Gompper

An important example is the one-order-parameter model invented by Gompper and Schick [77], which describes a ternary mixture in temis of the density difference between water and oil ... 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

Gompper G and Sohiok M 1994 Self-assembling amphiphilio systems Phase Transitions and Critical Phenomena vol 16, ed C Domb and J Lebowitz (New York Aoademio)... 

Gompper G and Sohiok M 1990 Correlation between struotural and interfaoial properties in amphiphilio systems Phys. Rev. Lett. 65 1116... 

Gompper G and Zsohooke S 1991 Elastio properties of interfaoes in a Ginzburg-Landau theory of swollen mioelles, droplet orystals and lamellar phases Euro. Phys. Lett. 16 731... 

Gompper G and Kroll D M 1995 Phase diagram and sealing behavior of fluid vesieles Phys. Rev. E 51 514... 

Theissen O, Gompper G and Kroll D M 1998 Lattice-Boltzmann model of amphiphilic systems Euro. Phys. Lett. 42 419... 

Cyclobutadiene itself is not stable at room temperature. Several derivatives with stabilizing groups have been prepared by the acid-catalyzed dimerization of alkjmes (R. Gompper, 1975). Less substituted cyclobutadienes could be obtained by photolytic reactions in solid matrix at low temperatures (G. Maier, 1973, 1974). 

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

A = 2k/ /3 for the case of cyhnders. In order to avoid this problem, Gompper and Kroll [241] have recently argued that a more appropriate discretization of the bending free energy should be based directly on the square of the local mean curvature ... 

As yet, models for fluid membranes have mostly been used to investigate the conformations and shapes of single, isolated membranes, or vesicles [237,239-244], In vesicles, a pressure increment p between the vesicle s interior and exterior is often introduced as an additional relevant variable. An impressive variety of different shapes has been found, including branched polymer-like conformations, inflated vesicles, dumbbell-shaped vesicles, and even stomatocytes. Fig. 15 shows some typical configuration snapshots, and Fig. 16 the phase diagram for vesicles of size N = 247, as calculated by Gompper and Kroll [243]. 

FIG. 15 Conformations of fluid vesicles for different values of the bending rigidity and pressure increment (a) branched polymer (b) inflated vesicle (c) prolate vesicle (d) stomatocyte. (From Gompper and Kroll [243]. Copyright 1995 APS.)... 

FIG. 16 Phase diagram of fluid vesicles as a function of pressure increment p and bending rigidity A. Solid lines denote first-order transitions, dotted lines compressibility maxima. The transition between the prolate vesicles and the stomatocytes shows strong hysteresis efifects, as indicated by the error bars. Dashed line (squares) indicates a transition from metastable prolate to metastable disk-shaped vesicles. (From Gompper and KroU 1995 [243]. Copyright 1995 APS.)... 

G. Gompper, M. Schick. In C. Domb, J. L. Lebowitz, eds. Phase Transitions and Critical Phenomena, Vol. 16, Self-assembling Amphiphilic Systems. London Academic Press, 1994. 

For a recent review on Ginzburg-Landau theories, see G. Gompper. Ber. Bunsenges Phys Chemie i00 264-271, 1996. 

G. Gompper, D. M. Kroll. Curr Opin Coll Interf Sci 2 373-381, 1997 G. Gompper, D. M. Kroll. J Phys Cond Matt (5 8795-8834, 1997. 

The second difference is related to the structure of the lamellar phase. The Euler characteristic has been assumed zero in the whole lamellar phase by Gompper and Kraus [47], whereas we show that it fluctuates strongly in the lamellar phase between the transition line and the topological disorder fine. The notion of the topological disorder line has not appeared in their paper. We think that the topological disorder line is much closer to the transition... 

In ternary mixtures of oil, water, and surfactant the ordering properties of the system follow from the vectorial character of the interactions of the surfactant molecules with both the oil and the water molecules. The typical size of the ordered domains, much larger than the molecular size, justifies the application of the mesoscopic Landau-Ginzburg approach to the ordering. In the simplest approach of Gompper and Schick [3,12], which we call here the basic Landau-Ginzburg model, the orientational degrees of free-... 

We would like to thank Gerhard Gompper, Johann Hoye, Andrzej Poniewierski, Michael Schick, and George Stell for fruitful collaboration which led to the results presented in this chapter. This work was partially supported by the Komitet Badah Naukowych under grants 2P03B12516 and 3T09A07316 and by the Maria Sklodowska Curie Joint Fund II. 

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