Langmuir 1995,11, 3928-3936
3928
A Confined Complex Liquid. Oscillatory Forces and Lamellae Formation from an L3 Phase Plamen Petrov," Stanley Miklavcic,? Ulf Olsson, and Hakan Wennerstrom Division of Physical Chemistry 1, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden Received March 6, 1995. I n Final Form: June 14, 1995@ In this paper, we present the results from measurements of forces between macroscopic mica surfaces immersed in an L3 (sponge) phase, Two different surfactant systems are investigated. As previously reported (Petrov,P.; et aZ. Langmuir 1994,10,988)for the AOThrineIwater L3 phase, two distinct regions in the force profile are observed: At large separations (40-120 nm), the force is oscillatory due t o the structure in the bulk solution. In a confined space, at separationssmaller than -30 nm, there is a topological transition to a layered, lamellar-like structure with a force having huge repulsive barriers with a period of 3-3.5 nm, weakly dependent on concentration. The same characteristic forces are observed for an L3 phase made of ClzEs/hexanol/water. The period of long-rangeoscillations, scaled by the thickness of the bilayer, is inversely proportional to the surfactantvolume fraction and shows a uniform hnctional dependence for both L3-phase systems, in accordance with the general scaling law. Model calculations of the force profile within the flexible surface model reproduce and rationalize all qualitative observations and give a reasonable quantitative estimate ofthe force. We derive generally applicable conditions for the appearance of a stable lens of lamellar phase between curved surfaces in a bulk L3 solution and calculate the force explicitly, without invoking the Derjaguin approximation.
Introduction The condensation of a liquid from its vapor in a wetting capillary is a common occurrence. Although a classical physical chemistry problem, it continues to be the focus of modern experimental and theoretical interest.lS2 Capillary condensation is, however, but one example of a broader class of phenomena which may be more generally referred to as surface-induced phase transition. For capillary condensation, the principal driving force is the lowering of the surface energy on liquid condensation. Under some circumstances, in particular for hydrophobic surfaces in water, one can have the reverse phenomenon of capillary vaporization, where the formation of a gassolid interface actually lowers the surface en erg^.^,^ A third possibility is a liquid-liquid surface included phase separation. This can, for example, be found when apolar liquids, containing some dissolved water, meet polar surfaces in a confined ge~metry.~" Yet another version of the phenomenon is encountered when a surface induces a transition between different condensed phases of essentially the same composition. The surface constraints can reduce fluctuations and thus favor more ordered phase^.^ However, one can find the reverse, that surface confinements are incompatible with the ordered structure, thus favoring a more disordered arrangement. These phenomena can all occur in simple liquids, and for a complex fluid, we should expect an even richer behavior. In the complex fluid, we have a structure on the mesoscopic length scale, and we can identify two Present address: I a n Wark Research Institute, University of South Australia, T h e Levels, S.A. 5095, Australia Abstract published i n Advance A C S Abstracts, September 1,
aspects of the surface confinements. The fluid structure can be more or less compatible with a single surface. Furthermore, in a confined geometry with two surfaces, there might be a compatibility problem between the intrinsic fluid structure and the geometrical constraints between the surfaces. In the present paper, we investigate the properties of an L3 or sponge phase between curved mica surfaces in the surface force apparatus. The L3 phase consists of a (single)surfactant bilayer of complex topology in a solvent. It is an isotropic liquid phase, and in the bulk, it forms two phase equilibria with a lamellar phase, an isotropic dilute phase, and, sometimes, a bicontinuous cubic phase. In our view, the driving force leading to the formation of the complex bilayer topology can be associated with a slight negative spontaneous curvature of the constituent monolayer.8 Recently, we have analyzed the thermodynamicg and light-scattering propertiesgJOusing such a description. In other theoretical approaches, more emphasis is put on random surface models" and the role of the renormalization of the bending constant.12 The present work should also be considered as part of a continuing effort to understand the properties of LBphases in general. We have already provided a preliminary report on force measurements in the L3 phase.13 The present work contains a wider experimental exploration with more concentrations studied and comparison between two chemically different systems. In addition, we make a detailed theoretical interpretation of the experimental findings. At the time of writing, we discovered that Antelmi et aZ.I4 had reported the results of a similar
+
@
1995. (1)Wanless, E. J.; Christenson, H. K. J. Chem. Phys. 1994,201, 4260. (2)Christenson, H.K. Phys. Reu. Lett. 1994,73,1821. (3)Yushchenko, V. S.;Yaminsky, V. V.; Shchukin, E. D. J . Colloid Interface Sei. 1983,96,307. (4)Christenson, H.K.; Claesson, P. M. Science 1988,239,390. (5) Christenson, H. K. J . Colloid Interface Sci.1986,104, 234. (6)Christenson, H.K.; Fang, J.; Israelachvili, J. N. Phys. Rev. B 1989,39,11750. ( 7 )Moreau, L.; Richetti, P.; Barois, P. Phys. Reu;Lett. 1994,73,3556.
0743-746319512411-3928$09.00/0
( 8 )Anderson, D.; Wennerstrom, H.; Olsson, U. J . Phys. Chem. 1989, 93. - - , 4243. ~~~( 9 ) Daicic, J.;Olsson, U.; Wennerstrom, H.; Jerke, G.; Schurtenberger, P. J . Phys. II. Fr. 1996,5,199.
(10)Daicic, J.; Olsson, U.; Wennerstrom, H.; Jerke, G.; Schurtenberger, P. Phys. Reu. E , in press. (11) Pieruschka, P.;Safran, S. A.Europhys. Lett. 1993,22,625. (12)Cates, M.E.;Roux, D.; Andelman, D.; Milner, S. T.; Safran, S. A.Europhys. Lett. 1988,5,733. (13)Petrov, P.; Olsson, U.;Christenson, H.; Miklavic, S.; Wennerstrom, H.Langmuir 1994,10, 988. (14)Antelmi, D. A,; KBkicheff, P.; Richetti, P. J . Phys. II Fr. 1995, 5,103.
0 1995 American Chemical Society
Langmuir, Vol. 11, No. 10, 1995 3929
A Confined Complex Liquid
2 '
'
20
10
I
I
I
30
40
50
wt% AOT
6
" 14
" 16
" ' 18 20
22
24
26
wt% C&
Figure 1. Phase diagrams for the pseudobinary systems AOT/ water/NaNOs (part a) and ClpE&exanol/water (part b). The L3-phaseregion is drawn shaded, neighboring two-phase regions with excess brine or lamellar (La)phase. The compositions, at which surface force measurements were performed, are denoted by crosses ( x ).
investigation. Our results agree to a large extent but not completely, and we comment on this issue throughout the text below.
Materials and Method Forces between mica surfaces were measured using Israelachvili's technique, described in full elsewhere.15J6 In brief, the procedure involves cleavingequally thin sheets of molecularly smooth ruby mica which are then silvered on one side. The surfaces are glued with an epoxy resin (Epikote 1004 from Shell ChemicalCo.), silvered side down, onto two supporting cylindrical silica disks which are then mounted, respectively, to a piezoelectric tube and double-cantilever spring in crossed cylinder geometry. As the spring has a finite elastic strength, an inherent mechanical instability occurs whenever the gradient of the force, aF/aD, overcomes the spring constant. The surfaces then jump to the next stable position, where aFlaD is again less than the spring constant. Between such two points, we only have a lower bound to the force. With the SFA assembled, the surface-surface contact in air was determined and an L3 phase injected (-30 cm3). The first force run was performed some hours after injection, as soon as the thermal drifts were sufficiently low to ensure a stable calibration (Le., Cf0.5 nm/min). Subsequent measurements made within intervals of several days indicated no time dependence of the force. Each individual study was repeated several times on different contact positions with excellent reproducibility found. All the measurements were carried out in a thermally insulated room, kept at a constant temperature of 22 "C. The first series of experiments were performed on the L3 phase formed in a system ofAOT/water/NaNOs. The nitrate salt, unlike the more common NaC1, does not degrade the silver layer on the back side of the mica surfaces. The phase diagram with NaN03, shown on Figure la, does not differ significantly from that observed with NaC1.17 Sodium bis(2-ethylhexy1)sulfosuccinate (AOT)was obtained from Sigma (99%pure) and was used without further purification. N d 0 3 was supplied by Merck (pro analysi, '99%). Deionized and distilled water used in all experiments was passed through a Millipore water system consisting of an ion-exchangecartridge, organex-Q,activated charcoal filter, and (15) Israelachvili, J. N. J . Colloid. Interface Sci. 1973,44, 259. (16) Parker, J. L.; Christenson, H. K.; Ninham, B. W. Rev. Sci. Instrum. 1989,'60, 3135. t 17) Fontell, K. AOT-NaC1-water. In Colloid Dispersions and Micellar Behavior; ACS Symposium Series; American Chemical Society: Washington, DC, 1975; p 270.
nucleopore filters. The samples were prepared by weighing the components and mixing until homogeneity. The sample compositions used (in weight percent) were as follows: (1)10%AOT, 2.8% NaN03; (2) 20% AOT, 3.15%NaN03; (3)30% AOT, 3.1% NaN03; (4) 30% AOT, 3.3% NaN03; (5) 30% AOT, 3.5% NaN03; (6) 30% AOT, 3.8% NaN03; (7) 40% AOT, 4% NaN03; (8)50% AOT, 4.4% NaN03. The positions on the phase diagram of these compositions are denoted by crosses in Figure la. The samples were inspected in transmitted light, in scattered light, and between crossed polarizers to confirm the existence of a single, homogeneous, and isotropic phase. The second experimental system studied was the pseudobinary C12E5/hexanol/water.~~ Pentakis(ethy1ene glycol)dodecyl ether (C12E5)was obtained from Nikko Chemicals Co. and used without purification. Hexanolwas supplied by BDH. The completephase diagram of this system is currently under investigation; however, we present in Figure l b the relevant part containing the L3phase region for high-volume fractions, at the constant temperature 22 "C. Experiments were performed on samples with the followingcompositions: (1)14%C12E5,7% hexanol; (2) 15.6% C12E5,8%hexanol; (3)18.1%C&5,9.4% hexanol; (4) 22%C12E5, 11.9% hexanol; (5) 25.8% C12E5, 14% hexanol. These two surfactant systems were chosen, among others exhibiting L3 phase regions, for two main reasons. First, it was essential for the L3 phase to be stable at room temperature. Second, the ability to detect structural effects with the SFA requires measurements a t distances up to several times the characteristic structural size in the system. Since it is only possible to measure at distances less than 400-500 nm and since the structural length is inversely proportional to the volume fraction, a lower limit on the surfactant concentration is inevitable. Therefore, to establish any meaningful dependence on concentration, the phase to be studied must extend across a sufficiently broad surfactant concentration range, at sufficiently high concentrations. (1.9 While the bilayer thickness in the AOT system is nm), we measured this quantity explicitly for the C12Es/hexanol samples, used in the SFA study, by small-angle X-ray scattering. Here, the samples were cooled from the L3 phase (22 "C) to 15 "C, where all samples were in a homogeneous lamellar phase. The bilayer thickness was determined from the first-order quasiBragg peak position. A similar value of 2.9 nm was found for all samples. The viscosity of the solution is a potential problem in surface force measurements in highly concentrated surfactant or polymer systems.20$21As we discussed previously,13 the low L3-phase viscosity and the absence of entanglements in the structure lead to fast equilibration rates. Nevertheless, the measurements were carried out very slowly (in steps of 1.5-2 nm per 30-60 s) to ensure that the system had sufficient time to establish equilibrium after a change of separation.
Experimental Results Figure 2 shows the measured forces between mica surfaces (normalized by their mean radius of curvature R ) in the presence of the 50%AOT (part a) and the 20% AOT (part b) LBphases. Each set of points in this and subsequent figures represents data from at least two separate force runs. The results at 40% AOT were presented in our previous communication,13while those at 30% AOT are given later and discussed in a different context. In comparing the results among the different volume fractions, one is naturally drawn to certain features common to all the force profiles. In all cases, a very weak (18) Jonstromer, M.; Strey, R. J . Phys. Chem. 1992, 96, 5993. (19) Strey, R.; Jahn, W.; Skouri, M.; Porte, G.; Marignan,J.; Olsson, U. Fluid Membranes in the WateriNaCl-AOT System: A Study Combining Small Angle Scattering, Electronic Microscopy and NMR Self-Diffusion. In Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution; Chen, S.-H., Huang, J. S., Tartaglia, P., Eds.; Kluwer Academic: Dordrecht;, The Netherlands, 1992; p 351. (20) Horn, R. G.;Hirz, S. J.; Hadziioannou, G.; Frank, C. W.; Catala, J. M. J . Chem. Phys. 1989,90,6767. (21) Kekicheff, P.; Richetti, P.; Christenson, H. K.Langmuir 1991, 7,1874.
3930 Langmuir, Vol. 11, No. 10, 1995
Petrov et al.
0.10 0.05
-0.104 0
50
0
50
Separation, nm
-0.101:': 0
.
50
100 150 Scparation, nm
100 150 Separation, nm
I 200
I
200
100
150
200
Separation, nm
Figure 2. Normalized force vs distance measured on approach (black circles) and separation (gray circles) for L3 phase of 50% (part a) and 20% (part b) AOT. Forces at short separations are shown in the inserts.
Figure 3. Normalized force vs distance for the L3 phase of 15.6%C12E5/S% hexanol (part a) and 25.8% CnEd14%hexanol (part b). Inward force runs are denoted by black circles and outward ones by gray circles.
oscillatory force is present at separations in the range from 30 to 60 nm, decreasing in amplitude and disappearing in the noise around 120- 140 nm. The periods of these oscillations are quite regular, being 28 f 0.5 nm at 20% AOT down to 8.5 f 0.2 nm at 50% AOT. Their magnitude decays with separation and generally decreases with decreasing surfactant volume fraction. The forces measured on separation coincide with those measured on approach. The small discrepancy seen in Figure 2b illustrates the typical experimental error incurred with this system. For all AOT concentrations, a transition at separations of about 30-50 nm delimits the regime ofweak oscillations, with a regime exhibiting strong repulsive force barriers (shown in the inserts to Figure 2). The true magnitudes of these repulsions are difficult to quantify since the inward jumps to successively stable positions take place at somewhat arbitrary force values. As an additional complication, it is well-known that with the application of such high loads, the mica surfaces tend to flatten since the supporting glue layers have finite e l a ~ t i c i t y . ~Some ~9~~ of the compressive work involved in making the measurement then goes into this deformation, while the remainder, the forces shown, are actually those between planes rather than cylinders. In this event, the concept of a mean radius of curvature is then inappropriate, which, consequently, leaves the magnitude of the normalized force all but arbitrary. During compression, it was quite often the case, particularly at low-volume fractions, that one or more of the force barriers were jumped over, especially near the outer extreme of the lamellar region. Above 40% AOT, the barriers appeared as almost hard walls, while in the 20% system, they were relatively soft, with the stiffness increasing with diminishingsurface separation. Similar force behavior was observed when a stack of lamellar bilayers was confined between mica surfaces.21 On each of the repulsive barriers, the forces measured on separating the surfaces complement the data measured on
approach; the outward profile in deeply attractive (down to -2 to -3 mN/m for the last barrier) until finally the instability sets in so that the surfacesjump out to hundreds of nanometers. In the case of 20%AOT, the slopes of such outward runs, along each successive barrier position, decrease monotonically with increasing separation. Likewise, the depths of the attractive minima diminish rapidly with separation. The 50% AOT system demonstrates an additional interesting feature. The long-range regime of weak oscillations does not have the same uniform behavior as found with low AOT volume fractions but exhibits two distinct subregions. After the first two or three measurable oscillations, the force suddenly becomes steeper and stronger, breaking up into discontinuous barriers resembling those present at short separations, although orders of magnitude weaker in strength. Note that the periods of the oscillations are one and the same in both subregions. Experiments made with the 10% AOT system show similar qualitative force behavior to that at any higher concentration. However, at this low volume fraction, we were unable to quantify the long-range oscillations to any accuracy with our setup. This is readily understood since by extrapolating the results for higher concentrations, we anticipate that the forces will lie well below the resolution limit. Figure 3 shows two representative results from the Cl2EFJhexanol/waterL3-phase system. We have observed the same two characteristic regions in the force profile found in the AOT system: the long-range weak oscillations and short-range discontinuous force barriers-again separated by an extended transition regime of mechanical instability. The results demonstrate that the structural force due to alignmedordering in the L3 phase and its subsequent transformations is a general phenomenon for such systems. There do, however, exist some small details, which highlight the differencebetween the AOT and Cl2E5 systems. For example, the forces in the latter ease are weaker, particularly in the lamellar region. Also, after the very last repulsive barrier is met on approach, the surfaces jump into adhesive mica-mica contact. The subsequent pull-off force is very high, about 50-60 mN/ m, which is greater than that found in water. While the
(22) Hughes, B. D.; White, L. R. Quart. J . Mech. Appl. Math. 1979, 32, 445. (23)Hughes, B. D.; White, L. R. J . Chem. SOC.,Faraday Trans. 1 1980, 76,963.
A Confined Complex Liquid
Langmuir, Vol. 11, No. 10, 1995 3931
0.1
0.2
0.3
0.4
.0.5
Volume fraction
Figure 5. Period (peak to peak) of the long-range force oscillations in the La phase, scaled by the bilayer thickness, as a function of surfactant volume fraction. The results for the AOT system are marked by 0 and for the system by 0. The lines are hyperbolic fits to the corresponding data points.
the variations in salt concentration, and the same is true for the onset of the force curve. Similarly, the period and the overall shape of the short-range repulsive barriers remain the same. The only feature which is obviously affected is the separation a t which the instability occurs. This point appears at increasing separations as one traverses the L3 phase in the direction toward the boundary with the lamellar phase.
Separation, nm Figure 4. Force vs distance for the L3 phase of 30%AOT and different salt contents: (a) 3.8%NaN03; (b) 3.5% NaN03; (c) 3.3% NaN03; (d) 3.1% NaN03. Forces on approach and separation are denoted by black and gray circles, respectively.
period of the short-range oscillation was again distinctly smaller than that found at large distances and is only weakly dependent on the bulk film volume fraction, in most cases it was difficult to quantify it accurately. The reproducibility of the forces measured in this lamellar region was poorer than in all other cases. This we mainly attribute to the high temperature sensitivity of the samples. The L3 phase in the C12E5 system is stable only in the very narrow temperature range of about 1"C. Any slight variation in temperature, which is inevitable under our experimental conditions, effectively shifts the system closer to either phase boundary. It must be noted that only the forces in the lamellar region are less reproducible; the reproducibility of the bulk oscillations is as good as in the other system. On the other hand, the smaller magnitude of the forces in the lamellar region makes access to them easier, and our ability to quantify them greater, than in the case ofAOT. Figure 3b shows forces measured in the system with the highest ClzE&exanol composition. In this case, reproducibility was excellent. Note that the positions of the minima in these forces, measured on separation, lie quite distinctly on a straight line. Our final task was to investigate the effect on the forces and on the lamellar transition of the relative position within the L3-phase boundaries, which is known to influence the properties of the L3 phase.g Figure 4 demonstrates this effect for the case of 30% AOT surfactant. The composition with the lowest salt content lies very close to the phase boundary with excess lamellar phase, while at the highest salt concentration one reaches the upper phase limit with excess brine. First of all, the period and the magnitude ofthe bulk long-range oscillation remain unaffected (to within experimental accuracy) by
Discussion We observe a range of regimes for the measured force. We discuss them separately and consider first: The Force at Large Separations. As the surfaces approach the first measurable force, appearing in our experiments at 120- 140-nm separation is an oscillatory force. The period of the force varies with concentration. In our previous publi~ation,'~ we interpreted this observation as being caused by the structural order in the bulk phase, an interpretation that was also subsequently made by Antelmi et aZ.14 A further confirmation of the interpretation can be found from the observed concentration dependence of the force. From scattering experiment^^^ and from the general scaling argument for Ls it is known that the basic structural length scale is inversely proportional to the volume fraction of the bilayer. In Figure 5, we have plotted the peak-to-peak period scaled by the bilayer thickness vs concentration. Not only do we find the expected 114 dependence but also measurements for the two chemically very different systems fall on virtually the same line. When extrapolated to 4 = 0.135, the fit also accounts for the period reported in ref 14 (the fit gives a period of 39.8 nm, while their experimental value is 40 f 4 nm). It is interesting to note that the magnitude of the observed period differs from the characteristic size ( 114determined by SANS in the AOTIwaterhJaCl system.lg ( = 2zIq,, where qmis the wave vector magnitude at the structure factor peak maximum. (Replacing NaCl with NaN03 as the electrolyte should have no consequence for the structure at a given volume fraction.) The characteristic lengths in the SANS measurements are a factor of 1.4 shorter than observed in the force measurements. The L3 phase has a macroscopically isotropic structure with a short-range order of a rather complex nature. Consequently, different methods may well pick up dif-
-
(24) Gazeau, D.; Bellocq, A. M.; R o w , D.; Zemb, T.Europhys. Lett. l989,9,447. (25)Porte, G . ; Delsant, M.; Billard, I.; Skouri, M.; Appell, J.; Marignan, J.;Debeauvais, F. J.Phys. II Fr. 1991,1, 1101.
Petrov et al.
3932 Langmuir, Vol. 11, No. 10, 1995 ferent periodicities. (We note that Antelmi et al.I4report a factor of 2.0 f 0.2 for the ratio of the characteristic length in the force experiments and in SAXS measurements ofthe same system. We see no obvious explanation for this difference between the results of the scattering experiments. ) One disputed way of conceptually understanding the structural properties of the L3 phase is to start from the ordered cubic analogue.s In the scattering of the ordered phase, the most intense peak is, for the gyroid structure, the 211 r e f l e c t i ~ n . ~We ~ . ~can ~ then assume that the scattering peak observed in the isotropic L3 phase is due to short-range order corresponding to the 211 direction.27 In the force measurements, however, we observe a period which is a factor of 1.4 f i longer, which would then correspond to a periodicity along the 111direction. (The = ratio between the 211 and 111 distances is &I& h.)The tentative conclusion is thus that the L3 phase orients preferentially with a 111plane of short-range order parallel to the mica surface. To obtain a quantitative interpretation ofthe magnitude of the force, let us consider a model system of two parallel surfaces of area A at a separation h immersed in a bath of the L3 phase with a total volume Vtot. The volume fraction of bilayer surfactant is &, and its thickness is L. We can identify three contributions to the force between the surfaces. First, there is the direct, mainly, van der Waals interaction. At separations of the order of 100 nm, it is very weak. Second, there is an ordering of the L3 phase at the surfaces. When ordered regions from two surfaces overlap, there is a synergetic effect, leading to a lowering of the free energy, and thus an attractive force. This effect can be analyzed within the Landau-Ginsburg formalism, which predicts an exponential decay with a characteristic length as a correlation length of the bulk liquid.28 Third, the structural periodicity in the liquid phase between the surfaces has to be compatible with the constraints due to the solid surfaces. In the L3 phase, there is an intimate connection between the composition and the characteristic structural length. Thus, as the gap size changes, there is a change in the structural periodicity and, thus, in the average density in the gap. There is a free-energy penalty associated with changing the density from the bulkvalue. This free-energypenalty oscillates between zero, when the gap size matches a multiple of the structural length in the bulk, to a maximum value, occurring when there is a change in the number of ordered layers in the gap. This effect can be analyzed quantitatively as follows. At large separations, the phase is isotropic in three dimensions and makes no resistance to the surfaces when they move closer. When h is such that the interaction between the solution and the hard walls is strong enough to induce alignment and orientation in the phase, we can treat the confined liquid as an L3 phase with volume fraction &, different from the one in the bulk. Since the volume of the gap, V L =~ A h , is small compared to Vtot,the compressiodexpansion of the L3 phase leads to infinitesimally small perturbation in the bulk volume fraction, @b & A@. The total free energy of the system is
-
GY* is the free energy of interaction between the sponge ana mica wall, which has an attractive component from the overlap of the order profile as discussed above, and we have neglected a possible energy associated with the interface between the bulk and the confined L3 phases. The bulk concentration & is defined as VfilmNtot.From material balance in the whole system, we find
Al$= We can expand gs(&
VL34S- VL3& Vtot - VL,
(1)
+ A@) around & as
and obtain
It remains to calculate 4, of the confined L3 phase. From simple scaling arguments, we have (for /i >> L)
l$ = GLIA
(4)
where /i is a structural periodicity, L is the bilayer thickness, and C is a geometric factor. Equation 4 is also taken to be valid for the disordered sponge phase, in this case A being some average bilayer-to-bilayer distance. When our Ls phase is oriented in the gap, at the same time compressed or dilated, we assume that the same equation still holds. Depending on the orientation of the unit cell with respect to the wall, /i could be the lattice parameters or one of the space diagonals. At any given h, we have some number ( n ) of unit cells between the surfaces. A minimum in the total free energy (3)would be achieved if n is the cloest integer to hll, A being the characteristic length of the undistributed bulk solution. In the disordered phase, the periodic order decays out from the surface with a characteristic length IC. An approximate expression for the density in the gap would then be
For h < /ic,@, is for a cubic phase in the gap, while for h ,Ic @, = ,&, and there is no contribution to the force. Inserting (5) into eq 3, we get our final expression for the free energy as a function of separation:
3
+
whereg,(@)is the free-energy density of the L3 phase and (26) Clerc, M.; Dubois-Violette, E. J . Phys. ZI Fr. 1994,4, 275. (27) Olsson, U.; Mortensen, K. J . Phys. II Fr. 1995, 5, 789. (28) MarEelja, S.; RadiC, N. Chem. Phys. Lett. 1976, 42, 129.
Equation 6 predicts an oscillating force with period A = GLl& and magnitude decaying as e-2hWn. Dividing by A, eq 6 gives the free energy per unit area as a function of the separation between two planar surfaces. According to the Derjaguin approximation, this also represents the measured force per unit area between two curved surfaces if one divides by 2 n R (radius of curvature of the cylindrical surfaces). We have previously
Langmuir, Vol. 11, No. 10, 1995 3933
A Confined Complex Liquid analyzed the thermodynamics of L3 phases and concluded that the free-energy density is of the
g,(+> = const
+ a+3+ p+5
(7)
where a < 0 and /?> 0. The thermodynamic calculations also provide an estimate of the coefficients. C is weakly topology dependent, and for a gyroid, C = 3.091.8,30The only remaining undetermined coefficient is A,, since n is chosen to minimize Gbt. The insert on Figure 9 shows a calculated force curve which qualitatively reproduces the measured curve at long range for the 20% sample, as well as giving the correct order of magnitude of the force. We don't expect accurate quantitative agreement primarily because we have neglected the fact that the L3 phase will deform to become anisotropic between the surfaces. Transition to an Ordered Cubic Phase. Above we have interpreted the oscillatory force at long range as caused by the short-range order in the isotropic L3 phase. However, for the sample containing 50%w/w surfactant, we observe a qualitative change in the nature of the oscillatory force at a separation of 80 nm. This sample has st composition close to the boundary with the twophase coexistence L3-ordered cubic. The surfaces induce an increased order in the L3 phase, while for a cubic phase this order is already present. As a consequence, the surface energy for the cubic phase should be lower than for the L3 phase, since free energy does not have to be spent ordering the system. Accordingly, we expect a confinement-induced L3-to-cubic transition sufficiently close to the phase boundary, and we tentatively interpret that this transition actually occurs for the 50%w/w sample at a separation around 80 nm. Further experiments are needed to verify this interpretation. We note that an analogous observation has recently been made by Moreau et aL7of a surface-induced ordering of a nematic-to-smectic lyotropic liquid crystal. Transition to a Lamellar Phase. In the range of separation between 40 and 70 nm, depending on the composition, we observe a mechanical instability and a transition to another regime. Here the force is again oscillatory but much stronger and with a shorter period. In our preliminary report, we interpreted these observations as being caused by a first-order transition, leading to a lens of lamellar phase between the curved surfaces. Antelmi et uZ.l4made the same quantitative observations and supported our interpretation. In our extended measurements reported here, all the new observations are consistent with a formation of a lamellar phase, and they also indicate that this transition is a generic feature since it is not only observed with AOT but also with the chemically rather different nonionic surfactant systems. Lyotropic lamellar phases are known to orient parallel to surfaces31presumably since the planar geometry of the surface fits well to the symmetry of the lamellar phase. In the L3 phase, on the other hand, the bilayer extends in three dimensions in a topologically complex way. To conform to a planar surface, one should expect local structural deformations, which cost some free energy. This leads to the expectation that the surface free-energy micalamellarphase,y ~ l ~is,lower ~ , than the surfacefree-energy mica-L3 phase, YLJmica. It is this difference in surface free energy that drives the La-to-La transition in the gap, analogously as for a capillary condensation. Let us now analyze the L3-to-lamellar phase transition in the confined geometry in more detail. In this case, we (29) Wennerstrom, H.; Olsson, U. Langmuir 1993,9, 365. (30)Anderson, D. M.;Davis, H. T.; Scriven, L. E.; Nitsche, J. C. C. Adv. Chem. Phys. 1990, 77,337. (31)Lindblom, G. Acta Chem. Scand. 1972,26, 1745.
+
c Hm
Figure 6. Schematicdrawing of a model system consisting of a lens of lamellar phase with radius R,, confined between a sphere (of radius Ro)and a plane, in equilibrium with the bulk L3 phase. The closest separation between the surfaces is ho.
cannot rely on the Derjaguin approximation, since a new phase is formed between the surfaces and the extent of the new phase depends in an elaborate way on the geometry. In the SFA, we have an approach of two crossed cylinders. Equivalent, from a curvature point of view, to two crossed cylinders is a sphere over a plane. We use this as a model system as illustrated in Figure 6, showing a sphere with radius Ro separated by a distance ho from a plane and confining lamellar lens with finite radius Rm in equilibrium with the surrounding L3 phase. A priori, we don't know the composition in the lamellar phase, the magnitude OfRm, or ho where a lamellar phase first appears. These variables are determined by minimizing the free energy of the system. The-Gbtof the system is a sum of the bulk free energy due to the L3 phase, Gb, and the free energy of La in the gap, GL,:
Gtot = GL3+ GL, These free energies are made of bulk and surface terms:
where Vbt is the total volume of the system, VL,is the volume of the La drop, g, is the bulk free-energy density of the L3 phase, and &,is the volume fraction of surfactant in the whole system. The surface term GCd can be divided into a constant and avariable part that we include in GC*. As argued above, we assume that the creation of a La phase affectsonly slightly the bulk sponge volume fraction, and we can expandg,(&, A@)around &, (eq 2). In a way similar to eqs 1and 2, we can relate the volume fraction in the La lens, 6,to the bulk &,. Inserting eqs 9 and 10 and 1and 2 into the total free-energy expression (eq 8), we get
+
where is the volume fraction of surfactant film in the La lens. Our first task will be to derive the volume fraction of the lamellar phase. In the curved geometry that we consider, the bilayers are in general no longer strictly parallel to each other and a set of defects appear to allow them to follow the curvature of the walls. This implies
Petrov et al.
3934 Langmuir, Vol. 11, No. 10, 1995
that the phase is inhomogeneous and we cap only talk about its volume-averaged volume fraction, 41:
The local volume fraction at each position within VL, will fluctuate around the averaged value 41. In a first-order approximation, we can take these local density fluctuations to be very small and write the G Y energy as
(13) -
41 has to be determined from the condition, giving a minimum in the total free energy. Equation 11 can be rewritten using (12) and (13):
L~ volume fraction, $1
Figure 7. Plot of &id+ as a function of 41, calculated according t o eqs 14 and 15. The parameters used in the calculations are & = 0.2,a = - 4 k ~ T ,p = 50k~T,CH = lo-' J/m2,LH = 0.2 nm, J/m2,LD = 0.5 nm, A = 9 x J, and K = 10k~T. CD=
6.
The surface free-energy term, GYf, is largely independent of & since the surfactant layers are oriented parallel to t_he mica surfaces. After differentiation with respect to 41, we get the following equation:
(14) If gl and g, are known functions, we can solve the above equation with respect to 41. This is a central result, providing a link between the properties of one phase confined in a small volume surrounded by a different phase. Before continuing to construct a model which accounts for the defects and for the local changes in density, we explore the consequences of eq 14. Above in eq 7 we presented an expression for the g, of the L3 phase. For lamellar phases, there exists a substantial amount of data on forces between bilayer^.^^^^^,^^ At large separations, an undulation force is present. For the AOT system, an electrostatic double-layer force is operating, but due to the high electrolyte concentration 0.3-0.5 M, there is a sharp screening with Debye lengths in the range 0.4-0.6 nm. Also operating is a van der Waals attraction on a short-range repulsion. This leads to an expression for the free energy per unit area of a lamellar phase:
where K is the bending rigidity, LDis the Debye screening length with amplitude CD,Ais the Hamaker constant, LH is the decay length of the short-range repulsion and CH is its amplitude, and L, and L are the thicknesses of the water layer and liquid bilayer. From this, we can obtain the expression for dgl/d$. Figure 7 shows a plot of @Id@ for typical values of the parameters. Since dg$d@varies from slightly negative to slightly positive over the stability range of the L3 phase, we-only find solutions to eq 14 at relatively large values of $1. Due to the steepness of the curve for dglId4, a relatively large change in (dgJd@)lh (32) Kbkicheff, P.;Christenson, H. K. Phys. Rev.Lett. 1989,63,2823. (33)Abillon, 0.;Perez, E. J.Phys. (Paris) 1990, 51, 2543. .
-
results in only a small change in The conclusion is then that the volume fraction surfactant in the lamellar phase in the lens between the surfaces is substantially larger than in the bulk. Furthermore, it is not expected to vary substantially when the composition in the bulk phase is changed, This prediction is consistent with the experimental observations, but it should be remembered that it is based on a description of the lamellar phase in terms of a mean density 41. Now let's go back to the general formulation ofthe freeenergy problem (eq 11)and account for the effect of the defects and the local density variations. In order to calculate G Y and we have to state explicitly the shape and number of defects. If we accept the situation as shown on Figure 6, then we have some (integer) number of lamellar bilayers between the tip of the sphere and the plane, no, and at certain radius rL,a defect appears. Although there is no direct experimental evidence, we assume for simplicity ofthe calculations that the number of layers changes from no 2(i - 1)to no 2i, where i goes from 1to some maximal number ofdefects, Ndef. As an approximation, we can assume that at any distance r out from the center, the bilayers are equally spaced and parallel (locally)to the planar surface; i.e., we approximate the area of each bilayer with its projected one. p" is then
e",
+
+
where L is the bilayer thickness and R, is the radial extent of the lamellar drop. The free energy G$lk can be written in a similar way, but it has to account also for the presence of defects. The simplest way to do this is to take circular defects with some line tension Kdef:
where G*(L,) is the interaction free energy per unit area between two bilayers as a function of the water layer thickness (L,) between them, eq 15. In our approximation, L,.,(r) is
Two surface energies contribute to Grf-one due to the contact between La and the solid mica surface and another from the cylindrical La& interface:
A Confined Complex Liquid
Langmuir, Vol. 11, No. 10, 1995 3935
where AI and A2 are the areas of the Ldmicaand L&3 interfaces, respectively. What remains is to calculate the integral of GA(L,(r)) in (17). The local water spacings, L&), are not expected to differ a lot from the average one, L,, so we can expand GA-around the average value. Up to second order in L,(r) - L,, it gives " 1
1
0
,
1
.
10
20
.
I
30
.
I
40
Separation, nm
(L,- LWl2d2G,
Working out the integration in G T and substituting (14), (16), and (18) into ( l l ) ,we end up with the following result for Gbt:
Figure 8. Numerical results for the optimal radius of lamellar lens, R,, between the sphere and a plane as a function of the separation ho. The same numerical parameters as in Figure m. 7 are used. The radius of the sphere is Ro = 2 x 0.5 7
"._
I
0
I
.
I
10
.
, 20
30
40
Separation, nm
Figure 9. Force vs distance for LS (insert)and La(main plot) phases between the sphere and a plane (see text for details). The numerical values used in the calculations are (inclusive those used in Figure 7) YL,/mica - y ~ g= ~-3 , x ~ ~ J/m2,~ L , I L ~ J/m, and ;i, = 100 nm. =2 x J/m2,Kdef =
i=l
4
where is the ayerage reticular spacing in the lamellar phase; d = L L,. The radial extent of L, drop, R,, the radii ri, and number Ndefofdefects are parameters which have to be adjusted to give the minimum in the total free energy (eq 19). In-eq 19, there are six different terms which play various roles in determining the overall force behavior. The second term provides the main driving force for the L3-to-La transition, while the fourth term, proportional to the amount lamellar phase, represents the free-energypenalty of transforming a bulk L3 to a bulk La phase. This term thus represents the main restoring effect. The surface energy in the third term is important since it causes the transition to be first order. For sufficiently small radii of the lens, the area A2 >> AI, and the surface energy terms can't drive the transition. However, since AI Rm2while A2 R,, essentially AI >> A2 at large R,. The fifth term appears since the spacing between bilayers deviates from L, due to the constraints from the surfaces. This results in an oscillatory component to the force which is further modified by the presence of the defects, whose free-energy contribution is found in the sixth term. There exists no independent measurements of the surface energies appearing in (19), nor do we have an estimate for K&f. Nevertheless, (19) can be used, if not to predict then to interpret the force measurements. Choosing reasonable values for the parameters, Figure 8 illustrates how the radius of the lens varies with separation. At = 37 nm, a lens of lamellar phase with radius ~5 pm appears which grows in size to around 35 pm at contact. The corresponding force curve, calculated by taking the derivative in (191, is shown in Figure 9. (Note that the Derjaguin approximation is not invoked.) It has a period of 3 nm. Note that repulsive barriers are much larger than in the L3 phase. Another feature of the calculation that is also seen in the experiments is the linear slope of the force minima (cf. Figures 2b and 3b).
-
+
-
The discontinuities in the force arise in the calculations from the creation of a new defect in the central area of the lens. In the experiment, such a bilayer breakthrough also has to occur. Nucleation of holes in the bilayer or thin and we don't films is often a slow or very slow expect to measure a true equilibrium force under these circumstances. Indeed, force curves are not fully reproducible in this regime, and we observe jumps of one, two, or more bilayers in individual experiments. In repeating under nominally identical conditions, barriers and jumps occur in a slightly different way. The curves shown are the combination of a number of separate measurements and reveal a pattern with nearly equidistant barriers, each of which is observed more than once. For samples with 30% surfactant, it was observed that the L3-to-L, instability occurred at larger separations the lower the salt content. The qualitative interpretation of this observation is that the closer to the L A , two-phase region, the less costly is the formation of the La phase. It is thus the fourth term in (19) that decreases with decreasing salt content. All our qualitative observations of the induced La phase agree with these made by Antelmi et al. However, they find a significantly larger spacing in the lamellar phase. We have no obvious explanation for this discrepancy but remark that the surfaces are different and that the authors report on the results at a rather dilute concentration.
Conclusions The main results of this investigation can be summarized as follows: (i)A n LBphase confined between two solid surfaces gives rise at large separations to an oscillatory force reflecting the presence of a short-range periodic order in the bulk phase. (34)Exerowa, D.; Kashchiev, D.; Platikanov, D.Adu. Colloidlnterface Sci. 1992,40, 201.
Petrov et al.
3936 Langmuir, Vol. 11, No. 10, 1995 (ii) The effect can be seen for chemically distinctly different L3phases, and the period of the oscillationsshows a universal 114 dependence. (iii)The observed period is longer by a factor of 1.4than the characteristic length associated with the peak in the structure factor from small-angle scattering measurements of corresponding systems. (iv)We tentatively interpret this as due to a preferential orientation of the 111plane of a gyroid structure in the L3 phase. (v) At high volume fractions (50% w/w), the surface confinement seems to induce a transition to an ordered bicontinuous phase. (vi) At shorter separations, we found for all samples studied a first-order transition, leading to the formation of a concave lens of lamellar phase between the curved surfaces. (vii) The L3-to-La transition results in a mechanically unstable region followed by a series offorce barriers caused by the lamellar structure. (viii) The repeat distance in the lamellar phase correspond to a higher volume fraction of the surfactant in the lens of lamella than in the bulk L3 phase. To first order, the equilibrium condition is given by (14).
(ix) The transition to a lamellar phase is driven by a lower surface energy La-mica than L3-mica caused primarily by the more complex topology of the L3 phase. (x) The first-order character of the transition is due to the non-zero interfacial energy L3-L, of the interface at the edge of the lens. (xi)In the lamellar regime, there is an attractive periodic minimum in the force that varies linearly with the separation. This attractive background in the oscillatory force is caused by the fact that the closer the surfaces, the smaller is the free-energy cost in converting bulk L3phase to bulk La phase.
Acknowledgment. We are indebted to Hugo Christenson for his encouragement of this work and many helpful discussions. We thank Vijay Rajagopalan for his help with the SAXS measurements on the ClzE&exanol/ water system. This work was supported by grants from the Swedish Natural Science Research Council (NFR), the Swedish Council for Planning and Coordination of Research (FRN), and the Goran Gustafsson Foundation. LA9501804
