American Institute of Mathematical Sciences

Advanced
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

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.  Google Scholar

[2]

R. Alben, Phase transitions in a fluid of biaxial particles,, Phys. Rev. Lett., 30 (1973), 778.  doi: 10.1103/PhysRevLett.30.778.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Ball,  , Private Communication., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. G. de Gennes, "The Physics of Liquid Crystals,", Clarendon Press, (1974).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. Ennis, "Pattern Formation in Liquid Crystals: The Saffman-Taylor Instability and the Dynamics of Phase Separation,", Ph.D. dissetation, (2004).   Google Scholar

[19]

M. J. Freiser, Ordered states of a nematic liquid,, Phys. Rev. Lett., 24 (1970), 1041.  doi: 10.1103/PhysRevLett.24.1041.  Google Scholar

[20]

M. J. Freiser, Successive transitions in a nematic liquid,, Mol. Cryst. Liq. Cryst., 14 (1971), 165.   Google Scholar

[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.  Google Scholar

[22]

J. Israelachvili, "Intermolecular & Surface Forces,", Academic Press, (1992).   Google Scholar

[23]

L. Longa and G. Pajak, Luckhurst-Romano model of thermotropic biaxial nematic phase,, Liq. Cryst., 32 (2005), 1409.  doi: 10.1080/02678290500167873.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

W. Maier and A. Saupe, A simple molecular theory of the namatic liquid-crystalline state,, Zeischrift Naturforschung, 13 (1958), 564.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

B. M. Mulder, Solution of the excluded volume problem for biaxial particles,, Liq. Cryst., 1 (1986), 539.  doi: 10.1080/02678298608086278.  Google Scholar

[31]

P. Palffy-Muhoray, The single particle potential in mean-field theory,, Am. J. Phys., 70 (2002), 433.  doi: 10.1119/1.1446860.  Google Scholar

[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.  Google Scholar

[33]

S. Sarman, Molecular dynamics of biaxial nematic liquid crystals,, J. Chem. Phys., 104 (1996), 342.  doi: 10.1063/1.470833.  Google Scholar

[34]

C. S. Shih and R. Alben, Lattice model for biaxial liquid crystals,, J. Chem. Phys., 57 (1972), 3055.  doi: 10.1063/1.1678719.  Google Scholar

[35]

S. Sircar and Q. Wang, Shear-induced mesostructures in biaxial liquid crystals,, Phy. Rev. E, 78 (2008).  doi: 10.1103/PhysRevE.78.061702.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

J. P. Straley, Ordered phases of a liquid of biaxial particles,, Phy. Rev. A, 10 (1974), 1881.  doi: 10.1103/PhysRevA.10.1881.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

R. Alben, Phase transitions in a fluid of biaxial particles,, Phys. Rev. Lett., 30 (1973), 778.  doi: 10.1103/PhysRevLett.30.778.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Ball,  , Private Communication., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. G. de Gennes, "The Physics of Liquid Crystals,", Clarendon Press, (1974).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. Ennis, "Pattern Formation in Liquid Crystals: The Saffman-Taylor Instability and the Dynamics of Phase Separation,", Ph.D. dissetation, (2004).   Google Scholar

[19]

M. J. Freiser, Ordered states of a nematic liquid,, Phys. Rev. Lett., 24 (1970), 1041.  doi: 10.1103/PhysRevLett.24.1041.  Google Scholar

[20]

M. J. Freiser, Successive transitions in a nematic liquid,, Mol. Cryst. Liq. Cryst., 14 (1971), 165.   Google Scholar

[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.  Google Scholar

[22]

J. Israelachvili, "Intermolecular & Surface Forces,", Academic Press, (1992).   Google Scholar

[23]

L. Longa and G. Pajak, Luckhurst-Romano model of thermotropic biaxial nematic phase,, Liq. Cryst., 32 (2005), 1409.  doi: 10.1080/02678290500167873.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

W. Maier and A. Saupe, A simple molecular theory of the namatic liquid-crystalline state,, Zeischrift Naturforschung, 13 (1958), 564.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

B. M. Mulder, Solution of the excluded volume problem for biaxial particles,, Liq. Cryst., 1 (1986), 539.  doi: 10.1080/02678298608086278.  Google Scholar

[31]

P. Palffy-Muhoray, The single particle potential in mean-field theory,, Am. J. Phys., 70 (2002), 433.  doi: 10.1119/1.1446860.  Google Scholar

[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.  Google Scholar

[33]

S. Sarman, Molecular dynamics of biaxial nematic liquid crystals,, J. Chem. Phys., 104 (1996), 342.  doi: 10.1063/1.470833.  Google Scholar

[34]

C. S. Shih and R. Alben, Lattice model for biaxial liquid crystals,, J. Chem. Phys., 57 (1972), 3055.  doi: 10.1063/1.1678719.  Google Scholar

[35]

S. Sircar and Q. Wang, Shear-induced mesostructures in biaxial liquid crystals,, Phy. Rev. E, 78 (2008).  doi: 10.1103/PhysRevE.78.061702.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

J. P. Straley, Ordered phases of a liquid of biaxial particles,, Phy. Rev. A, 10 (1974), 1881.  doi: 10.1103/PhysRevA.10.1881.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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