1 July 1989
EUROPHYSICS LETTERS
Europhys. Lett., 9 (5), pp. 495-500 (1989)
Buckling of Langmuir Monolayers.
S. T. MILNER(*),J.-F. JOANNY(**)and P. PINCUS(***)
(*) AT&T Bell Laboratories, Murray Hill,NJ 07974, USA
(**) Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07,
(***) Materials Department, University of Calt$ornia
France
Santa Barbara, C A 93106, USA
(received 24 January 1989; accepted 18 April 1989)
PACS. 82.70 - Disperse systems.
PACS. 36.203 - Constitution (chains and sequence).
PACS. 64.70 - Phase equilibria, phase transitions, and critical points.
Abstract. - We show that under compression on a Langmuir trough, a surfactant monolayer
may have a mechanical instability similar to the buckling instability of a beam or a plate. In the
buckled state, the wavelength of the surface undulations is the capillary len h resulting from
the balance between gravity and curvature energy and is in the range of lo4 For the specific
example of block copolymer surfactants, we discuss the competition between the buckling
instability and dissolution of the surfactant in the supporting liquid.
f
1. Introduction.
Many amphiphilic molecules adsorb at an air-water or oil-water interface where they
form insoluble monolayers [l]. On a Langmuir trough one can compress these monolayers by
imposing an external surface pressure 17. The interfacial tension between air or oil and
water then decreases to
A variation of the surface pressure allows a thermodynamic study of the phase diagram of
monolayers: a very rich variety of phases (gas, liquid, and solid) is in general observed,
which have been extensively studied both experimentally and theoretically.
In this letter, we are interested in the limit of high surface pressures where the surface
tension y is very small and even becomes negative. The interface is then unstable and the
system wants to increase its interfacial area. For an oil-water interface, this often results in
the dispersion of oil in water or of water in oil and leads to the formation of a microemulsion
phaseE21. In such cases, however, the interface does not remain flat and in addition to
surface tension effects, one must also consider curvature effects which have been first
introduced and emphasized by Helfrich [3]. If we consider surfactant molecules for which
the spontaneous curvature vanishes, the curvature energy per unit area is proportional to
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the square of the local curvature 1/R
The bending constant K is in general of the order of a few times kT for usual surfactants, and
can be larger for polymeric surfactants (i.e., block copolymers).
From a mechanical point of view, a flat monolayer is thus equivalent to a plate under
compression; as the pressure is increased, the monolayer undergoes a mechanical buckling
instability which we discuss below [4]. In order to give a more detailed description of the
monolayer, we will use as an example block copolymer surfactants, either at an air-water or
an oil-water interface. These molecules have an A block with NAmonomers soluble in oil (the
oil is a good solvent) and insoluble in water, and a B block with NB monomers soluble in
water and insoluble in oil. For simplicity we consider here only symmetric polymers
(NA= N B= N ) and strongly incompatible blocks (the Flory parameter xAB is large).
At an oil-water interface, each block makes a so-called polymer <<brush.in its good
solvent as soon as the chain concentration o at the interface is large enough (oN6">> 1). At
the air-water interface, the B blocks form a brush in the water and the A monomers form a
molten layer atop the water substrate. In both cases, the free energy per unit surface F and
the surface pressure of the monolayer IT are dominated by the polymer stretching energy in
the brushes. The scaling behavior of F and IT may be estimated using the simple blob model
introduced by Alexander[5]. (Though in this paper we will use the scaling exponents of
ref. [5], our conclusions are not qualitatively affected if we use the mean-field exponents
resulting from the Flory-type arguments of de Gennes [6].)
This model also allows a determination [7] of the bending constant K as a function of
coverage c and molecular weight N
K(o)= TN30
5
'
.
(4)
(The more refined model of Milner et al. [81 gives values for the omitted prefactors of order
unity in eqs. (3), (4),but the scaling description is sufficient for the present work.) We now
study the buckling instability of a surfactant monolayer, first assuming that the monolayer
is insoluble. In the final section we discuss the role of dissolution of the surfactant in the
water and the influence of the buckling instability on the pressure-area isotherms of the
monolayer.
2. Buckling of a monolayer.
The total area of the interface is A and contains p insoluble surfactant molecules, i.e. the
surface density is o =PIA. In a buckled state, the interface is not flat; its total area A is
given in terms of its projected area A. and its displacement from the plane T(x, y) by
ds (1 + (V@')"'
A=
.
(5)
Ao
When a constant external surface pressure ITex is exerted on the monolayer, the total
S. T. MILNER et al.: BUCKLING OF LANGMUIR MONOLAYERS
497
interfacial free energy (neglecting gravity effects) is written
The first term is the interfacial free energy between water and air or oil; the second term
is surfactant free energy (given, e.g., by eq. (3) for copolymer surfactants). The third term is
the curvature energy, where the curvature radius R may be expressed as a function of the
displacement C [in a one-dimensional geometry where the pressure ne,is exerted along the
The final term is the work done by the external
x-axis, 1/R = (1+ d</d~)~’~(d~C/dx~)].
pressure.
In thermodynamic equilibrium, the free energy of eq. (6) is minimized with respect to A ,
Ao,and the displacement C(x, y), with the constraint given by eq. (5). We thus introduce a
1 I
r
Lagrange multiplier A and minimize G‘ = G - A A -
I
1
ds (1 + (d</dx)2)1’2. Assuming
AQ
that the distortion of the interface C is small and kekping only the lowest-ord& terms in
we obtain
ne,-yo+A=O,
WA)
+yo-A=O,
dA
C,
d4C
d2C
K(0) -- A-==.
dx4
dx2
The interface has thus a sinusoidal deformation C = ho sin (xx/L) (where L is the size of
the monolayer in the x direction), which first appears at a threshold IIc = yo + K(a,)(x/L)’.
Here ac is the equilibrium concentration of the monolayer under a surface pressure ITc
(a2d(F/a)/da = ITc).Notice that the interfacial tension given by eq. (1)is slightly negative at
the buckling threshold; the monolayer is stabilized by the bending energy if the surface
tension y is such that - K ( X / L <
) ~y < 0.
The deformation of the interface in general modifies the gravitational energy of the
liquids. This is taken into account by adding to the free energy the gravitational contribution
where Ap is the density difference between the two liquids and g is the acceleration of
gravity.
In the same linear approximation as above, only the last of eq. (7) is modified, to
K (U)-
d4<
d2<
- AApg<=0
dx4
dx2
+
Looking for a periodic solution C = ho sin qx, the wave number q is related to the external
pressure ne,by
The first instability occws when the right-hand side is a minimum, i.e. for a wave number
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The wavelength of the instability is thus the capillary length obtained by balancing the
gravity and curvature energies. With reasonable values of the parameters K kT,
AF 1
the capillary length is of the order of lo4A and could be accessible by
standard optical techniques. It is increased a bit when we use monolayers with a higher
bending constant (for copolymer monolayers, q, decreases as N-3’4but N is never extremely
large) or by choosing an oil with a density nearly matched to that of water. The gravity
effect is important if q,L >> 1 which will always be the case in practical situations.
The threshold of the instability is obtained from eq. (10) as
-
-
ITc= yo + 2(K Apg)l”.
(12)
-
With the same orders of magnitude as above, and y (10 + 100) ergcm-2, we find that
(I7,
- yo)/yo=
which should not be a measurable quantity. Here again the interfacial
tension at threshold is negative, but extremely small, and the buckling transition occurs in
practice when the interfacial tension vanishes.
The linear theory that we have used thus far does not give any prediction for the
amplitude of the undulations ho. This is obtained by expanding the free energy G’ to higher
orders. (Notice that the curvature energy we used is an expansion in lIR, and that higherorder terms should be included in principle. However, for block copolymers this is an
expansion in DIR where D is the thickness of the monolayer, and we expect DIR to remain
small in practice.)
In the absence of gravity, we find a second-order buckling transition with a continuously
varying undulation amplitude h above 17,
n-n,
(y217,’
8
=
When gravity is included, an expansion of the free energy G’ to sixth order in C (assuming a
single wave of sinusoidal shape) shows that the transition is first order. The amplitude h
jumps from zero to a finite value (of order the unstable wavelength) for an external pressure
ITl slightly smaller than 17,
- yo = 1.95KqE). The wave number q, is also increased from
the linear theory value of eq. (11)by about 10 percent. The amplitude of the undulation is
such that (qch)2= 0.35.
In practice, the critical pressure is (as above) indistinguishable from lI1= yo, but the
order of the transition is perhaps observable. Notice that all these results are obtained by an
expansion in qh, and that qh is of order unity; nonetheless, we expect these results to be
qualitatively correct. The above numerical results would also be altered somewhat if the
unstable mode were not assumed to be sinusoidal, i.e. if couplings to higher harmonics were
retained. Finally, the first-order transition need not be to a surface distorted in only one
direction; the actual unstable pattern may be, e.g., square or triangular, which could in
principle be calculated by an expansion to sixth order in q, h for each possible pattern (again
neglecting higher harmonics).
3. Discussion.
An insoluble monolayer should thus always buckle under compression. In real
monolayers, however, the amphiphilic molecules are always slightly soluble, either in oil or
in water. When the surface pressure is increased, the number of solubilized molecules
increases and this could preempt the buckling transition which then becomes unobservable.
S. T. MILNER
et al.:
BUCKLING OF LANGMUIR MONOLAYERS
499
We will not give here a complete description of the dissolution of monolayers, but will
discuss it qualitatively.
As soon as the bulk solution (in water or oil) in equilibrium with the monolayer is below
its critical micelle concentration (CMC), dissolution is a small effect which may be neglected.
When the critical micelle concentration is reached, the interface may shed micelles readily
and the monolayer cannot be considered insoluble, The criterion for neglecting the
dissolution (and thus for observing the buckling transition) is that the chemical potential of
the surfactant molecules in the monolayer p, be smaller than pcMC,the chemical potential at
the CMC.
In order to make this more quantitative, we use the example of block copolymer
surfactants. Choosing as a reference state the melt for A monomers and a good solvent for B
monomers, the chemical potential at the CMC has been calculated in ref. [91 as
At the buckling transition, the chain concentration c is approximately given by y = O , or
where a is a monomer size.
For a monolayer at an air-water interface, the A monomers form a molten layer with
For a monolayer at an oil-water interface, there is an extra term in the monolayer chemical
potential which takes account of the fact that the A monomers are in a good solvent. This
contribution is proportional to the number of A monomers per chain, so that we have
yoa2
pm = - N T
+T(T)
5/11
N6l1l
The thermodynamic argument then shows that for large molecular weight N , a monolayer at
an air-water interface should dissolve before buckling, while at an oil-water interface, a
buckling transition should be observed. However, this does not consider the kinetics of
dissolution of the monolayer, which may make the buckling transition observable in a
metastable state.
Many of the experimental data on monolayers are pressure-area isotherms. The above
theory predicts a break in the isotherm at the buckling transition (see eq. (13)) in the
absence of gravity and a plateau in the presence of gravity; this plateau, however,
corresponds to a very small range of area and i s probably not observable in practice. Direct
observation of the sinusoidal layer distortion should thus be the easiest way to observe this
transition.
Finally, let us mention that we have assumed throughout the paper that the surfactant
remains in a fluid phase; solid phases are often formed under compression of monolayers,
and our model is not valid for such monolayers.
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***
We thank for its hospitality the Aspen Center for Physics, at which this work was begun.
We have benefited from several useful discussions with M. E. CATES,M. 0. ROBBINSand D.
Roux.
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