Untitled
Balance of forces in simulated bilayers.
J. Stecki
Department III, Institute of Physical Chemistry,
Polish Academy of Sciences,
ul. Kasprzaka 44/52, 01-224 Warszawa, Poland
November 21, 2018
Dedicated to Professor Keith Gubbins on the occasion of his 70th birthday
Abstract
Two kinds of simulated bilayers are described and the results are reported for lateral
tension and for partial contributions of intermolecular forces to it. Data for a widest
possible range of areas per surfactant head, from tunnel formation through tensionless
state, transition to floppy bilayer, and its disintegration, are reported and discussed.
The significance of the tensionless state, is discussed.
Keywords: bilayer, lateral tension, buckling transition, tensionless.
http://arxiv.org/abs/0704.1477v3
I. Introduction: two-dimensional sheets in three dimensions.
Great interest in the properties of membranes and bilayers is reflected in the very
large volume of work, reporting simulations of a variety of models and simulation meth-
ods. However, simulations including a range of areas are relativery rare and those
including a widest possible range of areas - rarer still1−6. Most papers concentrate on
the ”tensionless state” for which the lateral tension vanishes.
The latter is the direct counterpart of the surface (interfacial) tension between two
liquids and in fact it is computed according to the same Kirkwood-Buff formula7.
At this stage we must mention that the physico-chemical properties of a sheet of
surfactant molecules, forming a bilayer, are positively exotic. Compare with a planar
inerface between e.g. a liquid and its vapor or two immiscible liquids. Then the surface
tension γ is defined by
(∂F/∂A)V = γ (1.1)
where F is the free energy, A the area, V the volume and the constancy of particle
numbers and temperature T is understood. But γ itself is independent of the area; it
is a material property, a function of state. In a shocking contrast to that, the lateral
tension of a bilayer (again defined after (1.1)) is area-dependent; moreover in the same
system as to composition, density, and temperature, γ(A) can be positive or negative.
Its zero defines the ”tensionless state” which is of particular interest and some physical
significance.
Other properties, such as internal energy U or correlation functions including the
structure factor S(q), also display the area dependence, if a sufficiently wide interval of
areas is investigated.
The peculiar shape of the bilayer isotherm i.e. of the function and plot of γ(A), as
shown and discussed in Section II, raises the question as to how it originates. A partial
answer to that is provided by the split of γ into its components.
Not all simulations are ”atomistic” i.e. not all simulations construct the inter-
molecular total energy E({r}) from model intermolecular potentials depending only on
the spatial coordinates of the constituent particles (atoms or molecules), which energy
is then used in a Monte-Carlo or Molecular Dynamics simulation scheme. One of the
advantages of an atomistic simulation is the possibillity of examining the role of the con-
stituent components such as surfactant heads, solvent molecules, and surfactant tails.
This is put to use in this paper, in which we report the split of lateral tension.
Our results are reported for two kinds of simulated bilayers; these are defined in
Section II.
In Section III we show the split of lateral tension into components and we discuss
the physical meaning of the ”tensionless state”.
In Section IV we return to the discussion of bilayer properties and deeper dis-
tinctions between interfaces and membranes or bilayers. The discussion introduces the
concept of constraint.
It also appears to be necessary to point out that a very important category of
objects, called ”vesicles”, is entirely outside the realm of simulated bilayers. Merging
”membranes and vesicles” in one sentence greatly oversimplifies matters, because vesi-
cles, like spherical micelles floating in a solvent, enclose a finite volume. The bilayers
do not.
The ”widest possible range of areas” of simulated bilayer, mentioned above, refers
to the limits of existence of a stable bilayer. When the imposed area is extended too
much, the bilayer recedes creating a spherical hole or rather a tunnel filled with solvent
joining the two sides of the solvent volume. When compressed, the bilayer flips into
a ”floppy” state (see Section III) which then if compressed further, disintegrates into
structureless object(s) which cannot be called a bilayer any more.
Section II. Two kinds of bilayers.
A planar bilayer is formed by amphiphilic surfactant molecules made each of a
hydrophilic polar ”head” and hydrophobic non-polar hydrocarbon chains as ”tails”.
The tails form the center of the bilayer and the two outer layers of heads separate the
tails from the polar solvent (water). Therefore the cohesive energy density (CED)8 is
high (solvent), high (first layer of heads), low (tails), low (second layer of tails), high
(second layer of heads), high (solvent on the other side of the bilayer).
The same forces and the same preferences operate in the formation of micelles of
various shapes. Normally the solvent is polar, of high CED, most commonly water,
and the micelles shield their hydrocarbon tails by an outer layer of heads in contact
with water. However, there exist rare examples of reverse micelles which are formed
by amphiphilic molecules in a non-polar solvent of low CED. In reverse micelles, the
(hydrocarbon) tails form the outward shell and the (polar) heads are in the center of
the micelle. By analogy with these I have2,4 constructed in simulations reverse bilayers
which are formed in a non-polar solvent of low CED. On crossing the reverse bilayer
along the z-direction perpendicular to the x-y plane of the bilayer, the sequence is now:
solvent, tails, heads, heads, tails, solvent or, in terms of CED : low,low,high,high,low,low.
These cases can be and have been modelled with the use of Lennard-Jones (6-12)
potential (LJ) with the minimum of 3 kinds of particles: s(solvent), h (head), t (tail).
With 6 energies, 6 collision diameters, and 6 cut-off radii, various simplifications have
been used in bilayer simulations with the LJ potentials1−4,5,9−13.
In our work the solvent was made of structureless spherical particles; freely jointed
chains of such particles of the same size have modelled the surfactant molecules making
up the bilayer.
We found certain regularities in the stability of the model bilayers. We found
that the chain lengths of the tails can be shortened even down to unity, the molecule
becomes then a dimer (”h-t”) made of two particles. We also found that it is worthwhile
to keep the presence of the solvent; in some simulations very unusual, in fact unphysical,
intermolecular forces were required in order to ensure existence (in simulation) of stable
bilayers14 made of trimers. The reverse bilayers made of trimers were successfully
simulated without a solvent 15, although it appears this was not followed with further
work. We also found that longer chains stabilize the bilayer; of the tail lengths l =
1, 4, 8 the dimers were difficult to stabilize without an extra repulsion (replacing the
hydrophobic effect) from the solvent whereas, for longer chains, this extra repulsion was
not necessary for stability. Chains with l = 8 produced stable bilayers with great ease.
The modelling of solvent as made of spherical structureless particles interacting with
a central potential creates a certain conceptual difficulty because of the hydrophobic
effect. It has been partially resolved by an introduction9 of an additional repulsive
force between the solvent and the tails, e.g. of the form c/rn with c > 0 and n=10 or
more9−13. A temperature dependence would be needed too, to take into account the
entropic effects of the hydrophobicity.
In the case of a reverse bilayer, the reasons for its formation are mostly energetic:
a pair of heads (now inside the bilayer) attract each other more strongly than any other
pair and there is no hydrophobic effect since the solvent is non-polar with a weak CED.
Section III. The Lateral Tension and its Components.
A bilayer originally planar, lies in the x−y plane in the middle of a simulation box
which is a parallelepiped of volume V = Lx×Ly×Lz; its area is A = LxLy. It is made of
Nm molecules which contain Nm heads, originally Nm/2 in each of the two monolayers,
so that the ”area per head” a = A/(Nm/2). The bilayer is surrounded ”from above”
and ”from below” by Ns molecules of the solvent. The overall density is ρ = N/V with
N = Ns + (l + 1)Nm. The lateral tension is defined
7 as the response to the process of
increasing the area at constant volume. With Lx → Lx + δLx, Ly → Ly + δLy, Lz →
Lz + δLz, the constraint of constant volume requires δLx/Lx + δLy/Ly + δLz/Lz = 0
and the Kirkwood-Buff equation follows
γA = Lz(2pzz − pxx − pyy)/2 (3.1)
with the known definitions
pαα =<
> (3.2)
valid for a rectangular box. The average is a canonical average at given temperature T ,
all N ′s, A, and V . In the simulation, for each state point of T, V,Nm, Ns, A we obtain
one value of γ(A). In our work, Lx = Ly; changing Lx to any new value, we calculate
the new value of Lz needed to keep the volume V = LxLxLz at exactly the same value.
Each time γ is computed, it is a sum of binary terms and these are grouped into
partial sums so as to produce the split
γ = γss + γsb + γbs + γbb. (3.3)
Here the indices ss refer to the solvent-solvent part, the indices sb and bs to the solvent-
bilayer part, and bb refers to the bilayer-bilayer part. There is no approximation in-
volved.
Fig.1 shows an example of a bilayer made of chain molecules of tail length l = 4 plus
one more particle as the head, immersed in a solvent at high liquid-like density. Each
γαβ is plotted against the area per head a; the lines joining the data points are there
to guide the eye, only. The contribution of solvent-solvent pairs is about 0.2-0.3 and
almost constant i.e. independent of a. All pairs made of solvent particle and any particle
of any bilayer molecule make up the sb or bs contribution; it is also positive, varying
smoothly with a and taking values between 0.5 and 1.05 . The bb contribution shows the
characteristic break into two curves, nearly linear with a, but one with positive slope
and the other with a negative one. This pattern is transferred to the sum, eq.(3.3),
total γ(a).
Fig.2 shows the effect of size for the same chain-length l = 4. The solvent-solvent
contribution γss(a) is practically independent of size, as is nearly so the solvent-bilayer
part. The bb part and, consequently, the total γ display the break into two regions; the
floppy part strongly affected by size and the extended region negligibly so. The bigger
size shows flatter γ(a), still negative, but closer to zero. Lowest values of γ(a), which
may be taken as the border between the floppy region and the extended region, are
negative but lower value corresponds to smaller size.
These remarkable patterns are repeated for chain length l = 8, with breaks much
more pronounced. Fig.3 shows γαβ(a) for two sizes; the intermediate size is omitted,
just as was in Fig.2, in order to make the plot clearer. Again the ss part is constant and
size-independent, the sb part nearly so, and the total γ along with the sb part, show
these remarkable features: breaks and abrupt changes of slope. For large a the slope is
positive and the size-dependence negligible. The transition to the floppy region occurs
near a = 1.8 for the smaller system and the slope dγ/da changes sign; the break is very
pronounced. The transition is shifted to a = 1.9 in the bigger system, the break is more
pronounced in the bb part than in the total γ. Most characteristically in the bigger
system γ(a) is flat and near zero (though still always negative) in the floppy region.
The tensionless point appears to lie on the r.h.s. curve ( extended bilayer region) in
this case.
All bilayer isotherms i.e. all plots of γ(a) we have obtained1−4 show the same
pattern of two curves joined at some negative value of the lateral tension γ. We interpret
the region of lower a as the region of the floppy state of the bilayer which escapes into
the third dimension, buckles, and takes a fuzzy and irregular form. A gently fluctuating
planar bilayer is found in the other region at higher a’s. Generally, the region of extended
bilayer includes all states with positive γ. The crossing of the isotherm with the a-axis
marks the tensionless state . In the plots of Fig.1-3 the region of the extended bilayer
has a large positive slope of γ(a). Such positive slopes have been found independently5,6
and also earlier9−13. The entire range including the floppy region is found only in few
references2−6. The corresponding split of the lateral tension is found in this paper only.
The status of the tensionless state appears now in a new light, as an accidental event
resulting from the algebraic sum of positive and negative contributions after (3.3). The
partial contributions suggest that the break point dividing the two branches of γ(a) (or
γbb(a) ) is a truly characteristic point which ought to have a physical significance, rather
than γ(a0) = 0.
The shortest possible tail of one particle is present in a dimer. We have attempted
and partially succeeded1 in simulating bilayers made of dimers, but these appeared to
be only imperfectly stabilized by strong head-head and head-solvent interactions or/and
the extra-repulsion introduced artificially to mimick the hydrophobic effect, as invented
earlier9. Possibly at other densities and temperatures the simulations would have been
more encouraging. We do not show these results. As mentioned in Section II, we turned
to the new case of reverse bilayers formed in a weakly interacting solvent and succeeded
in producing (in simulation) stable reverse bilayers of amphiphilic dimers2,4. Fig.4 and
5 show a selection of several characteristic cases, each for two sizes. These cases are: (a)
extra repulsion added, (b) no extra repulsion, (c) no extra repulsion and longer range
of attractive forces. In cases (a) and (b) the cut-off was the generally used r = 2.5,
in the case (c) it was r = 3.2. For clarity, we split the data into two Figures: Fig.4
shows γ total only, for all 5 systems whereas Fig.5 shows the components ss, sb, and bb.
All curves for all 5 systems in Fig.4 show a pattern similar to γ(a) in Fig.3 (for l = 8).
This characteristic pattern is : almost linear rise with a for larger a and almost constant
negative γ for small a. Such a description fits bigger systems better. The biggest system
of an area about 100× 100 of reverse dimers case (a) is shown in Reference 4. As can
be seen from Fig.4, the plot again strongly suggests a transition between two forms of
the bilayer and certainly the existence of two regions1−5. In the floppy region γ < 0
and the bigger the system the closer to zero γ becomes. Fig.5 shows the components of
γ but without γ itself. The pattern is similar to those shown in Fig.1,2,3 except that
the ss contribution to γ is now negative, between -0.6 and -0.2, the sb contribution
much larger, between 2. and 3.6. The bb contribution is, for all sizes, practically linear
in a with again negative slope and negative value in the region of ”floppy” state. The
transition appears softer than for chain molecules, especially those with longer tails
where the breaks were sharp (see Fig.3) in small systems.
The status of the tensionless state appears to be relegated to the category of acci-
dental coincidences. An analogy may be drawn with the equation of state of liquid in
equilibrium with its saturated vapor; certainly P = 1 is a particular point on the vapor
pressure curve, but it has no particular physical meaning - in contrast with e.g. the
triple point. The only physical consequence of zero in lateral tension is the special form
of the structure factor10,1−4,5. However, even this has been questioned recently12,4. In a
detailed investigation of the structure factor S(q) of a simulated bilayer, which should,
for small q, asymptotically vary as 1/S ∼ kq4 + gq2 where k is the rigidity constant
and g - the surface tension, we found g ≥ γ systematically4. The explanation advanced
elsewhere12 is that γδA is not the correct work of deforming a bilayer initially planar,
but gδAtrue is. Here Atrue is the true area of the interface or bilayer, obtained by
following its surface, whereas A ≡ LxLy is the projected area. Since S(q) measures
the spontaneous fluctuations, gq2 is correct and γq2 is not. On the other hand, when
measuring the response of the system to an imposed change in the projected area, γδA
is the correct free energy increment. The increase in the projected area is under control
of the experimentalist; the change of the true area induced by the twists and bends of
the bilayer, is not. By taking these arguments into account, the correct definition of the
tensionless state was assumed12 to be g = 0 and interpolation procedures were used12
to determine the new correct tensionless state. Whether the details of the advanced
explanation12 are correct or not, it is obvious that g parametrizes the spectrum S(q)
of spontaneous shape fluctuations. The difference between g and γ was also discussed
elsewhere13,6,15. The zero of γ(a) is unambiguous in small systems; in large system of
reverse bilayers4 it falls within the narrow region of fastest change in slope, signalling
the transition of the bilayer to the floppy state. Then the precision of its determination
is doubtful. Some of this ambiguity is seen in Fig.4 and in Figures of Reference 4.
Section IV. Discussion.
We have noted in the Introduction that, to a physical chemist familiar with the
interfaces, some properties of a bilayer are truly exotic. Let us list these properties: (1)
the derivative given by eq.(1.1) depends on the specific area; (2) it can be negative;
(3) extension of the area leads to a hole or tunnel as the bilayer resists further extension;
(4) there is distinction between actual area and the projected area but the bilayer keeps
its true area nearly constant; (5) as there are two areas, there are two interfacial
tensions.
Definitely this is not what we learn from classical textbooks on thermodynamics.
The feature (1) has been seen in all simulations, including these over a wide range of
(projected) areas1−6,9−15. The feature (2) has also been seen2−6,11,12,13,15. The feature
(3) has not only been seen2,4,15 but also investigated15. Distinction (4) has been the
subject of papers13,5 specifically devoted to it. Finally, the existence of an(other) surface
tension(s), besides the lateral tension, has been repeatedly surmised5,6,11,13; equations
have even been derived14.
The explanation of such specific behaviour ultimately lies with intermolecular forces
and steric effects which any future theory will have to take into account. The bilayers are
made of amphiphilic molecules, each containing a hydrophilic part and a hydrophobic
part (or more generally, ”solvophilic” and ”solvophobic”). These molecules use their
very special setup of intermolecular forces to create a stable sheet of constant mass (and
approximately constant area). This creates a new situation with a constraint of given
total area almost conserved when the projected area is varied by varying the simulation
The constraint of constant mass (or anything related to it) is entirely absent in
fluctuating interfaces. The molecular mechanism of fluctuations of shape of e.g. a liquid-
vapor interface, involves diffusion from either phase; a local protrusion or excursion of
the interface takes place not only by deforming the existing surface but also by absorbing
or releasing molecules from/to either phase.
The surface of a perfect crystal in equilibrium with its vapor is a good example; all
shape fluctuations are due to evaporation or condensation.
Such processes clearly take part in shape fluctuations of a liquid surface as well.
Hence an interface is a system open with respect to particle number. It is not so
with the bilayer. Not only the solvent is virtually absent from the bilayer, but also the
surfactant molecules making the bilayer never leave it. In principle the surfactant must
be present in the solvent but the bilayer changes shape by changing the position of its
molecules, not by exchanging them with the solvent solution. The shape fluctuations
take place under the constraint of constant number of particles, i.e. constant mass.
A constant mass is implicit in the elastic theory of solids and the links to that field
have already been explored5,10. Conversely, application of the elastic theory of solids to
liquid interfaces, (the so called ”drumhead model”), is clearly not quite correct by not
allowing for diffusion described above.
We discuss now the new evidence for the transition. All plots of γbb(a) and γ(a)
show some kind of break into two branches, one corresponding to the ”floppy” bilayer.
Although the word ”floppy” or ”buckling” appeared occasionally4,5, in fact the status
of the floppy region and of the transition into that region are not very clearly estab-
lished. There are suggestive analogies with the crumpling transition of solid (tethered)
membranes or with coiling transitions of polymers. This would suggest it is a first-order
transition. The breaks in γbb(a) (see Section III) strongly support such a hypothesis
also suggesting it is an internall reorganization of the bilayer structure, in which the
solvent plays a secondary role.
Further support is found with the radius of gyration of the bilayer, shown in Fig.6;
the break in slope is very clear.
Finally, the issue of the other surface tension is not fully resolved. As we discussed
above in Section III, one may define two surface tensions, one coupled to the projected
area and the other coupled to something else, perhaps the true area. The first one is
the lateral tension and the other may be that given by the virial expression derived or
rather proposed recently14 or g parametrizing the structure factor S(q). (See the end
of Section III). These last two may be equal or may be not.
I hypothesize that g may be related to that given by the Zwanzig-Triezenberg (Z-
T) equation16,7 - because the latter is derived by considering a spontaneous fluctuation.
Now, in simulations of liquid-vapor interface, I have determined17 the surface tension
from both expressions and these two numbers were equal within 0.01 percent. Thus I
have confirmed the validity of the Schofield proof7 of the equivalence. Since the (Z-T)
equation is derived in a grand-canonical ensemble, g can only be ”related to it” as stated
above; the constraint would have to be introduced.
We finally remark that the constraint appears as a most natural thing in description
of polymers when the polymer string has a constant number of segments. Indeed a model
one-dimensional membrane/bilayer embedded in two dimensions, is identical to a model
of a polymer string on a plane.
Appendix. Details of the Model Used .
The simulations themselves were in the past and also recently based on the Verlet
leap-frog algorithm with Nose-Hoover thermostat and were never shorter than half a
million steps, about 0.9 million in almost all runs. The intermolecular potential energy
was a sum over pairs of particles; all pairs were interacting Lennard-Jones 6-12 potentials
u(r) = 4ǫ
(σ/r)12 − (σ/r)6
with different parameters ǫ and σ and with different cutoffs rc. A constant u0 may be
added at will and this makes no difference to forces. u(r) has a minimum at r = r∗
of depth u(r∗) = −ǫ. There are two kinds of spherical particles ”a” and ”b”. The
solvent was made of particles ”a” and the amphiphilic molecules of chains ”abb..b”.
The collision diameter σ was common to all pairs. The cutoff parameter was rc = 2.5σ
for all like pairs and rc = r
∗ for unlike-pair intermolecular force. All potentials were
cut-and-shifted to assure continuity of force. Molecular Dynamics algorithms use forces.
The bilayer-forming amphiphilic molecules were freely-jointed chains and each in-
tramolecular bond was made of two LJ halves thus:
ubond = u(r) r < r
ubond = u(2r
∗ − r) (r∗ < r < 2r∗)
ubond = +∞ r > 2r
for either ”a-b” or ”b-b” bond. As explained in the text, for the special case of ”reverse
bilayers” made of dimers, a− b, the ”bb” intermolecular attractions were enhanced by
making ǫbb >> ǫaa. Thus ”b”’s became ”heads” located in the center of the bilayer.
Acknowledgements. This paper is dedicated to Professor Keith Gubbins on
the occasion of his 70th birthday. The author gratefully acknowledges most useful
interactions with Professor Keith Gubbins while visiting him at Cornell University.
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Captions to Figures
Caption to Figure 1
The general pattern of splits of lateral tension γ according to (3.3) is shown as
plots against the area-per-head a; here for an intermediate size of simulated bilayer
with Nm = 1800 molecules with tails l=4 segments long, with all collision diameters σ
and all energies ǫ equal , at T = 1., Ns+(l+1)Nm = 49000, liquid density ρ = 0.89204.
The ss contribution is marked with squares, the sb contribution - with stars, the bb
contribution - with crosses , and the sum - with black circles . All data are quoted
everywhere in LJ units reduced by the energy depth ǫ and collision diameter σ. The
lines are there to guide the eye.
Caption to Figure 2
The split of lateral tension γ plotted against the area-per-head a for two sizes (SM)
and (B) of simulated bilayer, made of Nm = 450 (SM) and Nm = 3973 (B) molecules
with tails l=4 segments long, with all collision diameters σ and all energies ǫ equal, at
T = 1., Ns + (l + 1)Nm = 12250 (SM) and 105134 (B), liquid-like density ρ = 0.89204.
With the notation of eq.(3.3), the contribution ss is shown with triangles (SM) and
squares (B) near 0.2-0.3; sb - plus signs (SM) and stars (B) near 0.5-0.7; bb - open
circles (SM) and crosses (B) with negative slope for small a and positive slope for large
a. The sum after (3.3), γ, is shown with thick lines, black squares (SM), and black
circles (B). See text and Caption to Fig.1. Compare with Fig.3.
Caption to Figure 3
The split of lateral tension γ plotted against the area-per-head a for two sizes (SM)
and (B) of simulated bilayer, made of Nm = 450 (SM) and Nm = 3973 (B) molecules
with tails l=8 segments long, with all collision diameters σ and all energies ǫ equal , at
T = 1., N = Ns + (l+ 1)Nm = 14050 (SM) and 121026 (B), density ρ = 0.89204. With
the notation of eq.(3.3), the contribution ss is shown with triangles (SM) and squares
(big) near 0.2-0.3; sb - with plus signs (SM) and stars (big) near 0.5-1.1; bb - with open
circles (SM) and crosses (big). Note the negative slope for small a and positive slope for
large a. The sum after (3.3) is shown with thick lines, black squares (SM), and black
circles (big). See text. Compare with Fig.2 and note: the unique breaks in γsb, almost
constant slope in the region of floppy bilayer, translated into zero slope in total γ.
Caption to Figure 4
Total lateral tension γ of reverse bilayers made of dimer molecules plotted against
area-per-head a for biggest (B) or smallest (SM) size, and 3 cases: (a) extra repulsion
added and head-head attraction enhanced (H-H); (b) only (H-H); (c) the cutoff of at-
traction is a long-range rc = 3.2, not rc = 2.5; otherwise (a). The other parameters are:
T = 1.9, ρ = 0.89024, Nm = 2238 dimers, N = Ns+2Nm = 40000(SM) and Nm = 5760
dimers, N = Ns + 2Nm = 160000 (B). These had areas near 36× 36 and near 55× 55.
The points are: open squares - Ba; plus signs - SMa; black squares - Bb; crosses -SMb;
stars -SMc. The lines are there to guide the eye only. The deviation of data-point at
a ∼ 1.13 (black square) is due to the tunnel (hole). see Fig.5 and text.
Caption to Figure 5
The split of lateral tension after (3.3) of systems shown in Fig.4. See caption to
Fig.4 for parameters. Here the points are: case ”Ba” contribution ss - open squares,
sb - black triangles, bb - pentagons; case ”SMa” and ss - plus signs, sb - open circles,
bb -black triangles; case ”Bb” and ss - black squares, sb - open triangles, bb -black
pentagons; case ”SMb” and ss - crosses, sb - black circles, bb -diamonds; case ”SMc”
and ss - stars, sb - open triangles, bb - black diamonds. All ss contributions cluster
near -0.5, all sb are > 2., and all bb show two regions interpreted as floppy bilayer and
extended bilayer. See text.
Caption to Figure 6
One example of the radius of gyration for the same bilayer whose γ is plotted in
Fig.1, also plotted against area-per-head a. It appears to support, along with other
correlations3,4, the hypothesis of a discontinuous transition between the extended and
floppy bilayer.
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1
1.4 1.6 1.8 2 2.2 2.4 2.6
0.9 0.95 1 1.05 1.1 1.15 1.2
0.9 0.95 1 1.05 1.1 1.15 1.2
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

  Two kinds of simulated bilayers are described and the results are reported
for lateral tension and for partial contributions of intermolecular forces to
it.Data for a widest possible range of areas per surfactant head, from tunnel
formation through tensionless state, transition to floppy bilayer,to its
disintegration, are reported and discussed. The significance of the tensionless
state, is discussed. Conclusions: (1) the tensionless state is a
coincidence;(2) the transition from extended to floppy bilayer occurs nearby
and has hallmarks of a phase transition (3) there is no theory of that
transition.(4)The lateral tension of the floppy bilayer scales with size; that
of the extended bilayer does not depend on size. (4) The drumhead model not
appropriate for interfaces as these fluctuate via diffusion.(5) The radius of
gyration also! shows a discontinuity.

Introduction: two-dimensional sheets in three dimensions.
Great interest in the properties of membranes and bilayers is reflected in the very
large volume of work, reporting simulations of a variety of models and simulation meth-
ods. However, simulations including a range of areas are relativery rare and those
including a widest possible range of areas - rarer still1−6. Most papers concentrate on
the ”tensionless state” for which the lateral tension vanishes.
The latter is the direct counterpart of the surface (interfacial) tension between two
liquids and in fact it is computed according to the same Kirkwood-Buff formula7.
At this stage we must mention that the physico-chemical properties of a sheet of
surfactant molecules, forming a bilayer, are positively exotic. Compare with a planar
inerface between e.g. a liquid and its vapor or two immiscible liquids. Then the surface
tension γ is defined by
(∂F/∂A)V = γ (1.1)
where F is the free energy, A the area, V the volume and the constancy of particle
numbers and temperature T is understood. But γ itself is independent of the area; it
is a material property, a function of state. In a shocking contrast to that, the lateral
tension of a bilayer (again defined after (1.1)) is area-dependent; moreover in the same
system as to composition, density, and temperature, γ(A) can be positive or negative.
Its zero defines the ”tensionless state” which is of particular interest and some physical
significance.
Other properties, such as internal energy U or correlation functions including the
structure factor S(q), also display the area dependence, if a sufficiently wide interval of
areas is investigated.
The peculiar shape of the bilayer isotherm i.e. of the function and plot of γ(A), as
shown and discussed in Section II, raises the question as to how it originates. A partial
answer to that is provided by the split of γ into its components.
Not all simulations are ”atomistic” i.e. not all simulations construct the inter-
molecular total energy E({r}) from model intermolecular potentials depending only on
the spatial coordinates of the constituent particles (atoms or molecules), which energy
is then used in a Monte-Carlo or Molecular Dynamics simulation scheme. One of the
advantages of an atomistic simulation is the possibillity of examining the role of the con-
stituent components such as surfactant heads, solvent molecules, and surfactant tails.
This is put to use in this paper, in which we report the split of lateral tension.
Our results are reported for two kinds of simulated bilayers; these are defined in
Section II.
In Section III we show the split of lateral tension into components and we discuss
the physical meaning of the ”tensionless state”.
In Section IV we return to the discussion of bilayer properties and deeper dis-
tinctions between interfaces and membranes or bilayers. The discussion introduces the
concept of constraint.
It also appears to be necessary to point out that a very important category of
objects, called ”vesicles”, is entirely outside the realm of simulated bilayers. Merging
”membranes and vesicles” in one sentence greatly oversimplifies matters, because vesi-
cles, like spherical micelles floating in a solvent, enclose a finite volume. The bilayers
do not.
The ”widest possible range of areas” of simulated bilayer, mentioned above, refers
to the limits of existence of a stable bilayer. When the imposed area is extended too
much, the bilayer recedes creating a spherical hole or rather a tunnel filled with solvent
joining the two sides of the solvent volume. When compressed, the bilayer flips into
a ”floppy” state (see Section III) which then if compressed further, disintegrates into
structureless object(s) which cannot be called a bilayer any more.
Section II. Two kinds of bilayers.
A planar bilayer is formed by amphiphilic surfactant molecules made each of a
hydrophilic polar ”head” and hydrophobic non-polar hydrocarbon chains as ”tails”.
The tails form the center of the bilayer and the two outer layers of heads separate the
tails from the polar solvent (water). Therefore the cohesive energy density (CED)8 is
high (solvent), high (first layer of heads), low (tails), low (second layer of tails), high
(second layer of heads), high (solvent on the other side of the bilayer).
The same forces and the same preferences operate in the formation of micelles of
various shapes. Normally the solvent is polar, of high CED, most commonly water,
and the micelles shield their hydrocarbon tails by an outer layer of heads in contact
with water. However, there exist rare examples of reverse micelles which are formed
by amphiphilic molecules in a non-polar solvent of low CED. In reverse micelles, the
(hydrocarbon) tails form the outward shell and the (polar) heads are in the center of
the micelle. By analogy with these I have2,4 constructed in simulations reverse bilayers
which are formed in a non-polar solvent of low CED. On crossing the reverse bilayer
along the z-direction perpendicular to the x-y plane of the bilayer, the sequence is now:
solvent, tails, heads, heads, tails, solvent or, in terms of CED : low,low,high,high,low,low.
These cases can be and have been modelled with the use of Lennard-Jones (6-12)
potential (LJ) with the minimum of 3 kinds of particles: s(solvent), h (head), t (tail).
With 6 energies, 6 collision diameters, and 6 cut-off radii, various simplifications have
been used in bilayer simulations with the LJ potentials1−4,5,9−13.
In our work the solvent was made of structureless spherical particles; freely jointed
chains of such particles of the same size have modelled the surfactant molecules making
up the bilayer.
We found certain regularities in the stability of the model bilayers. We found
that the chain lengths of the tails can be shortened even down to unity, the molecule
becomes then a dimer (”h-t”) made of two particles. We also found that it is worthwhile
to keep the presence of the solvent; in some simulations very unusual, in fact unphysical,
intermolecular forces were required in order to ensure existence (in simulation) of stable
bilayers14 made of trimers. The reverse bilayers made of trimers were successfully
simulated without a solvent 15, although it appears this was not followed with further
work. We also found that longer chains stabilize the bilayer; of the tail lengths l =
1, 4, 8 the dimers were difficult to stabilize without an extra repulsion (replacing the
hydrophobic effect) from the solvent whereas, for longer chains, this extra repulsion was
not necessary for stability. Chains with l = 8 produced stable bilayers with great ease.
The modelling of solvent as made of spherical structureless particles interacting with
a central potential creates a certain conceptual difficulty because of the hydrophobic
effect. It has been partially resolved by an introduction9 of an additional repulsive
force between the solvent and the tails, e.g. of the form c/rn with c > 0 and n=10 or
more9−13. A temperature dependence would be needed too, to take into account the
entropic effects of the hydrophobicity.
In the case of a reverse bilayer, the reasons for its formation are mostly energetic:
a pair of heads (now inside the bilayer) attract each other more strongly than any other
pair and there is no hydrophobic effect since the solvent is non-polar with a weak CED.
Section III. The Lateral Tension and its Components.
A bilayer originally planar, lies in the x−y plane in the middle of a simulation box
which is a parallelepiped of volume V = Lx×Ly×Lz; its area is A = LxLy. It is made of
Nm molecules which contain Nm heads, originally Nm/2 in each of the two monolayers,
so that the ”area per head” a = A/(Nm/2). The bilayer is surrounded ”from above”
and ”from below” by Ns molecules of the solvent. The overall density is ρ = N/V with
N = Ns + (l + 1)Nm. The lateral tension is defined
7 as the response to the process of
increasing the area at constant volume. With Lx → Lx + δLx, Ly → Ly + δLy, Lz →
Lz + δLz, the constraint of constant volume requires δLx/Lx + δLy/Ly + δLz/Lz = 0
and the Kirkwood-Buff equation follows
γA = Lz(2pzz − pxx − pyy)/2 (3.1)
with the known definitions
pαα =<
> (3.2)
valid for a rectangular box. The average is a canonical average at given temperature T ,
all N ′s, A, and V . In the simulation, for each state point of T, V,Nm, Ns, A we obtain
one value of γ(A). In our work, Lx = Ly; changing Lx to any new value, we calculate
the new value of Lz needed to keep the volume V = LxLxLz at exactly the same value.
Each time γ is computed, it is a sum of binary terms and these are grouped into
partial sums so as to produce the split
γ = γss + γsb + γbs + γbb. (3.3)
Here the indices ss refer to the solvent-solvent part, the indices sb and bs to the solvent-
bilayer part, and bb refers to the bilayer-bilayer part. There is no approximation in-
volved.
Fig.1 shows an example of a bilayer made of chain molecules of tail length l = 4 plus
one more particle as the head, immersed in a solvent at high liquid-like density. Each
γαβ is plotted against the area per head a; the lines joining the data points are there
to guide the eye, only. The contribution of solvent-solvent pairs is about 0.2-0.3 and
almost constant i.e. independent of a. All pairs made of solvent particle and any particle
of any bilayer molecule make up the sb or bs contribution; it is also positive, varying
smoothly with a and taking values between 0.5 and 1.05 . The bb contribution shows the
characteristic break into two curves, nearly linear with a, but one with positive slope
and the other with a negative one. This pattern is transferred to the sum, eq.(3.3),
total γ(a).
Fig.2 shows the effect of size for the same chain-length l = 4. The solvent-solvent
contribution γss(a) is practically independent of size, as is nearly so the solvent-bilayer
part. The bb part and, consequently, the total γ display the break into two regions; the
floppy part strongly affected by size and the extended region negligibly so. The bigger
size shows flatter γ(a), still negative, but closer to zero. Lowest values of γ(a), which
may be taken as the border between the floppy region and the extended region, are
negative but lower value corresponds to smaller size.
These remarkable patterns are repeated for chain length l = 8, with breaks much
more pronounced. Fig.3 shows γαβ(a) for two sizes; the intermediate size is omitted,
just as was in Fig.2, in order to make the plot clearer. Again the ss part is constant and
size-independent, the sb part nearly so, and the total γ along with the sb part, show
these remarkable features: breaks and abrupt changes of slope. For large a the slope is
positive and the size-dependence negligible. The transition to the floppy region occurs
near a = 1.8 for the smaller system and the slope dγ/da changes sign; the break is very
pronounced. The transition is shifted to a = 1.9 in the bigger system, the break is more
pronounced in the bb part than in the total γ. Most characteristically in the bigger
system γ(a) is flat and near zero (though still always negative) in the floppy region.
The tensionless point appears to lie on the r.h.s. curve ( extended bilayer region) in
this case.
All bilayer isotherms i.e. all plots of γ(a) we have obtained1−4 show the same
pattern of two curves joined at some negative value of the lateral tension γ. We interpret
the region of lower a as the region of the floppy state of the bilayer which escapes into
the third dimension, buckles, and takes a fuzzy and irregular form. A gently fluctuating
planar bilayer is found in the other region at higher a’s. Generally, the region of extended
bilayer includes all states with positive γ. The crossing of the isotherm with the a-axis
marks the tensionless state . In the plots of Fig.1-3 the region of the extended bilayer
has a large positive slope of γ(a). Such positive slopes have been found independently5,6
and also earlier9−13. The entire range including the floppy region is found only in few
references2−6. The corresponding split of the lateral tension is found in this paper only.
The status of the tensionless state appears now in a new light, as an accidental event
resulting from the algebraic sum of positive and negative contributions after (3.3). The
partial contributions suggest that the break point dividing the two branches of γ(a) (or
γbb(a) ) is a truly characteristic point which ought to have a physical significance, rather
than γ(a0) = 0.
The shortest possible tail of one particle is present in a dimer. We have attempted
and partially succeeded1 in simulating bilayers made of dimers, but these appeared to
be only imperfectly stabilized by strong head-head and head-solvent interactions or/and
the extra-repulsion introduced artificially to mimick the hydrophobic effect, as invented
earlier9. Possibly at other densities and temperatures the simulations would have been
more encouraging. We do not show these results. As mentioned in Section II, we turned
to the new case of reverse bilayers formed in a weakly interacting solvent and succeeded
in producing (in simulation) stable reverse bilayers of amphiphilic dimers2,4. Fig.4 and
5 show a selection of several characteristic cases, each for two sizes. These cases are: (a)
extra repulsion added, (b) no extra repulsion, (c) no extra repulsion and longer range
of attractive forces. In cases (a) and (b) the cut-off was the generally used r = 2.5,
in the case (c) it was r = 3.2. For clarity, we split the data into two Figures: Fig.4
shows γ total only, for all 5 systems whereas Fig.5 shows the components ss, sb, and bb.
All curves for all 5 systems in Fig.4 show a pattern similar to γ(a) in Fig.3 (for l = 8).
This characteristic pattern is : almost linear rise with a for larger a and almost constant
negative γ for small a. Such a description fits bigger systems better. The biggest system
of an area about 100× 100 of reverse dimers case (a) is shown in Reference 4. As can
be seen from Fig.4, the plot again strongly suggests a transition between two forms of
the bilayer and certainly the existence of two regions1−5. In the floppy region γ < 0
and the bigger the system the closer to zero γ becomes. Fig.5 shows the components of
γ but without γ itself. The pattern is similar to those shown in Fig.1,2,3 except that
the ss contribution to γ is now negative, between -0.6 and -0.2, the sb contribution
much larger, between 2. and 3.6. The bb contribution is, for all sizes, practically linear
in a with again negative slope and negative value in the region of ”floppy” state. The
transition appears softer than for chain molecules, especially those with longer tails
where the breaks were sharp (see Fig.3) in small systems.
The status of the tensionless state appears to be relegated to the category of acci-
dental coincidences. An analogy may be drawn with the equation of state of liquid in
equilibrium with its saturated vapor; certainly P = 1 is a particular point on the vapor
pressure curve, but it has no particular physical meaning - in contrast with e.g. the
triple point. The only physical consequence of zero in lateral tension is the special form
of the structure factor10,1−4,5. However, even this has been questioned recently12,4. In a
detailed investigation of the structure factor S(q) of a simulated bilayer, which should,
for small q, asymptotically vary as 1/S ∼ kq4 + gq2 where k is the rigidity constant
and g - the surface tension, we found g ≥ γ systematically4. The explanation advanced
elsewhere12 is that γδA is not the correct work of deforming a bilayer initially planar,
but gδAtrue is. Here Atrue is the true area of the interface or bilayer, obtained by
following its surface, whereas A ≡ LxLy is the projected area. Since S(q) measures
the spontaneous fluctuations, gq2 is correct and γq2 is not. On the other hand, when
measuring the response of the system to an imposed change in the projected area, γδA
is the correct free energy increment. The increase in the projected area is under control
of the experimentalist; the change of the true area induced by the twists and bends of
the bilayer, is not. By taking these arguments into account, the correct definition of the
tensionless state was assumed12 to be g = 0 and interpolation procedures were used12
to determine the new correct tensionless state. Whether the details of the advanced
explanation12 are correct or not, it is obvious that g parametrizes the spectrum S(q)
of spontaneous shape fluctuations. The difference between g and γ was also discussed
elsewhere13,6,15. The zero of γ(a) is unambiguous in small systems; in large system of
reverse bilayers4 it falls within the narrow region of fastest change in slope, signalling
the transition of the bilayer to the floppy state. Then the precision of its determination
is doubtful. Some of this ambiguity is seen in Fig.4 and in Figures of Reference 4.
Section IV. Discussion.
We have noted in the Introduction that, to a physical chemist familiar with the
interfaces, some properties of a bilayer are truly exotic. Let us list these properties: (1)
the derivative given by eq.(1.1) depends on the specific area; (2) it can be negative;
(3) extension of the area leads to a hole or tunnel as the bilayer resists further extension;
(4) there is distinction between actual area and the projected area but the bilayer keeps
its true area nearly constant; (5) as there are two areas, there are two interfacial
tensions.
Definitely this is not what we learn from classical textbooks on thermodynamics.
The feature (1) has been seen in all simulations, including these over a wide range of
(projected) areas1−6,9−15. The feature (2) has also been seen2−6,11,12,13,15. The feature
(3) has not only been seen2,4,15 but also investigated15. Distinction (4) has been the
subject of papers13,5 specifically devoted to it. Finally, the existence of an(other) surface
tension(s), besides the lateral tension, has been repeatedly surmised5,6,11,13; equations
have even been derived14.
The explanation of such specific behaviour ultimately lies with intermolecular forces
and steric effects which any future theory will have to take into account. The bilayers are
made of amphiphilic molecules, each containing a hydrophilic part and a hydrophobic
part (or more generally, ”solvophilic” and ”solvophobic”). These molecules use their
very special setup of intermolecular forces to create a stable sheet of constant mass (and
approximately constant area). This creates a new situation with a constraint of given
total area almost conserved when the projected area is varied by varying the simulation
The constraint of constant mass (or anything related to it) is entirely absent in
fluctuating interfaces. The molecular mechanism of fluctuations of shape of e.g. a liquid-
vapor interface, involves diffusion from either phase; a local protrusion or excursion of
the interface takes place not only by deforming the existing surface but also by absorbing
or releasing molecules from/to either phase.
The surface of a perfect crystal in equilibrium with its vapor is a good example; all
shape fluctuations are due to evaporation or condensation.
Such processes clearly take part in shape fluctuations of a liquid surface as well.
Hence an interface is a system open with respect to particle number. It is not so
with the bilayer. Not only the solvent is virtually absent from the bilayer, but also the
surfactant molecules making the bilayer never leave it. In principle the surfactant must
be present in the solvent but the bilayer changes shape by changing the position of its
molecules, not by exchanging them with the solvent solution. The shape fluctuations
take place under the constraint of constant number of particles, i.e. constant mass.
A constant mass is implicit in the elastic theory of solids and the links to that field
have already been explored5,10. Conversely, application of the elastic theory of solids to
liquid interfaces, (the so called ”drumhead model”), is clearly not quite correct by not
allowing for diffusion described above.
We discuss now the new evidence for the transition. All plots of γbb(a) and γ(a)
show some kind of break into two branches, one corresponding to the ”floppy” bilayer.
Although the word ”floppy” or ”buckling” appeared occasionally4,5, in fact the status
of the floppy region and of the transition into that region are not very clearly estab-
lished. There are suggestive analogies with the crumpling transition of solid (tethered)
membranes or with coiling transitions of polymers. This would suggest it is a first-order
transition. The breaks in γbb(a) (see Section III) strongly support such a hypothesis
also suggesting it is an internall reorganization of the bilayer structure, in which the
solvent plays a secondary role.
Further support is found with the radius of gyration of the bilayer, shown in Fig.6;
the break in slope is very clear.
Finally, the issue of the other surface tension is not fully resolved. As we discussed
above in Section III, one may define two surface tensions, one coupled to the projected
area and the other coupled to something else, perhaps the true area. The first one is
the lateral tension and the other may be that given by the virial expression derived or
rather proposed recently14 or g parametrizing the structure factor S(q). (See the end
of Section III). These last two may be equal or may be not.
I hypothesize that g may be related to that given by the Zwanzig-Triezenberg (Z-
T) equation16,7 - because the latter is derived by considering a spontaneous fluctuation.
Now, in simulations of liquid-vapor interface, I have determined17 the surface tension
from both expressions and these two numbers were equal within 0.01 percent. Thus I
have confirmed the validity of the Schofield proof7 of the equivalence. Since the (Z-T)
equation is derived in a grand-canonical ensemble, g can only be ”related to it” as stated
above; the constraint would have to be introduced.
We finally remark that the constraint appears as a most natural thing in description
of polymers when the polymer string has a constant number of segments. Indeed a model
one-dimensional membrane/bilayer embedded in two dimensions, is identical to a model
of a polymer string on a plane.
Appendix. Details of the Model Used .
The simulations themselves were in the past and also recently based on the Verlet
leap-frog algorithm with Nose-Hoover thermostat and were never shorter than half a
million steps, about 0.9 million in almost all runs. The intermolecular potential energy
was a sum over pairs of particles; all pairs were interacting Lennard-Jones 6-12 potentials
u(r) = 4ǫ
(σ/r)12 − (σ/r)6
with different parameters ǫ and σ and with different cutoffs rc. A constant u0 may be
added at will and this makes no difference to forces. u(r) has a minimum at r = r∗
of depth u(r∗) = −ǫ. There are two kinds of spherical particles ”a” and ”b”. The
solvent was made of particles ”a” and the amphiphilic molecules of chains ”abb..b”.
The collision diameter σ was common to all pairs. The cutoff parameter was rc = 2.5σ
for all like pairs and rc = r
∗ for unlike-pair intermolecular force. All potentials were
cut-and-shifted to assure continuity of force. Molecular Dynamics algorithms use forces.
The bilayer-forming amphiphilic molecules were freely-jointed chains and each in-
tramolecular bond was made of two LJ halves thus:
ubond = u(r) r < r
ubond = u(2r
∗ − r) (r∗ < r < 2r∗)
ubond = +∞ r > 2r
for either ”a-b” or ”b-b” bond. As explained in the text, for the special case of ”reverse
bilayers” made of dimers, a− b, the ”bb” intermolecular attractions were enhanced by
making ǫbb >> ǫaa. Thus ”b”’s became ”heads” located in the center of the bilayer.
Acknowledgements. This paper is dedicated to Professor Keith Gubbins on
the occasion of his 70th birthday. The author gratefully acknowledges most useful
interactions with Professor Keith Gubbins while visiting him at Cornell University.
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http://arxiv.org/abs/cond-mat/0509218
Captions to Figures
Caption to Figure 1
The general pattern of splits of lateral tension γ according to (3.3) is shown as
plots against the area-per-head a; here for an intermediate size of simulated bilayer
with Nm = 1800 molecules with tails l=4 segments long, with all collision diameters σ
and all energies ǫ equal , at T = 1., Ns+(l+1)Nm = 49000, liquid density ρ = 0.89204.
The ss contribution is marked with squares, the sb contribution - with stars, the bb
contribution - with crosses , and the sum - with black circles . All data are quoted
everywhere in LJ units reduced by the energy depth ǫ and collision diameter σ. The
lines are there to guide the eye.
Caption to Figure 2
The split of lateral tension γ plotted against the area-per-head a for two sizes (SM)
and (B) of simulated bilayer, made of Nm = 450 (SM) and Nm = 3973 (B) molecules
with tails l=4 segments long, with all collision diameters σ and all energies ǫ equal, at
T = 1., Ns + (l + 1)Nm = 12250 (SM) and 105134 (B), liquid-like density ρ = 0.89204.
With the notation of eq.(3.3), the contribution ss is shown with triangles (SM) and
squares (B) near 0.2-0.3; sb - plus signs (SM) and stars (B) near 0.5-0.7; bb - open
circles (SM) and crosses (B) with negative slope for small a and positive slope for large
a. The sum after (3.3), γ, is shown with thick lines, black squares (SM), and black
circles (B). See text and Caption to Fig.1. Compare with Fig.3.
Caption to Figure 3
The split of lateral tension γ plotted against the area-per-head a for two sizes (SM)
and (B) of simulated bilayer, made of Nm = 450 (SM) and Nm = 3973 (B) molecules
with tails l=8 segments long, with all collision diameters σ and all energies ǫ equal , at
T = 1., N = Ns + (l+ 1)Nm = 14050 (SM) and 121026 (B), density ρ = 0.89204. With
the notation of eq.(3.3), the contribution ss is shown with triangles (SM) and squares
(big) near 0.2-0.3; sb - with plus signs (SM) and stars (big) near 0.5-1.1; bb - with open
circles (SM) and crosses (big). Note the negative slope for small a and positive slope for
large a. The sum after (3.3) is shown with thick lines, black squares (SM), and black
circles (big). See text. Compare with Fig.2 and note: the unique breaks in γsb, almost
constant slope in the region of floppy bilayer, translated into zero slope in total γ.
Caption to Figure 4
Total lateral tension γ of reverse bilayers made of dimer molecules plotted against
area-per-head a for biggest (B) or smallest (SM) size, and 3 cases: (a) extra repulsion
added and head-head attraction enhanced (H-H); (b) only (H-H); (c) the cutoff of at-
traction is a long-range rc = 3.2, not rc = 2.5; otherwise (a). The other parameters are:
T = 1.9, ρ = 0.89024, Nm = 2238 dimers, N = Ns+2Nm = 40000(SM) and Nm = 5760
dimers, N = Ns + 2Nm = 160000 (B). These had areas near 36× 36 and near 55× 55.
The points are: open squares - Ba; plus signs - SMa; black squares - Bb; crosses -SMb;
stars -SMc. The lines are there to guide the eye only. The deviation of data-point at
a ∼ 1.13 (black square) is due to the tunnel (hole). see Fig.5 and text.
Caption to Figure 5
The split of lateral tension after (3.3) of systems shown in Fig.4. See caption to
Fig.4 for parameters. Here the points are: case ”Ba” contribution ss - open squares,
sb - black triangles, bb - pentagons; case ”SMa” and ss - plus signs, sb - open circles,
bb -black triangles; case ”Bb” and ss - black squares, sb - open triangles, bb -black
pentagons; case ”SMb” and ss - crosses, sb - black circles, bb -diamonds; case ”SMc”
and ss - stars, sb - open triangles, bb - black diamonds. All ss contributions cluster
near -0.5, all sb are > 2., and all bb show two regions interpreted as floppy bilayer and
extended bilayer. See text.
Caption to Figure 6
One example of the radius of gyration for the same bilayer whose γ is plotted in
Fig.1, also plotted against area-per-head a. It appears to support, along with other
correlations3,4, the hypothesis of a discontinuous transition between the extended and
floppy bilayer.
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1
1.4 1.6 1.8 2 2.2 2.4 2.6
0.9 0.95 1 1.05 1.1 1.15 1.2
0.9 0.95 1 1.05 1.1 1.15 1.2
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1
