PHYSICAL REVIEW A VOLUME 30, NUMBER 5 NOVEMBER 1984 Liquid-expanded — liquid-condensed phase transition in amphiphilic monolayers: A renormalization-group approach to chiral-symmetry breaking of hydrocarbon-chain defects J. -P. Legre, G. Albinet, and J. -L. Firpo Departement de Physique des Liquides, Uniuersite de Provence, place Victor Hugo, F-13331 Marseille Cedex 3, France A. -M. S. Tremblay Departement de Physique et Centre de Recherche en Physique du Solide, &Jnil)ersite de Sherbrooke, SherbrooI. e, Quebec, Canada JIK2RI (Received 16 April 1984j This paper is concerned with the liquid-expanded (LE) — liquid-condensed (LC) transition in monolayers of amphiphilic molecules at the air-water interface. A model, which can be mapped into the Blume-Emery-Griffiths Hamiltonian, has been considered before within the (mean-field) Bragg-Williams approximation and it gave results which could be successfully compared with exper- iment. The LE-LC transition has been associated with a chiral-symmetry breaking of the hydrocarbon-chain defects. This model is treated here with a Migdal-Kadanoff approximate position-space renormalization group. Renormalization-group flows are consistent with those ob- tained by previous authors. The conrlection between experin&ental and Hamiltonian parameters is easiest for a particular choice of ensemble, which turns out to be rather subtle for this problem. As in the work of Lavis, Southern, and Bell, isotherms in the surface-pressure — molecular-area plane do not show a signature of the LE-LC transition. The better agreement between experiments (showing a compressibility jump at the LE-LC transition) and mean-field theory suggests that in these cases long-range forces depending on the nature of the polar head and on the water substrate pH are re- sponsible for the jump. I. INTRODUCTION Monolayers of simple amphiphilic molecules (e.g. , fatty acids or alcohols) at the air-water interface, exhibit a variety of phase transitions. One of these, at relatively high surface density, is the so-called liquid-expanded (LE) — liquid-condensed (LC) transition. A similar transition also occurs in more complex systems, such as those con- taining molecules with two hydrophobic chains (phospho- lipids and lecithins), two polar heads (hydroxyhexade- canoic acids, abbreviated HHA in the literature), or discotics (BH-n). In this paper we present a position- space renormalizatio~-group study (PSRG) of the LE-LC transition in the simplest amphiphilic monolayers. In contrast with the gas-liquid transition in monolayers whose experimental study is extremely difficult, the LE- LC transition offers the advantage of being relatively easy to observe. Since, moreover, the nature of the LE-LC transition is not yet completely clear (in particular the or- der of the transition is still a subject of controversy), this type of system has been subjected to a large number of theoretical as well as experimental investigations. ' Experiments have shown that the LE-LC transition is not an artifact caused by a small spreading pressure and a crossover to a three-dimensional state. However, a piece of perfectly horizontal isotherm in the surface-pressure (H) — molecular-area (0. ) diagram, which is the unmistak- able signature of a first-order transition, has never been observed despite very careful experimentation where the return to equilibrium of the system was monitored at each point through relaxation-time measurements. - Even the recent experiments of Von Tscharner and McConnel and of Losche, Sackmann, and Mohwald have not allowed clear conclusions to emerge. Indeed their very clever mi- croscopic observation technique of phopholipid mono- layers, which is based on fjuorescent probes, has led these groups to contradictory results. The first one does not find anything which can support the hypothesis of a first-order LE-LC transition, while the second group has found evidence for domains, a characteristic of phase coexistence. Finally, the results of Bois et al. , who have systemati- cally studied surface-pressure relaxation times, are quite clearly in favor of a continuous (so-called "second-order" ) transition: indeed, these times, measured along H-cr iso- therms, become very important near the transition, as one would expect when there is critical slowing down. More- over, Bois has never obtained a horizontal isotherm seg- ment, even when sufficient time was allowed for return to equilibrium. Note also that the experiment was per- formed below the spreading pressure and that there was no leak of the surfactant. The model whose predictions most closely resemble the observations is in our opinion that recently suggested by Firpo, Legre, Bois, and Baret iFI.BB). In the present pa- per we use a more sophisticated method to study their model and we extend their conclusions. Let us then dis- cuss the FLBB model in a bit more detail. Firpo eI. al. use the fact that, close to the LE-LC transitions, kinks in the hydrophobic chains are by far the dominant type of 2720 ~c 1984 The American Physical Society