ler NOVEMBRE 1990,

J. Phys. France 51 ( 1990) 2395-2398

2395

Classification

Physics Abstracts
05.40-36.20 - 68.10

Short Communication

Spectral dimension of fluid membranes
Shigeyuki Komura(*) and Artur Baumgärtner
Institut für Festkörperforschung, Forschungszentrum Jülich, D-5170 Jülich, F.R.G.
1(Received 13August 1990, accepted 20August 1990)
Abstract.
The spectral dimension ds of polymerized and fluid self-avoiding vesicles are investigated by Monte Carlo methods. For both cases we obtained ds=2, which indicates that these surfaces
belong to the same class of "microcanonical" surfaces.
2014

Properties of flexible sheet polymer networks have been explored in recent theoretical investigations [1]. Polymerized membranes have a nonzero shear modulus in the plane and are said tobe solid-like, whereas fluid membranes are not rigid in the plane and have zero in-plane modulus.
It has been shown using computer simulations [2,3] that polymerized self-avoiding membranes
are essentially flat and the corresponding in-plane squared radius of gyration is proportional to
the number of monomers on the surface,

Very recently, a model of fluid self-avoiding membranes has been proposed which exhibits, in contrast to polymerized membranes, crumpled shapes with v m 0.8 [3] . One important question withrespect to the differences between polymerized and fluid membranes is related to their internal
connectivity which is commonly characterised by the spectral dimension ds [4,5] . The spectral
dimensions of various classes of membranes have been discussed in detail by Cates [6] . For the
2. Very recently, it hascase of polymerized surfaces with random connectivity, it is expected d,been shown that even for fractal surfaces with hierarchical connectivity [7] the spectral dimensionis, somewhat unexpectedly, not changed and also ds 2.
In the present Communication, we report on Monte Carlo studies of the spectral dimension forfluid membranes. For comparison, we also include the results of ds for the corresponding modelof polymerized membranes.
The standard way to measure the spectral dimension is related to a random walk on the given
surface [4, 5] . Given a particular realization of the membrane, the corresponding mean-square==

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0199000510210239500

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displacements at time t of a random walk on the surface is expressed aswhere df = 2/v is the fractal dimension of the model membrane. In practice, r2(t) has to be averaged over various walks on one particular frozen realization of the surface as well over differentrealizations. For convenience, we used a spherically closed surface ("vesicle") in order to take
into account the periodic boundary conditions for the random walk properly.
The simulation technique for generating various conformations of self-avoiding vesicles is thesame as has been used previously [3] . In a polymerized vesicle, the connectivity at each monomer
is fixed. For a fluid vesicle, however, we relax the restriction on the fixed connectivity; we allow themonomers to exchange their neighbors, but keeping the rule that the topology and the integrity ofthe stucture should be preserved ("triangulation" procedure). This procedure provides for a given
monomer to escape after several bond exchanges from its original neighborhoods of monomers,
and hence represents a "fluid" particle.
The averaged mean-square displacement of a random walker on various sizes of vesicles areanalyzed by using the crossover scaling form (similar to that of diffusion on percolating clusters,e.g., Ref.[8] ),where f (x) is a scaling function with f(z) - x ds/df for x « 1 and f (x)With this scaling function, the mean-square displacement behaves as=const. for x » 1.

ids/df for t « T,
Nv for t » r, where T = N 2/ ds gives the crossover time. For very long times
and remains constant reflecting the fact that the displacement of the(t » T),
random walker are bounded by the finite size of the vesicles. The exponent v has been estimatedNV, which yields v = 0.98 f 0.02 andaccording to the relation (2) at x » 1, i.e.,
0.83 f 0.02 for polymerized and fluid vesicles, respectively. Since we expect that under the scalingof (2) all data should collapse to a single curve, we have estimated the spectral dimension ds fromseveral attempts to obtain optimal overlap of the curves for all N. This is presented in figure
1, where y =IN" is shown as a function of x =Fixing v =0.98 and 0.83 forthe two types of vesicles, we obtained the best fit using ds = 1.96 + 0.05 for polymerized and
ds= 2.02 ± 0.04 for fluid vesicles. Of course, these estimates of v = 2/d f and cas are consistent
with the slope f(z ) - xds/df for x « 1. Our conjecture is that ds = 2 for both polymerized andfluid vesicles.

(r2( t») "-1and(r2 ( t») "-1(r2(t») saturates(r2(t)) -t/N2/ds.

(r2(t»)

It is worthwhile to point out the next consideration on the crossover time r. The return probability P(t) that the random walker comes back to the starting point at time t scales as [4,5]Therefore the number of accessible sites E(t) increases as E(t) - (P(t) ) - 1 - tds/2. When t
N 2/ds holds. This means that the number of accessible
sites amounts to the order of E (t)tds/2~N. This can be interpreted that once the random
walker has visited all N sites, it will start to access the sites it has already visited and therefore themean-square displacement begins to saturate for t &#x3E; T.
Finally, it should be noted that ds = 2 of the present model for fluid membranes implies that thistype of model belongs to the class of "microcanonical" surfaces [6] , which is commonly restricted
comes close to the crossover time r, t N~

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Fig. 1. Scaling plot for average mean-squared displacement of a random walker on polymerized and fluid
setf-avoidingvesicles. Here x tIN 2/d -1 and y = r2(t)&#x3E; IN" with ds = 1.96 and v 0.98 for polymerized-==vesicles ds = 2.02 and v = 0.83 for fluid vesicles.

polymerized random surfaces [2, 3, 9] subjected to the constraints of constant surface area, S
const., and fixed "local" connectivity. With respect to the present results for fluid membranes,
df 2.5, it seems to be reasonable to extend the class of "microcanonical" surfaces by including
fluid surfaces with S
const., but with the weaker constraint of fixed "global" connectivity, orin other words, constant numbers of vertices, faces and edges. Moreover, with respect to the2.33 [7] , it is suggestive torecent findings of ds
2 for deterministic fractal surfaces with df
attribute ds 2 to the class of "microcanonical" manifolds in general, including (so far as we know
currently) random polymerized, fluid and fractal surfaces.
to======Acknowledgements.
One of us (S.K.) is very grateful for the hospitality of IFF at Forschungszentrum Jülich. He also
would like to express his appreciation to Dr. M.E. Cates, Dr. D. Kroll and Dr. Y. làguchi foruseful discussions. This work was done on CRAY X-MP/416.

References[1] See for instance Proceedings of the Fifth Jerusalem Winter School, Statistical Mechanics of Membranes and Surfaces, Eds. D.R. Nelson, T. Piran and S. Weinberg (World Scientific) 1989.
[2] PLISCHKE M. and BOAL D., Phys. Rev. A 38 (1988) 4943;
ABRAHAM F.F., RUDGE WE. and PLISCHKE M., Phys. Rev. Lett. 62 (1989) 1757;
HO J.-S. and BAUMGÄRTNER A., Phys. Rev. Lett. 63 (1989) 1324.
[3] BAUMGÄRTNER A. and HO J.-S., Phys. Rev. A 41 (1990) 5747;
HO J.-S. and BAUMGÄRTNER A., Europhys. lett. 12 (1990) 295.

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[4] ALEXANDER S. and ORBACH R., J. Phys. Lett. France 43 (1982) 625.
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[6] CATES M.E., Phys. Rev. Lett. 53 (1984) 926;
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[7] GIUGLIARELLI G., MARTIN A. and STELLA A.L., Europhys. Lett. 11 (1990) 101.
[8] GEFEN Y., AHARONY A. and ALEXANDER S., Phys. Rev. Lett. 50 (1983) 77.
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Cet article a été imprimé avec le Macro Package "Editions de Physique Avril 1990".