by R Nagarajan · Cited by 28 — He teaches the short courses “Surfactants and The interactions between surfactant molecules and synthetic polymers in aqueous solutions are.
19 pages

141 KB – 19 Pages

PAGE – 1 ============
POLYMERŒSURFACTANT INTERACTIONS R. Nagarajan Department of Chemical Engineering The Pennsylvania State University University Park, PA 16802 Phone: 814-863-1973; Fax: 814-865-7846 E-mail: URL: ABSTRACT Surfactants and polymers are two of the importan t components in detergent formulations. The interactions between polymers and surfactants in aque ous media give rise to the formation of association structures, thereby modifying the solution and interf acial properties. The morphologies of association complexes depend on the molecular properties of th e polymer and the surfactant. In this paper, we examine in detail, the association structure that form s in the presence of nonionic polymers, based on a quantitative theory. First we show that a variety of experimental observations are suggestive of the proposed structure of the complex. Then a detailed thermodynamic treatment is formulated to examine the competition between the formation of polymer-f ree surfactant aggregates and polymer-surfactant association complexes. The molecula r features controlling this competitio n are identified to be the steric interactions between surfactant head groups and polymer segments, the shielding of micelle-water contact by the polymer, and the polymer hydrophobi city. The theory is used to explain why the presence of the polymer lowers the critical micelle concentration, reduces the size of spherical micelles, allows formation of complexes with mixed surfactants, and transforms large rodlike micelles and vesicles into smaller globular micellar aggregates. We conclude by alluding to the impact of the polymer on other properties such as solubilization and microemulsification. BIOGRAPHICAL INFORMATION Nagarajan received his Ph.D. in chemical engineering from the State University of New York at Buffalo. He joined The Pennsylvania State University in 197 9, where he now holds the rank of Professor of Chemical Engineering. His research interests span the fundamentals and applications of surfactant and block copolymer systems. A major part of his resear ch concerns the creation of molecular theories to predict a priori the self-assembling behavior of surfactants an d block copolymers in a variety of aqueous, organic solvents and solvent mixtures. Phenomena su ch as micelles, mixed micelles, solubilization, microemulsions, surfactant-polymer interactions, ad sorption at solid-liquid interfaces are among the problems addressed in his work. The other major aspect of his research pertains to developing applications for surfactant and block copolymer system s. Examples of past work include new methods of chemical separations based on selective micellar so lubilization, forward and backward extraction of proteins using reverse micelles, removal of fermentati on products exploiting phase separation of block copolymers, steroid conversion and hydrolysis of na tural oils using enzymes immobilized in colloidal media, deinking of laser-printed paper using block copolymers and enzymes, synthesis, extraction and stabilization of metal nanoparticles using surfactant an d block copolymer solution media. Nagarajan is a Fellow of the American Institute of Chemical Engine ers and the American Institute of Chemists, Program Chairman of American Chemical Society Division of Colloid and Surface Chemistry, and editorial board member of the Journal of Colloid and Interface Scie nce. He teaches the short courses “Surfactants and Block Copolymers: Self-Assembly” and “Surfactants and Block Copolymers: Colloidal and Interfacial Properties”. Nagarajan, R. Polymer-Surfactant Interactions . In fiNew Horizons: Detergents for the New Millennium Conference Invited Papersfl, publis hed by American Oil Chemists Society and Consumer Specialty Products Association, Fort Myers, Florida (2001).

PAGE – 2 ============
1I. INTRODUCTION The interactions between surfactant molecules and synthetic polymers in aqueous solutions are of importance to many applications in detergents and personal care products, chemical, pharmaceutical, mineral processing and petroleum indu stries. In general, the mutual presence of polymer and surfactant molecules alter the rheological properties of solutions, adsorption characteristics at solid-liquid interfaces, stability of colloidal dispersions, the solubilization capa cities in water for sparingly soluble molecules, and liquid-liquid interfacial tensions [1-5]. All of these phenomena are relevant to detergency since surface tension lowering affects the foaming tendencies, solub ilization affects the oily dirt or food protein removal efficiency, adsorption affects the extent of soil rele ase from the fabric, and adsorption and stabilization are important for controlling soil re deposition onto the fabric. The ability of the surfactant and the polymer molecules to influence the solution and interfacial characteristics is controlled by the state of their occurrence in aqueous solutions. In the absence of the polymer, the surfactant molecules aggregate in aqueous solutions to form spherical, globular or rodlike micelles or spherical bilayer vesicles (Figure 1), at co ncentrations beyond a critical micelle concentration (CMC). The occurrence of such a critical phenomenon is a direct result of the cooperative process involving a large number of surfactant molecules which together form an aggregate. The nature of the surfactant head group (ionic, nonionic, zwitterionic , size, etc.) and the tail group (hydrocarbon or fluorocarbon, branching, unsaturation, aromaticity, etc.) determine which type of aggregate structure would form, what would be the average size and size dispersion of the aggregates, and the magnitude of the CMC. When a polymer is added to the aqueous surfactant solution, singly dispersed polymer molecules as well as intermolecular complexes between the po lymer and the surfactant can also be present as additional species in solution, besides the si ngly dispersed surfactant molecules and polymer Œfree surfactant aggregates. Equilibrium considerations de termine, which, if any, of the above species will be actually present at given solution conditions. Stud ies of aqueous polymer-surfactant solutions in the literature have sought to identify and quantify these species. Spherical or Globular Micelle Rodlike Micelle Spherical Bilayer Vesicle Figure 1. Structures of surfactant aggregates fo rmed in dilute aqueous solutions. Surfactant molecular structure and solution conditions co ntrol which aggregate structure actually forms.

PAGE – 3 ============
2One class of studies concerns the morphology of polymer-surfactant complexes in solution. Techniques such as nuclear magnetic resonance (N MR) [6,7], neutron scattering [8] and fluorescence spectroscopy [9,10] have been used to elucidate the structure of polymer-surfactant complexes and to estimate the size of the polymer-bound micelles. Th e second type of investigations has involved the quantitative measurement of the amount of surfactant associating with the polymer molecules. Also, the occurrence of critical phenomena in solution properti es have been examined. For these studies, classical techniques such as dialysis [11], surface tension [12- 14] , viscosity [15-20] , electrical conductivity [14] , dye solubilization [13,21,22], specific ion activity [23- 25], etc., have been employed. Results from these studies show that some surfactants do not associate at all with polymers while others do so significantly. Also, the solution properties exhibit critical behavio r at one or two surfactant concentrations in some systems but not in others. The third class of investigations has focused on the phase behavior of polymer-surfactant solutions with or without the presen ce of additional components like electrolytes and oil [26,27]. To develop a system atic understanding of polymer Œsurfactant interactions from any of the above perspectives, it is necessary to consider structural models of polymer Œsurfactant aggregates that may form in aqueous solutions. II. POLYMER-SURFACTANT ASSOCIATION STRUCTURES Various morphologies of polymer-surfactant comp lexes can be visualized [20,28] depending on the molecular structures of the polymer and the surfac tant and on the nature of the interaction forces operative between the solvent, the surfactant and the polymer. A schematic view of these morphologies is presented in Figure 2. Structure A denotes only the polymer implying that no polymer-surfactant association occurs. This could arise in a situation where both the polymer and the surfactant carry the same type of ionic charges. This could also occur when the polymer is relatively rigid and for steric reasons does not interact with ionic or nonionic surfacta nts. It could also be the situation when both the polymer and the surfactant are uncharged and no obvious attractive interactions, promoting association, exists between them. Structure B denotes a system where the polymer and the surfactant carry opposite electrical charges. Their mutual association is prom oted by electrostatic attractions. These cause the creation of a complex with reduced charge and hence, reduced hydrophilicity. Indeed, this eventually leads to the precipitation of these complexes from solu tion. Structure C also occurs in systems containing surfactant and polymer possessing op posite charges. In this case, the surfactant promotes intramolecular bridging within a polymer molecule by interacting with multiple sites on one molecule or intermolecular bridging by interacting simultaneously with sites on different polymer molecules. Structure D depicts a situation when the polymer is a random copolymer or multiblock copolymer with relatively short blocks. In this case, the po lymer molecule assumes a conformation in solution characterized by segregation of dissimilar segments or blocks of varying polarity. Depending upon whether the polymer is a random copolymer or a bl ock copolymer, the segregation in the polymer can take different forms, including the formation of poly meric micelles. In either case, one can imagine the surfactant molecules to locate themselves at the interfaces between the segregated regions. Structures E, F, and G pertain to hydrophobically mo dified polymers. In this case, the size of the hydrophobic modifier, its grafting density along th e polymer, and the relative concentrations of the surfactant and the polymer, all influence the nature of the association structure. In general, at low surfactant concentrations, structure E may be obtained with single surfactant molecules or very small surfactant clusters (dimers, trimers, etc.) interact ing with one or more hydrophobic modifiers, without causing any conformational changes on the polymer. When the surfactant concentration is increased, somewhat larger surfactant clusters form co-aggregat es with multiple hydrophobic modifiers belonging to the same polymer molecule, causing the polymer conformation to change significantly. At larger surfactant concentrations, it is possible to obtain the structure G where surfactant aggregates are formed around each of the hydrophobic modifier. Structure H denotes a complex consisting of th e polymer molecule wrapped around surfactant micelles with the polymer segments partially penetrating the polar head group region of the micelles and reducing the micelle core-water contact. Such a structure can describe a nonionic polymer interacting with surfactant micelles. Such a structure can al so be imagined in the case of an ionic polymer interacting with oppositely charged micelles.

PAGE – 4 ============
3 A. Polymer molecule does not interact with surfactants for electrostatic or steric reasons. No surfactant is bound to the polymer. For example, the surfactant and the polymer are both anionic or both cationic. B. The polymer and the surfactant are oppositely charged. Single surfacta nt molecules are bound linearl y along the len gth of the polymer molecules. C. The polymer and the surfactant are oppositely charged. A single surfactant molecule binds at multiple sites on a single polymer molecule, giving rise to intra- molecular bridging. Alternatively, it binds to more than one polymer molecule allowing intermolecular bridging. D. The polymer is an uncharged random or multiblock copolymer. The surfactant molecules orient themselves at domain boundaries separating the polymer segments of different polarities. E. Polymer is hydrophobically modified. Individual surfactant molecules associate with one or more of the hydrophobic modifiers on a single polymer molecule or multiple polymer molecules. F. Polymer is hydrophobically modified. Clusters of surfactant molecules associate with multiple hydrophobic modifiers on a sin gle polymer molecule. H. The polymer segments partially penetrate and wrap around the polar head group region of the surfactant micelles. A single polymer mo lecule can associate with one or more surfactant micelles. G. Polymer is hydrophobically modified. Clusters of surfactant molecules associate with each of the hydrophobic modifier on a single polymer molecule. Figure 2. Schematic visualization of various ty pes of polymer-surfactant association structures involving nonionic polymers, charged polymers , random or multiblock copolymers, and hydrophobically modified polymers.

PAGE – 5 ============
4For each type of polymer molecule such as thos e depicted in Figure 2, including the nonionic polymer, charged polymer, hydrophobically modified po lymer, star polymer, random or block copolymer, some of the unique features of the polymer molecule will have to be invoked in developing quantitative models of polymer Œsurfactant association. Given the widespread use of nonionic polymers, we explore in this paper, the association of surfactants with such polymers. The general ideas of molecular aggregation and the formation of competitive associatio n structures discussed in this paper are applicable to the various types of polymers mentioned above, but the details of the modeling would have to be modified in each case to account for the particularities of the polymer molecule. III. PROPERTIES OF SURFACTANT + NONIONIC POLYMER SOLUTIONS Inference about the formation of nonionic polymer-surfactant complex of the type shown in Figure 2 (Structure H) has been obtained from a vari ety of experimental observations. We will illustrate here only the measurement of surface tension, determination of singly dispersed surfactant concentration, solubilization of hydrophobic solutes, an d rheological behavior, since all of these properties are of importance to detergency. It is well Œknown that a classical method for determining the critical micelle concentration of surfactants is by measuring the surface tension at the air-water interface as a function of the surfactant concentration. The surface tension first decreases and when the concentration reaches the CMC, becomes more or less a constant. Th is is because micelles begin to form and if the micelle size is large enough (say 30 or more molecules), then the ac tivity of the singly dispersed surfactant becomes practically constant. This, in turn, leads to effective constancy of surface concentration of the surfactant and thus of the surface tension. The measured surface tension of solutions cont aining the anionic surfactant, sodium dodecyl sulfate (SDS), are shown in Figure 3 at various concentrations of added nonionic polymer, polyvinyl pyrrolidone (PVP) [13]. One can observe, that in the presence of the polymer, the critical concentration where the surface tension becomes a constant decr eases implying that polymer-micelle association structures begin to form at concentrations belo w that required for the formation of polymer-free surfactant aggregates. Further, the constancy of th e surface tension beyond the critical concentration suggests that the aggregates formed in the pres ence of the polymer are sufficiently large. Figure 3. Surface tension of aqueous solutions of an ionic surfactant SDS as a function of the total SDS concentration, C T. The different lines correspond to various concentrations of added nonionic polymer PVP (From Ref.13 as reproduced in Ref.2 ). T1 and T 2 are the critical concentrations signaling the formation of polymer Œbound and polymer Œfree aggregates. T 1 is practically constant (note that it is affected by the amount of polymer only at very low polymer concentrations) while T 2 increases with increasing polymer concentration.

PAGE – 6 ============
5A second type of measurement involves the direct determination of the concentration of singly dispersed surfactant molecules. The results from spec ific-ion-activity measurements carried out by Gilanyi and Wolfram [25] on solutions of the nonionic polymer polyethylene oxide (PEO) + anionic surfactant SDS in the presence of 0.1 M NaNO3 as electrolyte are shown in Figure 4. Four regions 0A, AB, BC and CD are identified on the expe rimental curves obtained at different concentrations of the polymer in solution. In the region from 0 to A, the surfactant molecules remain singly dispersed. In the region from A to B, th e formation of polymer-bound micelles occurs. Two features are to be noted in this region. Fi rst, the monomeric-surfactant concentration X 1 increases very little in this region. This is a consequence of the relatively large size of the polymer-bound micelle and reflects the cooperative nature of formation of bound micelles. If the mass concentration of the polymer is small, then the region AB is limited to a narrow range of surfactant concentrations and sometimes may even fail detection. Alternately, if the mass concen tration of the polymer is large, then for a given concentration range of the surfactant investigated, the saturation point B may not even be reached. At B, the polymer is saturated with bound mi celles. In the region BC, an increase in X t results in a corresponding increase in the singly dispersed surfactant concentration X 1. At C, the formation of free micelles becomes possible. The region CD denotes the range of surfactant concentrations over which any addition of surfactant yields free micelles in solution . Two features can again be noted in this region. The monomeric-surfactant concentration X 1 is almost constant implying that the size of the free micelles in solution is quite large. Secondly, the total surfactant concentration X t corresponding to the point C (denoted as the second critical concentration T 2 in Figure 3) depends upon the mass concentration of the polymer. This is in contrast to the first critical concentration (denoted as T 1 in Figure 3 which is the total surfactant concentration at A), which is independent of the concentration of the polymer in solution. One can see in Figure 4, that at low polymer concentr ation (0.25 g/l), all the four regions are present whereas, at high polymer concentration (4 g/l), the av ailable surfactant is inadequate for saturating the polymer molecules and only two of the four regions are present. Another classical method for determining the CMC is dye solubilization where the spectral change in surfactant solution induced by the uptake of dye molecules is monitored. Since hydrophobic dyes are incorporated within the micelle core, the formatio n of micelles is a precondition to observe dye absorbance. Results from such experiments are shown in Figure 5 for the anionic surfactants, sodium alkyl sulfates (tails in the range of 10 to 16 carbon at oms) in the absence of and in the presence of the nonionic polymer PVP. One can observe from the shar p transitions in the amount of dye solubilized, that in polymer containing systems, large micellar aggregat es must form as in polymer-free solutions. The Figure 4. Equilibrium concentration of the singly dispersed surfactant X 1 as a function of the total surfactant concentration X t in PEO+SDS solutions containing 0.1 M NaNO 3. The different lines correspond to different mass concentrations of the polymer PEO in solution. (From Ref.25)

PAGE – 8 ============
7however, the absence of a change in rela tive viscosity need not rule out polymer Œsurfactant association, since there is no particular reason for the polymer co il to expand on the binding of uncharged surfactant molecules. Results in Figure 6 show that for the anionic surfactant SDS, rel remains close to unity up to a surfactant concentration of 4 mM, indicating absence of any association or minimal association with the polymer molecule. Beyond 4 mM SDS, the relative vi scosity shows a large increase, up to about 26 mM SDS. This can be attributed to the association of multiple SDS micelles with the PEO segments and the resulting expansion of the PEO molecule. Beyond 26 mM SDS, a reduction in the relative viscosity is observed. This is because, when saturation bindin g of SDS to PEO is reached (at 26 mM SDS), further addition of SDS results in an increase in the concentration of singly dispersed SDS molecules and of free SDS micelles unattached to the polymer . This gives rise to an increase in the ionic strength of the solution and a consequent reduction in rel, as is expected for polyelectro lyte solutions [29]. The behavior of the other anionic surfactants sodium dodecyl benzene sulfonate (SDBS) and 8 Œphenyl hexadecane benzene sulfonate (UT-1) in solution s containing PEO are very similar to that of the PEO+SDS solution. As is to be expected, the critic al binding concentrations and the concentrations corresponding to saturation binding of the surfactant are however different from those for SDS. In contrast to the behavior of these anionic surfactants, all the cationic surfactants display relative viscosities close to unity implying low degrees of binding to the PEO, with ethyl hexadecyl dimethyl ammonium bromide (EHD) demonstrating virtually no binding. The relative viscosity is unity also for the case of the nonionic surfactant, isooctyl phenoxy polyoxyethanol (Tri ton X-100). As mentioned before, in this case of a nonionic surfactant, th e viscometric measurement is not adequate to draw conclusions about the binding of the surfactant to the polymer. Similar viscometric measurements have been cond ucted on another nonionic polymer dextran. Dextran is a nonionic polymer with a rigid backbone in contrast to the flexible PEO molecule. Figure 7 shows the viscosity ratio of the dextran-surfactant solution, ratio = (dextran + surfactant)/ [ (dextran) x (surfactant)], for solutions containing the anionic su rfactant SDBS and the cationic surfactant EHD [20]. The measurements yield relative viscosities close to uni ty, implying that no binding occurs in the case of both the anionic SDBS and the cationic EHD with the nonionic dextran polymer. The absence of any surfactant binding to the dext ran polymer is also confirmed by surface tension measurements (Figure 8) which show that the surface tensions of anionic SDBS or SDS solutions in the absence and presence of dextran are identical. Figure 7. Viscosity ratios of solutions containing the nonionic polymer dextran and the anionic surfactant sodium dodecyl benzene sulfonate (SDB S) or the cationic surfactant ethyl hexadecyl dimethyl ammonium bromide. In both cases, th e polymer concentration is 1000 ppm. (From Ref.20)

PAGE – 9 ============
8Figure 8. Surface tension of solutions containing the nonionic polymer dextran and the anionic surfactant sodium dodecyl benzene sulfonate (SDBS) or the cationic surfactant ethyl hexadecyl dimethyl ammonium bromide. In both cases the polymer concentration is 1000 ppm. (From Ref.20 ) The relative viscosity of SDS + PEO solutions is plotted in Figure 9, as a function of the concentration of n-butanol or the nonionic surfactant Triton X-IOO, which are used as additives. Both additives form mixed micelles with SDS and the co mposition of the mixed micelles changes as the concentration of the additives is altered. When no a dditives are present, the PEO + SDS solution is at a SDS concentration of 26 mM corresponding to the saturation binding of SDS to PEO (for the given polymer molecular weight and concentration). As th e concentration of the additive is increased, rel decreases indicating either a decrease in the polymer -micelle association, or a decrease in the polymer chain expansion because of the reduced charge dens ity at the mixed micelle (anionic SDS + nonionic alcohol or surfactant) surface . The viscosity data show that the non Œionic surfactant is more effective than the alcohol in causing a dramatic viscosity decrease. From the experimental results discussed in this section, it is clear that anionic surfactants associate with the nonionic polymer PEO to form micelle Œlike aggregates, with sufficiently large Figure 9. Relative viscosities of solutions co ntaining 1000 ppm of the nonionic polymer PEO and 26 mM of the anionic surfactant SDS as a function of the amount of the nonionic n-butanol or the nonionic surfactant Triton-X 100 added to the solution. (From ref.20)

PAGE – 10 ============
9aggregation numbers. The cationic and nonionic surfactants do not appear to form such polymer Œbound micelles. In the case of the more rigid nonionic polymer dextran, none of the types of surfactant molecules appear to form association structures. To proceed further and to understand the molecular origin of these observations, it is necessary to invoke thermodynamic methods. IV. COMPETITIVE FREE MICELLIZATION AND COMPLEXATION EQUILIBRIA The formation of polymer Œfree surfactant aggregates in aqueous solution in competition with the formation of surfactant aggregates that are associated with the polymer molecule can be formally represented by classical binding equations. The total surfactant concentration X t is partitioned into singly dispersed surfactant X 1, surfactant in free micelles X f, and surfactant in bound micelles X b [19,28,30 Œ32]. Xf and Xb can be represented in terms of th e aggregation equilibrium constants K f and Kb in the form: bbfg1bg1bbPg1ff1t)XK(1)XK(gXn)XK(gXX (1) In eq.(1), the second and the third terms represent X f and X b, respectively. In the second term, g f is the average aggregation number of the free micelles and K f is the intrinsic-equilibrium constant for their formation. In the third term, each po lymer molecule is assumed to have n binding sites for micelles of average size g b. Kb is the intrinsic-equilibrium constant for the binding of the surfactant on the polymer. It can also be visualized as the intrinsic-equilibrium constant for the formation of polymer-bound micelles. Xp is the total concentration of polymer molecules in solution. It has already been observed in Figure 6, that polymer-micelle co mplexation often gives rise to conformational changes in the polymer molecule because of interactions between polymer-bound surfactant molecules or surfactant aggregates. These changes, however, are assumed not to affect K b and gb since these quantities are related principally to th e local conditions at the polymer binding site rather than to the overall conformation of the po lymer molecule. However, the total number of binding sites n may possibly be affected by the conformation al changes in the polymer molecule. Quantitative accounting of such changes in the polymer conformation due to inter Œmicelle interactions, both electrostatic and steric, has been considered in Ref.[33]. The relative magnitudes of K b, Kf, gb and g f determine whether or not complexation with polymer occurs as well as the nature of the critical surf actant concentrations exhibited by the system. If K f > K b, and gb = gf, then the formation of free micelles occurs in preference to complexation. If K f < Kb, and g b = gf, then micelles bound to the polymer occur first and only upon saturation of the polymer, the formation of free micelles occurs. If K f < Kb, but gb is much smaller than g f, then formation of free micelles can occur even prior to the saturation of the polymer. A first critical surfactant concentration is observed close to X 1 =1/K b (indicated as T 1 on Figure 3) and a second critical concentration occurs near X 1 = 1/K f (indicated as T 2 on Figure 3). However, depending upon the magnitude of nX p, one may observe sometimes only the first critical concentration over a finite range of surfactant concentrations. The quantity n is approximately proportional to the size or the molecular weight of the polymer while X p is the molar concentration of the polymer. Therefore, nX p is approximately proportional to the mass concentration of the polymer. This means that for polymers of different molecular weights, but at the same mass concentration in solution, one should ob serve essentially similar surfactant binding behavior. In the following section, models are presente d to permit the estimation of parameters K b, Kf, gb and gf. V. THERMODYNAMICS OF POLYMER-FREE MICELLES The concentration of surfactant present in the form of free micelles (appearing as the second term in eq.1) can be calculated from the size dist ribution of micelles given by the expression [34-38] kTgexpXkTgexpXXogg1o1ogg1g (2) PAGE - 11 ============ 10In the above equation, X g is the mole fraction of micelles of size g , X1 is the mole fraction of singly dispersed surfactant molecules, og is the standard chemical potential of the micelle, and o1 is the standard chemical potential of the singly dispersed surfactant in water. The standard states for og and o1 are chosen to be infinitely dilute solutions in water. k is the Boltzmann constant and T is the absolute temperature. The factor og in eq.(2) is the standard free energy change on transferring the surfactant molecule from its singly dispersed state into an aggreg ate. This is the most important quantity needed for theoretical predictions since all interesting physical properties of the surfactant solution can be calculated if an expression for og is available. An estimate of the most probable micelle size g f is obtained [35-38] from the solution of fogggat0kTdgd (3) The critical micelle concentration can be approximately calculated [35-38] from fogf1ggatkTXln (4) The intrinsic-equilibrium constant for th e formation of free micelles of size g f is then given by fogfggatkTexpK (5) For cylindrical micelles of aggregation number g, having g cap molecules at the two endcaps of the spherocylinder (see Figure 1), the size distribution equation (eq.2) reduces [36,38-42] to kT g =K ln , kT exp X = Y , Y K1 =Xocylo cap capocyl1gg-- (6) Here, ocap and ocyl are the standard state free energy d ifferences per molecule corresponding to those on the endcaps and on the cylindrical part, respec tively, of the rodlike micelles, and K is referred to as the sphere-to-rod transition parameter. It is known that K must be in the range 10 8 to 10 12, if rodlike micelles are to form at physically realistic surfactant concentrations [36,38-42] . The rodlike aggregates are highly polydispersed and their weight and number average aggregation numbers (g w and g n) depend on total surfactant concentration X t in the form [36,38-42] ] )X X(K [ 2 +g = g ,] )X X(K [ +g = g2/11t capw2/11t capn-- (7) In the case of surfactant mixtures, similar free energy expressions can be formulated as a function of the composition of the mixed micelle. Such expressions have been developed earlier and are described in detail in ref.[43-44] To proceed with calculation of the shape, size di stribution and average size of the aggregate, and the CMC, it is necessary to start with an expression for og. The free energy change on aggregation og has a number of contributions that arise from the ch anges experienced by the singly dispersed surfactant molecule when it is transferred from water into th e aggregate [35,38-40]. Accounting for all of these changes phenomenologically, one can write ionicogdipoleogstericogintogdefogtrogog)()()()()()( (8) The six contributions account respectively for the transf er free energy of the surfactant tail in going from water to a hydrophobic core domain, the deformatio n free energy of the tail due to chain packing constraints inside surfactant aggregates, the free ener gy of formation of the aggregate-water interface, 141 KB – 19 Pages