Langmuir 1986,2,321-329 321

essential role for the unique character in the phase-tran- sit ion.'^^

In the present study we showed that an Na+/K+ con- centration gradient could cause excitation in a L-B film of dioleoyllecithin. This result is quite important in re- lation to the mechanism of excitation of biological systems at the molecular level and may also be significant in the development of a “molecular electronic device”. Further experimental and theoretical studies are awaited to make clear the interesting properties of the deposited membrane herein reported.

Note Added in Proof. Quite recently, we have observed that fluctuation of membrane potential and electrical

current was generated in a pipette-clamp bilayer mem- brane of dioleoyllecithin in the absence of any “channel” protein. It was found that the magnitude of the change of the conductivity through the membrane was ca. 60 pS, which is the same order as the values reported for the membranes embedded with so-called “channel” proteins. Details of this experiment will be published in a separate manuscript.

Acknowledgment. This work was partly supported by Grants-in-aid for Scientific Research to K.Y. (No. 59212029 and 59219017) from the Ministry of Education, Science and Culture of Japan, and by Nissan Science Foundation.

Registry No. NaCl, 7647-14-5; KC1, 7447-40-7.

Possible Mechanism for the Origin of Lamellar Liquid Crystalline Phases of Low Surfactant Content and Their

Breakup To Form Isotropic Phases

Clarence A. Miller* and Olina Ghosh Department of Chemical Engineering, Rice University, Houston, Texas 77251

Received July 30, 1985. In Final Form: January 6, 1986

It is shown that the observed existence of lamellar liquid crystals in many anionic surfactant- alcohol-sodium chloride-water systems containing less than 10% amphiphilic material cannot be explained in terms of DLVO theory if the surfactant bilayers are assumed to have infinite lateral extent. But if the bilayers are taken to exist as plates of appreciable but finite size, a simple model shows that the additional thermal motion increases the mean equilibrium spacing between plates and permits dilute lamellar phases to occur. This model further allows the transition of such phases to dilute isotropic phases containing platelike micelles to be explained in a straightforward way as the result of a decrease in plate size. Such transitions have been observed to occur in both anionic and nonionic surfactant systems as the surfactant becomes less hydrophilic. The mechanism for the necessary decrease in plate size is discussed and a simple model presented.

Introduction The existence of lamellar liquid crystals in many sur-

factant-water and surfactant-alcohol-water systems is well-known.’ In most cases the liquid crystals contain at least 30% amphiphilic material by weight and are not readily deformed owing to their high viscosities and/or elastic moduli. Such materials are useful for some pur- poses, e.g., for stabilizing emulsions,2 but they must be avoided in other applications such as enhanced oil recovery where they would prevent virtually all flow through the oil-bearing formation.

A few years ago lamellar phases of much lower surfactant content and viscosity were found in our laboratory in certain anionic surfactant-alcohol-brine systems of interest for enhanced oil re~overy .~-~ Indeed, their compositions were in the range of interest for injected fluids in such processes. Apparently these liquid crystals and their dispersions in brine had been used without difficulty in numerous laboratory experiments involving flow through

(1) Ekwall, P. Adu. Lig. Cryst. 1976, 1, 1. (2) Friberg, S.; Mandell, L.; Larason, M. J. Colloid Interface Sci. 1969,

29. 155. ~ I~

(3) Benton, W. J.; Fort, T., Jr.; Miller, C. A. Annu. Tech. Conf.-Soc. Petrol. Eng. 1978,36th, SPE 7579.

(4) Miller, C. A.; Mukherjee, S.; Benton, W. J.; Natoli, J.; Qutubuddin, S.; Fort, T., Jr. In ‘Interfacial Phenomena in Enhanced Oil Recovery”; Wasan, D. T., Payatakes, A., Eds.; American Institute of Chemical En- gineers: New York, 1982; AIChE Symp. Ser. No. 212, pp 28.

(5) Benton, W. J.; Miller, C. A. J. Phys. Chem. 1983,87, 4981.

0743-7463/86/2402-0321$01.50/0

sandstone cores and even in some field tests. Typical formulations contained only about 10% of an anionic surfactant-alcohol mixture, the remainder being an aqueous electrolyte solution.

The simplest system having such dilute lamellar phases is the aerosol OT (A0T)-water-NaC1 system. Ita phase behavior was studied by Fontell: and his basic findings have been verified in our laboratory.’ As Figure 1 shows, a lamellar liquid crystalline phase exists a t surfactant concentrations as low as 5 w t % when NaCl concentration is about 1.5 wt %.

One question that may be raised regarding such dilute lamellar structures is why they do not separate into a more concentrated phase and excess brine. After all, even if the bilayers are as thin as 1 nm, the brine layers in a dilute phase containing about 10% amphiphilic material must be about 9 nm thick. If instead the bilayers are 1.5 nm thick, the thickness of the brine layers is about 13.5 nm. Since both 9 and 13.5 nm are more than an order of magnitude greater than the Debye length in these systems, which have sodium chloride concentrations of at least 1% by weight (0.17 M), one might initially expect from DLVO theory that separation into a concentrated lamellar phase and brine would take place with brine layers in the former

(6). Fontell, K.; In “Colloidal Dispersions and Micellar Behavior”; American Chemical Society: Washington, DC, 1975; ACS Symp. Ser. 9, p 270.

(7) Ghosh, 0. Ph.D. Thesis, Rice University, Houston, TX, 1985.

1986 American Chemical Society

322 Langmuir, Vol. 2, No. 3, 1986 Miller and Ghosh

vature in the bilayers of some lamellar phases containing anionic surfactant. These could be holes or edges where local separation has occurred. Accordingly, it seems quite plausible that the lamellar phase does consist of individual plates near its transition to an isotropic phase also thought to contain plates, which is discussed next.

We have observed experimentally in several systems that upon addition of further electrolyte to the dilute lamellar phase an isotropic liquid phase of low viscosity forms which scatters light and exhibits streaming birefringen~e.~ This phase, termed L,, is seen at NaCl concentrations of about 2.0 w t % in the AOT system of Figure 1.' We suggest that this phase also contains platelike micelles made up of bilayers of finite lateral extent. But the plates are some- what smaller than in the lamellar phase and have random orientations. A similar phase occurs in mixtures of non- ionic surfactants and water. It forms when dilute dis- persions of the lamellar liquid crystal in water are heated in systems containing ethoxylated alcohols." Some evi- dence does exist which suggests that the L, phase contains platelike micelles in these s y t e m ~ . ~ ~ J ~ The two situations are analogous because heating the nonionic surfactants makes them less hydrophilic, as does increasing electrolyte content for the anionic surfactants.

The occurrence of the L, phase is, a t first glance, puzzling because the usual effect of making a surfactant less hydrophilic is to reduce the repulsive interaction be- tween adjacent head groups in a micelle and thus to allow closer packing at the micelle surface. The result is a de- crease in the area-to-volume ratio for micelles in aqueous phases and transformation in micelle shape from spheres to cylinders to bilayers.14J5 In contrast, the transformation of interest here involves a decrease in the lateral dimen- sions of platelike micelles, an increase in edge area, and hence an increase in the area-to-volume ratio. We present below a simple theory of how a decrease in the interfacial tension at the micelle surface, which makes the micelle more deformable, or an increase in temperature can bring about such a transformation. The entropy of dispersion also plays a key role, promoting the formation of a large number of small particles in spite of the increase in in- terfacial area involved.

The basic concept of surfactant films having finite size has been put forward in other contexts. De Gennes and Taupin16 discuss the "persistence length" of such films as a characteristic length for bicontinuous middle (or sur- factant) phase microemulsions. Indeed, the model de- scribed below has much in common with their approach. The chief novelty of the present work lies in formulating a particular mechanism for applying the concept of finite bilayers to explain the existence of dilute lamellar phases and their transformation to the isotropic phase discussed above.

Interaction between Thin Layers As indicated above, we have observed lamellar phases

in several systems where the thickness of the brine layers is over an order of magnitude greater than the Debye length. It is of interest to see whether these thicknesses can be explained in terms of DLVO theory.

Lf L3+ L, /-L

Figure 1. Phase behavior of aerosol OT-NaCl-H20 system near the water corner of diagram at 30 "C.

having thicknesses comparable to the Debye length. To address this question, we calculate below the inter-

action between thin hydrocarbon layers in brine using Lifshitz theory for the attractive forces and the usual electrical double layer theory for repulsion. With very thin layers the attractive forces are weak enough that brine layers can sometimes reach thicknesses of 8-10 nm in the systems of interest. Nevertheless, the theory is inadequate to explain many of the observations of dilute lamellar phases. Accordingly, we modify the assumption implicit in the theory that the bilayers are of infinite lateral extent. We present a simple model to demonstrate that a single dilute lamellar phase is possible for aligned plates (bilayers) of finite lateral extent. Basically, the entropy of dispersion of the finite plates opposes separation into a concentrated lamellar phase and brine and is able under certain practical conditions to prevent the separation from taking place.

We note that Helfrich* has proposed an alternate mechanism to explain the existence of thick water or brine layers in a lamellar phase. His analysis treats spontaneous undulations of the bilayers produced by random thermal motion. When the bilayers are quite flexible, a large spacing between bilayers allows a wider variety of undu- lations to occur, thus increasing the entropy of the system. As a result, undulations have the effect of increasing equilibrium spacing between flexible bilayers. Further development of this concept has been carried out recently by Evans and Parsegian? They showed that the predicted increase in equilibrium spacing was substantial for lamellar phases of charged phospholipid molecules having relatively thick aqueous layers (6-8 nm) with salt contents exceeding about 0.1 M. These systems are obviously closely related to those considered here except that the thicker, alco- hol-free phospholipid bilayers likely have greater cohesive energies and hence less tendency to break up into indi- vidual plates.

There seems to be no conclusive evidence at present to reveal whether large, fluctuating bilayers or smaller in- dividual plates are present in lamellar phases with thick water layers. It may be that each can be found under suitable conditions and that intermediate states are pos- sible as well. In any case the two theories are basically alternate schemes showing that thermal motion can effect significant increases in equilibrium spacing for bilayers of low rigidity.

In this connection, we note that di Meglio et al.1° have recently interpreted certain experiments involving electron resonance measurements of spin-labeled amphiphiles as demonstrating the existence of local regions of high cur-

(8) Helfrich, W. Z. Naturforch. A 1978, 33A, 305. (9) Evans, E. A.; Parsegian, V. A., unpublished results. (10) di Meglio, J. M.; Dvolaitzky, M.; Ober, R.; Taupin, C. J . Phys.

Lett . 1983, 44, L229.

(11) Mitchell, D. J.; Tiddy, G. J. T.; Waring, L.; Bostock, T.; McDo-

(12) Bostock, T. A.; Boyle, M. H.; McDonald, M. P.; Wood, R. A. J .

(13) Nilsson, P. G.; Lindman, B. J. Phys. Chem. 1984,20, 4764. (14) Tanford, C. "The Hydrophobic Effect"; Wiley: New York, 1973. (15) Israelachpili, J. N.; Marcelja, S.; Horn, R. G. Q. Reu. Biophysics

(16) De Gennes, P. G.; Taupin, C. J . Phys. Chem. 1982, 86, 2294.

nald, M. P. J. Chem. SOC., Faraday Trans. 1 1983, 79, 975.

Colloid Interface Sci. 1980, 73, 368.

1980, 13, 121.

Origin and Breakup of Lamellar Liquid Crystalline Phases

Accordingly, we have made calculations of the interac- tion energy per unit area between bilayers. London-van der Waals interactions were obtained by using equations based on the Lifshitz method given by Parsegian and Ninham" for the case of two thin hydrocarbon layers in an aqueous medium. The final equation used was

Here h is the thickness of the aqueous layer separating bilayers of thickness d , K - ~ is the Debye length, and r, is defined in the notation section. The frequency-dependent dielectric constant t h for the hydrocarbon layer was cal- culated by the expression given by Ninham and Parsegi- an,l* while tw for the aqueous phase was obtained from data in Parsegian's review article on intermolecular forces.lg

Repulsion due to interaction of the electrical double layers was calculated using the usual small-overlap ap- proximation valid for Kh >> 1:"

where q = tanh ( ~ e ~ + ~ ) / ( 4 k T ) , +o is the electrical potential at the surfaces of the bilayers, and other terms are defined in the notation section. With small overlap it is satisfac- tory for our purposes to use the salt concentration of the brine added to the surfactant in calculating the Debye length K - ~ even when all the brine is incorporated into the liquid crystalline phase and no additional reservoir of brine remains. Also the following equation valid for a single double layer may be used when double-layer overlap is small to relate surface potential +o and surface charge density u:

EwK kT zeo+o 2a zeo 2kT

Q = -- sinh (-) (3)

A key parameter in eq 1 is the thickness d of the hy- drocarbon bilayers. These contain both the surfactant and a short-chain alcohol in most situations of interest here. The alcohol can be present in relatively large amounts ranging from 35% to well over 50% of bilayer volume. Moreover, some of the surfactants involved have two short instead of one long hydrocarbon chain. Both these factors can cause hydrocarbon bilayers to be thin.

We can obtain some idea of bilayer thickness by a simple calculation. As an example, we consider a system con- sisting of Exxon's C12 orthoxylene sulfonate, tertiary amyl

(17) Parsegian, V. A.; Ninham, B. W. J. Theor. Biol. 1973, 38, 101. (la) Ninham, B. W.; Parsegian, V. A. Biophys. J . 1970,10, 646. (19) Parsegian, V. A. In "Physical Chemistry: Enriching Topics from

Colloid and Surface Science"; van Olvhen, H., Mysels, K. J., Eds.; The- orex: La Jolla, CA, 1975; p 27.

Lyophobic Colloids"; Elsevier: Amsterdam, 1948. (20) Verwey, E. J. W.; Overbeek, J. Th. G. "Theory of the Stability of

Langmuir, Vol. 2, No. 3, 1986 323

Table I. Position of the Secondary Minimum in the Interaction Curve between Surfactant-Alcohol Bilavers

~ ~~~

brine layer bilayer NaCl area per thickness

thickness, concn, surfactant a t secondary nm mol/L ion, nm2 minimum, nm 1 0.225 1 10.7 1 0.306 1 8.7 1 0.400 1 7.4 1 0.400 0.5 7.7 1.5 0.225 1 10.0 1.5 0.306 1 8.3 1.5 0.400 1 7.0 1.5 0.400 1 6.9" 5.0 0.225 1 9.0 5.0 0.306 1 7.3 5.0 0.400 1 6.2 m 0.225 1 8.5 m 0.306 1 7.0 m 0.400 1 6.0

Attractive forces arbitrarily increased by 15% to estimate the effect of including interaction between nonadjacent layers.

alcohol, and NaCl brine. In most of the molecules the benzene ring is attached to one of the interior atoms of the Clz chain, so that the molecules as oriented in the bilayers are of the double-chain type. The volumes of individual surfactant and alcohol molecules, calculated from the known densities and molecular weights, are about 0.6 and 0.2 nm3, respectively. If we assume that the surfactant and alcohol molecules occupy areas of 0.5 and 0.4 nm2, re- spectively, at the bilayer surface, we can readily calculate bilayer thickness as a function of its composition. For the system of our experimental studies the alcohol volume fraction in the bilayers is about 37 % , and calculated bilayer thickness is about 1.5 nm (15 A). The thickness of the hydrocarbon portion of the bilayer, i.e., excluding the polar groups, would be somewhat smaller. This estimate is, of course, a rough one, but it does indicate that the bilayers can be quite thin-much thinner than those of biological membranes which are in the range of 5 nm (50 A).

Table I gives values of the distance between bilayers at the secondary minimum position, which corresponds to the equilibrium separation distance, for various conditions. The present case of thin bilayers is compared there with thicker bilayers typical of biological membranes and with the limiting case of very thick layers. Clearly the brine layers can be quite thick. The reason is that the attractive forces turn out to be rather weak when appropriate values of the dielectric properties are used in eq 1 and when the effect of electrolyte on attraction is considered. If the calculated interaction energy between bilayers 1.5 nm thick separated by a brine layer 10 nm thick is forced into the simpler form obtained by Hamaker's procedure,21 for ex- ample, a value of only 6.8 X J is obtained for the Hamaker constant.

The results of Table I indicate that a simple balance between attractive and repulsive forces could account for some of our observations of lamellar phases of high water content. When bilayers are 1 nm thick and NaCl con- centration is 0.225 M, for instance, the brine layers are about 10.7 nm thick at the secondary minimum, which corresponds to a volume fraction of only 8.5% of bilayer material.

On the other hand, when bilayers are 1.5 nm thick and NaCl concentration is 0.4 M, the brine layers are only 7 nm thick and the volume fraction of bilayer material is 17.6%. With volume fractions below this value and cer-

(21) Hamaker, H. C. Physica 1937,4, 1058.

324 Langmuir, Vol. 2, No. 3, 1986 Miller and Ghosh

priate equation for the energy u, per unit volume is f h t

Figure 2. Schematic diagram of aligned plates in the anisotropic phase.

tainly with those under l o % , separation into a liquid crystal and excess brine would be expected on the basis of these simple calculations. We have, in fact, seen a single lamellar phase under these conditions in the orthoxylene sulfonate system mentioned a b ~ v e . ~ ~ ~ It thus appears that considerations beyond simple DLVO theory are required in order to explain many of our observations of a single lamellar phase.

It could be argued that the discrepancy is simply due to uncertainties in the values of the dielectric properties used in the calculations, especially those of the hydro- carbon portion of the bilayers. Such an explanation cannot be conclusively rejected, but it seems very unlikely that more accurate values of dielectric properties would produce an increase in equilibrium spacing from 7 to 15 nm in the above example. Such an increase would be required to explain our observation of a lamellar phase in a system containing 9% by volume of a mixture of C12 orthoxylene sulfonate and tertiary amyl alcohol.

The single lamellar phase is seen in systems having high salt and/or alcohol concentrations. Under these conditions the bilayers should be rather fluid and relatively easy to deform and break. The low viscosities found for these phases-typically only a few centipoise-support this s ~ g g e s t i o n . ~ , ~ ~ The existence of bilayer deformation and breakup suggests that an explanation of the observed behavior requires consideration of thermal motion of the bilayers in addition to the potential energy of the above calculations. In the following section we pursue this ap- proach.

Theory of Phase Separation of Aligned Plates We seek an approximate theory for phase separation in

an oriented suspension of interacting plates. That is, conditions are assumed to be such that an oriented rather than an isotropic suspension exists. Forsyth et ai.% outline these conditions for the case of noninteracting plates.

Our main purpose here is to get a basic idea of when phase separation does and does not occur and thus to determine whether the existence of individual plates can be a possible explanation for some of our observations described briefly above. Accordingly, we employ the sim- ple model shown in Figure 2. The upper and lower sur- faces of the plates are each assumed to have area A . Plate thickness is taken as d. The basic arrangement is in parallel layers separated by brine layers of thickness h. The area of the gaps between the plates within a layer is assumed to be small in comparison with the area of the plates themselves.

We shall approximate the interaction energy among the plates in the actual suspension by the energy of the ar- rangement in Figure 2. For London-van der Waals in- teractions we use Hamaker's methodz1 and, in particular, the closed form equation for interaction of an infinite array of layers given by In the present case the appro-

where AH = Hamaker constant and 4 = volume fraction of plates in the suspension. Electrical interaction can be calculated from eq 2, which for the present situation takes the form

We note that the brine layers have thickness h equal to (d(1- 4)/4).

To estimate the configurational entropy of the plates, we adapt the scheme used for spheres by Ruckenstein and Chi.25 Each plate is constrained to remain parallel to the others but it can assume various vertical positions between the midpoints of the brine layers immediately above and below its position. For example, each plate of Figure 2 can move either upward or downward from the average posi- tion shown by an amount not greater than (h /2 ) . The number of positions it can occupy is taken to be (h/ho), where ho is the dimension of a solvent molecule. With this scheme the configurational entropy S per unit volume is given by

k @ d(1 - 4) Ad hO4

(6)

Combining eq 4-6, we can calculate the Helmholtz free

S = -- In ~

energy F per unit volume:

Thermodynamics teaches that no phase separation oc- curs if d2F/&p2 is positive for all 4. If this condition is violated, phase separation does occur and the compositions of the phases in equilibrium are those falling on a common tangent to the curve giving the free energy F as a function of volume fraction 4,

The interfacial tension between separated phases can be obtained by using the basic method developed by Cahn and Hilliard26 for ordinary binary systems and employed by us previously to demonstrate that ultralow tensions are to be expected in certain microemulsion system^.^' A variational method is used to find the profile 4(x) in the interfacial region that minimizes system free energy. For small volume fractions 4, which is the case of interest here, the resulting equation for the interfacial tension y is given by

In this equation +a and 4s are the compositions of the bulk phases in equilibrium while F*(@) is the free energy per unit volume of a mixture of these bulk phases which has an overall plate volume fraction equal to 4.

(22) Benton, W. J.; Baijal, S. K.; Ghosh, 0.; Qutubuddin, S.; Miller, C. A. "Viscosity and Phase Behavior of Petroleum Sulfonate Solutions in the Liquid Crystalline Region With and Without Small Amounts of Added Hydrocarbons", April, 1982; SPE/DOE 12700.

(23) Forsyth, P. A.; Marcelja, S.; Mitchell, D. d.; Ninham, B W. A ~ I J . Colloid Interface Sci. 1978. 9, 37.

(24) Huh, C. J. Colloid Interface Sci. 1979, 71, 408. (25) Ruckenstein, E.; Chi, J. C. J. Chem. SOC., Faraday Trans. 2 1975,

(26) Cahn, J. W.; Hilliard, J. E. J . Chem. Phys. 1958,28, 258. (27) Miller, C. A,; Hwan, R. N.; Benton, W. J.; Fort, T., Jr. J. Colloid

71, 1690.

Interface Sci. 1977, 61, 554.

Origin and Breakup of Lamellar Liquid Crystalline Phases Langmuir, Vol. 2, No. 3, 1986 325

Table 11. Predicted Conditions for Phase Separation area per

K , surfactant phase cm-.' X ion. nm2 AM, J d , nm A, fim2 separation

~

1.5" 1 5 x 1 0.04 no 1 1.5 no 1 2 no

1.75b 1 1 no 1 1.5 yes (near

0.5 1.5 no 1 2 Yes

2.0c 1 1 no 1 1.5 Yes 0.5 1.5 Yes 1 2 Yes

1.5 1 1 x 10-21 1 no 1 1.5 Yes 1 2 yes

critical)

1.5 1 5 x 1.5 0.07 no 1 1.5 0.08 yes

2 1 1.5 0.02 no 1 1.5 0.03 yes

"NaCl concentration 0.225 M (1.32% by weight). bNaCl con- centration 0.306 M (1.79% by weight). CNaCl concentration 0.400 M (2.34% by weight).

Results on Phase Separation and Interfacial Tension

We have used eq 7 to study the conditions of phase separation. As might be expected, phase separation is favored by larger Hamaker constants AH, which mean greater attraction between plates, by larger plate areas A, and by higher salinities, Le., smaller values of the Debye length ( l /~) . It is opposed by higher values of the surface charge density, which lead to higher values of the surface potential q0 and thus to greater repulsion between plates. Information justifying these statements is given in Table 11.

The results presented in Table I1 show that no phase separation is predicted for conditions similar to those occurring during some of our experiments. In such a case a system containing only a few volume percent surfactant would have an anisotropic lamellar structure as in Figure 2 and would not separate into a phase with spacing cor- responding to the secondary minimum and an excess brine phase. For example, if the Hamaker constant AH is 5 X

J, if the area per surfactant ion on the bilayer surface is 1 nm2 (100 A2), and if each plate has an area of 4 X lO-'O cm2 (corresponding to a lateral dimension of about 0.2 pm), Table I1 shows that no phase separation occurs for bilayers 1.0 and 1.5 nm thick until NaCl concentration reaches about 0.3 M (1.8% by weight). Similar calculations lead to the same conclusion for AH = 1 X J and A = 1 X 10-lo cm2. We have, in fact, seen single lamellar phases in this salinity range which contain only a few percent of the surfactant and alcoh01.~-~ For the CI2 orthoxylene sulfonate system mentioned earlier, we see from the last two entries in the table that with bilayers 1.5 nm thick and with AH = 5 X J, phase separation at 0.4 M NaCl would not occur for plates having lateral dimensions below about 0.15 pm. Since this number is quite large in com- parison with bilayer thickness, it does not seem unrea- sonable that a lamellar phase of such plates could exist.

Some remarks about the case of phase separation should also be made. First of all, the point of incipient phase separation is a critical point where the phases in equilib- rium have identical compositions and where interfacial tension is zero. As one moves from this point into the two-phase region by, for example, increasing plate area, the composition of the more concentrated phase soon ap-

Table 111. Compositions of Phases in Equilibrium" volume

fraction of plate ~ - - plates interfacial

area A. um2 ah 6, tension Y, mN/m 0.072b 0.103 0.103 0 0.073 0.100 0.105 9.6 X 0.300 0.039 0.124 1.2 x 104 1.000 0.009 0.126 2.3 X lo4

0 0.129 3.0 X 10" ,c

"Parameters held constant: A H = 5 X J, d = 1.5 nm, K = 1.5 X lo7 cm-I, 1 nm2 per surfactant ion at bilayer surface. Critical point. cSpacing in the concentrated phase corresponds to

the secondary minimum.

proaches that in which plate spacing corresponds to the secondary minimum in the interaction curve between plates. Table I11 shows an example of this behavior. In the dilute phase the concentration of plates decreases gradually, as Table 111 indicates. The result is a range of plate sizes where both phases contain appreciable amounts of surfactant. Eventually, however, the volume fraction of plates in the dilute phase falls to very low values. Of course, a t some point during this process the spacing be- tween plates reaches dimensions comparable to plate di- ameter, the plates lose their alignment, and the dilute phase becomes isotropic. For the relatively large plates (diameters of order 0.1 pm) of interest here, the surfactant concentration of the dilute phase is already rather low when this transition occurs and the predictions of the theory should not be greatly affected. In the limiting case of plates of very large lateral extent, entropy effects are negligible and the phases in equilibrium are brine and a lamellar phase with spacing corresponding to the secondary minimum.

In this limit the theory is closely related to that of Huh,% who considered oil-containing bilayers of infinite lateral extent and calculated the conditions for occurrence of a lamellar phase in equilibrium with both excess brine and excess oil. A special case of this theory, viz., the case of no oil and of bilayers with a fixed thickness, is equivalent to this limit of our theory.

Table I11 also gives values of the interfacial tension between separated phases predicted by eq 8. Ultralow tensions would be expected in these systems in view of our previous calculations for micro emulsion^^^ and Huh's for lamellar phases in equilibrium with excess brine or However, the calculated tensions are very low indeed. Even far from the critical point a value of about 3 X lo4 mN/m is obtained for the particular system considered. No data are currently available to compare with these values.

It appears that tensions are much lower than calculated for micro emulsion^^^ because attractive forces are much lower in the present case where very thin bilayers are widely separated than for microemulsions where consid- erably larger drops are separated by smaller distances. Also contributing to the low values is a Hamaker constant lower by about a factor of 40 than that used in the mi- croemulsion calculations. In view of the above calculations using Lifshitz theory, the value of 5 X J for AH used here seems more realistic.

Huhz4 notes that his theory for interfacial tension be- tween a lamellar phase and excess oil or brine also gives very low values. His approximate method of calculating interfacial tensions is not equivalent to our variational approach and should yield tensions somewhat higher than but of the same order of magnitude as our values. Indeed, if we neglect his term accounting for entropy of the hy-

326 Langmuir, Vol. 2, No. 3, 1986 Miller and Ghosh

Figure 3. Schematic diagram of surfactant molecules in an oblate spheroid.

drocarbon chains, which is not considered in the present theory, we find that his theory gives a value of about 5 X lo4 mN/m for interfacial tension between brine and a lamellar phase for the same conditions where we find 3 X lo4 mN/m (Table 111).

Theory of Plate Size In the preceding section we considered plate diameter

a parameter to be specified. Here we develop a simple model for predicting variation of the diameter since it is obviously the major factor determining whether an iso- tropic or an anisotropic phase forms. Moreover, as indi- cated previously, it seems likely that a decrease in diameter is responsible for the transition from a lamellar to an isotropic phase which takes pIace as the surfactant be- comes less hydrophilic in both nonionic and anionic sur- factant systems. We do not attempt to predict here the precise conditions for the transition as this would require calculation of the attraction among a large number of plates having random orientations in the isotropic phase, a difficult problem.

We assume the bilayers to be oblate spheroids containing n, surfactant molecules and having semiaxes a and b (see Figure 3) . The smaller semiaxis b is constrained to be less than the fully extended length 1, of a surfactant molecule. Since a and b determine spheroid area and volume and since the volume u, of each surfactant molecule is fixed, it is readily shown that only two of the four quantities a, b, n,, and a, are independent, where a, is the average area per surfactant molecule a t the spheroid surface. Free energy is to be minimized with respect to the two inde- pendent parameters, taken here as a, and b for a system containing a fixed total number N, of surfactant molecules. Analysis shows the free energy to be a decreasing function of b under physically reasonable conditions.' Hence, we can set b = 1, and find the equilibrium condition using the remaining equilibrium equation which takes the following form for the case of nonionic surfactants, the situation we consider first:

The first two terms in this equation represent contri- butions from effects a t the surface of the oblate spheroid, viz., the tension yo of a surfactant-free oil-water interface and the surface pressure caused by thermal motion of the head groups at the bilayer surface. For the latter a com- mon form of the kinetic contribution to surface pressure is used

(10)

with ahg the actual area of each head group, i.e., the ex- cluded area. The third term in eq 9 is the contribution of the configurational entropy of the hydrocarbon chains which form the interior of the spheroid. It is derived

kT n h g = ~

- ahg

below. The fourth term is the effect of the free energy of mixing. Onsager's equation is used% in the form applicable for an isotropic phase containing plates whose diameters greatly exceed their thicknesses:

( 1 1 )

In this equation V, is the volume of a water molecule and V the total volume of the system.

The chain entropy term is derived as follows. We imagine the volume of the spheroid to be divided into cubic cells with the area of each face equal to the cross-sectional area of a hydrocarbon chain. As discussed by Dill and F10ry~~ such a cell contains about 3.6 methylene groups. If each surfactant chain consists of m, cells, the total number of cells in the spheroid (assuming no unoccupied cells) is n,m,. The number Nl of these cells at the surface can be obtained by dividing total spheroid area by the area per cell. Each chain must originate a t the surface. If we assume, along the lines of Flory-Huggins theory, that the fractional occupancy f k of cells remains uniform throughout the spheroid as the (12 + 1)st chain is placed, the number of independent configurations for this chain is

X k + 1 = N1(Om0-' ( 1 - fk)m* ( 1 2 )

where l is the number of neighboring cells accessible a t each step, assumed constant for all steps. It is assumed that the chains cannot reverse themselves, i.e., are not completely flexible, so that l is less than the total number of nearest neighbors z. The total number of ways W to place chains in the spheroid is

I * w = - i i l h k + l n,! k=O

Taking the logarithm of this quantity and approximating the sum by an integral in the usual way, we find the fol- lowing expression for the configurational contribution to the total free energy of the chains of all the spheroids: Gchain = -(N,/n,)kT In W

= N,kT[( l -m,/n,) In n, - In N1 - (m, - 1 ) In (t) + m,(l -1/n,) - 11

( 1 4 )

This expression is used in evaluating the third term in eq 9 above.

To apply this analysis to nonionic Surfactants, we require information on yo and a . For the former we use a linear equation to describe the 8ependence of oil-water interfacial tension on temperature on the basis of the data of Harkins and Cheng:30

(15) where T is in K .

Unfortunately, no reliable data are available for the excluded area of a partially hydrated ethylene oxide (EO) chain of given length. However, a rough idea of its mag- nitude can be obtained in the following way. The Gibbs adsorption equation can be used with surface tension data just below the cmc to calculate the area per molecule of the compact film at the air-water interface for surfactants of various EO chain lengths which form micellar solutions

7 0 = 52.0 - 0.032(T - 293)

(28) Onsager, L. Ann. N . Y . Acad. Sci. 1949,51, 627. (29) Dill, K. A.; Flory, P. J. Proc. Nut!. Acad. Sci. U.S.A. 1981, 78 (2),

(30) Harkins, W. D.; Cheng, Y. C. J. Am. Chem. SOC. J. 1921,43, 35. 676.

Origin and Breakup of Lamellar Liquid Crystalline Phases Langmuir, Vol. 2, No. 3, 1986 327

Table IV. Predicted Diameters of Platelike Micelles in the Ci2E8-H20 System"

excluded area per plate temp area molecule aggregation diam T, K ah., nm2 as, nm2 no. n. 2a, nm 288 0.3367 0.433 1.40 X lo4 61.6 288 0.3380 0.435 1.10 X lo4 54.6 298 0.3270 0.428 8.72 X lo4 154 298* 0.3315 0.433 1.40 X lo4 61.6 308 0.3220 0.429 6.90 X lo4 137 308 0.3280 0.435 1.04 X lo4 53.2 308 0.3315 0.438 6.74 X lo3 42.8

"Calculations based on eq 9 as supplemented by eq 10, 11, 14 and 15. Also u, = 0.579 nm3, b = 2.00 nm, V = 10 cm3, N , = 1.89 X lom molecules per cm3 of mixture (10 wt % C12E3): *Base case from experimental data and Gibbs adsorption equation (see text).

at the temperature of interest. For example, Lange3' has made such calculations using room temperature data for four surfactants with C12 hydrocarbon chains and with EO chain lengths between five and twelve. He found the area to be approximately proportional to the square root of the EO chain length. This relationship can be used to estimate the area per molecule of CI2E3, which forms a lamellar liquid crystal in dilute aqueous solutions at this temper- ature.

Now if this value, which is 0.433 nm2 (43.3 A2), is used as a, in eq 9 applicable to oblate spheroids along with suitable values for the other parameters, a value of about 0.33 nm2 (33 A? is found for aw As temperature increases, ah, should decrease since head-group hydration should diminish. As shown in Table IV, the size of this decrease controls the behavior of spheroid diameter. If the decrease is relatively large, spheroid diameter increases. Such be- havior is expected for substantial decreases in hydration, e.g., as might be expected at lower temperatures where the lamellar liquid crystalline phase first forms from the Ll phase, which is thought to consist of aggregated micelles. On the other hand, Table IV shows that spheroid diameter decreases when there is but a small decrease in ab, as would be expected at somewhat higher temperatures near the transition of interest here from liquid crystal to LB. Under these conditions hydration is already small and cannot decrease much more. In other words, this model provides a plausible mechanism for formation of the L3 phase with increasing temperature although sufficient data are not available to establish its validity conclusively or to make detailed comparisons with experiment.

As the two entries a t 298 K show, plate diameter 2a is rather sensitive to the value of a,. Since the estimate of a, in the base case using the Gibbs adsorption equation is very approximate, the actual values obtained for 2a should be considered an order of magnitude estimate only. Nevertheless, the order (about 100 nm) seems reasonable, and the model provides a rationale for a decrease in plate diameter leading to formation of the L3 phase.

A physical explanation of the results shown in Table N can be given as follows. Interfacial tension acts to minimize interfacial area, which, for a fixed value of b, means that it acts to maximize spheroid diameter 2a. The entropy of dispersion, however, is greatest when many spheroids are present, i.e., when a is small. An increase in temperature not only lowers the basic oil-water tension yo but also increases the importance of entropic effects. Accordingly, spheroid diameter 2a decreases for the simple case of constant ak In actuality, ah, probably decreases, but if the decrease is small enough, the net result will still be a

Table V. Predicted Diameters of Platelike Micelles in the Aerosol OT-NaCl-H20 System"

brine surfactant area per plate salinity concn in bulk molecule aggregation diam

C,, g/dL Cad, molecules/cm3 a,, A2 no. n, 2a, nm 1.0 1.88 X 10l8 101.0 0.42 X lo3 16.0 1.2 1.77 X 1Ol8 100.0 0.46 X lo3 16.8 1.5 1.65 X 10l8 99.2 0.67 X lo3 20.2 1.7 1.59 X 10l8 99.3 0.64 X lo3 19.8 2.0 1.48 X lo'* 100.0 0.48 X lo3 17.1

"Calculations based on eq 9 as supplemented by eq 11 and 16. Also u, = 0.653 nm3, b = 1.02 nm, V = 10 cm3, N , = 6.7744 X lom molecules per 10 cm3 of mixture (5 wt % aerosol OT), bo = 2 X 1015 molecules/cm3, T = 30 "C.

decrease in spheroid diameter. This proposed mechanism for formation of the L, phas;

differs somewhat from that suggested by Mitchell et al. These authors attribute the decrease in plate diameter leading to formation of the L3 phase to an increase in ab resulting from a rearrangement of the ethylene oxide chain. In contrast, our theory requires no such rearrangement and indeed even allows ab to continue to decrease slightly with increasing temperature.

Because electrical effects must be considered, the situ- ation is more complex for the transition from liquid crystal to L3 with increasing salinity in anionic surfactant systems. Following Ruckenstein and K r i ~ h n a n , ~ ~ we replace (yo + rk) in eq 9 by the following expression obtained by com- bining the Gibbs adsorption equation with a Langmuir- type adsorption isotherm for univalent surfactant ions:

reo2

Here Po is a constant and CsWf and C, are the concentra- tions of surfactant and univalent salt in the bulk aqueous phase. The surface potential +o and the area per surfactant ion a, (=eo/u) are further related by eq 3.

As inorganic salts are surface inactive at (surfactant-free) oil-water interfaces, yo increases slightly with increasing salinity, an effect that, taken alone, acts to increase spheroid diameter as may be verified by using eq 9. Moreover, the electrical contribution to surface pressure calculated from the usual Gouy-Chapman theory'* and given by the last term of eq 16 decreases in magnitude as salinity increases at constant area per surfactant ion. Here too the effect is to increase the total interfacial tension given by eq 16 and hence spheroid diameter as well. Nevertheless, total interfacial tension can decrease with increasing salinity owing to the second term which stems from the Langmuir isotherm.

When eq 16 is substituted for yo + IIb in eq 9, spheroid diameter can be calculated as a function of salinity. Table V gives results obtained for the AOT-water-NaC1 system. As Figure 1 shows, increasing brine salinity from 1.5 to 2.0 wt % with at least 6 wt % surfactant in the system pro- duces a transition from the lamellar to the L3 phase. Our calculations for this system employed a value of the ad- sorption constant Po selected from the range found to be of practical interest in the original paper.32 The variation of yo with salinity used in these calculations was based on the available Moreover, since AOT molecules

(31) Lange, H. Xolloid 2. Z. Polym. 1964, 201, 131. (32) Ruckenstein, E., Krishnan, R. J. Colloid Interface Sci. 1980, 76,

201.

328 Langmuir, Vol. 2, No. 3, 1986

have two hydrocarbon chains attached to the polar group, m8 in eq 14 must be replaced by 2m8. Finally, CsWf was assumed to be approximately equal to the cmc. The variation of cmc with salinity was estimated on the basis of the following equation by Shinoda et al. :35

In cmc = -K, In Ci + constant (17)

where Ci is the total concentration of counterions in the system and a value of 0.5 was used for the constant K g on the basis of experimental results for several anionic sur- f a c t a n t ~ . ~ ~

As Table V indicates, the model does predict a maxi- mum in plate diameter when NaCl concentration is about 1.5 wt %. This result is consistent with the observed existence of a lamellar phase in this salinity range and transformation to the L3 phase at higher salinities. Similar calculations for other systems also give results that are generally consistent with experimental observation^.^

Finally, we note that formation of the L3 phase with increasing salinity is not the only phenomenon involving aggregates of anionic surfactants that seems at variance with DLVO theory. Evans, Ninham, and c o - w ~ r k e r s ~ ~ ~ ~ ' found that even upon adding relatively large quantities of NaOH, the aggregation numbers of small, spherical mi- celles of quaternary ammonium hydroxide surfactants showed only small increases instead of the expected large increases. They considered that the hydroxide counterions induced changes in the water structure which lowered the local dielectric constant near the micelle surface and thereby increased electrical repulsion between adjacent surfactant ions. Whether this behavior is quite general like the formation of the L3 phase remains to be determined, however.

Summary A simple model has been proposed to explain the ex-

istence of lamellar liquid crystals of relatively low sur- factant content in anionic surfactant systems with ap- preciable electrolyte contents. The bilayers are assumed to exist as aligned plates of finite lateral dimensions. The increased system entropy resulting from the presence of many individual plates causes the equilibrium spacing between bilayers to be significantly greater than for the case of infinite sheets. The model also leads to a straightforward explanation of the observed transformation of the lamellar phase to an isotropic phase containing platelike micelles when the surfactant is made less hy- drophilic in both anionic and nonionic surfactant systems. Basically, plate diameter decreases so that alignment can no longer be maintained and the plates assume random orientations.

It is further suggested that the reason for this decrease in plate diameter, which is not predicted by the usual concepts of molecular packing in micelles, is that the tension at the bilayer surfaces is small. As a result, the increase in interfacial area and interfacial free energy ac- companying the decrease in plate diameter and corre- sponding increase in edge area is outweighed by the in- crease in entropy brought about by the increase in the number of plates. Simple models predicting such behavior

(33) Harkins, W. D., Cheng, Y. C. J . Am. Chem. SOC. 1921, 43, 35. (34) Dickinson, W. Trans. Faraday SOC. 1940, 36, 839. (35) Shinoda, K.; Tamamushi, B.; Nakagawa, T.; Isemura, T.

(36) Hashimoto, S.; Thomas, J. K.; Evans, D. F.; Mukherjee, S.; Nin-

(37) Ninham, B. W.; Evans, D. F.; Wei, G. J. J. Phys. Chem. 1983,87,

"Colloidal Surfactants"; Academic Press: New York, 1963; pp 58.

ham, B. W. J . Colloid Interface Sci. 1983, 95, 594.

5020.

Miller and Ghosh

are presented for nonionic and anionic surfactants. A similar phenomenon occurs for a microemulsion in equi- librium with an excess phase of the drop material, Le., for an oil-in-water microemulsion in equilibrium with excess

Drop radius is predicted to be somewhat smaller than that at which its interfacial film is unstressed because, by forming smaller drops, the system increases the total number of drops and the entropy of dispersion to an extent that outweighs the energy input required to deform the film from its unstressed state.

Notation semimajor axis of oblate spheroid actual area of each head group average area per surfactant molecule at the

spheroid surface plate area Hamaker constant semiminor axis of oblate spheroid velocity of light in vacuum bulk concentration of added electrolyte bulk concentration of surfactant total concentration of counterions in the system critical micelle concentration bilayer thickness charge per electron fractional occupancy of the cells after i surfactant

Helmholtz free energy per unit volume Helmholtz free energy per unit volume of a mixture

confgurational contribution to the total free energy

Gibbs free energy of system free energy of mixing thickness of aqueous layer separating the bilayers dimension of a solvent molecule Planck's constant/2r Boltzmann constant empirical constant in eq 17 fully extended length of the surfactant molecule isotropic micellar solution isotropic surfactant phase that scatters light and

anisotropic lamellar liquid crystalline phase number of cells occupied by each surfactant chain aggregation number number of cells at the surface total number of surfactant molecules (2ht,'/2~n)/c configurational entropy per unit volume temperature interaction energy per unit volume due to repulsion

of electrical double layers interaction energy per unit area due to repulsion

of electrical double layers London-van der Waals interaction energy per unit

volume London-van der Waals interaction energy per unit

area between bilayers volume of each surfactant molecule total volume of the system volume of a water molecule total number of ways to place chains in the spheroid valence of surfactant ion adsorption constant interfacial tension interfacial tension at surfactant-free oil-water in-

dielectric constant for hydrocarbon layers dielectric constant of water

molecules have been placed

of bulk phases

of the chains of all the spheroids

exhibits streaming birefringence

terface

(38) Miller, C. A.; Neogi, P. AIChE J. 1980, 26 (2), 212.

Langmuir 1986,2,329-331 329

number of neighboring cells accessible at each step, assumed constant for all steps

r 7 tanh (ze0$0)/(4kT) K inverse Debye length Xi

chain tn

4

number of independent configurations for the ith

(2ukTn)/h, n = 0, 1 , 2 ,... surface pressure surface charge density volume fraction of plates in the suspension

=h, U

$0

Acknowledgment. This research was supported by grants from Amoco Production Co., Arc0 Oil and Gas Co., Gulf Research and Development Co., Exxon Production Research Co., the Mobil Foundation, and Shell Develop- ment Co.

Registry No. AOT, 577-11-7; CI2E3, 3055-94-5; NaC1, 7647-

electrical potential at surface of bilayers

14-5.

Adsorption Model for Amphiphile Molecules at the Air/Water Interface

Nai-Fu Zhou* Department of Chemistry, East China Normal University, Shanghai 200062, China

Received October 14,1985. In Final Form: January 13, 1986

On the basis of the adsorption model for amphiphile molecules at the air/water interface in which the hydrocarbon group is exposed partially out of the water surface, the extent of exposure of the hydrocarbon group was calculated by using the data of standard free energy of adsorption and the principle of independent surface tension. It has been shown that for homologues the longer the hydrocarbon chain the more the group is exposed.

Introduction The adsorption-orientation effect of surface-active am-

phiphile molecules a t either an air/water or an oil/water interface has long been generally acknowledged. On the basis of the thermodynamic functions of adsorption, Wardl studied the configuration of the adsorbed monolayer composed of the most probable spheroidal amphiphile molecules. In a previous paper, Zhou and Gu2 calculated the extent of exposure of the hydrocarbon group of sur- face-active solute molecules at the adsorbed monolayer on a similar basis. In this brief report the author extends the calculation to more amphiphiles by using data on standard free energy of adsorption and the principle of independent surface t e n ~ i o n . ~

Thermodynamics of Adsorption As given by Ward and Tordai? and also by Posner,

Anderson, and Alexander,6 the standard free energy of adsorption, AGaho, for the surface-active solute from so- lution onto its surface can be expressed as

where 6 is the thickness of the surface phase, ( ~ 3 y / d C ~ ) ~ is the limiting slope of the surface tension isotherm for the infinite dilute solution, ,R the gas constant, and T the absolute temperature. The standard states selected in this work are the same as used by Posner, Anderson, and Al- e ~ a n d e r : ~ i.e., ( 1 ) for the bulk phase, the infinite dilution is referred to where the concentration of the solute is equal to ita activity and the standard state of the solute is a linearly extrapolated hypothetical state a2 = 1; ( 2 ) for the surface, using a film thickness of 6, we adopt the standard state r2/6 = 1, also an extrapolated hypothetical state. By

*Present address: Department of Chemical Engineering, Auburn University, Auburn, AL 36849.

eq 1 one can obtain the standard free energy of adsorption from the experimental surface tension isotherms. For the amphiphile electrolyte, a coefficient of 2 is necessary in eq 1 , so that

The change in surface energy in the adsorbing process can be evaluated, on the other hand, by the “principle of independent surface tension” and the molecular model assumed. The principle here considered implies that the surface (interfacial) tensions are applicable to individual molecules. Two different cases are now considered.

(1) For Cylindrical Molecules. If we assume the length of the molecule is 1 with its hydrocarbon group exposed a height h, then the exposed area is 2rrh + rr2; the area of water surface eliminated is rr2. Accordingly the change in surface energy in the adsorbing process is given by

AGadso = N[(2rrh + rr2)(yR - Y R ~ ) - rr2yw1 (3) where N is Avogadro’s number, r is the radius of the cross-sectional area of cylindrical hydrocarbon group, yR is the surface tension of hydrocarbons which is equal to 31 erg/cm2 at room temperature, y ~ / ~ is the interfacial tension of the hydrocarbon/water interface which is about 51 erg/cm2, and yw is the surface tension of water, about 72 erg/cm2.

(2) For Spherical Molecules. If we assume the spherical amphiphile molecules are exposed with their hydrocarbon groups out of the surface with a height h and the diameter of the spherical molecule is d, then the area

(1) Ward, A. F. H. Trans. Faraday SOC. 1946, 42, 399. (2) Zhou, N. F.; Gu, T. R. Scientia Sin. (Engl. Ed.) 1979,22, 1033. (3) Langmuir, I. Colloid Symp. Monogr. 1925, 3, 48. (4) Ward, A. F. H.; Tordai, L. Trans. Faraday SOC. 1946, 42, 413. (5) Posner, A. M.; Anderson, J. R.: Alexander, A. E. J. Colloid Sci.

1952, 7, 623.

0743-7463/86/2402-0329$01.50/0 0 1986 American Chemical Society