One order parameter tensor mean field theory for biaxial liquid crystals

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March 2011, 15(2): 475-490. doi: 10.3934/dcdsb.2011.15.475

One order parameter tensor mean field theory for biaxial liquid crystals

Xiaoyu Zheng ^1, and Peter Palffy-Muhoray ^2,

1.	Department of Mathematical Sciences, Kent State University, Kent, OH 44242, United States
2.	Liquid Crystal Institute, Kent State University, Kent, Oh 44242, United States

Received November 2009 Revised February 2010 Published December 2010

Abstract
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In this paper, we present a simple one tensor mean field model of biaxial nematic liquid crystals. The salient feature of our approach is that material parameters appear explicitly in the order parameter tensor. We construct the free energy from a mean field potential based on anisotropic dispersion interactions, identify the order parameter tensor and its elements, and obtain self-consistent equations, which are then solved numerically. The results are illustrated in a 3D ternary phase diagram. The phase behavior can be simply related to molecular parameters. The results may be useful for designing molecules that show a thermotropic biaxial phase.

Keywords: biaxial liquid crystals, Mean field, Landau expansion..

Mathematics Subject Classification: Primary: 82B2.

Citation: Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475

References:

[1]	B. R. Acharya, A. Primak and S. Kumar, Biaxial nematic phase in bent-core thermotropic mesogens,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.145506.
[2]	R. Alben, Phase transitions in a fluid of biaxial particles,, Phys. Rev. Lett., 30 (1973), 778. doi: 10.1103/PhysRevLett.30.778.
[3]	D. W. Allender and M. A. Lee, Landau theory of biaxial nematic liquid crystals,, Mol. Cryst. Liq. Cryst., 110 (1984), 331. doi: 10.1080/00268948408074514.
[4]	D. Allender and L. Longa, Landau-de Gennes theory for biaxial nematics reexamined,, Phy. Rev. E, 78 (2008). doi: 10.1103/PhysRevE.78.011704.
[5]	J. Ball, , Private Communication., ().
[6]	M. A. Bates, Influence of flexibility on the biaxial nematic phase of bent core liquid crystals: A Monte Carlo simulation study,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.061702.
[7]	R. Berardi and C. Zannoni, Do thermotropic biaxial nematics exist? A Monte Carlo study of biaxial Gay-Berne particles,, J. Chem. Phys., 113 (2000), 5971. doi: 10.1063/1.1290474.
[8]	B. Bergersen, P. Palffy-Muhoray and D. A. Dunmur, Uniaxial nematic phase in fluids of biaxial particles,, Liq. Cryst., 3 (1988), 347. doi: 10.1080/02678298808086380.
[9]	P. Biscari, M. C. Calderer and E. M. Terentjev, Landau-de Gennes theory of isotropic-nematic-smectic liquid crystal transitions,, Phys. Rev. E, 75 (2007). doi: 10.1103/PhysRevE.75.051707.
[10]	F. Biscarini, C. Chiccoli, P. Pasini, F. Semeria and C. Zannoni, Phase diagram and orientational order in a biaxial lattice model: A Monte Carlo study,, Phy. Rev. Lett., 75 (1995), 1803. doi: 10.1103/PhysRevLett.75.1803.
[11]	F. Bisi, E. G. Virga, E. C. Garland, Jr., G. De Matteis, A. M. Sonnet and G. E. Durand, Universal mean-field phase diagram for biaxial nematics obtained from a minmax priniciple,, Phys. Rev. E, 73 (2006). doi: 10.1103/PhysRevE.73.051709.
[12]	C. G. Broyden, A class of methods for solving nonlinear simultaneous equations,, Math. Comput., 19 (1965), 577. doi: 10.1090/S0025-5718-1965-0198670-6.
[13]	P. J. Camp and M. P. Allen, Phase diagram of the hard biaxial ellipsoid fluid,, J. Chem. Phys., 106 (1997), 6681. doi: 10.1063/1.473665.
[14]	C. Chiccoli, P. Pasini, F. Semeria and C. Zannoni, A detailed Monte Carlo investigation of the tricritical region of a biaxial liquid crystal system,, Int. J. Mod. Phys. C, 10 (1999), 469. doi: 10.1142/S0129183199000358.
[15]	P. G. de Gennes, "The Physics of Liquid Crystals,", Clarendon Press, (1974).
[16]	G. De Matties, S. Romano and E. G. Virga, Bifurcation analysis and computer simulation of biaxial liquid crystals,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.041706.
[17]	G. De Matties, A. M. Sonnet and E. G. Virga, Landau theory for biaxial nematic liquid crystals with two order parameter tensors,, Continuum Mech. Thermodyn., 20 (2008), 347. doi: 10.1007/s00161-008-0086-9.
[18]	R. Ennis, "Pattern Formation in Liquid Crystals: The Saffman-Taylor Instability and the Dynamics of Phase Separation,", Ph.D. dissetation, (2004).
[19]	M. J. Freiser, Ordered states of a nematic liquid,, Phys. Rev. Lett., 24 (1970), 1041. doi: 10.1103/PhysRevLett.24.1041.
[20]	M. J. Freiser, Successive transitions in a nematic liquid,, Mol. Cryst. Liq. Cryst., 14 (1971), 165.
[21]	W. M. Gelbart and B. A. Baron, Generalized van der Waals theory of the isotropic-nematic phase transition,, J. Chem. Phys., 66 (1977), 207. doi: 10.1063/1.433665.
[22]	J. Israelachvili, "Intermolecular & Surface Forces,", Academic Press, (1992).
[23]	L. Longa and G. Pajak, Luckhurst-Romano model of thermotropic biaxial nematic phase,, Liq. Cryst., 32 (2005), 1409. doi: 10.1080/02678290500167873.
[24]	L. Longa, P. Grzybowski, S. Romano and E. Virga, Minimal coupling model of the biaxial nematic phase,, Phys. Rev. E, 71 (2005). doi: 10.1103/PhysRevE.71.051714.
[25]	G. R. Luckhurst and S. Romano, Computer simulation studies of anisotropic systems II. Uniaxial and biaxial nematics formed by non-cylindrically symmetric molecules,, Mol. Phys., 40 (1980), 129. doi: 10.1080/00268978000101341.
[26]	L. A. Madsen, T. J. Dingemans, M. Nakata and E. T. Samulski, Thermotropic biaxial nematic liquid crystals,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.145505.
[27]	W. Maier and A. Saupe, A simple molecular theory of the namatic liquid-crystalline state,, Zeischrift Naturforschung, 13 (1958), 564.
[28]	K. Merkel, A. Kocot, J. K. Vij, R. Korlacki, G. H. Mehl and T. Meyer, Thermotropic biaxial nematic phase in liquid crystalline organo-siloxane tetrapodes,, Phys. Rev. Lett., 93 (2004). doi: 10.1103/PhysRevLett.93.237801.
[29]	P. K. Mukherjee, Improved analysis of the Landau theory of the uniaxail-biaxial nematic phase transition,, Liq. Cryst., 24 (1998), 519. doi: 10.1080/026782998206966.
[30]	B. M. Mulder, Solution of the excluded volume problem for biaxial particles,, Liq. Cryst., 1 (1986), 539. doi: 10.1080/02678298608086278.
[31]	P. Palffy-Muhoray, The single particle potential in mean-field theory,, Am. J. Phys., 70 (2002), 433. doi: 10.1119/1.1446860.
[32]	P. Palffy-Muhoray and B. Bergersen, van der Waals theory for nematic liquid crystals,, Phys. Rev. A, 35 (1987). doi: 10.1103/PhysRevA.35.2704.
[33]	S. Sarman, Molecular dynamics of biaxial nematic liquid crystals,, J. Chem. Phys., 104 (1996), 342. doi: 10.1063/1.470833.
[34]	C. S. Shih and R. Alben, Lattice model for biaxial liquid crystals,, J. Chem. Phys., 57 (1972), 3055. doi: 10.1063/1.1678719.
[35]	S. Sircar and Q. Wang, Shear-induced mesostructures in biaxial liquid crystals,, Phy. Rev. E, 78 (2008). doi: 10.1103/PhysRevE.78.061702.
[36]	S. Sircar and Q. Wang, Dynamics and rheology of biaxial liquid crystal polymers in shear flows,, J. Rheol., 53 (2009), 819. doi: 10.1122/1.3143788.
[37]	A. M. Sonnet, E. G. Virga and G. E. Durand, Dielectric shape dispersion and biaxial transitions in nematic liquid crystals,, Phys. Rev. E, 67 (2003). doi: 10.1103/PhysRevE.67.061701.
[38]	A. M. Sonnet and E. G. Virga, Steric effects in dispersion forces interactions,, Phys. Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.031704.
[39]	J. P. Straley, Ordered phases of a liquid of biaxial particles,, Phy. Rev. A, 10 (1974), 1881. doi: 10.1103/PhysRevA.10.1881.
[40]	M. C. J. M. Vissenberg, S. Stallinga and G. Vertogen, Generalized Landau-de Gennes theory of uniaxial and biaxial nematic liquid crystals,, Phy. Rev. E, 55 (1997), 4367. doi: 10.1103/PhysRevE.55.4367.
[41]	L. J. Yu and A. Saupe, Observation of a biaxial nematic Phase in potassium Laurate-1-Decanol-Water mixtures,, Phys. Rev. Lett., 45 (1980), 1000. doi: 10.1103/PhysRevLett.45.1000.

show all references

References:

[1]	B. R. Acharya, A. Primak and S. Kumar, Biaxial nematic phase in bent-core thermotropic mesogens,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.145506.
[2]	R. Alben, Phase transitions in a fluid of biaxial particles,, Phys. Rev. Lett., 30 (1973), 778. doi: 10.1103/PhysRevLett.30.778.
[3]	D. W. Allender and M. A. Lee, Landau theory of biaxial nematic liquid crystals,, Mol. Cryst. Liq. Cryst., 110 (1984), 331. doi: 10.1080/00268948408074514.
[4]	D. Allender and L. Longa, Landau-de Gennes theory for biaxial nematics reexamined,, Phy. Rev. E, 78 (2008). doi: 10.1103/PhysRevE.78.011704.
[5]	J. Ball, , Private Communication., ().
[6]	M. A. Bates, Influence of flexibility on the biaxial nematic phase of bent core liquid crystals: A Monte Carlo simulation study,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.061702.
[7]	R. Berardi and C. Zannoni, Do thermotropic biaxial nematics exist? A Monte Carlo study of biaxial Gay-Berne particles,, J. Chem. Phys., 113 (2000), 5971. doi: 10.1063/1.1290474.
[8]	B. Bergersen, P. Palffy-Muhoray and D. A. Dunmur, Uniaxial nematic phase in fluids of biaxial particles,, Liq. Cryst., 3 (1988), 347. doi: 10.1080/02678298808086380.
[9]	P. Biscari, M. C. Calderer and E. M. Terentjev, Landau-de Gennes theory of isotropic-nematic-smectic liquid crystal transitions,, Phys. Rev. E, 75 (2007). doi: 10.1103/PhysRevE.75.051707.
[10]	F. Biscarini, C. Chiccoli, P. Pasini, F. Semeria and C. Zannoni, Phase diagram and orientational order in a biaxial lattice model: A Monte Carlo study,, Phy. Rev. Lett., 75 (1995), 1803. doi: 10.1103/PhysRevLett.75.1803.
[11]	F. Bisi, E. G. Virga, E. C. Garland, Jr., G. De Matteis, A. M. Sonnet and G. E. Durand, Universal mean-field phase diagram for biaxial nematics obtained from a minmax priniciple,, Phys. Rev. E, 73 (2006). doi: 10.1103/PhysRevE.73.051709.
[12]	C. G. Broyden, A class of methods for solving nonlinear simultaneous equations,, Math. Comput., 19 (1965), 577. doi: 10.1090/S0025-5718-1965-0198670-6.
[13]	P. J. Camp and M. P. Allen, Phase diagram of the hard biaxial ellipsoid fluid,, J. Chem. Phys., 106 (1997), 6681. doi: 10.1063/1.473665.
[14]	C. Chiccoli, P. Pasini, F. Semeria and C. Zannoni, A detailed Monte Carlo investigation of the tricritical region of a biaxial liquid crystal system,, Int. J. Mod. Phys. C, 10 (1999), 469. doi: 10.1142/S0129183199000358.
[15]	P. G. de Gennes, "The Physics of Liquid Crystals,", Clarendon Press, (1974).
[16]	G. De Matties, S. Romano and E. G. Virga, Bifurcation analysis and computer simulation of biaxial liquid crystals,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.041706.
[17]	G. De Matties, A. M. Sonnet and E. G. Virga, Landau theory for biaxial nematic liquid crystals with two order parameter tensors,, Continuum Mech. Thermodyn., 20 (2008), 347. doi: 10.1007/s00161-008-0086-9.
[18]	R. Ennis, "Pattern Formation in Liquid Crystals: The Saffman-Taylor Instability and the Dynamics of Phase Separation,", Ph.D. dissetation, (2004).
[19]	M. J. Freiser, Ordered states of a nematic liquid,, Phys. Rev. Lett., 24 (1970), 1041. doi: 10.1103/PhysRevLett.24.1041.
[20]	M. J. Freiser, Successive transitions in a nematic liquid,, Mol. Cryst. Liq. Cryst., 14 (1971), 165.
[21]	W. M. Gelbart and B. A. Baron, Generalized van der Waals theory of the isotropic-nematic phase transition,, J. Chem. Phys., 66 (1977), 207. doi: 10.1063/1.433665.
[22]	J. Israelachvili, "Intermolecular & Surface Forces,", Academic Press, (1992).
[23]	L. Longa and G. Pajak, Luckhurst-Romano model of thermotropic biaxial nematic phase,, Liq. Cryst., 32 (2005), 1409. doi: 10.1080/02678290500167873.
[24]	L. Longa, P. Grzybowski, S. Romano and E. Virga, Minimal coupling model of the biaxial nematic phase,, Phys. Rev. E, 71 (2005). doi: 10.1103/PhysRevE.71.051714.
[25]	G. R. Luckhurst and S. Romano, Computer simulation studies of anisotropic systems II. Uniaxial and biaxial nematics formed by non-cylindrically symmetric molecules,, Mol. Phys., 40 (1980), 129. doi: 10.1080/00268978000101341.
[26]	L. A. Madsen, T. J. Dingemans, M. Nakata and E. T. Samulski, Thermotropic biaxial nematic liquid crystals,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.145505.
[27]	W. Maier and A. Saupe, A simple molecular theory of the namatic liquid-crystalline state,, Zeischrift Naturforschung, 13 (1958), 564.
[28]	K. Merkel, A. Kocot, J. K. Vij, R. Korlacki, G. H. Mehl and T. Meyer, Thermotropic biaxial nematic phase in liquid crystalline organo-siloxane tetrapodes,, Phys. Rev. Lett., 93 (2004). doi: 10.1103/PhysRevLett.93.237801.
[29]	P. K. Mukherjee, Improved analysis of the Landau theory of the uniaxail-biaxial nematic phase transition,, Liq. Cryst., 24 (1998), 519. doi: 10.1080/026782998206966.
[30]	B. M. Mulder, Solution of the excluded volume problem for biaxial particles,, Liq. Cryst., 1 (1986), 539. doi: 10.1080/02678298608086278.
[31]	P. Palffy-Muhoray, The single particle potential in mean-field theory,, Am. J. Phys., 70 (2002), 433. doi: 10.1119/1.1446860.
[32]	P. Palffy-Muhoray and B. Bergersen, van der Waals theory for nematic liquid crystals,, Phys. Rev. A, 35 (1987). doi: 10.1103/PhysRevA.35.2704.
[33]	S. Sarman, Molecular dynamics of biaxial nematic liquid crystals,, J. Chem. Phys., 104 (1996), 342. doi: 10.1063/1.470833.
[34]	C. S. Shih and R. Alben, Lattice model for biaxial liquid crystals,, J. Chem. Phys., 57 (1972), 3055. doi: 10.1063/1.1678719.
[35]	S. Sircar and Q. Wang, Shear-induced mesostructures in biaxial liquid crystals,, Phy. Rev. E, 78 (2008). doi: 10.1103/PhysRevE.78.061702.
[36]	S. Sircar and Q. Wang, Dynamics and rheology of biaxial liquid crystal polymers in shear flows,, J. Rheol., 53 (2009), 819. doi: 10.1122/1.3143788.
[37]	A. M. Sonnet, E. G. Virga and G. E. Durand, Dielectric shape dispersion and biaxial transitions in nematic liquid crystals,, Phys. Rev. E, 67 (2003). doi: 10.1103/PhysRevE.67.061701.
[38]	A. M. Sonnet and E. G. Virga, Steric effects in dispersion forces interactions,, Phys. Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.031704.
[39]	J. P. Straley, Ordered phases of a liquid of biaxial particles,, Phy. Rev. A, 10 (1974), 1881. doi: 10.1103/PhysRevA.10.1881.
[40]	M. C. J. M. Vissenberg, S. Stallinga and G. Vertogen, Generalized Landau-de Gennes theory of uniaxial and biaxial nematic liquid crystals,, Phy. Rev. E, 55 (1997), 4367. doi: 10.1103/PhysRevE.55.4367.
[41]	L. J. Yu and A. Saupe, Observation of a biaxial nematic Phase in potassium Laurate-1-Decanol-Water mixtures,, Phys. Rev. Lett., 45 (1980), 1000. doi: 10.1103/PhysRevLett.45.1000.

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