PHYSICAL REVIEW A VOLUME 35, NUMBER 10 MAY 15, 1987 Analysis of intermicellar structure factors with the mean spherical and hypernetted-chain approximations Dusan Bratko, Eric Y. Sheu, and Sow-Hsin Chen Nuclear Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received 18 August 1986) Applicability of the mean spherical approximation (MSA) and hypernetted-chain approximation (HNCA) to the analysis of intermicellar structure factors was examined. For a strongly coupled micelle-counterion system at high volume fraction the MSA partial structure factors are close to the corresponding HNCA solutions provided the micellar charge is appropriately reduced in the case of MSA. The same agreement for the micelle-micelle structure factor can still be maintained at low volume fraction by using the previously proposed rescaled MSA. In certain cases, the experimental intermicellar structure factor can be satisfactorily reproduced within the HNCA by assuming the micellar charge to be substantially lower than the mean aggregation number of the micelle. I. THEORETICAL BACKGROUND It is well known that ionic surfactants form micellar aggregates in aqueous solutions spontaneously when con- centration of the surfactant monomers exceeds a threshold value called the critical micellar concentration (CMC). ' The formation of micellar aggregates in aqueous solution is a cooperative process driven by strong hydrophobic in- teractions between the tails of the surfactants in the aque- ous environment. The energetically favorable self- association is, however, counterbalanced by a decreasing entropy of dispersion upon micellization. Thus near CMC micelles are formed with a well-defined minimum size consistent with the geometrical packing constraint of the tail groups of the surfactant molecules. For example, a surfactant such as sodium dodecyl sulfate (SDS) has a hydrocarbon tail of length i=16.7 A and a steric volume of U, =350 A which results in formation of a minimum compact spherical micelle of aggregation number 53. Similarly, sodium dodecyl orth oxylene sulfonate (NaC&zOXYS) has corresponding parameters 1=13.75 A and U, =370 A resulting in a minimum aggregation num- ber of 30. For surfactant concentrations not too far above the CMC (without electrolyte added) the micellar aggregates grow slowly in size while their shape changes from spherical to spheroidal. Take the case of NaC&2OXYS at a concentration of 4 gm/dl and tempera- ture of 53.6 'C, the measured mean aggregation number is 49.8 indicating micelles of spheroidal shape with an axial ratio 1.69 or an equivalent sphere of diameter 48 ~ 3 A. A small-angle neutron scattering (SANS) experiment shows that the micellar system at this condition is nearly mono- disperse so it can be considered as a two-component macroion-counterion system. In this case the micelles are charged macroions with surface charges equal to or less than the mean aggregation number n in equilibrium with the counterion (Na+) cloud surrounding them. The situa- tion is similar to an electrolyte solution except that in the micellar solutions the charge and size between micelle and counterion are highly asymmetric. coupled with the approximate closure relations for the direct correlation function c;j(r;j). In the case of the mean spherical approximation (MSA) one takes the clo- sure relation as and ctj(re ): /3 tj u(re ) for rj ) crt'j hjj (rtj ): 1 for re ( cr(j (3) (4) where /3=1/(ktt T), kjt is the Boltzmann constant, and T the absolute temperature. In the case of the hypernetted- chain approximation (HNCA), cj(r~ ) = gu&(r~)+h j(r J ) —— ln[hj(rj )+ I] . By comparisons with computer simulations of the primitive model electrolyte solutions, HNCA has been confirmed to be an accurate theory for the structural and thermodynamic properties of 1-1 and 1-2 electrolyte solu- For the theoretical treatment of this type of system the similarities with a simple electrolyte prompt the applica- tion of the primitive model of ionic solutions. According to this simple model, the charge aggregates and the small ions are treated as uniformly charged spheres immersed in a solvent of a dielectric continuum with dielectric con- stant e. The solvent-averaged pair potentials are then as- sumed to be given by 2 Z]. ZJ e C7l. + 0J. u;J — — for r &o. ;z er, , J where r;J is the distance between the particles i and j of valences z; and zJ, and diameters o. ; and cd, and e is the elementary charge. The most successful statistical mechanical treatments of this model have so far been based on the application of the Ornstein-Zernike integral equation for the total correlation functions h;j(rj) be- tween ion pair ij h, j(r&)=c;j(r&)+ g nk j c;k(r;k)hjk(r&k)d rk, k 35 4359 1987 The American Physical Society