Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T13:56:18.358Z Has data issue: false hasContentIssue false
  • English
  • Français

A criterion for symmetric tricritical points in condensed ordered phases

Published online by Cambridge University Press:  05 January 2011

F. BISI ,
E. C. GARTLAND JR.  and
E. G. VIRGA
F. BISI
Affiliation:
Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy email: eg.virga@unipv.it Laboratory of Applied Mathematics, Fondazione Università di Mantova, Via Scarsellini 2, 46100 Mantova, Italy email: fulvio.bisi@unipv.it
E. C. GARTLAND JR.
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA email: gartland@math.kent.edu
E. G. VIRGA
Affiliation:
Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy email: eg.virga@unipv.it
Get access Rights & Permissions [Opens in a new window]

Abstract

Basic methods from bifurcation theory are applied to derive a criterion that predicts when a symmetric tricritical point may occur in a transition between condensed ordered phases described by any finite number of scalar order parameters. At such a point, a change of order takes place in the phase transition, which passes from first to second order, or vice versa.

Keywords

Phase transitionsCritical points of functions and mappingsBifurcation problemsBifurcations and instabilityComputational methods for bifurcation problems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 1: Liquid Crystals , February 2012 , pp. 3 - 28
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bisi, F., Virga, E. G., Gartland, E. C. Jr., De Matteis, G., Sonnet, A. M. & Durand, G. E. (2006) Universal mean-field phase diagram for biaxial nematics obtained from a minimax principle. Phys. Rev. E 73, 051709.CrossRefGoogle ScholarPubMed
[2]Chow, S.-N. & Hale, J. K. (1982) Methods of Bifurcation Theory, Springer-Verlag, New York.CrossRefGoogle Scholar
[3]De Matteis, G., Bisi, F. & Virga, E. G. (2007) Constrained stability for biaxial nematic phases. Contin. Mech. Thermodyn. 19, 1–23.CrossRefGoogle Scholar
[4]De Matteis, G. & Virga, E. G. (2005) Tricritical points in biaxial liquid crystal phases. Phys. Rev. E 71, 061703.CrossRefGoogle ScholarPubMed
[5]Gartland, E. C. Jr., & Virga, E. G. (2010) Minimum principle for indefinite mean-field free energies. Arch. Ration. Mech. Anal. 196, 143–189.CrossRefGoogle Scholar
[6]Gibelli, L. & Turzi, S. (2009) A catastrophe-theoretic approach to tricritical points with applications to liquid crystals. SIAM J. Appl. Math. 70, 63–76.CrossRefGoogle Scholar
[7]Golubitsky, M. & Schaeffer, D. G. (1985) Singularities and Groups in Bifurcation Theory, Vol. I, Springer-Verlag, Berlin, Germany.CrossRefGoogle Scholar
[8]Golubitsky, M., Stewart, I. & Schaeffer, D. G. (1988) Singularities and Groups in Bifurcation Theory, Vol. II, Springer-Verlag, Berlin, Germany.CrossRefGoogle Scholar
[9]Govaerts, W. J. F. (2000) Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, PA.CrossRefGoogle Scholar
[10]Griffiths, R. B. (1973) Proposal for notation at tricritical points. Phys. Rev. B 7, 545–551.CrossRefGoogle Scholar
[11]Griffiths, R. B. (1974) Thermodynamic model for tricritical points in ternary and quaternary fluid mixtures. J. Chem. Phys. 60, 195–206.CrossRefGoogle Scholar
[12]Griffiths, R. B. & Widom, B. (1973) Multicomponent-fluid tricritical points. Phys. Rev. A 8, 2173–2175.CrossRefGoogle Scholar
[13]Healey, T. J. (1988) A group-theoretic approach to computational bifurcation problems with symmetry. Comput. Methods Appl. Mech. Eng. 67, 257–295.CrossRefGoogle Scholar
[14]Keller, H. B. & Langford, W. F. (1972) Iterations, perturbations and multiplicities for nonlinear bifurcation problems. Arch. Ration. Mech. Anal. 48, 83–108.CrossRefGoogle Scholar
[15]Kuznetsov, Y. A. (2004) Elements of Applied Bifurcation Theory, 3rd ed., Springer, New York.CrossRefGoogle Scholar
[16]Laurie, I. D. & Sarbach, S. (1984) Theory of tricritical points. In: Domb, C. & Lebowitz, J. L. (editors), Phase Transitions and Critical Phenomena, Vol. 9, Academic Press, New York.Google Scholar
[17]Longa, L. (1986) On the tricritical point of the nematic-smectic A phase transition in liquid crystals. J. Chem. Phys. 85, 2974–2985.CrossRefGoogle Scholar
[18]Longa, L. (1989) Order-parameter theories of phase diagrams for antiferroelectric smectic-A phases: Role of orientational degrees of freedom. Liq. Cryst. 5, 443–461.CrossRefGoogle Scholar
[19]Longa, L., Grzybowski, P., Romano, S. & Virga, E. G. (2005) Minimal coupling model of the biaxial nematic phase. Phys. Rev. E 71, 051714.CrossRefGoogle ScholarPubMed
[20]Luckhurst, G. R. (2004) Liquid crystals – a missing phase found at last? Nature (London) 430, 413–414.CrossRefGoogle Scholar
[21]Maier, W. & Saupe, A. (2004) A simple molecular theory of the nematic liquid-crystalline state. In: Sluckin, T. J., Dunmur, D. A. & Stegemeyer, H. (editors), Crystals That Flow: Classic Papers from the History of Liquid Crystals, Taylor & Francis, London, pp. 380–387 (1958 trans.).CrossRefGoogle Scholar
[22]Maier, W. & Saupe, A. (1958) Eine einfache molekulare. Theorie des nematischen kristallinflüssigen Zustandes. Z. Nat. Forsch. 13a, 564–566.Google Scholar
[23]Merkel, K., Kocot, A., Vij, J. K., Korlacki, R., Mehl, G. H. & Meyer, T. (2004) Thermotropic biaxial nematic phase in liquid crystalline organo-siloxane tetrapodes. Phys. Rev. Lett. 93, 237801.CrossRefGoogle ScholarPubMed
[24]McMillan, W. L. (1971) Simple molecular model for the smectic A phase of liquid crystals. Phys. Rev. A 4, 1238–1246.CrossRefGoogle Scholar
[25]Sonnet, A. M., Virga, E. G. & Durand, G. E. (2003) Dielectric shape dispersion and biaxial transitions in nematic liquid crystals. Phys. Rev. E 67, 061701.CrossRefGoogle ScholarPubMed
[26]Straley, J. P. (1974) Ordered phases of a liquid of biaxial particles. Phys. Rev. A 10, 1881–1887.CrossRefGoogle Scholar
[27]Triantafyllidis, N. & Peek, R. (1992) On stability and the worst imperfection shape in solids with nearly simultaneous eigenmodes. Int. J. Solids Struct. 29, 2281–2299.CrossRefGoogle Scholar
[28]Vainberg, M. M. & Trenogin, V. A. (1962) The methods of Lyapunov and Schmidt in the theory of non-linear equations and their further development. Russ. Math. Survs. 17, 1–60.CrossRefGoogle Scholar
[29]Vainberg, M. M. & Trenogin, V. A. (1974) Theory of Branching of Solutions of Non-Linear Equations, Noordhoff, Leyden, the Netherlands.Google Scholar
[30]Widom, B. (1996) Theory of phase equilibrium. J. Phys. Chem. 100, 13190–13199.CrossRefGoogle Scholar
[31]Yeomans, J. M. (1992) Statistical Mechanics of Phase Transitions, Oxford University Press, Oxford, UK.CrossRefGoogle Scholar
[32]Yu, L. J. & Saupe, A. (1980) Observation of a biaxial nematic phase in Potassium Laurate-1-Decanol-water mixtures. Phys. Rev. Lett. 45, 1000–1003.CrossRefGoogle Scholar