated from the largest non-thermal low-frequency component in the |
455 |
< |
fourier transform of $h(q_{\mathrm{rip}})$. Amplitudes can be |
455 |
> |
Fourier transform of $h(q_{\mathrm{rip}})$. Amplitudes can be |
456 |
|
estimated by measuring peak-to-trough distances in |
457 |
|
$h(q_{\mathrm{rip}})$ itself. |
458 |
|
|
492 |
|
\end{figure} |
493 |
|
|
494 |
|
The ripples can be made to disappear by increasing the internal |
495 |
< |
surface tension (i.e. by increasing $k_r$ or equivalently, reducing |
495 |
> |
elastic tension (i.e. by increasing $k_r$ or equivalently, reducing |
496 |
|
the dipole moment). The amplitude of the ripples depends critically |
497 |
|
on the strength of the dipole moments ($\mu^{*}$) in Eq. \ref{eq:pot}. |
498 |
|
If the dipoles are weakened substantially (below $\mu^{*}$ = 20) at a |
506 |
|
|
507 |
|
We have also investigated the effect of the lattice geometry by |
508 |
|
changing the ratio of lattice constants ($\gamma$) while keeping the |
509 |
< |
average nearest-neighbor spacing constant. The antiferroelectric state |
509 |
> |
average nearest-neighbor spacing constant. The anti-ferroelectric state |
510 |
|
is accessible for all $\gamma$ values we have used, although the |
511 |
|
distorted triangular lattices prefer a particular director axis due to |
512 |
|
the anisotropy of the lattice. |
513 |
|
|
514 |
|
Our observation of rippling behavior was not limited to the triangular |
515 |
< |
lattices. In distorted lattices the antiferroelectric phase can |
515 |
> |
lattices. In distorted lattices the anti-ferroelectric phase can |
516 |
|
develop nearly instantaneously in the Monte Carlo simulations, and |
517 |
|
these dipolar-ordered phases tend to be remarkably flat. Whenever |
518 |
|
rippling has been observed in these distorted lattices |
568 |
|
|
569 |
|
We have been able to show that a simple dipolar lattice model which |
570 |
|
contains only molecular packing (from the lattice), anisotropy (in the |
571 |
< |
form of electrostatic dipoles) and a weak surface tension (in the form |
571 |
> |
form of electrostatic dipoles) and a weak elastic tension (in the form |
572 |
|
of a nearest-neighbor harmonic potential) is capable of exhibiting |
573 |
|
stable long-wavelength non-thermal surface corrugations. The best |
574 |
|
explanation for this behavior is that the ability of the dipoles to |
575 |
|
translate out of the plane of the membrane is enough to break the |
576 |
< |
symmetry of the triangular lattice and allow the energetic benefit from |
577 |
< |
the formation of a bulk antiferroelectric phase. Were the weak |
578 |
< |
surface tension absent from our model, it would be possible for the |
576 |
> |
symmetry of the triangular lattice and allow the energetic benefit |
577 |
> |
from the formation of a bulk anti-ferroelectric phase. Were the weak |
578 |
> |
elastic tension absent from our model, it would be possible for the |
579 |
|
entire lattice to ``tilt'' using $z$-translation. Tilting the lattice |
580 |
|
in this way would yield an effectively non-triangular lattice which |
581 |
< |
would avoid dipolar frustration altogether. With the surface tension |
582 |
< |
in place, bulk tilt causes a large strain, and the simplest way to |
583 |
< |
release this strain is along line defects. Line defects will result |
584 |
< |
in rippled or sawtooth patterns in the membrane, and allow small |
585 |
< |
``stripes'' of membrane to form antiferroelectric regions that are |
586 |
< |
tilted relative to the averaged membrane normal. |
581 |
> |
would avoid dipolar frustration altogether. With the elastic tension |
582 |
> |
in place, bulk tilt causes a large strain, and the least costly way to |
583 |
> |
release this strain is between two rows of anti-aligned dipoles. |
584 |
> |
These ``breaks'' will result in rippled or sawtooth patterns in the |
585 |
> |
membrane, and allow small stripes of membrane to form |
586 |
> |
anti-ferroelectric regions that are tilted relative to the averaged |
587 |
> |
membrane normal. |
588 |
|
|
589 |
|
Although the dipole-dipole interaction is the major driving force for |
590 |
|
the long range orientational ordered state, the formation of the |
591 |
|
stable, smooth ripples is a result of the competition between the |
592 |
< |
surface tension and the dipole-dipole interactions. This statement is |
592 |
> |
elastic tension and the dipole-dipole interactions. This statement is |
593 |
|
supported by the variation in $\mu^{*}$. Substantially weaker dipoles |
594 |
|
relative to the surface tension can cause the corrugated phase to |
595 |
|
disappear. |
597 |
|
The packing of the dipoles into a nearly-triangular lattice is clearly |
598 |
|
an important piece of the puzzle. The dipolar head groups of lipid |
599 |
|
molecules are sterically (as well as electrostatically) anisotropic, |
600 |
< |
and would not be able to pack in triangular arrangements without the |
601 |
< |
steric interference of adjacent molecular bodies. Since we only see |
602 |
< |
rippled phases in the neighborhood of $\gamma=\sqrt{3}$, this implies |
603 |
< |
that there is a role played by the lipid chains in the organization of |
604 |
< |
the triangularly ordered phases which support ripples in realistic |
605 |
< |
lipid bilayers. |
600 |
> |
and would not pack in triangular arrangements without the steric |
601 |
> |
interference of adjacent molecular bodies. Since we only see rippled |
602 |
> |
phases in the neighborhood of $\gamma=\sqrt{3}$, this implies that |
603 |
> |
even if this dipolar mechanism is the correct explanation for the |
604 |
> |
ripple phase in realistic bilayers, there would still be a role played |
605 |
> |
by the lipid chains in the in-plane organization of the triangularly |
606 |
> |
ordered phases which could support ripples. The present model is |
607 |
> |
certainly not detailed enough to answer exactly what drives the |
608 |
> |
formation of the $P_{\beta'}$ phase in real lipids, but suggests some |
609 |
> |
avenues for further experiments. |
610 |
|
|
611 |
|
The most important prediction we can make using the results from this |
612 |
|
simple model is that if dipolar ordering is driving the surface |
630 |
|
behaviors. It would clearly be a closer approximation to the reality |
631 |
|
if we allowed greater translational freedom to the dipoles and |
632 |
|
replaced the somewhat artificial lattice packing and the harmonic |
633 |
< |
``surface tension'' with more realistic molecular modeling |
634 |
< |
potentials. What we have done is to present an extremely simple model |
635 |
< |
which exhibits bulk non-thermal corrugation, and our explanation of |
636 |
< |
this rippling phenomenon will help us design more accurate molecular |
637 |
< |
models for corrugated membranes and experiments to test whether |
638 |
< |
rippling is dipole-driven or not. |
633 |
> |
elastic tension with more realistic molecular modeling potentials. |
634 |
> |
What we have done is to present a simple model which exhibits bulk |
635 |
> |
non-thermal corrugation, and our explanation of this rippling |
636 |
> |
phenomenon will help us design more accurate molecular models for |
637 |
> |
corrugated membranes and experiments to test whether rippling is |
638 |
> |
dipole-driven or not. |
639 |
|
|
640 |
|
\begin{acknowledgments} |
641 |
|
Support for this project was provided by the National Science |