Extension of the ladder model of self-assembly from cylindrical to disclike surfactant micelles Peter A. Kralchevsky a, ⁎, Krassimir D. Danov a , Svetoslav E. Anachkov a , Gergana S. Georgieva a , Kavssery P. Ananthapadmanabhan b a Department of Chemical Engineering, Faculty of Chemistry and Pharmacy, Sofia University, Sofia 1164, Bulgaria b Unilever Research & Development, 40 Merritt Blvd., Trumbull, CT 06611, USA abstract article info Article history: Received 23 August 2013 Received in revised form 7 October 2013 Accepted 2 November 2013 Available online 7 November 2013 Keywords: Disclike micelles Nanodiscs Cylindrical micelles Self-assembly Radius of gyration Hydrodynamic radius The ladder model of growth of cylindrical micelles gives expressions for the micellar size distribution and for the mean aggregation number, which are in good agreement with the experiment. Here, we consider this model and its extension to the case of disclike micelles. In analogy with the modeling of elongated micelles as sphero- cylinders, the disclike micelles can be modeled as toro-discs. Upon micelle growth, the hemispherical caps of a cylindrical aggregate remain unchanged, whereas the semitoroidal periphery of a disclike micelle expands. This effect can be taken into account in the expression for the size distribution of the disclike micelles, which pre- dicts the dependence of the micelle mean aggregation number on the surfactant concentration. It turns out that disclike micelles could form in a limited range of surfactant concentrations, and that their mean aggregation number cannot exceed a certain maximal value. Large disclike micelles can exist only near the border with the domain of cylindrical micelles. Then, small variations in the experimental conditions could induce a transforma- tion of the disclike micelles into cylindrical ones. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction The disclike surfactant micelles can be considered as predecessors of the lamellar phase in the same way as the cylindrical micelles are prede- cessors of the formation of hexagonal phase. Then, a question arises: Why do the disclike micelles represent a rare form of self-assembly [1,2], despite the fact that lamellar phases are often observed? In the present article, we will try to answer this question on the basis of a recently developed model of the growth of disclike micelles [3], which upgrades the ladder model for cylindrical micelles [4]. Although disclike micelles are not so frequently observed, there is a considerable amount of accumulated experimental material from the investigations of such self-assemblies, termed also nanodiscs or bicelles. Single component disclike micelles have been detected in solutions of anionic [5]; nonionic [6] and fluorinated surfactants [7–10]. Nanodiscs have been observed and investigated in various binary mixtures of cat- ionic and anionic (catanionic) surfactant solutions [1,11–15]. Discoidal micelles and nematic phase from such micelles have been detected in ternary mixtures of lauric acid with anionic and zwitterionic surfactants [16]. Disc-shaped aggregates are formed also in solutions of diblock and triblock copolymers [17–22]. Such aggregates are formed also by phos- pholipids dispersed in water [23,24] and in aqueous surfactant/lipid systems [25]. The self-assembly of discoidal micelles has been found to be a transitional kinetic stage in the processes of formation and decomposition of liposomes [26,27]. Disc-shaped aggregates have been discovered also in solutions of bile salts [28–30] and their mixtures with phospholipids [31]. Shape polydispersity and shape fluctuations in ionic surfactant mi- celles have been analyzed and transitions from spherical micelles to prolate and oblate spheroids have been predicted in the frame of a the- oretical model [32] as well as by computer simulations [33]. Branching instabilities in growing cylindrical and disclike micelles have been also investigated [34]. The formation of such micelles and their transforma- tion into liquid crystalline phases was theoretically described in terms of the Helfrich's curvature moduli [35,36] and lattice Hamiltonian models [37]. The phase transitions between isotropic and columnar phases (for rodlike micelles), as well as between isotropic and lamellar phases (for disclike micelles) have been theoretically studied [38]. It was established that the size of the cylindrical aggregates increases continuously with concentration, while the size of the discs could jump from small to infinite [37,39]. For cylindrical micelles, there are molecular–thermodynamic models, coupled with geometrical- constraint considerations, which quantitatively predict the micelle growth with the rise of surfactant concentration [4,40,41]. A molecu- lar–thermodynamic model of disclike micelles was recently developed [3], which quantitatively describes the variation in the micelle size with the increase of surfactant concentration in agreement with the experiment. To answer the question formulated in the beginning, here we first compare expressions for the mean aggregation number and area per surfactant-molecule headgroup for different micellar geometries: Current Opinion in Colloid & Interface Science 18 (2013) 524–531 ⁎ Corresponding author. Fax: +359 2 9625643. E-mail address: pk@lcpe.uni-sofia.bg (P.A. Kralchevsky). 1359-0294/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cocis.2013.11.002 Contents lists available at ScienceDirect Current Opinion in Colloid & Interface Science journal homepage: www.elsevier.com/locate/cocis