Change in Micelle Form Induced by Cosurfactant Addition in Nematic

Sep 17, 1997 - The relative stability of these forms is measured by the micellar elastic bending energy, calculated in the surfactant parameter model,...
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Langmuir 1997, 13, 5016-5021

Change in Micelle Form Induced by Cosurfactant Addition in Nematic Lyotropic Phases L. Q. Amaral,*,† O. Santin Filho,‡ G. Taddei,§ and N. Vila-Romeu| Institute of Physics, University of Sa˜ o Paulo, C.P. 66318, 05315-970, Sa˜ o Paulo, SP, Brasil, Department of Chemistry, State University of Maringa´ , Parana´ , Brasil, Department of Chemistry, University of Florence, Florence, Italy, and Department of Chemical Physics, University of Santiago de Compostela, Spain Received January 2, 1997. In Final Form: June 16, 1997X

The nematic cylindrical (Nc)-nematic discotic (Nd) phase transitions in three amphiphile/decanol/water systems, occurring as a function of the decanol/amphiphile molecular ratio (Md), are correlated with a change of micellar form from spherocylinder (SC) to square tablet (ST), which occurs geometrically in a continuous way, with an intermediate biaxial object. The relative stability of these forms is measured by the micellar elastic bending energy, calculated in the surfactant parameter model, as a function of the molecular quantity p ) v/al (where v, l, and a are respectively the volume and length of the hydrophobic chain and the area per polar head at the apolar-polar interface) and of the decanol/amphiphile mixing. A SC-ST transition is predicted for increasing Md and p0 values, because the amphiphile (spontaneous p01) and decanol (p02) molecules prefer zones with different curvatures. A good agreement is obtained by comparison of the predicted SC-ST transformation with experimental results obtained at the Nc-Nd transitions for KL (potassium laurate), SDS (sodium decyl sulfate), and SLS (sodium dodecyl sulfate).

1. Introduction The polymorphism of ionic surfactant solutions is greatly enhanced with addition of a nonionic cosurfactant (e.g., alcohol), leading to rather complex ternary phase diagrams.1 In particular, lyotropic nematic phases with two basically different uniaxial symmetries have been discovered in ternary and quaternary systems [amphiphile/water/additives (alcohol and/or salt)] by means of their different orientations under applied magnetic fields.2 Such phases have been correlated with micelles of cylindrical (Nc) and discotic (Nd) shapes via X-ray diffraction studies.3,4 Biaxial nematic lyomesophases (Nbx)5 have been discovered in the system potassium laurate (KL)/water/ decanol with second-order transitions to Nc and Nd phases. The study of phase transitions as a function of concentration and temperature in this system5,6 and in the system sodium sulfate (SDS)/water/decanol7-9 did not lead however to a clear structural model at the micellar level, particularly in view of molecular theories that predict second-order transitions and Nbx phases for inherently biaxial objects.10 †

University of Sa˜o Paulo. State University of Maringa´. § University of Florence. | University of Santiago de Compostela. X Abstract published in Advance ACS Abstracts, August 15, 1997. ‡

(1) Ekwall, P. Adv. Liq. Cryst. 1975, 1, 1. (2) Lawson, K. D.; Flautt, T. J. J. Am. Chem. Soc. 1967, 89, 5489. Radley, K.; Reeves, L. W.; Tracey, A. S. J. Phys. Chem. 1976, 80, 174. Forrest, B. J.; Reeves, L. W. Chem. Rev. 1981, 1, 1. (3) Amaral, L. Q.; Pimentel, C. A.; Tavares, M. R.; Vanin, J. A. J. Chem. Phys. 1979, 71, 2940. (4) Charvolin, J.; Levelut, A. M.; Samulski, E. T. J. Phys. Lett. 1979, 40, L587. (5) Yu, L. J.; Saupe, A. Phys. Rev. Lett. 1980, 45, 1000. (6) Figueiredo Neto, A. M.; Liebert, L.; Galerne, Y. J. Phys. Chem. 1985, 89, 3737. (7) Yu, L. J.; Saupe, A. J. Am. Chem. Soc. 1980, 102, 4879. Boonbrahm, P.; Saupe, A. J. Chem. Phys. 1984, 81, 2076. (8) Hendrikx, Y.; Charvolin, J. J. Phys. 1981, 42, 1427. Hendrikx, Y.; Charvolin, J.; Rawiso, M.; Liebert, L.; Holmes, M. C. J. Phys. Chem. 1983, 87, 3991. (9) Bartolino, R.; Chiaranza, T.; Meuti, A.; Compagnoni, R. Phys. Rev. A 1982, 26, 1116.

S0743-7463(97)00007-3 CCC: $14.00

With the discovery of the nematic domain in the system sodium dodecyl (lauryl) sulfate (SLS)/water/decanol,11 the importance of studying these systems as a function of the decanol/amphiphile molecular ratio (Md) was stressed and the role of decanol in promoting a change of micellar form related to the Nc-Nd transition was emphasized.12 Evidence for the existence of a first-order transition between the uniaxial nematic phases was reported.12,13 The recent discovery of small islands of Nbx phase in the SLS phase diagram,14 with first-order transitions to Nc and Nd phases, gives further evidence for changes in the micellar shape. The necessity of taking into account changes in micellar form to reproduce experimental phase diagrams in lyonematics is clear. However, up to now, no modeling to explain a change in the symmetry of the micellar object has been worked out. The partitioning of cosurfactant in mixed micelles has been analyzed by Gelbart,15 showing that the cosurfactant goes preferentially into the body rather than the caps of spherocylinder aggregates. Preferential partitioning of decanol in the lower curvature zones of the micelles has also been experimentally observed.16 In this paper the experimentally observed Nc-Nd transitions are correlated with a change of micellar form, induced by the micellar bending energy. Such intrinsic change of micellar form may trigger the phase transition. In an earlier paper (by G.T. and L.Q.A.)17 the conditions for a particular change of micellar form, from prolate ellipsoid to spherocylinder (SC), in binary systems were (10) Alben, R. Phys. Rev. Lett. 1973, 30, 778. Shih, C. S.; Alben, R. J. Chem. Phys. 1972, 57, 3055. Freiser, M. J. Phys. Rev. Lett. 1970, 24, 1041. Straley, J. P. Phys. Rev. A 1974, 10, 1881. (11) Amaral, L. Q.; Helene, M. E. M.; Bittencourt, D. R.; Itri, R. J. Phys. Chem. 1987, 91, 5949. (12) Amaral, L. Q.; Marcondes Helene, M. E. J. Phys. Chem. 1988, 92, 6094. (13) Amaral, L. Q. Liq. Cryst. 1990, 7, 877. (14) Quist, P. O. Liq. Cryst. 1995, 18, 623. (15) Gelbart, W. M.; McMullen, W. E.; Masters, A.; Ben-Shaul, A. Langmuir 1985, 1, 101. (16) Hendrikx, Y.; Charvolin, J.; Rawiso, M. J. Colloid Interface Sci. 1984, 100, 597. Alperine, S.; Hendrikx, Y.; Charvolin, J. J. Phys. Lett. (Paris) 1985, 46, L27. (17) Taddei, G.; Amaral, L. Q. J. Phys. Chem. 1992, 96, 6102.

© 1997 American Chemical Society

Change in Micelle Form

analyzed. The two curves of bending energy as a function of micelle axial ratio µ cross around µ ∼ 2. Such form transition in the concentrated isotropic I phase may be a prerequisite for the occurrence of the I-hexagonal phase transition in lyotropic systems because SC allows polydisperse growth.18,19 The basis for calculation of the bending energy in a ternary system is first given, and then experimental results are discussed correlating the Nc-Nd phase transitions with the predicted change in micellar form. 2. Micellar Bending Energy for a Single Mixed Micelle It is well-known that the membranes of amphiphilic aggregates in colloidal systems can be elastically deformed. The phenomenological theory of Helfrich20 gives the bending energy as a function of the principal and spontaneous curvatures of the membrane interfaces via the elastic constants (bending rigidity and elastic modulus of Gaussian curvature). Expressions for the bending moduli have been derived21-23 and the first principles basis of the theory has been discussed.24 Hyde has developed a semiempirical model,25-27 where the same bending energy is expressed in terms of the surfactant parameter p and spontaneous surfactant parameter p0 introduced by Israelachvili28 (p ) v/al where v, l, and a are respectively the hydrophobic chain volume, effective chain length, and polar head areasthe area available per hydrophilic group at the hydrocarbon/water interfacesof the amphiphilic molecule). Hyde’s model has the advantage of being easily applicable to the case of surfactant monolayers, even with radius of curvature equal to the chain length. It has been explicitly discussed29 how Helfrich’s formulation is suitable for the case of spontaneously planar surfaces, while Hyde’s model accounts naturally for the case of spontaneously curved interfaces. The first application of Hyde’s model to micelles was made by two of us (G.T. and L.Q.A.17). The relative stability of several micellar forms has been further discussed by Taddei.30 In the case of micelles it is not possible to use the conditions of equal surface and volume to deduce the shape that minimizes the energy, as can be done for vesicles.31 The constant paraffin volume criterion implies instead necessarily a change in the surface. Hyde’s model is a convenient single parameter model to test the relative stabilities of different micellar forms, chosen from criteria of molecular packing conditions. The surfactant spontaneous parameter p0 is a molecular property, and its numerical value can be transferred among different aggregate forms of the same amphiphilic molecule. Such transferability allows analysis of the relative stability of such different forms, even if the micellar shape cannot be rigorously predicted. (18) Taylor, M. P.; Herzfeld, J. Phys. Rev. A 1991, 43, 1892. (19) Itri, R.; Amaral, L. Q. Phys. Rev. E 1993, 47, 2551. (20) Helfrich, W. Z. Naturforsch., Teil C 1973, 28, 693. (21) Mitchell, D. J.; Ninham, B. W. Langmuir 1989, 5, 1121. (22) Fogden, A.; Ninham, B. W. Langmuir 1991, 7, 590. (23) Fogden, A.; Daicic, J.; Mitchell, D. J.; Ninham, B. W. Physica A, in press. (24) Robledo, A.; Varea, C. Physica A 1996, 231, 178. (25) Hyde, S. T. Colloq. Phys. 1990, 51, C7-209. (26) Hyde, S. T. J. Phys. Chem. 1989, 93, 1458. (27) Hyde, S. T.; Barnes, I. S.; Ninham, B. W. Langmuir 1990, 6, 1055. (28) Israelachvili, J. N.; Micthell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (29) Fogden, A.; Hyde, S. T.; Lundberg, G. J. Chem. Soc., Faraday Trans. 1991, 87, 949. (30) Taddei, G. Colloid Polym. Sci. 1994, 272,1300. (31) Deuling, A. J.; Helfrich, W. J. Phys. (Paris) 1976, 37, 1335.

Langmuir, Vol. 13, No. 19, 1997 5017

The Hyde model assumes that the elastic bending energy is given by the equation:

Ebend ) k

∫ (p - p0)2 dS

(1)

where k is an elastic constant with dimension energy/ surface area, and the integral is extended to the whole micelle interface (considered as a continuous interface between apolar paraffin moiety and polar heads). p is the actual surfactant parameter, related to the curvatures of the micellar interface, and p0 is the spontaneous molecular value, in absence of external bending stress. The physical meaning of Hyde’s constant k has been discussed in ref 29. There it was shown that in a more consistent formulation of the Helfrich model, to account for spontaneously curved interfaces, the saddle splay contribution (Gaussian curvature) is vanishing to the first order and the bending energy per unit area is

g ) 2kc*(H - H0)2 + ...

(2)

with

H02 1 1 kc* ) kc + k hc - k hc 2 2 (H 2 - K ) 0 0 where H is the average curvature, H0 and K0 are the spontaneous average and Gaussian curvatures, and kc and kh c are the bending rigidity and the elastic modulus of the Gaussian curvature. It was also shown29 that in the case of a surfactant monolayer the bending energy per unit area can be writen as

[

1 l0 1 + H0l0 2 g)k 3(1 + H0l0)

(

)

]

2

(H - H0)2 + ...

(3)

where k is Hyde’s constant and l0 is the chain length (equal to the radius of curvature). Here the absence of a linear saddle splay term is due to an additional constraint that couples the mean and Gaussian curvatures, so that the area of the parallel surface to the interface at the chain ends vanishes. Comparison of eqs 3 and 2 gives a correlation between Helfrich and Hyde constants in cases where both equations can be applied:

[(

k ) 2kc*

3(1 + H0l0) 1 l0 1 + H0l0 2

)]

2

(4)

It is clear from eq 4 that in the case of spontaneously curved interfaces the elastic constants of Helfrich and Hyde formalisms (kc* and k) are interconnected but depend also on the values of spontaneous curvatures (H0, K0). Since in the present paper the important point is the relative magnitude of Ebend calculated for different micellar forms, it will be adopted for the estimation of Hyde’s constant the same criterion used previously:17 the equality of the bending energy calculated via Helfrich original equation and Hyde equation in the particular case of a spherical micelle with p0 ) 1 (H0 ) K0 ) 0). Using the relation kh c ) -2kc/3 obtained by Mitchell and Ninham21 we obtain

k ) 3kc/l02

(5)

Equation 5 gives a clear meaning to Hyde’s constant. Estimates of k for other particular cases change only the numerical factor (up to an order of magnitude).

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Amaral et al.

The theoretical method of analysis of different micellar forms is described in ref 30. Here the spherocylinder (SC) was chosen as the form of the elongated micelle and the square tablet (ST) as the flat micelle form. SC is constituted by a cylinder with two hemispherical caps at the ends and ST by a square parallelepiped with spherical quadrants at the four corners and four half-cylinders along its borders. The diameter of SC and the thickness of ST are both always equal to 2l, where l is the effective length of the hydrocarbon chains. The surfaces of both forms are closed and without cusps. The choice of these two particular forms depends on their particular geometrical properties: (a) both forms degenerate to a spherical micelle in the lower limit case. In the upper limit case ST becomes a lamella of infinite extension, whereas SC takes the form of a cylinder of infinite length; (b) both forms can grow indefinitely in size without involving any bending stress, differently from ellipsoids, toruses, and ovoids in which the growth of the micelle introduces bending stresses and chain overlappings in the micelle bulk;30 (c) it is easy to transform geometrically ST into SC or vice versa when one considers a gradual growth of SC along its transverse axis. The corresponding sequence is: spherocylinder f rectangular tablet (RT) f square tablet. This geometrical transformation does not introduce any cusp in the micelle surfaces. However, the most important property of SC and ST when they occur in a ternary system is that the zones of different curvatures of these micelles can be occupied preferentially by distinct kind of amphiphilic molecules which fit best in these zones so that the whole micelle is strongly stabilized. Note that in a binary system ST is a micelle form of very low stability with respect to many other forms.30 The partitioning of surfactant/cosurfactant in the SC form has been already analyzed explicitly.15 The electrostatic strain is minimized when the cosurfactant concentrates in the cylindrical body of SC, leaving only surfactant in the spherical caps. This will be therefore the starting point of the present modeling. Some basic assumptions must be made to evaluate Ebend from eq 1 in the case of a micelle with mixed surfactant and cosurfactant molecules with different spontaneous surfactant parameter values, p01 and p02, respectively. The initial assumption is that in the mixed region (body of SC) the surfactant and cosurfactant are statistically at random. There are two ways of viewing the problem: (i) Since the paraffin chains have approximately the same cross section area, the interface is partitioned according to the number of molecules of each type. In such a case surfaces are simply added, in an “independent molecules model”. (ii) An “effective unity” consisting of surfactant/cosurfactant in the average proportion can be considered as a new molecular unity, that alone fills the whole interface. In case (i) of independent molecules, eq 1 can be factorized for SC and ST since the segments of SC and ST have constant curvatures and then constant p’s. Ebend is given by the following equations (6)

ESC/k ) 4πl2(1/3 - p01)2 + 4πl(L - l)[(x(1/2 - p01)2 + (1 - x)(1/2 - p02)2] (6) EST/k ) 4πl2(1/3 - p01)2 + 8πl(D - l)[(y(1/2 - p01)2 + (1 - y)(1/2 - p02)2] + 8(D - l)2[(z(1 - p01)2 + (1 - z) (1 - p02)2] where 2L is the total length of SC and 2D is the total length of ST. L and D are related to each other since SC

and ST have equal paraffin volumes (i.e., the same aggregation number) and the same concentration. The equality of volumes assures that the aggregation entropies of the two forms are practically equal so that ∆Ebend ) ESC - EST measures the relative micelle stability. x, y, and z are the fractions of micellar surface of hydrocarbon chains related to the surfactant, as measured at the interface between apolar paraffin moiety and polar heads, in the segments of the micelle where the mixing among amphiphile and decanol occurs. The fraction in the cylindrical body of SC is x, while y and z refer to the fractions in the cylindrical and planar regions of ST. Note that these fractions coincide with the molecular fraction amphiphile/(amphiphile + decanol) in the segments when the paraffin chains are equal. In the case of mixing k is no longer a constant, varying with x, y, and z (see later). The transition between the two micelle forms occurs when ESC ) EST. At the transition point it is possible to consider x ) y ) z (equal mixing) since the demixing in the ST form is not an instantaneous process. In such conditions there is no difference in the entropy of mixing (except for eventual changes of absolute entropies in the SC and ST form). The SC-ST transition occurs when the following equation is satisfied:

xp012 + (1 - x)p022 ) 1/2

(7)

It is seen therefore that the SC-ST transition is not expected for arbitrary (p01, p02) pairs. The solution p02 > x2/2 and p01 < p02 holds usually for the systems considered. The equivalent inequalities p02 < x2/2 and p01 > p02 would apply to systems where the cosurfactant has a larger polar head than the surfactant. It should be stressed that although only SC and ST forms were considered in eq 6, the model takes automatically into account any intermediate biaxial RT form, since at the transition ESC ) ERT ) EST for equal mixing and volume. Therefore eq 7 is also valid for any RT intermediate form. To illustrate the expected transitions, calculations of Ebend have been made in the particular case of SLS, using l ) 16.7 Å (extended chain value19,33) and k ) 0.01075 × 10-12 erg/A2.17 Changes in k of the order of 20%-30% may be expected when decanol is present,32 but since the results are only illustrative, the k value adopted for the binary system17 was kept. Also, k has a smooth variation with x, y, and z, but such variation is not expected to change eq 7, which determines the SC-ST transition. From eq 7 it is seen that the transition between the two micellar forms occurs for interconnected values of p01, p02, and x. Note that the transition point does not depend on the micelle size. However, the total decanol/amphiphile molecular ratio is related to x via the micellar size, under the assumption that the SC caps contain only amphiphile and no decanol. Figure 1 shows Ebend for the two micellar forms as a function of x, for selected combinations of p0 and L values. Note that the figures of Ebend as a function of x, p01 and p02 represent different micellar states at constant volume. The actual transition may occur through a way in which the changes in x and p0 values imply a continuous change in volume. For this reason these figures of Ebend are illustrative of the behavior of the function, but quantitative comparison with results will be made in the next section using eq 7. Here the general behavior of Ebend is discussed. (32) Kegel, W. K.; Bodnar, I.; Lehkerkerker, H. N. W. J. Phys. Chem. 1995, 99, 3272. (33) Tanford, C. J. Phys. Chem. 1972, 76, 3020; The Hydrophobic Effect; Wiley: New York, 1980.

Change in Micelle Form

Langmuir, Vol. 13, No. 19, 1997 5019

Figure 1. Ebend as a function of x (eq 6) for selected values (p01, p02), with L ) 50 Å.

Figure 1 shows the transitions that occur in all cases: SC is more stable for higher amphiphile molecular fractions (less decanol). Addition of decanol promotes the transition to the ST form. For p02 ) x2/2 the transition occurs at x ) 0 (no mixing), and the most stable form is always SC for x > 0. For p01 ) 1/3 and p02 ) 1 the transition occurs at x ) 0.562, while for p01 ) 1/2 and p02 ) 1 it occurs at x ) 0.67. Figures 2 and 3 explore the transitions that occur as a function of p01 and p02 at these three x values. Increase in both p01 and p02 values promote also a SC-ST transition. The problem was solved for the case (i) of independent surfactant/cosurfactant molecules. The quadratic form of eqs 6 and 7 shows that in such case the molecules cannot “compensate” opposed deviations between the actual p and the spontaneous p0 values. An alternative view (ii) is to consider that 1 molecule of type 1 plus (1 - x)/x molecules of type 2 form an “effective unity” in the mixing zones, sharing a total area and volume, with an “effective” spontaneous surfactant parameter (p0)ef ) total volume/(total area × effective chain length). If such molecules have equal paraffin volume and chain length, there results

( )

( )

1 1 1 )x + (1 - x) p01 p02 (po)ef

(8)

The SC-ST transformation will be then given by eq 7 for the particular case p01 ) p02 ) (p0)ef, leading to

(p0)ef ) x2/2

(9)

The combination of eqs 8 and 9 leads to the condition for occurrence of the SC-ST transformation in such case:

( )

x

( )

1 1 + (1 - x) ) x2 p01 p02

(10)

The “effective unity” model allows molecules to compensate opposed deviations between the actual p and the spontaneous p0 values. As a result, this model implies stability of the SC form for higher cosurfactant content than the “independent” model. Such “effective unity” model seems more applicable in the case of SLS/decanol, since the anhydrous polar heads are of similar sizes and the differences in polar head area

Figure 2. Ebend as a function of p01 (eq 6) for selected values (x, p02), with L ) 50 Å.

Figure 3. Ebend as a function of p02 (eq 6) for selected values (x, p01), with L ) 50 Å.

are due to hydration water. In such a case the excess in polar head area of the amphiphile may naturally expand in the available area of the cosurfactant. However, in cases of polar heads well-defined by inner chemical bonds and with rigid attachment to the chain, such accommodation may be difficult, and then the “independent molecules” model may be better. It should be stressed that the Hyde model ignores the stretching energy related to changes in surface area. Physically this corresponds to assuming that removal/ addition of water from the interface at the form transformation does not cost energy. This is true for “free” water that is just filling space. One final comment should be made in relation to the effect of temperature at the individual micellar level. A small temperature effect occurs due to the negative contraction linear coefficient of the monolayer thickness. The SC-ST balance may also be temperature dependent if the absolute entropies in the two forms are not equal; in such a case an entropic effect may exist even if the relative entropy of mixing Smix is equal in both forms.

5020 Langmuir, Vol. 13, No. 19, 1997

3. Correlation between SC-ST Transformation and Nc-Nd Phase Transitions The micelles in the nematic liquid-crystalline phase have strong intermicellar interactions and their size is defined by such interactions and by availability of space. The size dependence has been explicitly analyzed in the works of Gelbart et al.15,34 and Herzfeld et al.18 However, no treatment exists up to now that predicts changes in micellar form. The micellar symmetry is always fixed as an “input to the theories”. The model developed in the previous section shows that Ebend of a single micelle can be considered as the dominant energy component to understand the change in the micellar symmetry. The basic argument justifying the use of this model for the Nc-Nd transition is that the SC-ST transformation does not depend on the micellar size, which is the quantity determined by intermicellar interactions. Furthermore, there are problems in adding Ebend to other energy terms, both because of its phenomenological nature and the fact that the constant k is correct in order of magnitude, but has not been precisely determined for the systems in discussion. It should be stressed that this model for the SC-ST transformation of the individual micelle can be correlated only with phase transitions induced by changes in concentration. Phase transitions as a function of temperature require a statistical mechanical treatment of the ensemble of micelles. Our focus will be on the Nc-Nd transitions occurring as a function of Md, the molecular fraction decanol/amphiphile of the sample. In an earlier paper12 a detailed analysis of reported transitions for the systems with KL,5,6 SDS,7-9 and SLS12 was made, in order to express the transitions as a function of Md, since many phase diagrams were published as a function of the weight percentage of one component having another fixed also in weight percentage. This procedure in a ternary system masks the important variable of the transition. Changes in the water/amphiphile molecular ratio (Mw) are not the dominant feature in the Nc-Nd transition, showing that the dominant effect occurs inside the micelle. In the SLS system, for instance,12 changes of 11% in Mw leave the Md value at which the transition occurs practically unchanged (within 3%). This paper focuses therefore in changes in x as the dominant effect for the SC-ST form transformation and correlates it with the phase transition Nc-Nd. The Nc-Nd transitions observed in lyotropic liquid crystals occur precisely in the same directions as the SCST transition, for changes in concentration (decanol/ amphiphile and water/amphiphile molecular ratios). The model for Ebend gives therefore a good basis for a more detailed comparison between experimental results and predictions based on eqs 7 and 10. Continuous changes in p0 values (related to changes in the water/amphiphile molecular ratio in the whole sample or to change in temperature) may occur. Also, changes in x are usually accompanied by changes in micellar aggregation number and volume, since the added decanol inserts itself in the cylindrical part of SC. But eqs 7 and 10 would remain valid at the transition if such changes are continuous and negligible at the transition itself. The variable x of eqs 6 and 7 is therefore related to the Md value at the transition and to the anisometry µ ) l/L (34) McMullen, W. E.; Gelbart, W. M.; Ben-Shaul, A. B. J. Chem. Phys. 1985, 82, 5616.

Amaral et al. Table 1. Experimental Md Values (Decanol/Amphiphile Molecular Ratio at the Nc-Nd Transition),12 Estimated Anisometries µ of the SC (See Text), and Related Surface Ratios (x from Eqs 11) Md µ x

SDS

KL

SLS

0.30 ( 0.03 2.3 ( 0.3 0.65 ( 0.03

0.38 ( 0.02 3.0 ( 0.3 0.64 ( 0.03

0.38 ( 0.01 3.0 ( 0.3 0.64 ( 0.03

of the SC through:

x)

u 1+u

(11)

with

u)

(µµ--1/31 )M1

+ d

v2 µ - 1 -1 v1 µ - 1/3

(

)

where v1 and v2 are the paraffin chain volumes of surfactant and cosurfactant, respectively. The dependence on v2/v1 ) l2/l1 (extended chain lengths of decanol and amphiphile, respectively) appears because the surfactant paraffin volume fraction in the mixing zones (denoted here by vx) does not coincide with x when the chain length of surfactant and cosurfactant is different

vx v2 x ) 1 - x 1 - vx v1

(12)

The anisometries at the phase transitions are poorly known. Results from X-ray diffraction8,12,19,35 give only the average distance between micelles, from which anisometries can be inferred only with hypothesis regarding water distribution. Conductivity measurements36 require knowledge of the order parameter to obtain micellar anisometries. Other molecular techniques37 give more direct estimates. From all available data, µ ) 3.0 ( 0.3 is considered a reasonable estimate of the anisometry for the systems SLS and KL at the transition, while a value 2.4 ( 0.3 can be considered for the SDS system. The estimate of p02 was made from molecular dimensions of decanol, compatible with p02 ) 1.0 ( 0.1. The chain volume v2 ) 297 Å3 from Tanford’s formula33 was used. Table 1 shows the experimental values of Md at the Nc-Nd transition for the systems with SDS, SLS, and KL, of the anisometries µ of the SC near the transition, and the corresponding values of x obtained from eq 11. For the three systems the x values are equal within experimental errors. Figure 4 shows the experimental values of x at the NcNd transition confronted with the curves of x as a function of p01, from eq 7 (assuming independent molecules) and from eq 10 (effective unity) with p02 ) 1. It is seen that a very good agreement between experimental values and prediction from the model is obtained for p01 ) 0.47 (independent molecules) and p01 ) 0.6 (effective unity). The exact value of p01 in these nematic phases is not known, but a value of the order of 1/2 can be expected from the amount of available water and from the fact that the nematic domain occurs between hexagonal and lamellar phases. The indication is thefore that some degree of “sharing” in areas may be occurring. The SC-ST transformation at the individual micellar level produces the population of micellar forms that (35) Figueiredo Neto, A. M.; Galerne, Y.; Levelut, A. M.; Liebert, L. J. Phys. Lett (Paris) 1985, 46, L499. Galerne, Y.; Figueiredo Neto, A. M.; Liebert, L. J. Chem. Phys. 1987, 87, 1851. (36) Photinos, P.; Melnik, G.; Saupe, A. J. Chem. Phys. 1968, 84, 6928. (37) Quist, P. O.; Halle, B.; Furo, I. J. Chem. Phys. 1992, 96, 3875.

Change in Micelle Form

Langmuir, Vol. 13, No. 19, 1997 5021

chemical potential, as already done by Gelbart.15 Such growth in the Nd phase after the Nc-Nd transition has been detected in the SLS system, with anisometries of the order of 4-6.39 4. Conclusions

Figure 4. Experimental values of the surfactant fraction in the cylindrical body of SC (x) at the Nc-Nd transition (for SDS, SLS, and KL) and predicted curves as a function of p01 for the SC-ST transformation: - - -, for independent molecule (eq 7); ..., for effective unity (eq 10).

induces the Nc-Nd phase transition,38 and it is a necessary condition for its occurrence. After the ST form has been generated by elastic bending energy requirements, further partitioning of surfactant/ cosurfactant may occur (y * z in eq 6), coupled with micellar growth. However such a process can no longer be analyzed solely in terms of Ebend, but rather in terms of the overall

The Hyde model for the bending energy has been used to analyze the bending energy of mixed micelles. Results show that a form transition from spherocylinder to planar micelles is expected with increase of the cosurfactant/ surfactant molecular ratio. Defined equations for SCST transformation have been derived for the limiting cases of independent molecules and of “effective unity”. There is good agreement between predictions of the model and experimental results for three different micellar systems at the Nc-Nd nematic liquid crystal phase transitions. The single micelle elastic bending energy is the dominant effect because the SC-ST transition does not depend on the size of particles, which is defined by the intermicellar interactions. Acknowledgment. We want to thank financial support of FAPESP and CNPq Brazilian agencies. O. Santin Filho had a doctorate fellowship at the Institute of Chemistry of the University of Sa˜o Paulo. LA9700073 (38) Henriques, E. F.; Henriques, V. B. J. Chem. Phys., in press. (39) Furo, I.; Halle, B. Phys. Rev. E 1995, 51, 466.