Fluctuations in the flat and collapsed phases of polymerized membranes

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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�

Fluctuations in the flat and collapsed phases ^of polymerized

membranes

Farid F. Abraham (1) and David R. Nelson (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)

Abstract.

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 structure function of oriented membranes.

Classification

Physics ^Abstracts

68.1OC 82.70K - 87.20C

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 in-

plane ^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

Fig. ^1.

Configurations ^{of a} self-avoiding tethered membrane of 4 219 particles (L

75). ^{Time is}

measured in molecular dynamics time-steps. Tethering ^{bonds are} drawn between neighboring

monomers, whose hard core size is not shown.

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.

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.

The prefactor mo is ^an order parameter (12) ^{which measures the} shrinkage of the surface caused by thermal fluctuations. The free _energy of a nearly flat tethered membrane is a sum of

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 configuration parametrized by u (xl, x2) ^and f (xl, x2) ^is proportional ^to e ^{kob T}

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 epsilon-

expansion, ^{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

straightforward generalization ^{of the} integral equation ^for K R(q ) ^derived in reference [6]

gives

where KR(q ) ^{is a} function of the renormalized elastic constants,

and PJ(k)

& ii - ^ki kjlk 2 ^{is the} transverse projection operator. Upon inserting ^the scaling

ansatzes (1.5) ^and (1.6) ^{into the} integral equation (1.7), ^{we obtain an} important scaling relation, first derived to all orders in an epsilon expansion by Aronovitz and Lubensky (10),

The exponent C determines the size of out-of-plane fluctuations ; using (1.4), ^{we find}

so that the membrane thickness is J"(j2) ’" ^{L e. We} introduced an upper cutoff a-1, ^{where a}

is the lattice spacing. In-plane phonon fluctuations, ^{on the other} hand, are. determined by ^W.

Equation (1.4) ^{leads to}

When tethered surfaces with a perfect ^six-fold triangular coordination topology ^are confined ^{to a} plane, ^one expects power law Bragg peaks ^{in the} in-plane scattering ^function (15),

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) = 4 Eij ^{2 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.

We ^now summarize the details of the simulation procedure. Our model for the molecular

dynamics 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

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 ;

o-

1 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 ^molecular- dynamics 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

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 ni 1 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 (ni n2> ^is

Here, cf> 0 ^{= ’TT - cos -1} ( 1 /3) ^{is the} largest angle .0 between normals permitted by ^{the excluded}

volume constraint. We can model this effect by adding ^{a term 8 H = - K} (n 1 . n 2 - 1 ) ^{to the} microscopic Hamiltonian, ^{which leads to}

Fig. 2. - Pictoral definition of the normals fil and n2 ^{of the two} triangular plaquettes spanned by ^the

four hard spheres ^{in the} figure.

where 10 (x) ^and I1 (x) ^are Bessel functions. Equations (2.1 ) ^and (2.2) agree provided ^{we take}

K /kB ^{T;:- 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 consisting ^{of N monomers. Also}

shown in figure ^{3 is the} squared ^{radius of} gyration RG,

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.

Fig. ^4. - Temporal sequence of configurations ^{of L}

75 tethered membrane often removing ^the

attractive part of the distant neighbor interaction and starting ^{from a} highly convoluted compact object

and time t

0.

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}

temperature ^of kB T kB T

_1.4, where e is the Lennard-Jones well-depth parameter. ^A scaling

e

analysis ^{of the} orientationally averaged ^{structure function,}

along ^{the lines taken} by ^{Kantor et} al. in reference [4], ^{is shown in} figure ^{5. Surfaces with}

L

13, 25, 49 and 75 all collapse ^{onto a} single ^{curve when} plotted ^{as a function of} qL v, ^{with v}

2/3, showing ^that RG - ^L L 2/3 the result characteristic of a collapsed object

with self-avoidance (4). ^The slope ^{of this} log-log plot ^{in the} region RG « q « a ^{1 is}

approximately - 3, consistent with the theoretical expectation ^that (4) S(q) ’- Ilq d,

^where

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.

Fig. ^{5. - The} directionally average ^structure factors for collapsed ^{membranes of size L}

^{13, 25, 49 and}

75 plotted ^{as a} function of qL ’’ ^{where v}

2/3.

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.

Fig. ^{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.

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.

In the remainder of this _paper we focus on membranes in the flat phase with distant self- avoidance 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.

We start with ^a discussion of membrane density profiles, ^{which are} averages of the

microscopic density ^function

over the time evolution of N monomer positions {rj}. ^The density profile perpendicular ^to

the surface is defined by

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 monomer density distribution along ^{î is} characterized by ^a single length ^scale

B/ (T> - L e ^(see the discussion in Sect. 1) ^we expect ^that p (z, L ) obeys

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 profiles ^{for L}

⁴⁹ and 75 tethered

membranes with the appropriate scaling described in the text.

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 phonon fluctuations, this width should scale like /(U2> _ ^{L CI) / 2 so that the} interfacial profile

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) > ^{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 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 time-} averaged projection ^of the membrane normal along the z-axis (12). ^In figure 10, ^we plot ^the angular average of

as a function of the internal membrane coordinates for an L

75 surface, ^where n (x) ^{is a unit normal erected} perpendicular ^{to each} triangular plaquette. The z-axis is the 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

Fig. 8. - Contour and surface plots of frms = J i2(x» ^{as a} function of the in-plane coordinate system defined by ^{the axes of the} principal ^moment of inertia 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.

Fig. ^10.

A surface of Q = ([ô. i]2) - 1/3 ^{as a} function of the internal coordinates for a

L

75 membrane. This ^« order pa-rameter » ^{tends to zero near} the membrane edge, ^{due to the} curling

fluctuations discussed in the text.

our view, simulations of the larger ^L

^{49 and L}

^{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» > , ^where 0 (x) ^{is the} angle ^{of near-} neighbor ^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

1 (e6(X» 12 0.005, ^{we see} that the order parameter is, however, very small,

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

polymerized 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

where rz(x) ^{is the monomer} coordinate along the direction of the smallest eigenvalue ^{of the}

moment of inertia tensor, and r, (x) ^{is the} corresponding perpendicular component. ^{The z-}

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), ^we find that the exponentiated averages ^must take the form

and

Fig. ^{12. - The structure} function for an oriented membrane of L = 75 from simulation (a) ^and theory (b).

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

where each integration ^{is confined to a} disk of radius L/2, ^and

A general method for simplifying ^such integrals ^{is described in the} Appendix, ^{and leads to the}

result

where Jo (x) ^{is a} Bessel function. The parameters b and b2 ^{have been} introduced to normalize the behavior of the structure function for small _q. We choose units of q, and q1 ^{such that}

A3 = ³ min is the smallest eigenvalue ^{of the moment} of inertia tensor, and A.L = à ^{(7ii + A2) is}

the average of the remaining ^two eigenvalues. ^{We then have} b 1 - lim -J A3(L)jI ^{L2l, and}

L -. ao

for C

^{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 for x - x’ [ ^small or x - x’ 1 . ^{L, the factor} s[cos-1 ^{s - 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

profile ^{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 > 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 in-} plane 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 order 1 (i6(X» ^{0.07 (19», we did not search}

for it.

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 DMR88- 17291 and through the Harvard Materials Research Laboratory.

Appendix.

Evaluation of an integral.

We want to simplify ^the expression (4.4) ^for S(qz, q 1-’ L), ^{which is a} constrained four- dimensional integral. ^Since S(q,, q,, L) clearly ^cannot depend ^{on the} direction of q 1

^{we can} average ^over orientations of q, in the plane ^{to obtain}

with

We first fix the direction and magnitude ^of

and integrate ^{over the center of mass coordinate}

As shown in figure 13, the constraint that both x, and x2 be inside a common disk of radius

Z./2 confines the center of mass integration ^{to a} lens-shaped region. ^Since f(x) ^is

independent ^{of the center of mass,} integration ^{over X amounts to the} computation ^{of the area}

of ihis _lens, which is

Fig. ^13.

Integration ^{over the center of mass} coordinate X = 1 _(X1 + x2) ^{for a} fixed direction and 2

magnitude ^{of x}

x, - x2. As illustrated in (a), ^{the set of} possible locations for X becomes increasingly

constrained by ^the requirements that 1 Xl 1 ^- L /2 and 1 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 ^one-

dimensional integral ^is easy ^to evaluate numerically. ^As explained ^{in the text, the constants} b ^{1 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 ;

See ELGSAETER A., STOKKE B., MIKKELSEN A. and BRANTON D., Science 234 (1986) ^1217.

[2] ^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.

[3] ^{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.

[4] ^KANTOR Y., KARDAR M. and NELSON D. R., Phys. Rev. Lett. 57 (1986) 791 ; Phys. ^{Rev. A 35} (1987) ^3056.

[5] ^{Eds. D. R.} Nelson, ^T. Piran and S. Weinberg, Statistical Mechanics of Membranes and Interfaces

(World Scientific, Singapore) ^1989.

[6] NELSON D. R. and PELITI L., ^J. Phys. ^{France 48} (1987) ^1085.

[7] ABRAHAM F. F., RUDGE W. E. and PLISCHKE M., Phys. Rev. Lett. 62 (1989) ^1757.

[8] ^{For earlier} speculations along ^{these lines, see PLISCHKE M. and BOAL} D., Phys. ^{Rev. A 38} (1988) 4943 ;

BOAL D., ^LEVINSON E., ^LIU D. and PLISCHKE M., Phys. ^{Rev. A 40} (1989) ^{3292. In our} view, ^{it is} difficult to distinguish between the isotropic crumpling hypothesis of reference [4] ^{and the} hypothesis ^of flat, but _very rough phase ^{in these more} modest simulations.

[9] ^{For a} summary which focuses on results for the structure function, ^see ABRAHAM F. F. and NELSON D. R., Science 249 (1990) ^393.

[10] ARONOVITZ J. A. and LUBENSKY T. C., Phys. Rev. Lett. 60 (1988) ^2634.

[11] Although references [6] ^and [10] ^{treat the flat} phase of membranes without explicit distant self- avoidance, 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.

[12] ^{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.

[13] LANDAU L. D. and LIFSHITZ E. M., Theory ^of Elasticity (Pergamon, ^New York) ^1970.

[14] See reference [6], and the article on the crumpling transition by ^{NELSON D. R.} in reference [5].

[15] See, e.g., NELSON D. R. and HALPERIN B. I., Phys. ^{Rev. B 19} (1979) ^2457.

[16] Similar conclusions have been reached by ^LEIBLER S. and MAGGS A. C., Phys. Rev. Lett. 63 (1989)

406.

[17] ^{We are indebted to KARDAR M.} for discussions on this point.

[18] ^{In reference} [9], ^we argued ^{that the L}

^{13 and L}

25 membranes were too small to give

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.

[19] See, ^for example, ^{BRUINSMA R.} and NELSON D. R., Phys. Rev. B 23 (1981) ^402.