Growth of wormlike micelles in nonionic surfactant solutions: Quantitative theory vs. experiment.

Despite the considerable advances of molecular-thermodynamic theory of micelle growth, agreement between theory and experiment has been achieved only in isolated cases. A general theory that can provide self-consistent quantitative description of the growth of wormlike micelles in mixed surfactant solutions, including the experimentally observed high peaks in viscosity and aggregation number, is still missing. As a step toward the creation of such theory, here we consider the simplest system - nonionic wormlike surfactant micelles from polyoxyethylene alkyl ethers, CiEj. Our goal is to construct a molecular-thermodynamic model that is in agreement with the available experimental data. For this goal, we systematized data for the micelle mean mass aggregation number, from which the micelle growth parameter was determined at various temperatures. None of the available models can give a quantitative description of these data. We constructed a new model, which is based on theoretical expressions for the interfacial-tension, headgroup-steric and chain-conformation components of micelle free energy, along with appropriate expressions for the parameters of the model, including their temperature and curvature dependencies. Special attention was paid to the surfactant chain-conformation free energy, for which a new more general formula was derived. As a result, relatively simple theoretical expressions are obtained. All parameters that enter these expressions are known, which facilitates the theoretical modeling of micelle growth for various nonionic surfactants in excellent agreement with the experiment. The constructed model can serve as a basis that can be further upgraded to obtain quantitative description of micelle growth in more complicated systems, including binary and ternary mixtures of nonionic, ionic and zwitterionic surfactants, which determines the viscosity and stability of various formulations in personal-care and house-hold detergency.


Micellization and micelle shape transformations
In 1913, the existence of self-assembled surfactant aggregates was suggested by McBain [1], who introduced the term "micelles" in relation to the interpretation of experimental data for the electrolytic conductivity and osmotic activity of carboxylate solutions [2,3]. McBain considered two types of aggregates: (i) small oligomeric surfactant clusters with approximately 10 molecules, and (ii) lamellar (disclike) aggregates consisting of two layers of 50 to 100 soap molecules. Harkins et al. [4] recognized that the orientations and packing of surfactant molecules at an interface depend on the molecular shape, which is closely related to the possible shapes of the micelles in the bulk. The micellar structures proposed by Hess, Philippoff et al. [5][6][7] represent McBain lamellar aggregates separated by layers of water. Harkins et al. [8][9][10] used lamellar (disclike) micelles to interpret their X-ray and solubility data. They introduced the idea that the micelles could have also cylindrical and ellipsoidal shapes [11]. The proposed small lamellar micelles are energetically unfavorable because of the contact of surfactant tails with water at their periphery. Adam [12] proposed the idea that "molecules larger at their polar ends will naturally pack into a curved film having the hydrocarbon side concave and the water attracting side convex, and such a film will fit the surface of an emulsion of oil dispersed in water", which in fact postulates the possibility to have spherical micelles (Fig. 1a). Hartley [13][14][15] suggested that the surfactant aggregates tend to present the minimum surface to the water, they are "roughly spherical of the largest radius with none of the heads being submerged in the paraffin interior". Corrin and Harkins [16,17] interpreted their experimental data assuming spherical shape of the micelles. The surfactant concentration, at which spherical micelles appear, was called critical micelle concentration (CMC). The definite answer of the question about the shapes of micelles and self-assembly of surfactant in solutions was not found from experimental and theoretical viewpoints up to 1951, when Philippoff [18] concluded that "all standard methods for determining the shape of dissolved particles, indicate that there is no significant departure of the shape of soap micelles from a sphere…. A practically spherical shape does not mean the micelles are true spheres: cubes, short cylinders, prisms or spheroids are all indistinguishable". A comprehensive historical review on the early studies on micellization was published by Vincent [19]. The development of powerful experimental techniques (static and dynamic light scattering, small angle X-ray and neutron scattering, electron paramagnetic resonance, nuclear magnetic resonance, etc.) gave the possibility to study complex self-assembly, phase behavior and structural behavior of surfactant solutions and microemulsions. It was established that the micelles can have different shapes, sizes, aggregation numbers, and compositions depending on surfactant and salt concentrations: small spherical and ellipsoidal micelles; discs; cylindrical and wormlike micelles; branched micelles; vesicles, and different types of lamellar structures. The real microstructure of micellar systems in solutions were revealed by using small angle scattering [20] and cryogenic transmission electron microscopy (cryo-TEM) [21].
The concentrated surfactant solutions show complex phase behavior and nonmonotonic trends in their viscoelastic properties, with one or two peaks in the zero-shear viscosity as a function of surfactant concentration, temperature, pH, and concentration of added salt . The viscosity of micellar solutions is directly related to their microstructure. The viscosity increases with the rise of concentrations of surfactant and added salt -the micelles transform from spheres to small elongated ellipsoids, cylinders (rods); the longer cylindrical (rodlike) aggregates are termed wormlike micelles because of their flexibility. The concentration, at which the micelles transform from spheres to cylinders is known as the second CMC. Upon variation of composition of the micellar system, the wormlike micelles may undergo a subsequent shape transformation, which leads to decrease in viscosity at higher concentrations of surfactant and/or electrolyte. Thus, a peak appears in the concentration dependence of viscosity. Several possible reasons for the decrease of viscosity at higher concentrations have been reported: (i) shortening of the wormlike micelles; (ii) formation of multiconnected (branched) micelles; (iii) shape transition to discs; (iv) phase transition to lamellar or liquid crystal mesostructures; (v) formation of spherical swollen micelles, and (vi) phase separation [60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78]. From practical viewpoint it is important to control the micelle shape, size and aggregation number in order to produce formulations with well-defined physicochemical and rheological properties.
The transition region between spherical and wormlike micelles deserves a special attention. With the nonionic surfactant C 12 E 5 , Talmon et al. [79] observed coexistence of spheroidal and threadlike micelles, but short and medium length cylindrical micelles were absent.

Theoretical modelling of micellization and micelle growth
The theoretical physicochemical reasons for micellization were first considered by Debye [80,81]. Hobbs [84] extended the Debye theory to electrolyte solutions and obtained semi-empirical expressions for the increase of micelle size and decrease of the CMC with the rise of salt concentration. Halsey [83] studied one-dimensional growth and formation of platelike micelles. Ooshika and Reich [84,85] showed that the surface energy of the water/micellecore interface is that, which opposes the headgroup repulsion. These authors proposed the minimization of the free energy of the whole solution (containing surfactant monomers, water molecules, and micellar aggregates) as a general principle for determining the micelle size and the CMC. Debye and Anacker [86] showed that at high salt concentrations the ionic micelles should be rodlike. Statistical mechanical theory was first applied for description of the micelle formation and CMC in Refs. [87,88], where spherical and ellipsoidal shapes of the micelles were considered. Contributions from the molecular degrees of freedom (translation, rotation, and vibration), solvent interactions, and electrostatic forces were taken into account [87][88][89].
The Tanford free energy model combined concepts formulated in the simpler earlier theories and included the most important physicochemical factors that control the micelle formation and growth [90][91][92]. Israelachvili,Mitchel,and Ninham [93] developed the first geometry-based approach to predict the self-assembly of surfactant molecules in aggregates of different shapes. They introduced the packing parameter, p, which characterizes the geometry of the various possible aggregates: p = 1/3 for spheres; 1/3 < p < 1/2 for ellipsoidal, globular and toroidal shapes; p = 1/2 for cylinders; 1/2 < p < 1 for vesicles and p = 1 for bilayers (lamellas). They considered the wormlike micelles as "cylindrical micelles with globular ends", in which the radius of the cylindrical part is less than, or equal to, the radius of the endcaps (truncated spheres; see Fig. 1b), and introduced the concept of "linear aggregates". The Tanford energy approach was applied to aggregates of different shapes to explain the formation of micelles and their shape transformations on the basis of general thermodynamic principles [93]. Review on the theoretical models that predict the CMC and electric conductivity of micellar surfactant solutions (phase separation model; mass action model; molecular thermodynamic models, and the quantitative structure-property relationship approach) can be found in Ref. [94].
The current picture of the rodlike micelles was introduced by Debye and Anacker [86] and experimentally confirmed by Scheraga and Backus [95], who pointed out that the micellar size distribution should be highly polydisperse. Using a stepwise association model, Mukerjee derived the popular formula showing that the mean mass aggregation number of the cylindrical micelles, n M , is proportional to the square root of surfactant concentration [96][97][98]: Here, X S is the molar fraction of surfactant in the solution; o S X is a constant parameter close to the value of X S at the CMC; the proportionality coefficient, K, was called "the micellar growth parameter" [93,[99][100][101]. For simplicity, Missel et al. [100] considered the rodlike micelles as cylinders with hemispherical caps and equal radii corresponding to the extended length of surfactant tail. They assumed that the chemical potentials of molecules in the micelle cylindrical part, µ c , and in the spherical end-caps, µ s , are different and derived a simple formula for the micellar growth parameter, where k B is the Boltzmann constant, T is the absolute temperature, and n s is the total number of surfactant molecules in the caps. Analogous result was obtained also for disclike micelles [101]. Detailed review on the different thermodynamic models used in the literature for the description of micellar solutions can be found in Ref. [102]. The thermodynamic theory of growth of nonionic cylindrical micelles is described in Refs. [93,[99][100][101].
The value of CMC is determined by the difference between the chemical potentials of a surfactant molecule in the state of free monomer in the bulk and when incorporated in a micelle. This difference is typically of the order of 10-16 k B T, so that small inaccuracies in the calculated chemical potentials do not essentially affect the predicted value of the CMC. In contrast, according to Eq. (1.2) errors in the calculated µ c and µ s , of the order of 0.1k B T, multiplied by (e.g.) n s = 70, lead to enormous relative errors in the micellar growth parameter, K. This circumstance makes the quantitative prediction of K extremely difficult and sensitive to the precise calculations of the different contributions to the micelle excess free energy.
First, the theoretical expressions used to quantify contributions of different origin to the micelle free energy are different in the BS and NR models. For example, the BS model takes into account the Tolman effect on the micelle hydrocarbon/water interfacial tension, s, which makes s curvature dependent, whereas s is assumed curvature independent in the NR model. Moreover, in the NR model the conformational free energy of surfactant tails in the micellar core is calculated by using the mean-field theory due to Semenov [153], while the BS model uses a different theory of chain packing in amphiphilic aggregates [154][155][156][157][158][159][160][161], which leads to different curvature dependence. In the electrostatic component of free energy of ionic surfactant micelles, the effect of counterion binding is taken into account in the BS approach [150,151], whereas this effect is missing in the NR approach.
Second, both the NR and BS approaches use minimization of the total free energy of the wormlike micelles with respect to the radius of their cylindrical parts. Furthermore, in the NR approach the free energy of the spherical end-caps is also minimized with respect to their radius and the most probable (optimal) micelle shape is calculated. In the BS approach, µ s is estimated in a different way, viz. as the value of the free energy per molecule in a separate spherical micelle of aggregation number n s . The equilibrium value of n s refers to the minimal micellar free energy. The latter could correspond either to a local minimum at a certain micellar radius, R, in the physical interval 0 < R ≤ l (with l being the length of the extended surfactant molecule), or to the free energy value at R = l, if the free energy decreases with the rise of R. In the cases considered in Refs. [162,163], computations based on the chain-packing theory used by these authors show that the optimal spherical micelle corresponds to the case The NR and BS approaches have been applied in different studies for molecularthermodynamic modeling of specific ion effects on the micellization and growth of ionic surfactant micelles [164][165][166].
A theoretical approach, which is alternative to the molecular-thermodynamic modeling, is the computer simulation using, e.g., molecular dynamics. Cross-grained molecular dynamic simulations have been used to investigate the mechanisms of self-assembly of surfactant molecules; transitions from spherical to rodlike micellar shapes; branching of micelles, and the formation of more complex micellar structures [167][168][169][170][171][172][173][174][175][176][177][178][179][180][181][182][183][184]. These calculations give impressive qualitative pictures, which help to visualize and clarify the structural assumptions made in the molecular thermodynamic approach. At present, quantitative information about the values of the micellar growth parameter, K, of large wormlike micelles cannot be obtained using cross-grained molecular dynamics because of computer limitations. Another promising method for computer simulation of self-assembled structures in concentrated surfactant solutions is the dissipative particle dynamics (DPD), which is a mesoscale simulation technique developed in the works by Warren et al. [185][186][187].

Goal and structure of the article
As already mentioned, a general molecular-thermodynamic theory that provides a selfconsistent quantitative description of the growth of wormlike micelles in solutions of nonionic, ionic, zwitterionic, and mixed surfactant solutions, including the experimentally observed peaks in viscosity and aggregation number, is still missing. As a step toward the creation of such theory, here we focus our attention on the simplest system -nonionic wormlike surfactant micelles. Our goal is to construct a molecular-thermodynamic model, which is in agreement with the available experimental data for nonionic wormlike micelles. For this goal, Here, N W is the number of water molecules; N S is the total number of surfactant molecules in the solution; N 1 is the number of surfactant molecules in monomeric form; N k (s) is the number of aggregates that are composed of k surfactant molecules; s denotes the shape of the aggregate -for example, we may have cylindrical and disclike micelles that have the same aggregation number, k, but different shapes. The total free energy of the micellar solution, G, can be expressed in the form: Here, o W µ is the standard chemical potential of the water molecules; o 1 µ is the standard chemical potential of free monomers in the aqueous solution, and ) ( o s g k is the standard free energy of an aggregate of given shape s and aggregation number k. The free energy of interactions between the micelles, G int , is proportional to the micelle volume fraction [143,144]; its contribution to G is important for sufficiently high micelle volume fractions.
Here, we consider the cases, in which the contribution of G int to the total free energy G is negligible.
At equilibrium, the free energy G has a minimum with respect to the variations of the numbers of all components, N 1 and N k (s), k > 1. At that, the total number of surfactant molecules in the solution must be constant: 3) The minimization of G under the constraint given by Eq. (2.3) leads to: where λ is a Lagrange multiplier.
Apparently, λ is the chemical potential of the free surfactant monomers. Eliminating λ between the two expressions in Eq. (2.5), we obtain: One possible simplification is to specify the shape of the micellar aggregate, which for wormlike micelles can be spherocylindrical, and in the limiting case of small micellesspherical. Then, in Eq. (2.3) the sum with respect to s has to be replaced with its maximal addend, which corresponds to the optimal (most probable) micellar shape: Here, the subscript 'opt,k' denotes that the respective quantity refers to the optimal shape of a micelle with aggregation number k. In view of Eq. (2.8), the mean mass (weight) micelle aggregation number, n M , can be calculated from the expression [100]: Likewise, the number-average micelle aggregation number, n N , is [100]: The above general equations can be used for theoretical predictions of the CMC, the micelle size distribution, and the growth of single component surfactant micelles. Their application to rodlike micelles is described in the next section.

Growth of rodlike micelles -linear aggregate model
Here, we consider surfactant solutions, in which the optimal aggregate shape is spherical or spherocylindrical (rodlike, wormlike); see Fig. 1. Let n t be the aggregation number, at which the transition from spherical to spherocylindrical aggregates takes place. In other words, for k ≥ n t the micelles possess a well pronounced cylindrical part (Fig. 1b). The free energy of such aggregates can be expressed in the form [188]: where µ c (R opt ) is the chemical potential of a surfactant molecule in the micelle cylindrical part of radius R opt , which corresponds to the optimal shape s opt,k ; E sc is the dimensionless excess energy due to the spherical caps. In Eq. (2.11), ) ( opt, o k k s g grows linearly with k. For this reason, the respective aggregates have been called "linear aggregates" [188]. The substitution of Eq. (2.11) into Eq. (2.6) yields: where the micelle growth parameter, K, is defined as follows: Because of its direct relation to K, the parameter E sc can be also termed "micelle growth parameter". Furthermore, using the definitions as well as Eq. (2.12) and the mathematical formula we can represent Eq. (2.8) in the form: On the one hand, the necessary condition for convergence of the series is q < 1, i.e. the mole fraction of free monomers, X 1 , should be smaller than X B . On the other hand, n t is a large number (e.g. n t = 200); then, q should be close to 1 -otherwise t n q would be a small number and then the mass fraction of the rodlike micelles would be negligible. Hence, we should have X 1 ≈ X B . Some authors call X B the second CMC [162,163].
One of the aforementioned difficulties, related to numerical computations using the distribution Eq. (2.12), is due to the fact that it contains exponentiation with base, which is very close to 1, and an exponent k that is a large number. Fortunately, for linear aggregates the series can be summed up analytically using Eq. (2.15). Thus, the series in Eqs. (2.9) and (2.10) acquire the form: If the predominant part of the surfactant exists in the form of large rodlike micelles, then q→1 and one can use power expansions for 1−q << 1. Then, the mean micelle aggregation numbers in Eqs. (2.9) and (2.10) can be expressed in the form [93,100,101]: at high surfactant concentrations could be due to micelle-micelle interactions.
The micelle growth was investigated for polyoxyethylene alkyl ethers, C i E j , at various i and j, and at various temperatures. Experimental data for the growth of C 12 E 6 and C 12 E 8 micelles are published also in Ref. [119], where the data for C 12 E 6 are in good agreement with those reported in Ref. [123]. In Fig. 2a and b, experimental data from Ref. [127] for the mean mass aggregation numbers, n M , of the micelles in C 10 E 5 and C 10 E 6 aqueous solutions are plotted vs. The effect of temperature, T, is also significant. For example, for C 14 E 7 , the values of E sc are 17.0, 20.5 and 23.5, respectively, for T = 25, 35 and 45 ºC (Table 1). This effect could be explained with the intersegment attraction in the polyoxyethylene headgroup [188,190], which leads to decrease of its volume and area per molecule at the micelle surface. This effect is quantified in Section 7.1.   It should be noted that in Refs. [123][124][125][126][127] values of the free energy parameter g 2 are reported. The parameter g 2 is approximately related to the dimensionless excess free energy E sc by the equation: where M is the molecular mass of the respective surfactant expressed in g/mol. The values of g 2 are with about 5-6 k B T smaller than those of E sc . In addition to the data for aggregation number n M , experimental data for the viscosity of C 12 E 5 , C 12 E 6 , C 12 E 7 , and C 14 E 7 micellar solutions have been obtained in Ref. [128], where it is demonstrated how the viscosity of these solutions increases with the rise of the length of the respective wormlike micelles.

Molecular geometric parameters
First, let us consider some geometrical relations needed for the modeling of spherical and spherocylindrical micelles. All volumes, surface areas and radii refer to the micelle hydrocarbon core.
Spherical micelle (Fig. 1a): The volume, V s , and the surface area, A s , are related to the radius, R s , as follows: where p is the packing parameter [93,188].
Cylindrical part of a spherocylindrical micelle (Fig. 1b): The cylinder volume, V c , the lateral surface area, A c , and the packing parameter, p, are related to the cylinder radius, R c , and length, L c , as follows: For prolate ellipsoids or caps of spherocylindrical micelles, the values of p are between 1/3 and 1/2 see below.
Spherical caps of a spherocylindrical micelle (Fig. 1b): The end-caps have the shape of truncated sphere. The total volume of the two truncated spheres, V sc , and the total area of their spherical surfaces, A sc , are related to the sphere radius, R s and the radius of the cylindrical part (of the truncated circle), R c , as follows: The packing parameter of the caps is defined as follows [136,142] The minimal value, p = 1/3, corresponds to hemispherical caps (R c /R s = 1), whereas the maximal value, p = 3/8, is realized at The numbers of surfactant molecules contained in the cylindrical part and in the spherical end-caps of a spherocylindrical micelle, n c and n s , are: where n C is the number of carbon atoms in the alkyl chain and v(n C ) is its volume. The extended chainlength, l, and the chain volume, v, can be calculated from the Tanford expressions [191]: The volumes of the CH 3 where T is the absolute temperature. For the lengths per CH 3 and CH 2 group, the following values have been used [191]: l(CH 3 ) = 0.280 nm; l(CH 2 ) = 0.1265 nm (4.11) Geometrical parameters for other types of surfactant tails can be found in Ref. [136].

Components of micelle free energy
As mentioned in the Introduction, we tried different versions of the theoretical model with different expressions for the free energy components and various values of the parameters that enter these expressions. Here, we present only the best model, which is in quantitative agreement with the data for single-component nonionic wormlike micelles in Section 3, as well as with data for mixed nonionic wormlike micelles that will be considered in a subsequent paper. Here, o m µ is a standard chemical potential of the surfactant molecule in the micelle, which accounts for molecular internal degrees of freedom; f s is the contribution of the interfacial tension, s, of the boundary between the micelle hydrocarbon core and the surrounding water phase at the micelle surface (Fig. 5a); f hs accounts for the steric repulsion between the headgroups of surfactant molecules; f conf accounts for the conformational free energy of the surfactant hydrocarbon chains inside the micelle (Fig. 5b). Except o m µ , all other terms in Eq.
(4.12) depend on the micelle shape. Interfacial tension component. The interfacial free energy per molecule, f s , is due to the contact area of the micelle hydrocarbon interior with the water phase at the micelle surface -see the "oil"-water interface in Fig. 5a; f s can be calculated from the expression [162,188]: where a is the area per surfactant molecule at the boundary between the micelle hydrocarbon core and the outer water phase (Fig. 5c) and a 0 is the surface area excluded by the surfactant headgroup, i.e. the geometric cross-sectional area of the headgroup. The area per molecule has been calculated from the expression: where v is to be determined from Eqs.  In comparison with the emulsion and microemulsion drops, the surfactant micelles are the objects of the highest surface curvature. Following Ref. [143], we estimated the micelle interfacial tension taking into account its curvature dependence described by the Tolman equation [192][193][194]: s ow = s ow (n C ) is the interfacial tension between the bulk oil and water phases, p is the packing parameter, and δ T is the Tolman length that has been calculated from the expression We obtained Eq. (4.17) by fitting a set of experimental data for s ow from Refs. [195][196][197]; see Headgroup steric repulsion component. This is the contribution from the repulsion between the surfactant headgroups at the micelle surface due to their finite size (Fig. 5a). This effect has been taken into account using the repulsion term in the two-dimensional van der Waals equation; see Ref. [102] and the references cited therein: Here, a and a 0 have the same meaning as in Eq. (4.13); see Fig. 5c. In principle, the excluded area a 0 could be determined by fitting surface-tension isotherms with the van der Waals model; see e.g. Refs. [198][199][200][201][202].
Chain-conformation component. This contribution to the micelle free energy is related to the variety of conformations of surfactant hydrocarbon chains in the restricted space of the micellar interior (Fig. 5b). The first statistical theory of this effect was proposed by Dill and Flory [203][204][205][206], who used lattice statistics. Ben-Shaul and coauthors [154,[157][158][159][160]163] extended the Dill-Flory theory and developed a mean field theory of amphiphile chain organization in micellar aggregates. An alternative approach was developed by Semenov [153] and applied to micellar aggregates by Nagarajan and Ruckenstein [136]. In our computations, we found that the best agreement between theory and experiment can be achieved if an expression for the chain-conformation free energy per molecule, f conf , which generalizes the Semenov [153] equations, is used. This expression reads (Appendix A): where c conf (p) is a dimensionless coefficient:  [203][204][205] suggested that a suitable value is l sg = 0.46 nm (for paraffin chains), which was used in our calculations. In Section 5 and Appendix A, we have derived Eqs.

Model of spherocylindrical micelles
The standard free energy of a spherocylindrical micelle (Fig. 1b)   where the values of s, a, R and p have to be substituted for the respective geometries (cylinder of spherical cap); for details -see Sections 6.1 and 6.2.
According to Eq. (4.24), E sc is proportional to the difference between two quantities of close magnitude, ∆f s and ∆f c . As seen in Table 1 As mentioned in Section 2, the experimental mean aggregation number n M , as well as the growth parameter E sc = lnK determined from the slopes of the experimental lines (Section 3), refer to the optimal micelle shape (denoted by the subscript 'opt' in the equations).
In view of Eq. (2.11), the optimal micelle shape should correspond to the minimal values of µ c (= f c ) and E sc . As described in Section 6.1, the condition for minimum of ∆f c , along with Eq. (4.25), can be used to determine the radius of the cylindrical part of the micelle, R c .
Likewise, in Section 6.2 it is demonstrated that the condition for minimum of E sc , along with Eqs. (4.24) and (4.25), can be used to determine the radius of the spherical end-caps, R s .

Formulation of the variational problem
Here, the considerations are general and applicable to micelles of spherical, cylindrical and lamellar shape, as well as for self-assemblies of other values of the packing parameter, p.
In his remarkable paper [153], by minimization of the free energy of the system of chains Semenov derived expressions describing the equilibrium chain-conformation in the hydrocarbon core of a micelle. In Ref. [153], only the formulation of the problem and the final results for sphere (p = 1/3), cylinder (p = 1/2) and lamella (p = 1) are given. The derivation of some of the basic equations and the solution of the minimization problem (which is rather non-trivial) was not published. We succeeded to reproduce Semenov's derivation, and further to generalize his result for any p values, including the range 1/3 ≤ p ≤ 3/8 corresponding to the micelle end-caps, and the range 1/2 < p < 1 corresponding to vesicles. whereas the inner end of the chain is located in the micelle interior, at r = r 0 .
The distribution of the chain free ends inside the micelle is characterized by the function G(r 0 ). By definition, G(r 0 )dr 0 gives the number of chains, whose ends are located in the interval (r 0 , r 0 +dr 0 ). The integration of G(r 0 ), which is equivalent to summation over all surfactant molecules in the micelle, yields: where N agg is the aggregation number of the considered micellar aggregate. As before, V is the Here, 3 sg l is the volume per segment; (dn) is the number of segments (from a given molecule) located in the considered layer of thickness dr and, finally, the integration with respect to r 0 is equivalent to summation over all surfactant molecules, whose chain-ends are located in the interval (r, R) and which contribute to the total number of segments contained in the elementary volume S(r)dr. It has been assumed that the segment density in the micellar core is uniform. In view of Eq. (5.2), we can represent Eq. (5.4) in the form [153]: where S(r) is the area of the surface r = const., which is placed at distance r from the respective interface. The standard definition of the packing parameter, p, is: where S(0) is the area of the surface of the micelle hydrocarbon core at r = 0; see Fig. 7a. For spherical, cylindrical and lamellar geometries, we have: where 0 ≤ r ≤ R; L c is the length of the cylinder, and S lam is the area of the lamella.
The unperturbed end-to-end distance of a chain containing dn segments is (dn) 1/2 l sg [206]. Inside the micelle, this chain could be extended, so that its ends lie at a distance dr from one another. This corresponds to a local conformation elastic free energy of the considered chain given by the Flory expression [136,153,206]:

Expression for the free energy per molecule
It is convenient to introduce dimensionless variables as follows: Then, the constraints expressed by Eqs. As an additional boundary condition, which is necessary to solve the problem, one can use [153]: which means that the free ends of the molecules are not extended.
The dimensionless form of the expression for the conformational energy, Eq. (5.11), reads: The conformational free energy per molecule is f conf = F conf /N agg = F conf v/V and the extended length of the surfactant tail is l = N sg l sg . Then, Eq. (5.16) acquires the form: The variational problem is related to the minimization of the functional in Eq. (5.17  results have been obtained also in Ref. [153]. We assume that Eq. (4.20) is applicable also for the spherical caps, for which 1/3 ≤ p ≤ 3/8. Fig. 7b shows that c conf is greater for the cylindrical part than for the spherical caps.
Contrariwise, R is greater for the spherical caps than for the cylindrical part. Insofar as f conf ∝ R 2 c conf , the chain-conformation component of micelle-growth energy, can be both positive and negative; see Fig. 8.
As seen in Fig. 7b (the dashed line), the latter empirical expression leads to smaller values of the chain-conformation free energy, which would essentially affect the theoretical predictions in view of the fine balance of the different components in E sc .
Further generalization of the Semenov's approach can give expressions for the chainconformation free energy in the case of mixed micelles from two and more surfactants with different chainlengths. Such generalization will be reported in a subsequent study.

Determination of the radius of micelle cylindrical part
The input parameters are the temperature, T, the number of carbon atoms in the surfactant alkyl chainlength, n C , and the length of a segment, l sg = 0.46 nm. The headgroup excluded area, a 0 , can be determined either by molecular size considerations, or it can be determined from the experimental value of E sc (see below). Next, the length of the extended surfactant chain, l, and its volume, v, are calculated by using Eqs. (4.7)-(4.11).
For cylindrical geometry, we have p = 1/2. At a given radius R = R c , the interfacial tension, s, is calculated from Eq. (4.15), along with Eqs. (4.16) and (4.17). The area per molecule, a, is calculated from Eq. (4.14) and the chain-conformation coefficient, c conf -from Eq. (4.20). Finally, ∆f c is computed from Eq. (4.25) as a function of R c . The optimal cylinder radius is identified with the value of R c that corresponds to the minimum of ∆f c in the interval It turned out that for all surfactants, for which data are given in Table 1, the function ∆f c (R c ) has a local minimum in the interval 1 < R c < l, from which the optimal (equilibrium) value of R c is determined (see Fig. 8a). In principle, there could be no local minimum, but instead, the minimal value of ∆f c could be at the boundary point R c = l (global minimum).
However, the latter case was not observed in our computations.

Determination of the spherical cap radius
The input parameters are the same as in the first paragraph of Section 6.1 and the value of the optimal cylinder radius, R c , has been determined as explained above.
At a given value of the cap radius, R s , we calculate the volume V sc ; the area A sc , and the packing parameter p from Eqs. For all surfactants, for which data are given in Table 1, it turned out that the function E sc (R s ) has a local minimum in the interval 1 < R s < l (see Fig. 8b), from which the optimal value of R s was determined. In principle, it could happen that the minimal value of E sc is at the boundary point, R s = l, however, this case was not observed in our computations.
The cross-sectional area of the ethylene-oxide headgroups, a 0 , is temperature dependent: a 0 decreases with the rise of temperature, T, because of enhanced segmentsegment attraction in the polyoxyethylene chains [188,190], which leads to headgroup compaction. There are no theoretical expressions for the a 0 (T) dependence. Because of that, using the system of equations described above, we determined a 0 from the experimental values of E sc in Table 1 for each temperature, T. For this goal, a 0 was varied until the value of E sc at the minimum of the E sc (R s ) dependence (Fig. 8b) coincides with the respective E sc value in Table 1. The value of R s at that minimum is identified with the optimal radius of the spherical end-caps.

Chemical potential vs. free energy per molecule
The system of equations described in Section 6.2 defines the number of molecules in the end-caps and their excess free energy as functions of R s , viz. n s (R s ) and E sc (R s ). The requirement for local minimum of E sc (Fig. 8b) We have taken into account that d(n s f s )/dn s = dg s /dn s = µ s , where µ s is the chemical potential of a molecule in the spherical end-caps. In addition, for the cylindrical part of the considered long spherocylindrical micelles we have g c = n c f c , where the free energy per molecule, f c , is independent of n c ; hence, µ c = dg c /dn c = f c . The fact that Eq. (6.2) yields µ s =µ c has an important physical meaning, viz. for the optimal micelle, which corresponds to the minimum of the E sc (R s ) dependence (Fig. 8b), the surfactant molecules in the spherical caps and in the micelle cylindrical part coexist in chemical equilibrium.
For the end-caps, g s is not simply proportional to n s , so that we have: If there is no local minimum of E sc , but instead, ∆f c takes its minimal value in the boundary point, R c = l, Eq. (6.1) does not hold. In such case, µ s ≠µ c and the micelle spherical caps are not in chemical equilibrium with the cylindrical part of the micelle. As already mentioned, this case was not observed when applying the model from Section 4 to the systems studied in the present article.

Numerical data for the free-energy components
The computational procedures described in Sections 6.1 and 6.2 have been applied to calculate the dependencies of the free energy per surfactant molecule in the cylindrical part, ∆f c vs. R c , and of the excess free energy of the spherical caps, E sc vs. R s . As an illustration, in Figures 8a and b we present results for three of nonionic surfactants, C 12 E 6 , C 14 E 6 , and C 16 E 8 .
For these surfactants (as well as for all other studied surfactants in Table 1), the optimal values of R c and R s , corresponding to the minima of the respective free energy curves, satisfy the inequality R c < R s < l. In Table 2, this is illustrated with data from Figures 8a and 8b. In view of Eq. and (4.6). Despite that, (E sc ) s decreases with the rise of R s (Fig. 8d) because the non-shielded area of hydrocarbon/water contact decreases with the decrease of surface curvature. In contrast, the steric-headgroup and chain-conformation contributions, (E sc ) hs and (E sc ) conf , are increasing functions of R s , and are negative for the smaller R s , for which (f hs ) s and (f conf ) s turn out to be smaller than (f hs ) c and (f conf ) c , respectively; see Eq. (6.4). The total excess free energy, E sc , turns out to be a result of a fine balance of components, whose range of variation is considerably greater than that of E sc (Fig. 8d). For the considered surfactant and temperature, the value of E sc at its minimum is determined from the balance of the positive (E sc ) s and the negative (E sc ) hs . The value of (E sc ) conf at the minimum of E sc turns out to be close to zero, but the increase of (E sc ) conf at greater R s blocks the growth of larger end-caps.

Temperature dependence of the headgroup cross-sectional area
As explained in the last paragraph of Section 6.2, for each nonionic surfactant at each temperature, T, the headgroup cross-sectional area, a 0 , was determined in such a way that the minimal value of E sc (Fig. 8b) coincides with the respective experimental value in Table 1. In   Fig. 9, the values of a 0 determined in this way are plotted vs. T. It is remarkable that the values of a 0 for surfactants with identical headgroups collapse on the same master curve, irrespective of their different chainlengths. This result confirms the correctness of our theoretical model, including its part related to the chain-confirmation free energy. Fig. 9. Plot of the geometric cross-sectional area per surfactant headgroup, a 0 , vs. temperature, T, for C i E j micelles; a 0 (T) is scaled with its value at 303 K (30 ºC). The points are determined from the experimental data for E sc (see the text). The lines are fits with Eq. (6.5).
The data for a 0 (T) in Fig. 9 can be fitted with a quadratic function: where a 0 (303) is the value of a 0 at 303 K (30 ºC); b 1 and b 2 are adjustable parameters. The obtained values of b 1 and b 2 are given in Table 3. In the case of C 10 E 6 , the last two experimental points (those at 55 and 60 ºC, which deviate from quadratic dependence) have not been taken into account when fitting the data. For the surfactants of longer polyoxyethylene chain, C i E 7 and C i E 8 , the data are fitted with straight line (b 2 ≡ 0). Table 3. Coefficients determined by fitting the data for a 0 (T) in Fig. 9 with Eq. (6.5).
Surfactant Using the temperature dependence of the headgroup cross-sectional area a 0 given by Eq. (6.5) with the parameter values in Table 3 and following the procedure described in Section 6.2, we calculated the theoretical dependence of E sc on T for each given surfactantsee the solid lines in Fig. 10. The points in Fig. 10 are the data from Table 1. As it could be expected, the experimental points and the calculated curves are in good agreement because of the fact that the curves are based on the fits in Fig. 9. Fig. 10

Cylinder radius and size of the end-caps
The model (the minimization illustrated in Figs Fig. 10 by means of the theoretical model from Section 4. The dashed lines are guides to the eye. As seen in Fig. 12, the calculated total aggregation number of the two spherical caps varies between n s = 54 (for C 10 E 6 at 60 ºC) and n s = 93 (for C 18 E 8 at 40 ºC). For the smaller chainlengths, C10 and C12, n s slightly decreases with the rise of temperature (Fig. 12a). For the larger chainlengths, C14, C16 and C18, the dependence n s (T) has a maximum at a certain temperature (Fig. 12b). In general, for nonionic surfactant micelles the equilibrium values of R c , R s and n s are result of a fine balance of the three free-energy components related (i) to the interfacial tension; (ii) to the steric headgroup repulsion, and (iii) to the chain conformations, as illustrated in Fig. 8c and d.
The temperature dependence of the geometric cross-sectional area per surfactant headgroup a 0 (T), see Eq. (6.5), has been determined from fits of experimental data by Einaga et al. [123][124][125][126][127][128][129][130][131]. In Fig. 13, our theoretical model is tested against an independent set of experimental data obtained by Talmon et al. [79]. In Fig. 13, the symbols are data for the mean mass aggregation number, n M , of C 12 E 5 micelles obtained from graphical data for the average micelle length L in Ref. [79] using Eq. (9) in the same paper along with parameter values given therein. In Fig. 13, the continuous lines are calculated by means of our theory for the respective C 12 E 5 concentrations with a 0 (T) from Eq. (6.5) and without using any adjustable parameters. The obtained excellent agreement between theory and experiment indicates that our theoretical model successfully passes this test. Fig. 13. Plots of the experimental micelle mean mass aggregation number, n M , vs. temperature, T, for four different concentrations of the nonionic surfactant C 12 E 5 denoted in the figure. The symbols are experimental data from Ref. [79], whereas the solid lines are predicted by our theoretical model for the respective C 12 E 5 concentrations without using any adjustable parameters.

Summary and conclusions
This article presents the first analytical molecular-thermodynamic model, which is in is defined in the triangular domain 0 ≤ x ≤ 1 and 0 ≤ y ≤ x. Integrating Eq.
(5.14) with respect to y from 0 to 1 and changing the order of integration we obtain: From equations (5.13) and (A.1), we obtain: In view of Eq. ( where µ E (x) and µ G (y) are weight functions, which represent Lagrange multipliers related to the constraints in Eqs. (5.13) and (5.14), respectively. The first variation of the extended functional, Eq. (A.3), reads: The necessary condition for stationary point in the functional space, δΦ = 0 with respect to the variations δ ( ) G x  , δ ( , ) E x y  , δµ E (y) and δµ G (x), leads to equations for determining the four unknown functions, as follows.
(ii) Setting zero the terms containing the independent variation ) ( δ x G , we obtain: (iii) Setting zero the terms containing the independent variation δ ( , ) E x y  , we obtain: Changing the order of integration in the last term, we obtain:  This result for ( , ) E x y  was published by Semenov [153] without proof.
Furthermore, we substitute ( , ) E x y  from Eq. (A.14) into Eq. (5.13) and obtain the following integral equation: The following notations have been used: z = µ G (y); b = µ G (x), and a = µ G (0 Thus, the solution for the weight function µ G (y) is: