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Thermodynamic defects, instabilities and mobility processes in the lamellar phase of a non-ionic surfactant
M. Allain, M. Kléman
To cite this version:
M. Allain, M. Kléman. Thermodynamic defects, instabilities and mobility processes in the lamellar phase of a non-ionic surfactant. Journal de Physique, 1987, 48 (10), pp.1799-1807.
�10.1051/jphys:0198700480100179900�. �jpa-00210621�
1799
Thermodynamic defects, instabilities and mobility processes in the lamellar phase of a non-ionic surfactant
M. Allain (a) (b, +) and M. Kléman (c)
(a) Institut Curie, Laboratoire de Physique et Chimie, 75231 Paris Cedex 05, France
(b) CEA/Section d’Etudes des Solides Irradiés, B.P. 6, 92265 Fontenay-aux-Roses, France
(c) Laboratoire de Physique des Solides, Université Paris-Sud, Bât. 510, 91405 Orsay Cedex, France
(Reçu le 19 janvier 1987, accepté le 4 juin 1987)
Résumé. 2014 Les phases lamellaires L03B1 de solutions aqueuses de surfactants non ioniques C12E5 et C12E6 présentent des caractéristiques qui n’existent pas dans les phases lamellaires usuelles : mobilité élevée des défauts non thermodynamiques, boucles de dislocation d’équilibre qui induisent la transition vers la phase isotrope à haute température, instabilité de courbure des couches à basse température. Nous discutons ces
phénomènes en fonction de la faible valeur du module élastique de courbure K1 (comparable à celui des
microémulsions) et nous présentons des modèles moléculaires qui montrent l’influence de l’eau d’hydratation
des têtes hydrophiles.
Abstract. 2014 The lamellar phase L03B1 of the aqueous solution of the non-ionic surfactants C12E5 and C12E6 shows up many features which do not exist in usual lamellar phases : high mobility of non- thermodynamic defects, structural dislocation loops whose density increases at the approach of the transition towards the high temperature isotropic phase, low temperature curvature instabilities. We discuss these
phenomena with regard to the low value of the curvature splay modulus K1 (comparable to that of microemulsions) and molecular models which stress the influence of water linked to the hydrophilic heads.
J. Physique 48 (1987) 1799-1807 OCTOBRE 1987,
Classification
Physics Abstracts.
61.30 - 61.70 - 64.70
It has been demonstrated recently, first by ESR spectroscopy [1], then by electron microscopy obser-
vations of replicas of freeze fractured samples [2, 3]
that the lamellar-isotopic transition in aqueous solu- tions of some non ionic surfactants ( ClzES and C12E6, i.e. penta and hexaethyleneglycol dodecyl ether) is driven by the formation of defects related to very highly curved regions of the interfacial film.
These curved regions appeared to be twist walls of
screw dislocations organized as shown in figure 1. In
a wall, dislocations have the same sign. These twist
walls appear in pairs of opposite sign, thus defining
blocks of layers tilted with respect to the lamellar matrix. It is our guess that these screw dislocations
pairs are vertical segments of dislocation loops which
cross layers. These defects are thermodynamically stable, their density increases with temperature ;
(+) Present address : ATOCHEM, Physico-chimie de l’Application, B.P. 108, 92303 Levallois-Perret, France.
their Arrhenius plot yields an energy Ea of the order of 0.4 eV (1).
In addition to these thermodynamically stable,
structural defects, usual metastable defects, like oily streaks, focal domains etc. are observed by optical microscopy. Contrarily to what is known of meta- stable defects in most ionic surfactant systems, they
anneal very easily when the sample is submitted to gently thermomechanical treatment. Then it is poss- ible to obtain very good homeotropic samples within
half an hour. Viscoelastic measurements performed
on such samples are highly reproducible. They
measure the mobility of the edge dislocations which
gather in the grain boundary of the mid-plane of the
(1) Although, owing to quenching effects, electron microscopy observed defects probably do not show up with their equilibrium size nor their equilibrium density,
their number, clearly and undoubtfully, varies with tem-
perature in a reproducible way (see Ref. [2]).
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480100179900
1800
Fig. 1. - C12E6-water lamellar phase 73 % of surfactant in weight. a) Group of parallel and finite steps. Arrows show ends of steps where screw dislocations emerge. Screws, at each end of a step, have opposite signs. These steps are the trace, in the fracture surface, of « vertical » dislocation loops which cross layers. From reference [3]. b) A dislocation
loop, made of two screw of opposite sign linked by edge dislocations, cut by a mediator plane. Note the twist angle of
lamellae inside the loop. c) Pile of vertical loops. Screws of the same sign build a twist wall.
sample, and they show that these dislocations reach their equilibrium density. The measured mobility is high (m = u/cr = 106 cm2 s g-1 ) and depends on temperature with an activation energy E. of the
order of 0.4 eV [4].
All these results point out the existence of two different classes of defects : first, usual metastable
defects which relax, and secondly, thermodynamical- ly stable, structural defects, which do not relax and play a role in the lamellar isotropic transition. This remark urged us to a new study of the physics of
defects in lyotropic lamellar phases. We discovered that two factors are relevant to this issue, the
smallness of the splay modulus Kl and the dehydra-
tion of the polar heads when T increases.
a) Kl, the splay modulus, is unusually small
(K=z 6 x 10-8 dyn), i.e. 20 times smaller than in most smectic phases (for example than in lecithin- water lamellar phases [5], which are besides well- known to anneal extremely slowly [6]) in fact as
small as the one claimed to be at the origin of the
strong layer fluctuations in microemulsions [7]. In
this paper we argue that a small K1 leads to a very essential feature : the core of edge dislocations is
widely spread on a large distance ç - dJ/ À (2) in the
glide plane of the dislocation. This favours easy
glide, in contradistinction with usual lamellar phases
where climb is easy. It also explains the existence of dislocation loops which cross layers, and the absence of large pores i.e. edge dislocation loops of large
radius.
b) Dehydration of polyoxyethylene headgroups.
At low temperatures, headgroups are surrounded by
H-bound water molecules which induce a large
effective area per molecule at the interface. As temperature increases, dehydration occurs. It is, for example, well known that dilute systems of non- ionic surfactants show up the cloud point
(2) do = lamellar repeat distance ; À = (K1/ B)lf2 ; B =
compression modulus.
1801
phenomenon : when heated these solutions demix
into a water-rich solution and a surfactant-rich solution. This dehydration of headgroups occurs at
any concentration and leads to a variation of interac- tions between hydrophilic groups at the interface,
which modifies the equilibrium curvature of the
interface and promotes transition from one mesoph-
ase to another as temperature varies [8, 9].
We believe that this phenomenon explains the
existence of two regimes of defects. At high tem-
peratures, we observe a proliferation of twist walls, while at low temperatures, the density of screw
dislocations is small, and layers show long range instabilities which vanish as temperature increases
[10] ; since this transition is driven by the appearance of stresses in the layers which are locally relaxed by
these instabilities, we compare the onset to a marten- sitic transformation.
This paper is divided as follows : we first discuss the anisotropic spreading of the edge dislocation ;
then we propose a model for the large mobility of
these defects, which involves the creation of pairs of
pores ; the elastic energy of a unique vertical loop of
the type observed by freeze-fracture, is then cal- culated, and we show that they can nucleate from an
instability of pores. We then address the question of
the formation of clusters of vertical loops, and why
their size varies with temperature. In the section which follows, we describe the specific features of
the long-range instabilities which exist in the low-
temperature region of the lamellar phase, when the
head groups are hydrated. The last section is devoted to a possible microscopic model which tends to put in a unified perspective the appearance of ther-
modynamical defects of a different nature when one
goes through the whole temperature domain of the lamellar phase at constant concentration, as indi- cated above.
1. Anisotropy of the core of edge dislocations.
The existence of an extended core for edge disloca-
tions is easy to establish : since K, is so small (A = (Kl/B)ll2 = do/10) the layers curve easily ;
the core size along the z axis (see Fig. 2) is defined as
the distance to the dislocation line at which the width of the parabola x2 = 4 Az reaches do (in this region
the curvature of layers is larger than 1 /do and the
linear analysis of the distortions fails down). There- fore )z = do/A (see Appendix I). This splitting does
not depend on the Burgers vector in this approach.
The size g x along the x axis is of the order of b = ndo (b is the Burgers vector and n is integer). In
our system where n = 1 and A is very small, it yields gz =-10 do and gx = do (see Fig. 2). The core is very
anisotropic. Such a configuration has in fact already
been observed in 2d layered systems formed by the
Fig. 2. - Picture of the spreading of the extended core
showing distorted layers along the direction normal to the
layers.
roll electrohydrodynamical instabilities of nematics
[11] : the quantity analogous to A is the frequency of
the ac electric voltage which drives the instability.
When A is small, the extended dislocation core is observed to play a major role in the shear of layers
which is at the origin of the « varicose » instability.
Outside the core region the elastic energy of the line writes [12, 13] :
with x - b (§ does not depend much on KII B = A2 ), Wi scales like 1
( 2013 )
A K 1 and is of the order of5 K, here. The energy of the core spread along the z
axis is a surface energy times 2 ,. Assuming the
curvature energy per unit length of a highly curved layer in the core to be of the order of K1/2, the
energy of the spread core writes :
which yields re - 10 Ki ; this value is, to our opinion, an upper estimate since the energy is
equally shared between line energy and core energy.
Therefore we put Wl = T e = 5 K1. Note that, in
usual lamellar phases where the core is not extended,
re and Wl are of the order of Kl but Kl is 20 times larger.
2. Mobility of edge dislocations : glide and climb.
The strong distortion of the layers in the core suggests that the formation of pores in this region plays a role in the mobility process, After nucleating,
a pore will extend in opposite directions along the
dislocation line, and create a kink. On geometrical grounds, there is a possibility that such a process does not require either removal or addition of
1802
matter coming from far away (3) : however it is
probable that there is a diffusion process, helped 1) by the presence of screw dislocations which are in the vicinity (matter flows along their core), 2) by the
nucleation of pores of opposite signs along the core.
By opposite signs, we mean that there are pores filled with water and pores filled with the aliphatic
chains. Each pore, whatever its sign may be, makes
the dislocation glide by half a lamella (a monolayer) (Fig. 3). In fact, we have never observed kinks
Fig. 3. - One pore between the extrabilayer and an adjacent one makes the dislocation line glide of one monolayer.
smaller than a bilayer, which means that both pores of opposite signs contribute simultaneously to the glide and therefore nucleate simultaneously, in a
process which does not require any diffusion. There- fore the growth (by climb) of the kinks along the
dislocation core is a cooperative process which should not be too difficult. Note that the small sides of the kinks are elements of screw dislocations
(Fig. 4). The activation energy of glide is therefore
the energy of nucleation of a double pore. As we shall argue in a later paragraph, the two types of pores have probably nearly equal energies, because
after dehydration the two sides of the monolayer (polar side and aliphatic side) occupy similar volum-
es. This energy is large, of the order of 0.5 eV, and is probably the activation energy which is measured in the mobility experiments alluded to above.
Similar arguments (but diffusion is necessary) can
be developed for climb of edge dislocations. Hence in general, the activation energy of the mobility of
line defects is that of the nucleation of pores.
(3) It is easy to convince oneself that there is an optimal
diameter of the pore where no new matter is involved.
Fig. 4. - A kink in the edge line (equivalent to two pores
of opposite sign) makes the line glide of one bilayer.
Locally the line has a screw character. Such a kink is alike
a partial vertical loop. See figure 1.
3. Vertical dislocation loops.
Let us now analyse the dislocation loops perpendicu-
lar to the layers (in the sequel, we call them vertical loops for short). Since the screw segments have a
very large line tension (but a small line energy) [12],
these loops are rectangular, with two screw segments and two edge segments (Fig. 1). We calculate their elastic self-energy as a double line integral [13] :
function of the linear smectic elasticity in an infinite
medium :
We find :
with u = L/2(fA )1/2, L is the length of each edge
segment, and t the length of each screw segment.
kc is a cut off wave number which has to be introduced to insure convergency in the dky inte- gration and is of the order of
2 b ir
(see Appendix 1).When integrated, Wt reads :
1803
To Wt, we add a core energy, Fe L for the edge part
and TS Q for the screw part. The total energy reads :
Let
Lumping the kc 2 core term in re, the first derivatives
2
write :
Except if F,, the screw core energy, is negative,
which is an assumption we shall not do (4), the
derivative with respect to the screw segment length
cannot go to zero and is always positive. Therefore
the screw part takes a length as short as possible compatible with the linear analysis we are making here, and this yields f - z, where § is the size of
the extended core. Conversely, aW can vanish ;
8L
however, since the e equilibrium condition aW = 0 aL obtains physically by motion of the screw parts, which are very little mobile compared to the edge parts, it is not certain that we get this derivative to
vanish actually.
Since we can take erf u = 1 as soon as u > 1 i.e.
L > 2 b, which is experimentally the case, we see
that aW
vanishes for J- (JJ ’IT 1/2 , i.e. for
that
aL vanishes for 2 -,/TA-
Te , i.e. for of the order of b2 for T = 5 Ki. This yields ( )
7T
e 1
f o--o §z, as inferred above by another argument. Note that, even if aw is slightly positive, it cannot be
aL
positive enough to provide a configurational force
on the screw parts which would be able to move them. Experimentally we also have L = 20 b = f. In other words the assumption r s > 0 does not play a
crucial role : the orders of magnitude we obtain for
L and f would also be obtained with I rsl - Fe - We (L, its plotted in figure 5 as a function of L and
f, with Fe = 5 Kl and Fs =z Kl. We see that the total energy for small loops ( L 20 b ) is of the order
(4) This would not be a completely extravagant assump- tion if, for example, the high temperature phase would
favour negative Gaussian curvature for the monolayers.
This would have necessarily to be explained in terms of entropy effects ; as pointed to us by F. C. Frank (private communication), W is a free energy (not a pure internal elastic energy) and could contain important entropy con- tributions, especially in the core of defects.
Fig. 5. - Energy of a vertical loop as a function of the
edge part length L and of the screw part length f.
of 1 2 kT and that, indeed, aw aL and
a W
at are positive,which would tend to close the loops. As the loops do
exist and their measured activation energy is equal
to 0.4 eV i. e. 15 kT, we believe that their activation energy does not originate in the self energy of loops.
Moreover, note that the collapse of such loops,
under sizes of the order of 2 do, would require the
same change in topology as that requested for nucleation, so that these loops do not disappear :
because of topology, the edge components suffer
repulsive interactions at short distance.
How do these loops nucleate ? We show in the
next paragraph that they might originate from a
fundamental instability of pores (prismatic disloca-
tion loops) whose energy is greatly superior to that
of the vertical loops. The formation of a pore itself involves a breaking of the layers and a reor- ganization ; such processes of breaking of layers and reorganization might also directly lead to vertical loops. Which path is followed indeed is difficult to
assess since, as we shall see, all the energies in play
are of the same order of magnitude and much larger
than Wt. However, since the density of curvature
measured by ESR is reversible with temperature,
Ea must be interpreted as an internal energy, i.e. the standard exact energy of an object, which has to be
the pore we discuss. In other words, it is most probable that the vertical loops originate from thermodynamical equilibrium density of pores.
4. Pores (prismatic dislocation loops).
The self elastic energy of a pore, i.e. a prismatic edge dislocation loop, can be calculated by the
method devised in references [12, 13] as a line integral :
1804
where R is the radius of the pore, i.e. the radius of the circle outside of which the monolayer is no longer highly curved, and qc is a cut off of the order of 2 iT where x is the characteristic size of the core
Ex
in the plane of the loop i.e. 6x = b.
One can easily see that W increases with R, and
that for a small pore, R = b one obtains :
which yields We = 0.3 eV ~ 10 kT.
The energy is of the same order of magnitude : re L = 5 K x 2 7rR i.e. Fe L = 0.5 eV ; therefore
the total energy Wp = W + Fe L of the pore is roughly 1 eV, i. e. 40 kT.
Such a high value in comparison with that obtained
in the former section for vertical dislocation loops
shows that pores are unstable with regard to a
transformation into a vertical loop which obtains by glide of some parts of the loop, with formation of
screw segments, by the process indicated in the section on mobility of edge lines.
Let us now compare Wp with the molecular energy which occurs in any transient state involving changes
in the topology of bilayers. In order to build any pore or loop, one must first of all make a hole in the
layer i. e. break intermolecular energy. The smallest structure is a pore made by the half inner part of a
torus whose both radii of curvature are of the order of b/2. The area of the surface is :
Since the area per hydrophilic head is u = 50 Å 2 [1]
we find roughly 100 molecules in such a structure.
Let E. be the intermolecular binding energy. We
assume E. = K1 e per molecule where E is a molecu- lar distance in the interface Le. e = u 1/2 = 7 Å.
Therefore we find E. = 3 x 10- 3 eV per molecule which gives 0.3 eV for a minimal structure.
This estimate, as well as the estimate of the self- energy of the pore, are very rough. While they are of
the same order of magnitude as the energy of activation measured in mobility experiments and
curvature measurements, it is not possible to say which process (either formation of a pore and
instability towards a vertical loop, or direct forma-
’tion of a vertical loop) takes place in reality.
Coming back to the energy of vertical loops, we represent in figure 5 the line energy Wr to which we
have added a 0.4 eV peak at the origin as an
activation barrier. If we take
ar
anda W
aw aL positive but small, this defines a flat minimum whose energy is around kT and which then allows quite a wide spectrum of sizes. In fact, the size of the loop isdefined by the z extension of the anisotropic core of edge dislocations, already discussed.
5. Clusters of vertical loops.
Up to this point, we have only considered isolated dislocation loops. We now discuss dislocation loops piling up so that screw dislocations build twist walls.
Let 6 be the distance between the loops, and
v = 5/2(fA )1/2 ; their interaction energy writes :
Then the interaction energy for a pile of n loops
situated at a distance 8 from one another is :
A first remark is that the introduction of the interaction energy in the total energy of a cluster does not allow us to make T > 0 to get
ar
aw= 0.Hence the above remarks on the sign of TS are still
valid. The general look of the surface W = W (L, l)
does not change and the conclusions obtained for an
isolated loop are valid. One can see that WI is negative : to gather the loops in a stack decreases their energy expected from experimental observa-
tions [2, 3]. But lim W1 (n ) exists (see Appen-
n - oo n
dix II). For loops with (L, l ~ 10 b) WI does not
decrease significantly for n > 20. When L increases,
this limit value of n increases. This is shown in
figure 6 where the total energy of a loop in a pile of
size n is plotted against n. We see that for
n > ncri, W no longer decreases. It is then useless to
n
increase the size of the pile. This defines the optimal
number of loops in a wall, and our figures match our experimental values well.
Fig. 6. - Mean interaction of a square vertical loop as a
function of the number of loops in the pile.
