J. Phys. A: Math. Gen. 29 (1996) 7439–7449. Printed in the UK

Monte Carlo study of a self-avoiding polymerized
membrane with negative bending rigidity
Shintaro Mori†§ and Shigeyuki Komura‡k† Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1
Tokyo 113, Japan‡ Department of Mechanical System Engineering, Kyushu Institute of Technology, Iizuka 820,
Japan
Received 26 April 1996, in final form 24 July 1996
Abstract. We study the possibility of a cancellation between the entropic bending rigidity due
to self-avoiding interaction and negative bending rigidity. Results of Monte Carlo simulation
show that a self-avoiding polymerized membrane with negative bending rigidity exhibits a phasetransition from the usual floppy flat phase to a rigid folded flat (thin plate) phase. Even for flexible
membranes, the complete cancellation does not occur. Based on these results, we discuss the
problems in theoretical approaches to the phase transitions of polymerized membranes.

1. Introduction
There has been considerable interest in the understanding of statistical properties ofpolymerized (or tethered) membranes [1–3]. One of the surprising features is that the
self-avoiding polymerized membrane is flat when it is embedded in three dimensional space[4–6]. The flat phase observed in the computer simulations was partly explained in termsof the crumpling transition induced by the implicit entropic bending rigidity due to theself-avoidance [7]. In fact, the restriction of the excluded volume interactions between thenext-nearest-neighbour particles through the core potential inevitably induces the bending
rigidity and prevents the membranes from bending freely. It is claimed in [7] that theprevious simulations of the hard sphere model did not treat the pure self-avoiding effectand attributed the flatness to the model-dependent problems.
On the other hand, there is an interesting report insisting that polymerized membranes
with self-avoiding interactions are always flat even for small diameters of hard spheres[5]. By reducing the size of the hard spheres, it is possible to remove the effect of theentropic bending rigidity from the next-nearest-neighbour core potential and one can study
the pure self-avoiding effect. We also note a different computational method of enforcing
self-avoidance in a model system by using a triangulated surface at which no vertex cancross a plane of a triangle created by any three neighbouring atoms [8, 9]. There is adiscrepancy between the results of these papers and we cannot draw definite conclusions.
Here, we assume that the self-avoiding membrane is flat even with this model, although thecrossovers are extremely slow [10].†§ E-mail address: mori@monet.phys.s.u-tokyo.ac.jp
k E-mail address: komura@iizuka.isc.kyutech.ac.jp† The results obtained in [8, 9] and in [10] are different. As in [11], we rely on the latter one.
0305-4470/96/237439+11$19.50

c 1996 IOP Publishing Ltd

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S Mori and S Komura

According to the paper by Kantor and Kremer [11], the model of a polymerized
membrane was described by a local Hamiltonian with repulsive interactions acting only
between hard spheres whose degree of neighbourhood does not exceed a certain value l.
They found that the hard sphere diameter σ has a l dependent critical value σc (l) at which
the membrane undergoes a crumpling transition. Moreover, a scaling relation such thatσc (l)3 ∼ l −4 has been found. This indicates that the self-avoiding membrane becomes flat
even for hard spheres of small diameter. Hence, the flatness is considered to be an inherent
property of self-avoiding polymerized membranes. In addition, these authors proposed
the following picture of the behaviour of the self-avoiding membrane. On a very shortlength scale, the self-avoiding interaction is irrelevant and the membrane is crumpled. Up
to a certain length scale lcrmp , self-avoiding interactions produce an effective rigidity thatsuffices to keep the entire membrane flat. Within this picture, the crumpling transition isbrought about by the effective rigidity. Beyond the scale lcrmp , the membrane is flat and theself-avoiding interaction no longer plays an essential role.
While it was established that the self-avoiding polymerized membrane is flat in computer
simulations, there is still no satisfactory theory which explains this flatness. By applying
a Gaussian variational technique to the generalized Edwards model, it was found that themembranes are crumpled above space dimension d = 4 and the radius of gyration Rg scaleswith the linear membrane size L as Rg ∼ Lν where ν = 4/d [12–14]. This exponent gives
a good estimate for the results of computer simulations in higher space dimensions d > 4[15]. Their theory, however, can only treat the crumpled phase and does not tell anything
about the crumpling transition induced by the entropic rigidity. In addition, one cannot
answer why the Gaussian variational technique, instead of the Flory approximation [2] orthe perturbative renormalization method [16], does work in the membrane case.
Recently, the phase transition of polymerized membranes with attractive interactions
has been investigated by several people [7, 17–21]. Extensive computer simulations haveshown that there are two types of phase transition, depending on the flexibility of themembrane [19]. When the membrane is less flexible, it undergoes a first-order phasetransition separating a high-temperature flat phase and a low-temperature collapsed phase. In
the presence of an open boundary, the membrane exhibits a well defined sequence of folding
transitions [17]. When the membrane is flexible, on the other hand, the transition becomes
either continuous or weakly first order. By assuming that the transition is continuous, theexponent for the radius of gyration ν at the transition point is estimated as ν = 0.85.
Phenomenological aspects of these phase transitions can be described by the Landaumodel proposed by Abraham and Kardar [17, 20]. The above-explained different behaviours
depending on the flexibility of the membrane are discussed in the following way [20, 21].
An attractive interaction induces negative bending rigidity when the membrane is flatand cancellation can occur between this negative rigidity and the entropic rigidity due tothe self-avoiding interactions. When the membrane is less flexible, the flat phase becomes
unstable before complete cancellation occurs and the transition becomes discontinuous or
first order. As for flexible membranes, the above cancellation can be complete and the
crumpling transition might occur. In this continuous transition, an intermediate crumpled
phase appears before the membrane becomes a collapsed phase. However, these theoretical
approaches incorporate the effect of a self-avoiding interaction through an effective bending
rigidity and only treat the instability of flat polymerized membranes without self-avoidance.
In this paper, we discuss the problem of cancellation between the entropic rigidity andthe negative bending rigidity both by theoretical arguments and Monte Carlo simulation.
We especially pay attention to the possibility of complete cancellation.
The outline of the present paper is as follows. In section 2, we use the generalizedSelf-avoiding polymerized membrane with negative bending rigidity

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Edwards model to describe the flat phase of the membrane. We obtain the relation between
the σc (l) and l according to the scaling argument. In section 3, we explain the model
used for the Monte Carlo simulation. The total Hamiltonian includes both self-avoiding
interactions and local bending energy with negative bending rigidity. The flexibility of themembrane can be changed by using different diameters of the hard spheres. Section 4contains the results of our simulation. Our study shows that there occurs a phase transition
from the usual flat phase to another rigid flat phase which we call the ‘folded flat phase’. An
intermediate crumpled phase does not appear even for small hard spheres, which indicatesthat complete cancellation does not occur.
2. Excluded volume effects in the flat phase
In this section, we shall use the Gaussian variational approximation [12–14] in order todiscuss the properties of a self-avoiding polymerized membrane being in the flat phase.
So far the self-avoiding interaction has always been considered to be irrelevant when themembrane is flat [1]. This picture seems correct intuitively. However, if the self-avoidanceis irrelevant for the flat membrane, then what makes the membrane flat? To answer
this question, we use the generalized Edwards model for a flat self-avoiding polymerized
membrane.
According to the discussion by Kantor and Kremer [11], the membrane is crumpled atshort length scales. Hence, at this length scale, we can use the generalized Edwards model,
Z
Z
Z
(2.1)
H[X(s)] = 12 dD s ∂α Xi ∂α Xi + u dD s dD s 0 δ d [Xi (s) − X i (s0 )].
Here, X i (s α ) (i = 1, . . . , d, α = 1, . . . , D) describes the position of the membrane withinternal coordinates s α . u represents the strength of the excluded volume interaction. We
assume that the interaction δ d [Xi (s) − X i (s0 )] is valid only when |s − s0 | is smaller thansome length scale lcrmp . When |s − s0 | is greater than lcrmp , the self-avoiding interaction
becomes irrelevant.
In order to estimate the effect of the self-avoidance at short length scales, we employ theusual variational method with the quadratic Hamiltonian in Fourier space with wavenumberk [12, 14]:Z
(2.2)
H0 = dD k X i (−k)g(k)X i (k).
From the variational equation with respect to g(k), it is possible to obtain the effective
rigidity κeff due to the self-avoidance of strength u asZκeff = κbare + c1 u dD s s 4 h|X(s) − X(0)|2 i−1−d/2(2.3)
where h|X(s)−X(0)|2 i is the two-point function averaged over the variational Hamiltonian
equation (2.2) and c1 is some constant. We have also added the constant bare bending rigidity
κbare for later convenience. We note that this equation is valid only in the crumpled partof the membrane, i.e. up to some length scale lcrmp . Beyond this length scale, the effectof the self-avoidance turns out to be irrelevant. Thus the length scale lcrmp is determinedby the condition that the effective bending rigidity becomes large such that the crumplingtransition to the flat phase occurs. Therefore, the value of the effective rigidity equation (2.3)
is almost fixed at the critical value κc of the crumpling transition of the membrane without
self-avoidance.

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S Mori and S Komura

In [11], the range of the neighbourhood l and the critical diameter of the hard sphereσc are determined by the condition at which the crumpling transition occurs. In thiscase, l can be identified with lcrmp and u corresponds to σcd . Although the behaviourof h|X(s) − X(0)|2 i near the length scale lcrmp is not clear, we simply assume here thath|X(s) − X(0)|2 i = c2 s 2 with some constant c2 since the membrane is almost flat near thelength scale lcrmp . Then the integral in equation (2.3) can be estimated to behave as
Z
4−d
(2.4)
d2 s s 4 (c2 s 2 )−1−d/2 ∼ lcrmp
for d < 4 and we obtain the following relation,4−dκc = κbare + c3 ulcrmp(2.5)

where c3 is a positive constant. If we put κbare = 0, equation (2.5) corresponds to therelation obtained by Kantor and Kremer†. In fact, what they have obtained is ul 4 ∼ 1 at
d = 3.
Next we consider the effect of the negative bending rigidity (κbare < 0). If we assume
that the system exhibits the crumpling transition as a result of the complete cancellation
between the effective entropic rigidity and the negative rigidity, the length scale lcrmp should0
be the order of membrane size L. Therefore, the critical value of the negative rigidity κbareat which the crumpling transition occurs is determined by0= κc − c3 uL4−d .
κbare

(2.6)

0
depends both on u and L, as long as the crumpling
From this equation, we see that κbaretransition occurs. In the following section, we examine this possibility by Monte Carlosimulation.
3. Model and simulation method
In this section we describe the model for the polymerized membrane used in the computer
simulations and summarize the computational procedure. The model membrane being
embedded in three-dimensional space consists of hard spheres connected with flexible
bonds. As shown in figure 1, hard spheres are connected in a two-dimensional triangulararray and the polymerized membrane of finite size forms a hexagonal cluster. There are
N = (3L2 + 1)/4 spheres in such a membrane, where L is the number of spheres on the
diameter. For the present simulation, we employ a commonly used tethering interaction,i.e. the spheres interact with their nearest neighbours through the potential(√0
σ < r < 3σ
(3.1)
VNN (r) =∞otherwise.
This potential consists of a hard core repulsive region which prevents the neighbouring
spheres from occupying the same space and a finite region where the spheres can movefreely. In addition to this neighbouring potential, self-avoiding interactions have been
included by restricting the distance between all the pairs of the spheres to be greater thanσ . In order to change the flexibility of the membrane, we allow σ to take several values(< 1.0) although it is usually set equal to 1.0. In this case the model has the drawback thatthe self-intersection is not necessarily prohibited. The membrane can be flexible and wewill study such a case.
† The discrepancy between theoretical and numerical results is considered to be from the simple assumption for
the two-point function. If we assume that h|X(s) − X(0)|2 i = c2 s ν with ν < 2.0, the correspondence becomes
better.

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Figure 1. A model of the polymerized membrane with linear size L = 9.

We also take into account the bending energy in the model by introducing the repulsive
spring between next-nearest-neighbour spheres in the network [4]κ0 X|X(σ ) − X(σ 0 )|2 .
(3.2)
Hb = −
2 NNN
In this case, the summation is taken over all next-nearest neighbours in the network. So
far, the bending rigidity parameter κ 0 has been set as positive numbers, which implies thatthe bending energy forces the membranes to become flat. In contrast, we consider her ananti-ferromagnetic-like interaction, namely, the rigidity parameter κ 0 takes negative values.
If this interaction cancels the entropic bending energy completely, one might expect theoccurrence of a crumpling transition separating the flat phase and the crumpled phase. At
sufficiently large values of |κ 0 |, the membranes are expected to be in the folded state. We
have used the standard Monte Carlo procedure (Metropolis algorithm) to investigate thethermodynamic properties of self-avoiding polymerized membranes with negative bending
rigidity. The total Hamiltonian is the sum of equations (3.1) and (3.2) with the self-avoiding
interactions.
The random displacement of step size s 6 0.2 is attempted on a randomly selected
sphere in the network and this procedure is repeated until the system reaches the thermal
equilibrium state. Let T denote the absolute temperature and kB the Boltzmann constant.
The attempted movement is always accepted if the energy Et of the trial movement does notexceed the original energy E0 . If Et exceeds E0 , the move is accepted with the probability
exp[(E0 − Et )/kB T ]. The sequence of this procedure depends on the dimensionlessparameter κ = |κ 0 |/kB T which is the measure for the temperature scale. Since the tethering
potential equation (3.1) has zero or infinite energy, the corresponding probability is either
1 (accepted) or 0 (rejected). For the bending energy equation (3.2), one has to evaluate
the probability exp[(E0 − Et )/kB T ] for each value of κ. Due to the restriction of selfavoidance, the distances between all the pairs in the network must be checked during thesimulation. Thus the procedure takes quite a long time for the system to reach equilibrium
configurations.
The relaxation time τ is determined by the slowest mode corresponding to large-scale
motions. For phantom membranes, τ is given by the Rouse relaxation time τ ≈ N/s 2 .

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S Mori and S Komura

It takes a much longer time for self-avoiding membranes to reach the equilibrium state,
which is one of the difficulties in carrying out the simulation. In order to know the typicalrelaxation time, we have traced the behaviour of the eigenvalues of the inertia tensor whose
sum is equal to the square of the radius of gyration. Although each eigenvalue has different
relaxation time, the dynamics of the radius of gyration is dominated by the slowest time.
Through the analysis of auto-correlation functions, we found that the relaxation time is theorder of a few hundred Rouse times for L = 11, 13 and 17. For all of the runs, we havecollected at least 100 samples and in most cases 500 samples separated in time by such arelaxation time.
4. Results
We have calculated the statistical properties of polymerized membranes over the range
0.0 6 κ 6 9.0 (−9.0 6 κ 0 /kB T 6 0). We have first investigated the membranes of sizes
L = 11, 13 and 17 with the conventional diameter σ = 1.0. For the membranes of size
L = 17, unfortunately, equilibrium states were not realized for κ > 7.0 in our computer
system. To investigate the features of the membrane conformations, we have calculated theinertia tensor Iαβ defined by
Iαβ =

1 X
1 X
Xiα Xiβ − 2
Xiα Xjβ
N i
N i,j

(4.1)

where the summation is over the spheres for each configuration and the Greek subscripts
α and β denote the components of three-dimensional coordinates of the sphere. The
summation of three eigenvalues of the inertia tensor denoted by λ1 , λ2 and λ3 (λ1 6 λ2 6 λ3 )is equal to the squared radius of gyration Rg2 defined by
Rg2 =

1 X
|Xi − Xj |2 .
2N 2 i,j(4.2)

In the flat phase, the eigenvalues scale with the linear membrane size L as λ1 ∼ L2ν1 , andλ − 2 ∼ λ3 ∼ L2ν3 where ν3 ' 1.0. Hence the radius of gyration which is dominated by
the largest eigenvalue λ3 scales as Rg2 ∼ L2 . ν1 corresponds to the roughness exponent andsome computer simulations estimated its value as 0.5 ∼ 0.8 [4, 6].
In contrast to the flat phase, the membranes are isotropic in the crumpled or compact
phase. In this case, all the three eigenvalues and the squared radius of gyration scale withthe same exponent ν1 = ν2 = ν3 = ν < 1.0. We have looked at each eigenvalue to findthe scaling behaviour of the membranes because the used system sizes are rather small.
Figure 2 shows three eigenvalues of the inertia tensor as a function of κ for the membrane
size L = 13. It can be seen that all three eigenvalues decrease as κ increases. For small
κ, it is expected that the self-avoiding interaction dominates the negative bending forces.
Thus the membranes should be asymptotically flat. As κ increases, the flat phase becomes
unstable and the membranes are expected to be driven into folded states. for large κ, the twolargest eigenvalues λ2 and λ3 converges to the same value, while the smallest eigenvalue
λ1 approaches a constant value.
To investigate the qualitative measure of anisotropic configurations of membranes, wehave calculated the asphericity A defined as the ratio of the largest to the smallest eigenvalue
such thatA(κ, L) = hλ1 /λ3 i.

(4.3)

Self-avoiding polymerized membrane with negative bending rigidity

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Figure 2. Three eigenvalues of inertia tensor as a function of rigidity for the size L = 13.

Hereafter h· · ·i denotes the configurational average. Figure 3 depicts A(κ, L) for L = 11, 13
and 17. One can see that the asphericity is less than 0.2 for all κ. For fixed L, the asphericity
decreases to a minimal value and then increases as κ is increased. For all values of κ, A
decreases with increasing L. In the limit of L → ∞, we expect that A tends to zero
independent of κ. These results suggest that the membranes are anisotropic for all valuesof κ and the isotropically crumpled or collapsed phase will not appear.

Figure 3. Asphericity A as a function of rigidity parameter κ for L = 11, 13 and 17.

We have also attempted to determine the exponent sν1 and ν3 for each different valueof κ. The results are summarized in table 1. We find that ν3 is almost equal to 1.0 for all κvalues within numerical uncertainty. On the other hand, ν1 approaches zero as κ increases,
indicating that the smallest eigenvalue is independent of L and approaches a universal
non-zero constant at large κ. We have also checked that the relation ν2 = ν3 holds.

7446

S Mori and S Komura
Table 1. The exponents ν3 and ν1 obtained from data L = 11, 13 and 17.
κ

ν3

ν1

0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0

0.94 ± 0.01
0.95 ± 0.01
0.91 ± 0.01
0.97 ± 0.02
0.94 ± 0.03
1.03 ± 0.01
0.99 ± 0.01
0.84 ± 0.01

0.45 ± 0.09
0.45 ± 0.12
0.41 ± 0.11
0.19 ± 0.05
0.30 ± 0.07
0.05 ± 0.03
0.01 ± 0.01
0.07 ± 0.02

Figure 4. The equilibrium configuration of the hexagonal membrane with L = 17 for κ = 1.0.
This picture corresponds to the flat phase.

In figure 4, a typical equilibrium configuration of membrane size L = 17 for κ = 1.0 is
depicted. This picture corresponds to the flat phase. Figure 5 represents the case for κ = 7.0.
Now the membrane is anisotropically folded and forms a hexagonal thin plate with a finite
width. From these observations, fixing σ = 1.0, membranes with negative bending rigidity
exhibit a transition from the flat phase to the folded rigid flat states by increasing κ. Since
an intermediate crumpled phase does not appear, the crumpling transition does not occur.
In view of the above results, we could not find any evidence for the existence of theintermediate crumpled phase. In order to investigate the more flexible case, we have studied
membranes with smaller spheres, i.e. σ = 0.5 and 0.3. We have determined the exponents
ν3 and ν1 from the system sizes L = 11, 13 and 17. Table 2 summarizes the results forseveral values of κ. Again, ν2 is close to ν3 . Similar to the case of σ = 1.0, there is no
isotropic intermediate crumpled phase even if the size of hard spheres is reduced.
In our simulation, we have also obtained other thermodynamic functions such as internalenergy or specific heat. In contrast to the situation of phantom membranes with bendingSelf-avoiding polymerized membrane with negative bending rigidity

7447

Figure 5. Typical configuration of L = 17 membrane with κ = 7.0. The perspectives except
for the upper right picture are along the directions of the principal axes of the three eigenvalues.

energy [2], we only found a vague peak for the case of σ = 1.0 and no clear singularity
in the smaller sphere cases. Much more investigations are required in order to discuss thenature of the phase transitions.

5. Discussion
In the theoretical part of this paper, we have used the generalized Edwards model to describe
the flat phase of the polymerized membrane. Within the picture by Kantor and Kremer,
we have obtained the scaling relation. In addition, we have discussed the possibility of thecancellation between the entropic rigidity and the negative bending rigidity, which mightcause the crumpling transition.

7448

S Mori and S Komura
Table 2. Scaling exponents of maximum and minimum eigenvalues for diameter σ = 0.5 and
0.3.
σ = 0.5

σ = 0.3

κ

ν3

ν1

ν3

ν1

1.0
2.0
3.0
4.0
5.0
6.0

0.87 ± 0.02
0.74 ± 0.02
0.87 ± 0.03
0.90 ± 0.01
0.90 ± 0.02
0.89 ± 0.01

0.47 ± 0.12
0.46 ± 0.09
0.30 ± 0.11
0.12 ± 0.03
0.08 ± 0.02
0.08 ± 0.01

0.82 ± 0.01
0.73 ± 0.03
0.80 ± 0.01
0.78 ± 0.01
0.78 ± 0.02
0.81 ± 0.01

0.38 ± 0.11
0.30 ± 0.07
0.24 ± 0.08
0.18 ± 0.04
0.21 ± 0.02
0.23 ± 0.04

However, the results of the Monte Carlo simulation showed that, instead of the crumplingtransition, the membrane exhibits a phase transition from the flat phase to the rigid folded
flat phase. An intermediate crumpled phase does not appear. We have also investigated themembranes with more local flexibility consisting of smaller hard spheres (σ > 0.3). Evenin this case, the crumpled phase has not been observed within our simulation. From theseresults, we conclude that the phase transition from the flat phase to the rigid folded flat
phase occurs before the negative rigidity cancels the entropic rigidity completely. At thispoint, the flat phase of the self-avoiding membrane is different from that of the phantom
membrane, which shows the crumpling transition.
We thing that complete cancellation occurs only in the very flexible membrane with an
attractive interaction and hence the continuous transition is expected. Theoretical approaches
to describe the transition have incorporated the effect of the self-avoiding interaction throughthe bending rigidity and treated only the stability of the flat phase of the phantom membrane[20]. In such a treatment, both negative rigidity and an attractive interaction can cause the
crumpling transition. Our computer simulation, however, has made it clear that we cannot
incorporate the effect of the self-avoiding interaction through the effective bending rigidity
even if the membrane is flexible. Therefore, one must be careful about the conclusion
which is based on the identification of the flat self-avoiding membrane with the flat phantom
membrane.
Up to now, theoretical arguments concerning the stability of the flat phase have been
limited to phantom membranes. As far as the self-avoidance is concerned, we can only
discuss the existence region of the crumpled phase. These points are the most difficult
aspects in studying the polymerized membranes theoretically. That is, if one predicts someinstability in the flat phantom membrane, such phenomenon may not occur in the selfavoiding membrane.
Finally, we comment that the phase transition under consideration is different from theone discussed by David, Francesco and Guitter [22, 23]. What they studied is the phasetransition of the folding square lattice with negative bending rigidity and they showed that
there is a first-order transition separating the flat phase and the folded compact state. Since
hard cores are of finite size in our simulation, the membrane takes the folded flat state rather
than the folded compact state. However, since our simulations have been performed onrather small system sizes, we could not accurately determine the properties of the transition.
In addition, the size of the hard spheres should be smaller in order to conclude whether
or not complete cancellation occurs for a very flexible membrane. We think that more
extensive simulations should be performed in the future.

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Acknowledgments
The authors thank K Kawanishi for collaborations and useful discussions. One of the authors(SM) thanks M Wadati for useful discussions and critical reading of the manuscript. He
also thanks J Nishimura for useful discussions that motivated this work.
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