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Fluid membranes in the “semi-rigid regime”: scale invariance
M. Skouri, J. Marignan, J. Appell, G. Porte
To cite this version:
M. Skouri, J. Marignan, J. Appell, G. Porte. Fluid membranes in the “semi-rigid regime”: scale invariance. Journal de Physique II, EDP Sciences, 1991, 1 (9), pp.1121-1132. �10.1051/jp2:1991208�.
�jpa-00247579�
Classificauon Physics Abstracts
05 40 82.70K
Fluid membranes in the « semi-rigid regime » : scale invariance
M. Skoun, J. Mangnan, J Appell and G. Porte
Groupe de Dynanuque des Phases Condensdes (*), USTL Case 26, 34095 Montpelher Cedex 05, France
(Received 26 March 1991, accepted 23 May 1991)
Abstract. We investigate here the swollen lamellar phase L~ and the anomalous isotropic (sponge) phase L~ of the temary system AOT/bnne using neutron and light scattenng We first
measure the excess area of bilayers from the deviations to the simple swelling behavior and so
obtain an evaluation of the mean curvature ngldity K of the bilayers K is found quite high
K = 3 k~ T and is basically the same in both L~ and L~ Correspondingly, the persistence length
§~ of the bilayer is of the order of 10~ h
so that the samples are in the « semi-ngld regime » at all
practical dilutions The scale invariance of the statistics of the fluid bilayers expected for such regime is then checked by companng the neutron and light scattenng pattems of samples of different concentrations
Inwoduction.
Both the Swollen lamellar phase L~ and the anomalous isotropic phase LJ (sponge phase) in
amphiphihc systems have been well characterized as phases of fluid bilayers [1-4]. They usually coexists in the phase diagrams of many common surfactant systems [5] and have very different large scale structures. The swollen lamellar phase L~ corresponds to the situation where the infinite bilayers are stacked regularly parallel to each other so that the sample shows long range smectic order [6] In the L~ structure, all amphiphilic molecules are incorporated into one infinite bilayer randomly multiconnected throughout the sample in the three directions of space (Fig. I) : the bilayer separates the space in two equivalent strongly
interwoven subvolumes.
Both structures can easily be obtained in the dilute range (low volume fraction of
membrane) where the structural charactenstic distance 1is very large (1is the smectlc
penodicity in L~ and corresponds to the average size of the « passages in LJ). In this limit, it
(*) UA 233 et GDR Films Molkculaires FlexJbles du CNRS
Fig. I. Schematic dravnng of the L~ structure.
is conunonly adm~tted that the respective stability of the two structures is mainly determined by the ben&ng elasticity of the bilayers [7~ :
H
= 1[ K(c, + c~)~ + kci ~j dA (1)
2
where ci and c~ are the pnncipal curvatures of the membrane of total area A and
K and k
are the rigl&ty modull associated with respectively the mean curvature
(ci + c~) and tile Gaussian curvature (ci c~) of the membrane.
In this respect, two different approachs have been proposed to explain the L~ to
L~ phase transformation. The first one [8] emphasizes the role of the mean curvature rigidity
modulus K: it is based on the idea that, since it shows no long range order, the
LJ structure corresponds to a larger configurational entropy for the membrane than the
L~ phase. But this entropic gain has to be pa~d by some mean curvature elastic energy (the
membrane is more distorted in L~ than in L~) indeed proportional to K, so that lower K values should favor the L~ to L~ structural trans1tlon. More specifically, model calculations
are performed in [8, 9] where the value of k is set to zero : as a common prediction of these calculation, the phase trans1tlon takes place when the smectlc periodicitylis increased up to
some value comparable in magnitude to the persistence length fK of the membrane [10] (or conversely when K is decreased so that fK becomes of the order of dj
:
fK = a exp
~'~~ (2)
3kI1T
where a is a molecular size. In other words, fK corresponds to the scale length beyond which the effective rigidity K~R(f)
3 k~ T f
~~~ ~
4 «
~~
a
~~~
renorrnahzed by the short wavelength thermal curvature fluctuations [I1-13] is decreased well below k~ T. At that scale, the energy paid upon distorting strongly the L~ structure becomes smaller than the entropic gain and the phase transformation takes place.
In the second approach [14], the emphasis is rather put on the difserence m topology of the membrane when going from L~ to L~. Each bilayer is singly connected in L~ while it is
multiply connected in L~. The integral of the Gaussian curvature being a topological invanant for any given closed surface (Gauss Bonnet theorem).
lcic~dA=-4«(1-N)
A
where N is the Euler characteristic of the surface (i.e. the number of passages),
k acts as the chemical potential for the topological complexity of the phase The Gaussian contnbution to the difference 6F~~
~ ~~ in free energy density between L~ and LJ is therefore
simply :
6F)~fl[(~~ = 4 « kn (4)
where n is the number density of the connectivity in L~ (n
= 0 for L~). So, higher values of k w~ll decrease that contnbution and make the L~ - L~ transformation easier. In the general
case, k is not zero and one expects that both the entropy difserence and the topology
difserence are actually involved and therefore both K and k play a significant role.
In the present article, we deal with the L~ and L~ phases of the AOT/brine system. The phase diagram of this system has been investigated by several authors [15-17] (Fig. 2). A
particular advantage is that only one amphiphihc chemical compound is involved (AOT) so that the bilayer consists of one component only. Moreover the phase transformations are pnmarily tnggered by the salt concentration and therefore determined by the properties of the bnne solvent only.
% Nacl
2 '~
L~ ',,
j L~+ L~ ,- L~+C
Li+La+L3 '
L~+La I
,, "----______
Li+La '~, La ""'"~,,_
l "~,
l 0 20 30 40 50 60 70 % AOT
Fig 2. Phase diagram of the AOT/D~O/Nacl system from [16] D~O is here used for the purpose of
neutron scattenng expenments
In section I, we measure the excess area of membrane upon dilution for both the
L~ and LJ phase. The mean curvature rigidity constant K is then derived : it happens to be rather large and basically identical in botll cases.
The corresponding persistence length fK (expression (2)) is very large compared to the
largest investigated structural distances 1(~#~~ where # is the volume fraction of membrane). In that1imlt, the renormalizations of the ngidities K and k with the scale length
are small and the efsective elastic Hainiltonian H (expression (I)) remains almost scale invariant. As a particular consequence, the statistics of the L~ phase should be scale invanant
as well. We check this prediction m sectlon2, mating a quantitative analysis of neutron
scattenng data.
In section 3, the L~ phase of the AOT system is investigated using static light scattering. The q-dependence of the scattered light is analysed in terms of the fluctuations around the so- called «inside/outside» symmetry first introduced in [18]. The corresponding correlation length is measured accurately. Its variations with the concentration # is found in agreement witll tile above mentioned scale invariance of the statistical propemes.
Excess area in L~ and I~ [3, 19].
In the swollen larnellar phase L~, increasing tile dilution basically leads to an increase of the
&stance1between tile bilayers. Assuming perfectly flat bilayers the conservation relation is
siJnply :
#
= 8A/V
=
8/1 (5)
where 8 is tile thickness of tile bilayers. Actually, due to the finite temperature T, the bilayers are submttted to tllerrnal curvature fluctuations. They undulate rather titan
being totally flat. As a result the total area A of a given piece of membrane w~ll be on the average somewhat larger titan its projected value Ao onto a plane normal to the director n of tile smectlc stacking. The so-called excess area AA :
AA=A-Ao (6)
has been evaluated in a perturbation scheme [lI]
i~ ~~i~~~ ~i ~~~
The high wave vector cut ofs q~~ is of the order of the inverse molecular size
a. The low q cut ofs q~~ corresponds to the inverse correlation length ii for the free membrane fluctuations. ii actually depends on the efsective interactions between the membranes in the lamellar stacking. Due to the rather hJgh salinity of the bnne solvent of the
AOT/bnne system we reasonably consider that the bilayers basically interact through only
stenc repulsion. iii has been evaluated for that situation by several authors [9, II, 20] and the
follow~ng expression has been obtained :
q~~ =
~ '~
with iii
= c
Jfi.
d (8)ii B
where c is a constant of order unity.
In this frame, the dilution relation [5] has to be somewhat modified and we get a relation of the form [19]
#.1=8(A+BIn#) (9)
where :
k~ T
/j
A
= 8 1+ In c- (9a)
4 "K ~ ~I1 T
and
B
=
~~ ~
(9b)
So the product #1should exhibit a # dependence linear in In # w~th a slope invo1vlng only
8 and the ratio KIT. Relation [9] therefore allows a very simple procedure to measure the
mean curvature ngidity K provided that 1can be measured -accurately which is easily done
using X-rays or neutron scattering [3,19] ~position of the Bragg maximum on the
I(q) profile)
The data obtained for the lamellar phase L~ of the temary AOT/bnne system are plotted in figure 3 The expected linear behavior of #las
a function of In # is actually observed. From its slope the ngidity K is denved and we obtain (taking 8
= 17 h
as reported in [19]) : K=3k~T.
We used the same procedure in order to investigate the efsect of addition of oil (I e. decane)
into the hydrophobic part of the bilayer. The data shown in figure 4 corresponds to lamellar samples where one molecule of decane has been added to each six molecules of AOT. As a
pnmary effect, the addition of decane results into a thlckenlng of the membrane [17] and we have 8
= 28 h. The vanation of #1as a function of In # is again linear (Fig. 4) and its slope
indicates a mean curvature ngldity still close to 3 k~ T. So, in spite of the larger thickness 8 of the bilayer, its rigidity K remains quite insensitive to the addition of small amounts of oil
suggesting that the decane is incorporated into the membrane in a rather aJnorphous liquid
state.
T o (I)
T4~IA)
15
'~ ~~
In O ° '~ '~
In O
~~ °
Fig 3. Fig 4
Fig 3 WI versus In 4 for the L~ phase of the ternary system (I*
=
?) full tnangles,
4fl 5 versus In 4 for the L~ phase (I*
=
fl 5) for the same system empty circles.
Fig 4. Same plot as in figure 3 bu~ for the case where one molecule of decane is added to every six molecules of AOT.
The L~ phase being a &lute phase of infiilite fluid membranes, it also shows a simple dilution behavior :
#1= 1.5 8 (10)
where the constant prefactor 1.5 is related to the multiconnected structure as discussed at
length in [1,2]. (10) is indeed obtained neglecting the small wavelength thermal curvature fluctuations. Taking them into account we again expect the excess area to be of the form
lAA
k~ T q
= In (11)
Ao L3 4 "K qnun
with the same prefactor (k~ TM «K) and the same q~~ as in case of the L~ phase. But q~~ has a difserent origin. In L~, ii corresponds to the scale length beyond which the efsective
rigidity of a given bilayer is « st1fSened » by the repulsion (stenc) of the adjacent ones. In L~, the situation is difserent : the relative displacements Au of two parts of the membrane
facing each other in one given « cell » or « passage » is of order (k~ T/K)~~~1[7j (which is
<1if K
> k~ T~. So, they very rarely come to close contact and the efsective stenc interaction is here presumably irrelevant. However, we still expect a stifsening efsect but ansmg from the
multiconnected nature of the L~ structure in all three directions of the space : two points, separated by a distance of order din the 1nltlally flat undulating bilayer, come close to each other in the L3 structure Their displacements are therefore constrained and over correlated.
So, it is again natural to think of a low q cut ofs length ii still scaling as 1but with a
presumably difserent prefactor than in (8).
As a result, we expect tile dilution relation corrected for the thermal excess area to be again of the form.
~~=8(A'+B'ln#) (12)
the interesting point here is that B' only depends on the prefactor in tile expression of AA/Ao (11) so that it is identical to B in (9b) for the L~ phase
k~ T
~' ~
fi ~~~~
On the other hand, A' is likely different from A in (9a) (different ii). (12) gives us the
possibility to measure K in L~ just as we did in L~ provided that1is accurately measured at difserent #'s. In general, this is not possible in the L~ phase : the ngldity K is often low
(K = kB T or less [19]) and the position correlation very loose The corresponding bump in the I(q) pattem is broad and weak (less well defined scale length ~lj and the position
qM of the maximum in I(q) cannot be pointed w~th sufficient accuracy specially for the most dilute samples. In the present case of the AOT system, the situation is more favorable. As shown m figure 5, the correlation maximum in the neutron scattenng pattem is clear and
sharp even for the most dilute sample ~presumably due to a h~gher K titan in other systems [19]). For the sake of companson to the swelling behavior in L~, tile data (#@1.5)
corresponding to L~ in tile temary AOT/bnne system are plotted in the same figure 3. The evolution is stnkingly s1mllar to the one we have obtained for L~ The linear dependence with In# strongly supports our analysis and the s1mllarity in slope clearly indicates that
~~~ P
400 o
o
fi ~
A
300 A
(q) ~&~~~
~ ~ ~
~~
A
~
i § § § §
0.00 O 02 0 04 0 06
q (A-I
Fig 5 Neutron scattenng patterns of L~ samples in the temary system with volume fractions 4
=
0 0474 tnangles
,
~b = 0.0675 circles
, ~b = 0.0897 squares. Data collected on line Dl I at the ILL
(Grenoble, France)
K is much the same in L~ and in L~ K = 3 k~ T. We also made the same expenment after incorporation of one molecule of decane every six molecule of AOT and again the
4@1.5 as function of In 4 (Fig. 4) is very s1mllar to the corresponding lamellar phase indicating that K is still close to 3 k~ T.
IIigh Hgidity K and scale invariance.
The main finding of the preceding section is that the AOT/brine system exhibits a disordered L~ phase w~th rather h~gh mean curvature ngldity K for the bilayer. As a matter of facts, using
expression (2) with K
= 3 k~ T provides a value for the persistence length i~ of the order of 10~h far beyond the largest 1measured for the most dilute samples (4
= 0 0474 so that
1=700h) So at all dilutions, we always have I«iK so that the efsects of the renormahzatlons related to small wavelength curvature fluctuations remain moderate amongst the here investigated 4 range. This is indeed true for the renormalization of the total
area A (excess area AA) since, as we saw above, the variations of 4d are small Using relation
(3), we see that the vanations of the effective rigidity K(I ) is also rather small in relative units
(about 18% amongst the dilution range ie amongst one order of magnitude for
~lj. Th~s feature makes the AOT system a good candidate for checking the scale invanance
expected for 2D fluid membranes [14, 22, 23]
A unique property of 2D fluid membranes is that their bare elastic Hamiltonian (I) is
invanant with respect to any isotropic dilation A dilation of ratio A changes ci and
c~ into ci/A and CJA and dA into A ~dA
so that H~ remains identical In the rigid limit
(T/K « I), i e. neglecting the renormalizations of K, k and dA, this invariance is preserved at
the level of the efsective Hamiltonian H~~ (since in that limit K~~=K, k~~=k and
dA =dAo) As a main consequence [14, 21, 22] any given configuration of a piece of
membrane of area A confined in a volume V of a sample w~th volume fraction 4 (characteristic distance dj
can be associated w~th a dual configuration of a piece of membrane of area ~A confined in A V of a sample with concentration 4 IA(A1) these
dual configurations having the same elastic energy they have the same statistical weight. Th~s
means that, except for small ripples (irrelevant in the ngid limit), We statistics of send.rigid
fluid surface are, in a rust approxbnation, scale u1vaHant just like the efsective elastic Hamlltonlan. Therefore, all averaged quantity characterizing the structure of L~ such as the
spatial density-density correlation function g(r) should be scale invariant when expressed in
the appropnate reduced units (i e. as function of the dimensionless ratio r/~lj. This can be checked in the reciprocal space when companng the neutron scattenng pattem I(q) (Founer
transform of g(r)) of L~ samples at difserent dilutions 4 (I.e. different dj. To illustrate that point, we have plotted in figure 6 the scattering pattems of three difserent samples (~# = 0.0474, ~#
= 0.0675, ~#
= 0.0897) as a function of q/~# (which is proportional to the dimensionless unit q/~lj. As a matter of facts, the tllree plots are remarkably similar.
14
.1
.01 .I I lo
(/@(11)
Fig. 6 Same data as in figure 5 but plotted against reduced coordinates 4. I(q) versus q/~ 4
=
0 0474 empty circles
,
4
=
0 0675 full tnangles
,
4 = 0 0897 empty squares
However, similar plots performed for another system (CPCI/hexanol/bnne [1, 14]), for which K is known to be significantly lower [19] K = I k~ T, have rather shown strong difserences in
a similar concentration range So clearly, the nice scale invariance shown by the structure factor is, in the present case, related to the rather h~gh value of K Note also that in the h~gh
q range, I(q) shows the q~~ dependence consistent with the local structure of th~n bilayers 13,4j.
However, more carefull attention focused in figure 6 indicates that the scattenng pattems
are not exactly superimposed That of the more concentrated sample is systematically slightly
below that of the more dilute one We guess that th~s discrepancy is related to weak renormalization efsects breaking, up to a small extent, the scale invanance derived in the ngJd limit only. This issue is addressed in next section.
Light scatteHng and low q mean curvature fluctuations.
In this section, we investigate L~ properties at still larger scales using light scattering. In figure 7 we have plotted the structure factor I(q) of two AOT/brine L~ samples in the light scattering q-range using the same reduced coordinates as for the neutron scattenng data in the
preceding section. Actually, they appear very parallel but with some difserence in magnitude
