Topology and nematic ordering. II. Observable critical behavior

John Toner; Paul E. Lammert; Daniel S. Rokhsar

doi:10.1103/PhysRevE.52.1801

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Topology and nematic ordering. II. Observable critical behavior

John Toner

Paul E. Lammert

Daniel S. Rokhsar

IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598
Department of Physics, University of Oregon, Eugene, Oregon 97403-1274

Department of Physics, University of California, Berkeley, California 94720
Department of Physics, Simon Fraser University, Burnaby British Columbia V5A 156, Canada

Department of Physics, University of California, Berkeley, California 94720

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Phys. Rev. E 52, 1801 – Published 1 August, 1995

DOI: https://doi.org/10.1103/PhysRevE.52.1801

Abstract

This paper is the second in a pair treating a lattice model for nematic media. In addition to the familiar isotropic (I) and nematically ordered (N) phases, the phase diagram established in the previous paper (paper I) contains a new, topologically ordered phase (T) occurring at large suppression of topological defects and weak nematic interactions. This paper (paper II) is concerned with the experimental signatures of the proposed phase diagram. Specific heat, light scattering, and magnetic susceptibility near both the N-T and I-T transitions are studied and critical behavior is determined. The singular dependences of the Frank constants $K_{1}$ , $K_{2}$ , $K_{3}$ and the dielectric tensor anisotropy Δε on temperature as T→ $T_{N T}^{-}$ are also found.

References (11)

P. E. Lammert, D. S. Rokhsar and J. Toner, preceding paper, Phys. Rev. E 52, 1778 (1995).
P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Clarendon, Oxford, 1993).
J. C. Le Guillou and J. Zinn Justin, J. Phys. Lett. 46, L137 (1985).
We are concerned here with single scattering. There are also both polarized and depolarized contributions from multiple scattering, even for a simple isotropic fluid [see W. H. Gelbart, Adv. Chem. Phys. 26, 1, (1974)]. The magnitudes of these effects can be strongly dependent on experimental geometry: the sizes of the sample and the volumes illuminated by the source and visible to the detector. In the case where the density fluctuations are relatively small in magnitude, multiple scattering will be highly suppressed.
S. K. Ma, Modern Theory of Critical Phenomena (Addison Wesley, Redwood City, CA, 1976).
D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Addison Wesley, Redwood City, CA, 1975).
B. D. Josephson, Phys. Lett. 21, 608 (1966).
A. Aharony, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz (Academic, London, 1983), Vol. 8.
It might appear that in systems without microscopic S to - S symmetry, one could add a term linear in bf H, namely, $m_{0}$ ${sum}_{i}$ H⋅ $S_{i}$ , where $m_{0}$ in a constant permanent molecular dipole moment, which would dominate the quadratic (in bf H) term ( refmag e). For this to make sense, we have to rewrite it in a gauge invariant form just as in paper I with the likewise gauge noninvariant $S_{i}$ ⋅ $S_{j}$ perturbation. To wit, we introduce additional variables $σ_{i}$ = pm 1 on the sites that keep track of whether Ssub i points along or opposite the local dipole and transform like S under a gauge transformation so that the combination $σ_{i}$ $S_{i}$ is invariant. Then the perturbation becomes $m_{0}$ ${sum}_{i}$ ( $σ_{i}$ H)⋅ $S_{i}$ . Since σ's on different sites are decoupled from each other, it is trivial to sum over them, yielding a magnetic perturbation of $H_{mathrm mag}^{mathrm eff}$ = ${sum}_{i}$ ln [ cosh( $m_{0}$ H⋅ $S_{i}$ ) ], an explicitly even function of each Ssub i. Expanding to lowest order, we recover the earlier expression ( refmag e) with Δ ${chi}_{0}$ = 1/2 $m_{0}$ sup 2.
The absence of a quadratic term is quite clear from a more general perspective. From Eq. ( refQ0I/T), we see that if $K_{mathrm eff}$ did contain an O( $h^{2}$ ) correction, $Q_{0}$ would not vanish at H=0, i.e., the system would have spontaneous nematic order. Since such order does not exist in the vicinity of the I T transition, that term must disappear.
P. E. Lammert, D. S. Rokhsar and J. Toner, Phys. Rev. Lett. 70, 1650 (1993).

[c1] P. E. Lammert, D. S. Rokhsar and J. Toner, preceding paper, Phys. Rev. E 52, 1778 (1995).

[c2] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Clarendon, Oxford, 1993).

[c3] J. C. Le Guillou and J. Zinn Justin, J. Phys. Lett. 46, L137 (1985).

[c4] We are concerned here with single scattering. There are also both polarized and depolarized contributions from multiple scattering, even for a simple isotropic fluid [see W. H. Gelbart, Adv. Chem. Phys. 26, 1, (1974)]. The magnitudes of these effects can be strongly dependent on experimental geometry: the sizes of the sample and the volumes illuminated by the source and visible to the detector. In the case where the density fluctuations are relatively small in magnitude, multiple scattering will be highly suppressed.

[c5] S. K. Ma, Modern Theory of Critical Phenomena (Addison Wesley, Redwood City, CA, 1976).

[c6] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Addison Wesley, Redwood City, CA, 1975).

[c7] B. D. Josephson, Phys. Lett. 21, 608 (1966).

[c8] A. Aharony, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz (Academic, London, 1983), Vol. 8.

[c9] It might appear that in systems without microscopic S to - S symmetry, one could add a term linear in bf H, namely, $m_{0}$ ${sum}_{i}$ H⋅ $S_{i}$ , where $m_{0}$ in a constant permanent molecular dipole moment, which would dominate the quadratic (in bf H) term ( refmag e). For this to make sense, we have to rewrite it in a gauge invariant form just as in paper I with the likewise gauge noninvariant $S_{i}$ ⋅ $S_{j}$ perturbation. To wit, we introduce additional variables $σ_{i}$ = pm 1 on the sites that keep track of whether Ssub i points along or opposite the local dipole and transform like S under a gauge transformation so that the combination $σ_{i}$ $S_{i}$ is invariant. Then the perturbation becomes $m_{0}$ ${sum}_{i}$ ( $σ_{i}$ H)⋅ $S_{i}$ . Since σ's on different sites are decoupled from each other, it is trivial to sum over them, yielding a magnetic perturbation of $H_{mathrm mag}^{mathrm eff}$ = ${sum}_{i}$ ln [ cosh( $m_{0}$ H⋅ $S_{i}$ ) ], an explicitly even function of each Ssub i. Expanding to lowest order, we recover the earlier expression ( refmag e) with Δ ${chi}_{0}$ = 1/2 $m_{0}$ sup 2.

[c10] The absence of a quadratic term is quite clear from a more general perspective. From Eq. ( refQ0I/T), we see that if $K_{mathrm eff}$ did contain an O( $h^{2}$ ) correction, $Q_{0}$ would not vanish at H=0, i.e., the system would have spontaneous nematic order. Since such order does not exist in the vicinity of the I T transition, that term must disappear.

[c11] P. E. Lammert, D. S. Rokhsar and J. Toner, Phys. Rev. Lett. 70, 1650 (1993).

Physical Review E

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Topology and nematic ordering. II. Observable critical behavior

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Topology and nematic ordering. I. A gauge theory

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