Roughening transition and formation of bicontinuous structures of immiscible solvents embedded in surfactant diblock copolymers

2014 The roughening transition of the interface in a model mixture of amphiphilic diblock copolymers, two immiscible solvents and a monomeric cosurfactant is examined. The equivalence of this model with the solid-on-solid Ising magnet implies the existence of a critical cosurfactant composition XRAB for the transformation of a lamellar microemulsion phase into a bicontinuous disordered structure. At a higher composition 1 XRAB another transition of the same nature produces again an ordered phase. J. Physique Lett. 46 (1985) L-967 L-972 15 OCTOBRE 1985, Classification Physics Abstracts 64.60 68.10 82.70 In this Letter we address the problem of the formation of bicontinuous microemulsion phases where the regions of each immiscible solvent are connected over macroscopic distances in multicomponent fluid mixtures where amphiphilic species are present. We show that the Kosterlitz-Thouless (KT) [1, 2] theory of phase transitions in two-dimensional systems provides a mechanism, at least within a particular model system where one amphiphile is polymeric, of how a disordered bicontinuous phase can be produced from phases with multilamellar arrangements. Our contention here is that the nature of a bicontinuous microemulsion phase is that of a set of delocalized (or rough) [3] interfaces, so that a multisurface arrangement, obtained when the surfactant concentration is many times that needed to saturate one interface, is necessarily (*) J. S. Guggenheim fellow. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019850046020096700 L-968 JOURNAL DE PHYSIQUE LETTRES disordered; i.e. the surfactant(s) film has a random nonperiodic structure throughout the sample volume. The roughening transition associated with the disappearance of a sharply defined orientation of an interface constitutes~ a mechanism by means of which ordered phases of immiscible solvents separated by localized interfaces transform into disordered structures. The appearance of disorder at a critical cosurfactant composition XAB is signalled by the vanishing of the free-energy cost associated to provide curvature to a planar interface. We find, for XAB > XAB, that the positional correlation function between two pieces of the same wrinkled (or rough) interface, separated by a distance r, decays algebraically like the spin correlations in the low temperature phase of the X Y magnet. As a consequence, a positional persistence length ÇR’ analogous to that introduced by de Gennes and Taupin [4] for orientations of the surfactant film, acquires a well defined finite value. Thus, the rough interface is approximately flat only at scales smaller than ~R and consecutive « pieces » of area ~R have independent positions. We find, as in reference [4], that ~R depends exponentially on the parameter that measures interfacial rigidity. Our model mixture is closely related to that discussed by Cantor [5] in his study of the interfacial properties of non-ionic diblock copolymer amphiphiles. These kind of surfactants consist of two different polymeric chains joined together, each of which strongly prefers one solvent. A water/oil interface saturated with such polymer chains has the hydrophilic blocks embedded, or swollen, by water and, similarly, the hydrophobic portions by oil. Cantor considered the formation of an ordered « middle » phase in a three component system. The particular arrangement studied was a multilamellar phase between two solvent phases composed of alternating layers of the two solvents stabilized at their interfaces by the chains. Here we analyse the possibility of transforming, isothermally, such ordered structure into a disordered bicontinuous one through the addition of a fourth component, a monomeric cosurfactant that increases the flexibility [4] of the layers embedded with solvent. We assume that the cosurfactant species is always preferentially located at solvent-solvent interfacial regions. But also, our model mixture is a variant of the Wheeler-Widom (WW) lattice mixture of bifunctional molecules [6]. The WW model abstracts some of the essential features of a ternary system composed of water, oil and surfactant. The interactions amongst molecules are simplified to a maximum that only retains (through infinite repulsions or no attractions) the basic character of each constituent. In its simplest version [6] the model is isomorphous to the Ising model in the same lattice, and displays as corresponding states both ferromagnetic and antiferromagnetic behaviour. The binodal curve and plait point of the mixture are images of the spontaneous magnetization and Curie point of the ferromagnet, respectively. At high concentrations of surfactant antiferromagnetic-like ordering appears. The model is composed of bifunctional molecules AA ( ), BB ( ) and AB ( ), and it is constructed by : (i) confining them to the bonds of a regular (in our case, three-dimensional) lattice; (ii) letting only A atoms or only B atoms meet at any lattice site (A ends of one molecule and B ends of another exclude each other); and (iii) filling every bond of the lattice with one, and only one, molecule. Since for all allowed configurations every site can be identified as either an A or a B site, the equivalence with the Ising model follows. The complete transcription to fluid mixture language, the model’s phase diagram and the mean-field description of the interface between coexisting phases is given in reference [6]. There, the consideration of micellar structures in different regions of the phase diagram, the behaviour of the liquid-liquid interfacial tension and other matters pertaining to the WW model are discussed. Here we shall focus attention on other issues. Because every bond in the lattice is filled by one molecule the chemical potentials ~AA, ~BB and ~cAB of the three species are all infinite, but the diffidence of any pair is in general finite. In particular, when the mixture and the magnet are set at the same temperature, the chemical potential differences are related to the Ising coupling J by