Langmuir 1991, 7, 1855-1860
1855
Bending Elasticity Measurements of a Surfactant Monolayer by Ellipsometry and X-ray Reflectivity J. Meunier* and L. T. Lee+ Laboratoire de Physique Statistique de l’ENS, 24 rue Lhomond, 75231 Paris cedex 05, France Submitted to Symposium Chairman April 13, 1990. Received November 15,1990
Low scale thermal fluctuations of flat surfactant monolayers can be studied by optical techniques to deduce the mean bending elasticities of the monolayers. There are two techniques: The first is ellipsometry, which is very sensitive to fluctuations of short wavelengths and therefore is an accurate method for measuring the bending elasticitybut requires large thermal fluctuations. It is limited to liquid interfaces of low surface tension (y 10-2 mN/m) and low bending elasticity (K kBT). Our analysis takes into account coupling between thermal modes, through a renormalization of the surface tension and the bending elasticity with the scale of measurement, and allows the Gaussian bending elasticity of the monolayer to be deduced. The second method consists of X-ray reflectivity measurements, which are sensitive to the whole spectrum of thermal fluctuations. This method is consequently less accurate than ellipsometry but permits the study of monolayers with large surface tension at the free surface of a liquid. It has furnished interesting information on the jumps of the bending elasticity of a Langmuir film at the “liquid-solid” phase transition. We compare our results obtained by optical measurements with those obtained with other techniques.
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Background The deformation of a film at a liquid interface requires energy to overcome the restoring forces. For instance, in the case of a liquid film, for an increase in area of the film, energy is needed to overcome the interfacial tension, y, and a bending energy is needed to bend the film, such that’ E=fS(&+k-2Co)
2
dS+KSm 1dS
(1)
1 2
where K and K are the mean and the Gaussian bending elastic moduli, respectively,R1 and Rz the radii of curvature of the film, and 2Co is the spontaneous curvature. The value of the mean bending elasticity of surfactant monolayers at oil-water interfaces is the most important parameter in the phenomenological theory of microemulsions.2 The Gaussian bending modulus, which has not been taken into account until recently, could play an important role in the structure of bicontinuous microem~lsions.~ For these reasons, we have developed an optical technique (ellipsometry) to study the thermal fluctuations of surfactant monolayers with low surface tensions at an oil-water interface and to deduce the mean bending elastic modulus. Another case where the mean bending elasticity gives interesting information is the study of “liquid-solid” phase transitions in monolayers at the free surface of water. For this case, another technique for the study of the thermal fluctuations (X-ray reflectivity measurements) was developed. From these experiments, it is also possible to obtain some information about the Gaussian bending elastic modulus of monolayers at oil-water interfaces. The energy needed to create a sinusoidal deformation of wave vector q and amplitude f, at a plane liquid-liquid or liquid-gas interface of area A is the sum of three terms, ~~
+ Present address:
~
LLB, CEN-Saclay, 91191 Gif sur Yvette Ce-
dex. France. (1) Helfrich, W. Z. Naturforsch., C: Bichem., Biophys. Biol., Virol. 1973, ZSC, 693-703. (2) De Gennes, P. G.; Taupin, C. J.Phys. Chem. 1982,86,2294-2304. (3) Porte, G.; Appell, J.; Bassereau, P.; Marignan, J. J. Phys. (Paris) 1989,50, 1335-1347.
corresponding to three different restoring forces The first term is the gravitational energy (Ap is the density difference between the bulk phases and g the gravity acceleration), the second term is the capillary energy (y is the interfacial tension), and the third term is the mean curvature energy ( K is the mean bending elastic modulus). The amplitude f, for agiven excitation energy E provides information on the restoring forces at various scales: At a macroscopic scale (>1mm), i.e. at small q, the gravity term dominates, as in the case of waves on the sea. A measurement of f, can only give information about Ap. At a microscopic scale (10-100 pm), i.e. at larger q, the capillary energy dominates. These q values allow the measurement of the surface tension y. One possibility of such a measurement is to use the thermal motion wave excitation. In this case, the average shape of the interface is planar but is constantly distorted by thermal motion. If the distance from a surface point to the average plane z = 0 is f ( r , t ) where , {f,r(x,y))are the coordinates of the point, one can write f as a sum of Fourier components
where f, is the thermal mode of wave vector q (this procedure is correct as long as the continuum hypothesis holds, i.e., q < qm=,where q- is a cutoff value determined by the typical molecular size). The average energy E, of each mode is E, = kgT/2 (where k g is the Boltzman constant). The surface waves can be studied by a surface light scattering technique4 because the scale of the wavelength of the light is close to the scale of the wavelength of these surface waves. A t an ultramicroscopic scale (typically 100 A), Le. at very large q, the curvature energy term (eq 2 ) dominates and the measurement of the amplitude f, of a thermally excited surface wave (E, = kgT/2) should allow one to determine the bending elastic modulus. This is typically (4) Langevin,D.; Meunier,J.;Chatenay,D. In Surfactants in Solution: Mittal, K. L., Lindman, B., Ed.; Plenum Press: New York, 1983; Vol3, pp 1991-2014.
0743-7463/91/2407-1855$02.50/0 0 1991 American Chemical Society
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1856 Langmuir, Vol. 7, No. 9, 1991
the scale that could be studied by surface X-ray or neutron scattering experiments, but in practice the refractive indices for these two types of beams are close to one and the intensity of the beams scattered by thermally excited surface waves is too low to be measured. The optical techniques, which are sensitive to ultramicroscopic scales, are reflectivity measurements such as ellipsometry, reflected intensity measurements of X-rays and neutrons. These measurements are sensitive to a large spectrum of surface fluctuations and to the structure of the interface; therefore the interpretation of experimental results is a little more complicated than that for surface scattering experiments. At a scale smaller than few angstroms (q > qmm) the continuum model, which gives eqs 1 and 2, is not valid because it is the scale of the molecules.
Reflectivity Measurements Optical techniques such as ellipsometry and reflected intensity measurements are well adapted to the study of liquid interfaces since they introduce very little pertubation. Light is an interesting radiation because many liquids are reasonably optically transparent. X-rays or neutrons can take the place of light, giving more information at low scale because of their small wavelengths. However X-rays have a disadvantage of being strongly absorbed by liquids. Neutron beams tend to have weak flux, but they have two advantages: they are weakly adsorbed by most of the materials and the contrast can be varied by isotopic substitution. In a reflectivity measurement (light, X-rays, neutron reflectivity, or ellipsometry), the scattering vector Q is normal to the interface allowing one to probe the refractive index along the normal. The origins of the index variation along the normal to the interface are the structure (monolayer at the interface) and the roughness of the interface. In the case of an interface between two structureless fluid phases with or without monolayer, the roughness at a scale larger than the molecular scale only originates from thermal fluctuations (or spontaneous fluctuations). Ellipsometry. The reflected field of polarization i, Ei;, for an incident field Ej of polarization j is
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interface,5and upon a possible anisotropy of the refractive index;6 a roughness term,7+' pgR Cq q(rq2). This relationship assumes that 1/X > r]I. The variation of
(18)Rukeinstein, E.In Surfactants in Solution; Mittal, K. L., Bothorel, P., Eds.; Plenum Press: New York; Vol. 4. (19)Verhoeckx, G. J.; de Bruyn, P. L.; Overbeek, J. Th. J. Colloid Interface Sci. 1987, 119, 409-421. (20)Binks, B. P.; Meunier, J.; Abillon, 0.; Langevin, D. Langmuir 1989,5, 415-421.
Bending Elasticity Measurements of a Surfactant Monolayer
Langmuir, Vol. 7,No. 9, 1991 1859
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72
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18
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22
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26
l
l
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34
Figure 3. Ellipsometric parameter r) versus the temperature t measured at the oil-water interface in a mixture of octane, water, and ClJ34, just above the cmc.
t Y
F
32 "/m
I
Figure 5. Tilt angle of the molecules in a behenic acid monolayer and the parameter X = (y(P))ll2deduced from X-ray reflectivity measurements versus the surface tension.
very weak as soon as the incident angle 8 is smaller than the critical total reflection angle 8,. For this reason the incident beam is always grazing: 0.001 < (a/2 - 8) < 0.1 radian. A monolayer of behenic acid on water has been studied in detail with this method.21 The tilt angle of the molecules, which is deduced from the thickness of the interface (using a model), and the quantity X = (y( F))lI2 versus the surface tension have been measured (Figure 5). A phase transition is observed for y 52 mN/m with a jump in the tilt angle and in the X value. It is the liquidcondensed solid phase transition that can be identified on the surface pressure-molecular area isotherm of the monolayer. The bending elasticity increases from a low value (that of a liquid monolayer, K kBT) to a large value (that of a solid monolayer: 1 2 0 k ~ T < K < 300k~T).The accuracy on K is low because it appears in a logarithm (X In ( K ) ) but the jump in K at the phase transition is so large that it is easely observed in X. A second jump is observed on the last experimental point for y 30 mN/ m, which might be a tilted-nontilted phase transition or an artifact due to the vicinity of the collapse. Comparison of Bending Elasticity of Monolayers Measured by Different Techniques. Three other methods have been used by others to measure the mean bending elasticity of monolayers at the oil-water interface. The surfactant is an ionicsurfactant, AOT, which is studied versus the salinity at a constant temperature, 20 "C. The different results obtained for these monolayers are compared to our measurements on the same monolayers: We have found K = l.lkBT in the approximation of independent modes and 1.65k~T at the scale 2 X lo7cm-l in the approximation of coupled modes.l1S20 Huang et al. have studied the shape fluctuation of droplets in microemulsion by neutron spin echo spectroscopy and have deduced K = 5 k ~ T More . ~ ~ recently, Huang et al. have studied the polydispersity of the droplets in the microemulsion by neutron scattering. They deduced K = 0.5k~T,a result incompatible with the previous SafranZ4has recently reanalyzed these two previous experimental results introducing the Gaussian bending
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Figure 4. Ellipsometric parameter versus l/yl/* for the oilwater interface in a mixture of octane, water, and Cl&, just above the cmc. t) versus l/y1/2 is shown in Figure 4. The experimental points fall on a straight line as is specified by the independent thermal mode approximation. The domain of the scale of measurement qe, which is proportional to the experimental domain of l / ~ ' / ~is , small and the logarithmic deviation from this straight line due to the coupling between modes is not observed; it is less than the incertainty on eachmeasurement. In the indepedent mode approximation, we found K = 0.51k~T,while in the coupling mode approximation we found K = 0.76k~Tat the molecular scale q = 1.3 X lo7 cm-', which is the scale length of the cross section of the surfactant molecule. Monolayers at the Free Surface of Water. The ellipsometric measurements of K fail for monolayers of large surface tensions because the roughness is too small and the structural term dominates. A technique using radiation with a wavelength smaller than that of light, such as X-rays, must be used. The refractive indices of liquids for X-rays are close to one and the reflected intensity is
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(21) Daillant, J.; Bosio, L.; Benattar, J. J.;Meunier, J. Europhys. Lett. 1989,8,453-458. (22) Huang, J. S.; Milner, S. T.; Farago, B.; Richter, D. Phys. Rev. Lett. 1987,59,2600-2603. (23) Farago,B.; Richter, D.; Huang, J. S. Submitted for publication
in Prog. Colloid Polym. Sci.
Meunier and Lee
1860 Langmuir, Vol. 7, No. 9, 1991
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elasticity in the calculation of the polydispersity of the T droplets. His preliminary results give K 2 k ~ andK/K -2. Van der Linden et al. have deduced the mean bending elasticity from optical anisotropy induced by an electric field, of a water in oil microemulsion at very small droplet volume fracti~ns.~s The anisotropy is a measurement of the ratio of electrostaticforces,which elongate the droplets, and the bending elastic forces, which favor spherical droplets. They deduced a small value, K = 0.46k~T.In this first analysis, they had not taken into account the polydispersity of the droplets. Since the Kerr constant is proportional to the sixth power of the droplet radius and the experiment is very sensitive to droplets of large radii, they have recently taken into account this effect and estimated K l k B T at the scale of the droplet.m These measurements, however, are very sensitive to the value of the polydispersity introduced in the calculation. In conclusion there are now three sets of experimental results on the mean bending elasticity of AOT monolayers at the oil-water interface, obtained with the different techniques. These results appear to be converging (the analysis from Van der Linden et al. is too sensitive to the polydispersity to give an accurate value and at this moment we do not have the exact value given by the new analysis from Safran to give more comments). The Gaussian bending elasticity of AOT monolayers was also deduced from the results of Huang et al. by Safran’s new analysis. This bending elasticity can also be deduced using equations 18and 19from the measurements of the interfacial tension y, the mean bending elasticity K, and the radii of the droplets in bulk. We have not yet measured the droplet radii for the C l a d system. In the case of the AOT surfactant, the mean bending elasticity was measured at 20 0C11120and the surface tension and droplet radii at 25 0C.27 The variation of the bending elasticity between 20 and 25 OC is expected to be small. The bending elasticity measured was l.lkBT in the approximation of independent modes and 1.65k~T at the scale 2 X lo7cm-l in the approximation of coupled modes. This latter value gives approximately 1 . 3 k ~ T at the scale of the droplets (typically 100 A). The measured radii of the droplets are the externalradii. However, the radius R, which must be introduced in equations 18and 19, is the radius of an internal surface (inside the monolayer), where the area per molecule is not changed when the monolayer is unfolded. The precise location of this surface is not known, but we have assumed that it is in the middle of the monolayer thickness, which is approximately 13 A. Half this thickness was thus subtracted from the measured radii
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(24) Safran, S. A,; Huang,J. S. Private communication. Safran, S. A. Presented at the 198th National Meeting of the American Chemical Society, Miami Beach, FL, 1989. (25) van der Linden, E.; Geiger, S.; Bedeaux, D. Physica A (Amsterdam) 1989,156, 130-143. (26) van der Linden, E. Private communication. (27) Aveyard, R.; Binks, B.; Lawless, T. A.; Mead, J. Can. J. Chem. 1989,66, 3031-3037.
Table I. Evaluation of the Entropic Term and 2K an AOT Monolayer
+ I? for
measd radius,
A
R,A
Y-Y~,
(effective) Y, cgs
yS,cgs
cgs
1/R2, W+@/ WSm-2 ksT
Winsor I
28 58 69 118
21.5 51.5 62.5 111.5
0.252 0.088 0.054 0.12
176 132 114 95 89
169.5 125.5 107.5 88.5 82.5
0.0178 0.0316 0.0562 0.0708 0.0794
KIK KIK
--
0.162 0.0279 0.0195 0.00605
0.09
0.06 0.034 0.0055 Winsor I1 0.00259 0.015 0.00416 0.016 0.00652 0.05 0.00956 0.06 0.0110 0.068
0.216 0.0377 0.0256 0.008
0.10 0.40 0.33 0.17
0.00348 0.00635 0.00865 0.0128 0.0146
1.08 0.63 1.45 1.17 1.16
-1.75 in the Winaor I domain -1.2 in the Winsor I1 domain
of the droplets to get the effective values of R. Taking a = 1,the entropic term yeand the sum of the Gaussian and the mean bending elasticities have been evaluated (Table I). The ratio K/K can be deduced from these calculations in the Winsor I and Winsor I1domain. There is no evidence of a variation of this ratio versus the salinity, but the accuracy is poor. The values are of the same order of magnitude as the one given by Safran (R/K -2). The low experimental value of this ratio K/K and the large K value ( K > kBT) explain the fact that for the AOT surfactant, the middle phase in the Winsor I11 domain is a lamellar phase.
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Conclusions The mean bending elasticity of a flat monolayer at a liquid interface can be deduced from the thermal fluctuations of the interface by optical techniques. Ellipsometry is a more accurate method because it is only sensitive to small wavelength thermal fluctuations whose energy is dominated by the curvature energy. It is well adapted to the study of monolayers of ultralow surface tensions and low bending elasticities, and roughness of which is large, where coupling between modes is important. For monolayers with large surface tensions, thermal fluctuations are small and short wavelength radiations such as X-rays are more appropriate. The sensitivity of X-ray reflectivity to thermal fluctuations is independent of their wavelength; consequently, the accuracy of the bending elasticitymeasured by this method is poor. However,these measurements furnish interesting information on twodimension phase transitions. The results obtained by ellipsometry on AOT monolayers at the oil-water interface are similar to those obtained by two other methods: neutron scattering and Kerr effect. The Gaussian bending elasticitydeduced from our experiments agrees with the one recently deduced by Safran from the experiments of Huang et al. using neutron scattering and neutron spin-echo spectroscopy.
