Fluctuations in the flat and collapsed phases of
polymerized membranes
Farid F. Abraham, David R. Nelson
To cite this version:
Farid F. Abraham, David R. Nelson. Fluctuations in the flat and collapsed phases
of polymerized membranes.
Journal de Physique, 1990, 51 (23), pp.2653-2672.
<10.1051/jphys:0199000510230265300>. <jpa-00212561>
HAL Id: jpa-00212561
https://hal.archives-ouvertes.fr/jpa-00212561
Submitted on 1 Jan 1990
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J.
Phys.
France 51
(1990)
2653-2672
ler DÉCEMBRE
1990,
2653
Classification
Physics Abstracts
68.1OC 82.70K - 87.20C
Fluctuations in the flat and
membranes
Farid F. Abraham
(1)
collapsed phases
and David R. Nelson
of
polymerized
(2)
(1) IBM Research Division, Almaden Research Center, 650 Harry Road, San Jose, California
95120-6099, U.S.A.
(2) Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138, U.S.A.
(Received
22 June 1990,
accepted
22
August 1990)
Fluctuations in polymerized membranes are explored via extensive molecular
dynamics simulations of simplified « tethered surface » models. The entropic rigidity associated
with repulsive second-nearest-neighbor interactions leads to a flattening of « phantom surfaces ».
An attractive interaction in the presence of distant self-avoidance leads to a collapsed membrane
with fractal dimension three at sufficiently low temperatures. When the attractive interaction is
turned off, the surface returns to the flat phase found in earlier simulations. A study of density
profiles and hexatic internal order allows a simple physical interpretation of results for the
Abstract.
2014
structure function of oriented membranes.
1. Introduction.
Polymerized networks appear naturally in a biological context (1), and can be made
artificially by, for example, modifying traditional methods of polymer synthesis (2), or by
polymerizing amphiphillic bilayers and monolayers (3). The statistical mechanics of these
« tethered surfaces » (4) has recently attracted intense theoretical interest (5), in part
because, unlike conventional linear polymers, they are expected to exhibit a low temperature
flat phase with an infinité persistence length. The flat phase arises because the resistance to inplane shear deformations leads to an anomalous stiffening of the surface in the presence of
thermal fluctuations (6). Although the first simulations of such tethered surfaces were
interpreted in terms of a high temperature crumpled phase (4), simulations of much larger
surfaces with a very similar potential revealed that these objects were in fact flat (7, 8) with
very large fluctuations in the direction parallel to the average surface normal (see Fig. 1).
In this paper we present the results of extensive computer simulations of the flat phase (9).
We discuss the entropic origin of the bending rigidity introduced by distant neighbor
interactions, and show explicitly that second neighbor repulsion alone is sufficient to produce
a flat phase in an initially crumpled « phantom » membrane for sufficiently short tethered
lengths. By introducing an attractive distant interaction, one can produce a compact or
collapsed self-avoiding tethered surface. In contrast to flat membranes, whose fractal
dimension is two, and crumpled membranes, whose fractal dimension is expected to be about
2.5 (4), the fractal dimension of this compact object is three. With the compact membrane as
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0199000510230265300
2654
Fig. 1.
Configurations of a self-avoiding tethered membrane of 4 219 particles (L
measured in molecular dynamics time-steps. Tethering bonds are drawn between
monomers, whose hard core size is not shown.
-
=
75). Time is
neighboring
an initial condition, we turn off the attractive part of the interaction and show that the system
relaxes to the same strongly fluctuating flat phase found with a « stretched » initial condition
in reference [7].
We also study the internal structure of membranes in the flat phase. Density profiles
perpendicular to the plane of the surface are characterized by a single exponent C, which
describes the divergence of the membrane thickness as the system size tends to infinity.
Density profiles in the plane, however, are characterized by at least two different diverging
length scales : an overall membrane diameter, and the width of the density distribution at the
membrane edge. Both in-plane phonon fluctuations and locally transverse roughing contribute
to the apparent width of the density profile at the edge of the membrane. Although these
fluctuations destroy extended translational order in monomers with the bonding topology of a
triangular lattice, we find that a small amount of long-range bond orientational order is
preserved, consistent with a prediction of Aronovitz and Lubensky (10).
This analysis of surface fluctuations in real space allows us to better understand the
remarkable signature of tethered surfaces in the reciprocal space (9). A simple theoretical
approximation to the equilibrium structure function is derived in detail, and provides a good
fit to the largest tethered surfaces simulated to date.
2655
Before proceeding further, we briefly summarize theoretical expectations (6, 10) for the
flat phase (11). In the flat phase, in-plane phonon displacements u (xl, x2 ) and an out-of-plane
displacement f (xl, X2) are defined by the equation
which gives the three-dimensional position vector r (xl, x2) of an atom in the membrane as a
function of internal membrane coordinates xl and x2 attached to the monomers. These
internal parameters multiply orthogonal unit vectors êl and ê2 which span a flat zero
temperature reference state (typically, a hexagonal piece of triangular lattice with lattice
constant a) of characteristic linear dimension L. The unit vector ê3 is given by ê3
êl x ê2.
of
the
surface
The prefactor mo is an order parameter (12) which measures the shrinkage
is
a
of
membrane
sum of
a
flat
tethered
caused by thermal fluctuations. The free energy
nearly
bending and stretching energies,
=
where K is a bending elastic constant, the elastic stretching energy has been expanded in
powers of the strain matrix, and » and À are elastic constants. The probability of a particular
kob T
configuration parametrized by u (xl, x2) and f (xl, x2) is proportional to e
Nonlinearities in the out-of-plane displacement enter via the strain matrix, given by
Because of such nonlinear couplings, the renormalized long-wavelength bending rigidity
and elastic constants differ considerably from their microscopic values. These renormalized
elastic constants enter an effective, long-wavelength free energy for the Fourier transformed
phonon variables u (q ) and f (q ), namely
where f2 is the membrane surface area in the initial, stretched state. The probability of a
particular fluctuation is now proportional to exp[-Feff/kB T]. Unlike the conventional
elastic theory of thin plates (13), the renormalized wave-vector-dependent bending rigidity
K R ( q ) and in-plane elastic parameters ftR(q) and ÀR(q) are singular for small q (6, 10). The
bending rigidity diverges according
to
(6, 10)
while the elastic constants vanish as q tends to
zero
(10)
The singular, small q behavior of these elastic constants can be calculated via an epsilonexpansion, for D 4 - e-dimensional manifolds embedded in a d-dimensional space (10), or
directly for the physically relevant case D 2, d 3 by an integral equation approach (14). A
=
=
=
2656
straightforward generalization
gives
where
and
KR(q )
is
a
of the
integral equation
for
K R(q ) derived
in reference
[6]
function of the renormalized elastic constants,
PJ(k) &#x26; ii - ki kjlk 2 is the transverse projection operator.
ansatzes
relation,
=
into the integral equation (1.7),
first derived to all orders in an epsilon expansion
(1.5) and (1.6)
The exponent C determines the size of
we
by
Upon inserting the scaling
obtain an important scaling
Aronovitz and Lubensky (10),
out-of-plane fluctuations ; using (1.4),
we
find
J"(j2) ’" L e. We introduced
so that the membrane thickness is
is the lattice spacing. In-plane phonon
Equation (1.4) leads to
When tethered surfaces with
confined
(15),
to a
plane,
one
a
fluctuations,
on
an upper cutoff a-1, where a
the other hand, are. determined by W.
perfect six-fold triangular coordination topology
are
expects power law Bragg peaks in the in-plane scattering function
The quasi-long-range translational order embodied in these peaks is destroyed when the
surface is allowed to fluctuate out of the plane, provided w z 0 (10). Long-range bond
orientational order, however, is preserved. Indeed, fluctuations in the local bond angle field
(15) 03B8(x)
= 42
Eij a 1 uj(x)
are
given by
The integral converges as Lu oo provided £o z 2, which is equivalent via equation (1.9) to
the inequality (clearly necessary in any flat phase), ( - 1. Thus tethered surfaces with the
topology of a triangular lattice are in fact « tethered hexatics » (10), a prediction we shall test
in section 3.
2657
summarize the details of the simulation procedure. Our model for the molecular
simulations is the same as reference [7], where the tethering is enforced by a
continuous potential. The particles on the network are arranged in a triangular array and
interact with their nearest neighbors through the potential
We
now
dynamics
where r’ = 2 (2’/’) + f - r. The region 21/6 - r - 21/6 + f is thus force free and equivalent to
the flexible string of other models (4) of tethered membranes. In our calculations we take
f 0.5 unless explicitly stated otherwise. Self-avoidance is generated by the interaction
between particles which are not nearest neighbors on the network. We take this interaction to
be
=
with a a parameter. The parameter u is a measure of an « effective » hard-core radius of the
distant-neighbor particles. The case a 0 corresponds to the phantom membrane ;
o1 to the self-avoiding membrane in which self-intersections are impossible. We have also
considered the effects of attraction by replacing Vd(r) with the simple Lennard-Jones
potential with an attractive well,
=
=
with a
1, and a smoothing procedure to eliminate the small discontinuity at r 2.5 a. We
have considered finite systems that are hexagonal in shape and characterized by a linear
dimension L. A hexagonal sheet of size L contains (3 L 2 + 1 ) /4 particles. We have simulated
membranes up to size L
75 (4 219 particles) and for 106 to 107 total time steps, the longer
times corresponding to the larger clusters. The procedure is a straightforward moleculardynamics calculation. Unless stated otherwise (see Sect. 2), the membrane is initially in a flat
configuration and the particles are given random displacements and zero velocities. The
clusters have zero total linear and angular momentum. The classical equations of motion are
integrated forward in time, and the appropriate microcanonical ensemble averages are
calculated. Time is expressed in molecular dynamics time-steps. In simulations without the
attractive distant neighbor potential ( 1.16), the temperature was typically kB T -- 0.6-0.7 s.
When the attractive potential was turned on, the temperature increased to kB T .-: 1.4
After the attractive potential was turned off in the « compact » or collapsed state, the kinetic
energy was rescaled upward to accelerate the retum to equilibrium. The temperature of the
flat phase which was eventually recovered was kB T -. 3.5 s. The properties of the flat phase
depend only weakly on temperature when the Lennard-Jones pair potential (1.16) is turned
off.
The self-avoiding membranes are characterized by long relaxation times and large
fluctuations in equilibrium. Figure 1 shows snapshots of molecular-dynamics configurations
which display the fluctuations in a 4 219-atom particle membrane, an effect which contributes
to long relaxation times. These snapshots show the « folding » motion, which typically has a
period of = 105 time steps for this large size cluster. Between these folding configurations, the
membrane returns to a nearly « flat » hexagonal form. In order to determine if our results are
=
=
=
2658
characteristic of equilibrium, we have calculated the autocorrelation functions of the
macroscopic variables of interest and have estimated the relaxation times. In all cases, the
molecular-dynamics simulation was carried out for at least 100 such relaxation times and,
except for L 75, for a much longer period.
=
2.
Entropic origin
of the
bending rigidity.
We first address the issue why triangulated tethered surfaces are flat (7). The isotropic
tethering potentials of references [4] and [7] lead to very flexible membranes with no explicit
microscopic bending rigidity. A priori, one might have expected such surfaces to crumple (4).
One natural explanation of the results of reference [7] is that a bending rigidity proportional
to temperature is generated for entropic reasons by excluded volume interactions, even if
there is no such term in the microscopic Hamiltonian (16). In fact, such a term is generated
immediately upon introducing next nearest-neighbor excluded volume constraints into a
tethered network.
To see this, note first that a repulsive next-nearest-neighbor interaction tends to align the
normals ni1 and n2 of the two triangular plaquettes spanned by the four hard spheres in
figure 2. Upon assuming for simplicity that the maximum sphere separation is just the sphere
diameter itself (i.e., the tether length is zero) we find that the average of (nin2&#x3E; is
Here, cf&#x3E; 0
= ’TT -
cos -1(
volume constraint. We
microscopic
1 /3) is the largest angle .0 between normals permitted by the excluded
model this effect
which
leads to
Hamiltonian,
can
2. - Pictoral definition of the normals
four hard spheres in the figure.
Fig.
by adding
filand n2
a
term 8 H
of the two
= - K
(n 1 . n 2 - 1 )
to the
triangular plaquettes spanned by
the
2659
where
K
/kB
10 (x)
T;:-
and
I1 (x)
are
Bessel functions.
Equations (2.1 )
and
(2.2)
agree
provided we take
1.13, which is larger than the ratio which produced a flat phase in the simulations of
reference [12].
To test the hypothesis of an entropic bending rigidity further, we repeated the simulations
of reference [7] with first and second neighbor excluded volume interactions only. By
systematically shortening the tether length (f in Eqs. ( 1.14) and ( 1.15)) relative to the range
of confining potential, we were able to induce a transition from a crumpled, « phantom »
membrane (whose squared radius of gyration varies as the logarithm of the linear dimension
L) to a flat phase with long-range order in the surface normals. As shown in figure 3, a change
in behavior occurs for a tether length of about 0.6. We have plotted here the time-averaged
eigenvalues Aa of the moment of inertia tensor,
Fig. 3.
Time-averaged eigenvalues of the moment of inertia tensor, equation (2.3), as a function of
tether length f for a tethered membrane with only first and second neighbor excluded volume
interactions.
-
where ri is the position of the i-th monomer in a membrane
shown in figure 3 is the squared radius of gyration RG,
consisting of N monomers. Also
For tethering lengths of order unity, the eigenvalues A1, A2 and A3 are approximately equal
and in ratios consistent with the logarithmically crumpled phantom membrane phase
discussed in reference [12]. For shorter tethering lengths, however, two of these eigenvalues
become much larger than the remaining one, indicative of a flat phase induced by the entropic
mechanism discussed above. Although we have not investigated this crossover in detail, the
transition which occurs in figure 3 for a tether length of about 0.6 is presumably the entropic
analogue of the crumpling transition driven by an energetic coupling found by Kantor and
Nelson in reference [12]. Here, the transition is driven by an entropic effective rigidity which
is increased by shortening the tethers.
2660
Fig.
4.
- Temporal
sequence of
attractive part of the distant
and time t
0.
configurations of L 75 tethered membrane often removing the
neighbor interaction and starting from a highly convoluted compact object
=
=
We have also simulated distant neighbor interactions with an attractive potential minimum
using the Lennard-Jones potential (Eq. ( 1.16)). This preserves, however, the distant neighbor
repulsive excluded volume interaction of reference [7]. As shown in figure 4 (for time 0),
the resulting membrane collapsed into a highly convoluted compact object at a reduced
=
2661
temperature of
analysis
of the
kB TT
kB
e
=
1.4, where e is the Lennard-Jones well-depth parameter. A scaling
orientationally averaged
structure
function,
Kantor et al. in reference [4], is shown in figure 5. Surfaces with
and
75
all
49
13, 25,
collapse onto a single curve when plotted as a function of
that
with
v
2/3, showing
qL v,
RG - L L 2/3the result characteristic of a collapsed object
1
The
self-avoidance
with
slope of this log-log plot in the region RG« q « a is
(4).
where
approximately - 3, consistent with the theoretical expectation that (4) S(q) ’the fractal dimension is dF 2/ v. It is remarkable that the simulation can find such a compact
configuration : as discussed in reference [4], folding of a surface with self-avoidance and a
finite thickness into a compact object is a nontrivial task.
along the lines taken by
L
=
=
Ilq d,
1
=
Fig. 5. - The directionally average structure factors for collapsed membranes of size L
75 plotted as a function of qL ’’ where v
2/3.
=
=
13, 25, 49 and
2662
It is interesting to take one of these collapsed compact manifolds as an initial condition for
the original model, after removing the attractive part of the distant neighbor interaction. The
time evolution of an L
75 membrane with this initial condition is shown pictorially in
figure 4, and the radius of gyration and moment of inertia eigenvalues, are shown as a
function of molecular dynamics time-steps in figure 6. The molecular dynamics produces
oscillations in size at short times (Fig. 6a), but eventually leads to an equilibrated membrane
in its flat phase. For long times, these are very close to the results of reference [7], obtained
from a completely different « stretched » initial condition.
=
6. - The early (a) and long (b) time evolution of the moment of inertia eigenvalues upon removal
of attraction for the collapsed membrane, eventually leading to an equilibrated flat phase.
Fig.
Our interpretation of this result is as follows : the attractive distant neighbor interaction
leads to a collapsed surface because it drives the effective bending rigidity negative (17). When
this attraction is turned off, the positive, entropically generated bending rigidity which
remains produces a xnembrane in its flat phase, essentially indistinguishable from the
equilibrated membranes in reference [7].
3. Membrane fluctuations in real space.
we focus on membranes in the flat phase with distant selfavoidance and with no attractive interactions. These simulations were carried out on
hexagonal sheets excised from a triangular lattice containing L monomers along the diagonal.
The Fourier-transformed density correlations associated with these manifolds were analyzed
in detail in reference [9]. The basic results will be summarized in section 4. In this section, we
study monomer distribution functions in real space. Our analysis leads to a better
understanding of diffraction data, and contains information not readily accessible in
conventional diffraction measurements.
In the remainder of this paper
2663
We start with
a
discussion of membrane
density profiles, which
are
averages of the
microscopic density function
over the time evolution of N
the surface is defined by
monomer
positions {rj}.
The
density profile perpendicular
to
where z is a coordinate aligned with the smallest eigenvalue of the moment of inertia tensor
and ... ) denotes a time average. We have integrated over the in-plane coordinates x and y,
and normalized p (z, L ) such that
If the
density distribution along î is characterized by a single length scale
(see the discussion in Sect. 1) we expect that p (z, L ) obeys
monomer
B/ (T&#x3E; - L e
Here, 0 (w) is a scaling function describing the density profile normal to the surface. As
shown in figure 7a, L
49 and 75 surfaces collapse to a single universal function with
C 0.65, the value of C determined from the diffraction analysis in reference [9] (18).
The angularly averaged in-plane density profile
=
=
Fig. 7. The normal (a) and angularly averaged in-plane (b) density
membranes with the appropriate scaling described in the text.
profiles for L
=
49 and 75 tethered
2664
normalized
so
that
is shown in figure 7b. Because the dominant in-plane length scale for flat membranes is clearly
L, we have plotted L 2 p 1 (R, L ) vs. R/L, for different values of L. The density distribution is
approximately flat until it drops precipitously at the edge of the surface. Unlike profiles in the
z-direction, at least two additional length scales come into play near this interface. If we
assume that the interfacial width at the edge of the membrane is dominated by the in-plane
L CI) / 2 so that the interfacial profile
phonon fluctuations, this width should scale like
should sharpen like 1 /L 1- s with à = w /2 when plotted as a function of R/L. The results of
reference [9], obtained by an analysis of in-plane phonon fluctuations near the center of
polymerized membranes, suggest that w 0.66. (Note that the values C 0.65 and
ca
0.66 are in good agreement with the scaling relation (1.4).) The interfaces in figure 7b do
indeed sharpen, but with the larger exponent of 8 "’V 0.7 instead of w /2
0.33. In-plane
phonon fluctuations alone are evidently inadequate as a model of the interface. As we show
below, the out-of-plane fluctuations (f2(X) &#x3E; are most pronounced near the perimeter,
suggesting that the membranes curl up significantly near the edge. We conjecture that this
curling introduces an additional contribution - Le to the interfacial width, leading to
/(U2&#x3E; _
=
=
=
=
8 ===== l.
We now consider a new in-plane coordinate system which has an instantaneous orientation
aligned with the orthogonal axes with coordinates, xmax and xd, determined by the two
largest eigenvalues, Amax and Amid, of the moment of inertial tensor for a membrane
configuration. Time averages of various state variables are obtained in this « dynamical »
coordinate system. The plot frms = J (j2(x) for this coordinate system is presented in
figure 8 for the L 75 membrane and suggests that fluctuations are particularly strong near
the membrane boundary. Use of the « dynamical » coordinate system leads to the two-fold
anisotropy evident in the figure. A related anisotropy is clearly demonstrated in the in-plane
one-particle density functions in figure 9. While there is a very narrow interface in the
direction of the largest eigenvalue, the interface is much more diffuse in the transverse
direction because the predominately one-dimensional « curling » fluctuations evident in
figure 8. The width of this interface has as a contribution which scales like Le because of the
curling. The interfacial width in the direction of the largest eigenvalue, on the other hand,
should scale like L CI) / 2.
Another measure of how fluctuations vary spatially within the flat phase is the timeaveraged projection of the membrane normal along the z-axis (12). In figure 10, we plot the
angular average of
=
75 surface, where
function of the internal membrane coordinates for an L
is
a
unit
normal
to
each
erected
n (x)
triangular plaquette. The z-axis is the
perpendicular
instantaneous direction of the smallest eigenvalue of the moment of inertia tensor. Although
Q (x) is large in the interior of the surface, it decreases near the perimeter, due to the wild
fluctuations associated with the free boundary conditions. Q (x) rises to half its value at the
membrane center in about 5 or 6 interparticle spacings, which can be interpreted as a
correlation length for the flat phase. For the smaller membranes simulated in references [4]
and [8], well over half of the monomers are within a correlation length of the boundary ! In
as
a
=
2665
Fig.
of frms = J i2(x»
principal moment of inertia
8. - Contour and surface plots
defined
by
the
axes
of the
as a
function of the
in-plane coordinate system
tensor.
Fig. 9. - The one-particle density profiles for the in-plane orthogonal coordinates defined by the axes of
the principal moment of inertia tensor. The densities are normalized by replacing L with the length of
the cross-sectional cord of the membrane at each point along the respective axes.
2666
Q = ([ô. i]2) - 1/3
10.
A surface of
as
L
75 membrane. This « order pa-rameter » tends to
fluctuations discussed in the text.
Fig.
-
=
view, simulations of the larger L
a
function of the internal coordinates for a
the membrane edge, due to the curling
zero near
75 surfaces are essential to clearly
demonstrate the existence of a flat phase uncontaminated by these edge fluctuations.
As pointed out by Aronovitz and Lubensky (10), a special feature of membranes with a
regular triangular coordination topology (provided w 2) is that long-range bond
orientational order is preserved despite the large fluctuations in the flat phase. Although
these fluctuations are sufficient to destroy the algebraic Bragg peaks which would be present
in a surface confined to a plane, there should be long-range order in the bond angle
correlation function G 6 (X) = (exp [6 i ( e (x) - 0 (0» &#x3E; , where 0 (x) is the angle of nearneighbor bond projected onto the average membrane plane makes relative to some in-plane
reference axis. As shown in figure 11, our membranes do indeed appear to be « tethered
hexatics » : G 6 (x) decays very slowly, with an orientational correlation length comparable to
the membrane dimensions. By identifying the asymptote of this correlation function with
we
see
that the order parameter is, however, very small,
our
=
49 and L
=
1 (e6(X» 12 0.005,
1 (f/1 6(X) 1 0.07.
4.
Approximate
form of the structure function.
In this section
we present a more detailed derivation of the predictions for diffraction from
membranes summarized in reference [9]. We compare our approximate
theoretical form for the structure function with a simulation of L
75 membranes.
Qualitative features of our results can be understood in terms of the results for membrane
fluctuations in real space tabulated in section 3.
Figure 12a shows the structure function for an oriented membrane with L 75 (i.e., 4 219
monomers). To calculate this structure function, we rewrite equation (2.5) as
polymerized
=
=
where
rz(x) is the
moment of inertia
coordinate along the direction of the smallest eigenvalue of the
tensor, and r, (x) is the corresponding perpendicular component. The z-
monomer
2667
Fig.
11. - Evidence for the
« fixed-connectivity
hexatic » from the
long-range
order in
Gr,.
axis is thus aligned with the average normal to the surface ; the structure function is averaged
over directions perpendicular to z. Experimental realizations of oriented tethered surfaces are
possible by, e.g., confining membranes between parallel glass plates. The function
S(q,, q , L ) can then be probed directly via, e.g., light scattering or X-ray diffraction
experiments.
The structure function can be calculated theoretically using the long wavelength description
of the flat phase embodied in the Gaussian-free energy equation (1.4), in terms of the
exponents ’and w defined by equations (1.5) and (1.6). The calculations are similar to those
which produced equations (1.10) and (1.11). Upon inserting the decomposition (1.1) into
equation (4.1) and using properties of Gaussian averages we find that
Upon using equations (1 . 5) and (1.6),
form
and
we
find that the
exponentiated
averages must take the
2668
Fig.
(b).
12. - The structure function for
an
oriented membrane of L = 75 from simulation
(a)
and
theory
where the coefficients A, B, and B’ depend on the coefficients in equations (1.5) and (1.6).
We shall take B = B’ for simplicity, although this assumption is easily relaxed. Upon taking
the continuum limit in (4.2) and approximating the hexagonal integration domains by disks,
we find we must evaluate
2669
where each
integration
A general method for
result
is confined to
a
disk of radius
L/2,
and
simplifying such integrals is described in the Appendix,
and leads to the
where Jo (x) is a Bessel function. The parameters band b2 have been introduced to normalize
the behavior of the structure function for small q. We choose units of q, and q1 such that
A33 = min is the smallest eigenvalue of
the average of the
the moment of inertia tensor, and
remaining two eigenvalues.
We then have
lim
b1 -
L -. ao
for C
=
A.L = à (7ii
+
A2) is
-J A3(L)jI L2l, and
0.65 and
The remaining parameter B in (12) (or, more generally B and B’, if Eq. (4.3b) is used) must
be fit to experiment.
The asymptotic large L structure function is determined once these parameters are known.
We expect equation (4.6) to be accurate for all wavelengths large compared to typical
monomer dimensions including, in particular, wavelengths large compared to either the
transverse or in-plane membrane size. Although we do not expect equations (4.3a, b) to be
reliable
forx - x’[
small
orx - x’ 1 . L,
the factor
s[cos-1s - s -., /1 _ S2 1 deemphasizes
these regions in equation (4.6) in favor of regimes where equations (4.3a, b) are valid.
The structure function S(q,, q.l’ L ) obtained from (4.6) for L
75 is plotted in figure 12b,
and provides a reasonable description of the simulation data. The theory predicts a
breakdown of scaling with L for q in the transverse direction : S(o, q.l’ L ) is not a function
only of the product q 1 L over a wide range of intermediate wave vectors. The physical reason
for this peculiar behavior is the large in-plane phonon fluctuations : scaling is restored in
equation (4.6) if we suppress these fluctuations by arbitrarily setting B 0. A related
breakdown of scaling occurs near the interface in figure 7b : For large L, the in-plane
interface sharpens like 1 /L 1- s when plotted vs. r.l / L. Sharp interfaces lead to the
oscillations in S( q z, q.l ; L ) along q.l. These oscillations are also visible in figure 5 of
reference [9], but are less pronounced for small L, reflecting to a more gradual interfacial
=
=
2670
in this case. In the simple theory of S( q Z’ q 1- , L ) sketched above, the damping of the
oscillations for small L is controlled by the exponent 8 = w /2. The « curling » fluctuations
near the edge which lead to 8 = C are not taken into account. It is this larger value of 8 which
makes the oscillations in the simulation less sharp than those predicted by the theory (9).
Although this hydrodynamic theory cannot describe the interesting structure in the
simulation for qd &#x3E; 1, where d is an interparticle spacing, the overall shape and folds in the
structure function in figures 12 are accounted for rather well. Had we not averaged over inplane directions, the atomic-scale oscillations for qd ± 1 would have had a six-fold modulation
reflecting the long-range hexatic order discussed in section 3. Because the expected
modulation is small (it should be of relative order1 (i6(X» 0.07 (19», we did not search
for it.
profile
Acknowledgements.
It is a pleasure to acknowledge helpful conversation with I. P. Batra, G. Grest, Y. Kantor,
M. Kardar and M. Plischke during the course of this investigation. One of us (DRN), would
like to acknowledge the support of the National Science Foundation, through Grant DMR8817291 and through the Harvard Materials Research Laboratory.
Appendix.
Evaluation of
We want to
dimensional
q 1
,
we can
an
integral.
the expression (4.4) for S(qz, q 1-’ L), which is a constrained fourSince S(q,, q,, L) clearly cannot depend on the direction of
over
orientations of q, in the plane to obtain
average
simplify
integral.
with
We first fix the direction and
and
integrate
over
magnitude of
the center of
mass
coordinate
figure 13, the constraint that both x, and x2 be inside a common disk of radius
confines
the center of mass integration to a lens-shaped region. Since f(x) is
Z./2
of
independent the center of mass, integration over X amounts to the computation of the area
of ihis lens, which is
As shown in
2671
Fig.
13.
-
Integration
over
the center of
mass
coordinate X
= 1 (X1 + x2)
2
for
a
fixed direction and
magnitude of x x, - x2. As illustrated in (a), the set of possible locations for X becomes increasingly
constrained by the requirements that1 Xl1 - L /2 and1 x21[
L /2 for large x. This leads to the lens-like
shaded domain of X-integration shown in (b).
=
The
remaining angular integral for
x
is trivial and
so we
obtain
Upon defining a new variable s x/L we recover equation (4.6). The remaining onedimensional integral is easy to evaluate numerically. As explained in the text, the constants
b1 and b2 have been introduced into equation (4.6) to normalize the small q, and
q, of the structure function in the scaling (large L) limit. The constant B must be fit to
experiment.
=
References
[1]
ALBERTS B., BRAY D., LEWIS J., RAFF M., ROBERTS K. and WATSON J. D., The Molecular
Biology of the Cell (Garland, New York) 1983. The best example of a biological tethered
surface is probably the spectrin protein skeleton of eurythrocytes, separated from its natural
lipid environment ;
[2]
[3]
[4]
[5]
See ELGSAETER A., STOKKE B., MIKKELSEN A. and BRANTON D., Science 234 (1986) 1217.
BLUMSTEIN A., BLUMSTEIN R. and VANDERSPURT T. H., J. Colloid Interface Sci. 31 (1969) 236 ;
REGEN S. L., SHIN J.-S., HAINFIELD J. F. and WALL J. S., J. Am. Chem. Soc. 106 (1984) 5756.
BEREDJICK N. and BURLANT W. J.,J. Polymer Sci. A 8 (1970) 2807 ;
FENDLER J. H. and TUNDO P., Acc. Chem. Res. 17 (1984) 3.
KANTOR Y., KARDAR M. and NELSON D. R., Phys. Rev. Lett. 57 (1986) 791 ; Phys. Rev. A 35
(1987) 3056.
Eds. D. R. Nelson, T. Piran and S. Weinberg, Statistical Mechanics of Membranes and Interfaces
(World Scientific, Singapore) 1989.
2672
[6]
[7]
[8]
NELSON D. R. and PELITI L., J. Phys. France 48 (1987) 1085.
ABRAHAM F. F., RUDGE W. E. and PLISCHKE M., Phys. Rev. Lett. 62 (1989) 1757.
For earlier speculations along these lines, see PLISCHKE M. and BOAL D., Phys. Rev. A 38
4943 ;
BOAL D., LEVINSON E., LIU D. and PLISCHKE M., Phys. Rev. A 40 (1989) 3292. In
difficult to
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
distinguish
=
=
[19]
our
view, it is
isotropic crumpling hypothesis of reference [4]
rough phase in these more modest simulations.
and the
of flat, but very
For a summary which focuses on results for the structure function, see ABRAHAM F. F. and
NELSON D. R., Science 249 (1990) 393.
ARONOVITZ J. A. and LUBENSKY T. C., Phys. Rev. Lett. 60 (1988) 2634.
Although references [6] and [10] treat the flat phase of membranes without explicit distant selfavoidance, self-avoidance is believed to be unimportant for flat surfaces, provided one
introduces a bending rigidity into the theory. See reference [5] and section 2.
KANTOR Y. and NELSON D. R., Phys. Rev. A 38 (1987) 4020 ;
See also PACZUSKI M., KARDAR M. and NELSON D. R., Phys. Rev. Lett. 60 (1988) 2638.
LANDAU L. D. and LIFSHITZ E. M., Theory of Elasticity (Pergamon, New York) 1970.
See reference [6], and the article on the crumpling transition by NELSON D. R. in reference [5].
See, e.g., NELSON D. R. and HALPERIN B. I., Phys. Rev. B 19 (1979) 2457.
Similar conclusions have been reached by LEIBLER S. and MAGGS A. C., Phys. Rev. Lett. 63 (1989)
406.
We are indebted to KARDAR M. for discussions on this point.
25 membranes were too small to give
13 and L
In reference [9], we argued that the L
reliable results for the flat phase. This is especially true for density profiles. See also the
discussion below equation (3.7). If the analysis which led to 03B6 in reference [9] is repeated with
L
13, 15, 49, and 75, we find 03B6 0.76 with a poor scaling fit.
See, for example, BRUINSMA R. and NELSON D. R., Phys. Rev. B 23 (1981) 402.
hypothesis
[9]
between the
(1988)
=
=