Liquid Crystals - Iam-Choon Khoo

Скачать книгу

normal upper Delta chi Superscript m Baseline n Subscript alpha Baseline n Subscript beta Baseline upper H Subscript alpha Baseline upper H Subscript beta Baseline period"/>

      (2.29)upper F Subscript i n t Superscript upper H Baseline equals minus one half sigma-summation Underscript alpha comma beta Endscripts upper Q Subscript italic alpha beta Baseline upper H Subscript alpha Baseline upper H Subscript beta Baseline period

      Therefore, the total free energy of a liquid crystal in the isotropic phase, under the action of an externally applied magnetic field, is given by

      (2.30)StartLayout 1st Row upper F equals upper F prime 0 plus one half upper A left-parenthesis upper T right-parenthesis sigma-summation Underscript alpha comma beta Endscripts upper Q Subscript italic alpha beta Baseline upper Q Subscript italic beta alpha Baseline plus one third upper B left-parenthesis upper T right-parenthesis sigma-summation Underscript alpha comma beta comma gamma Endscripts upper Q Subscript italic alpha beta Baseline upper Q Subscript italic beta gamma Baseline upper Q Subscript italic gamma alpha Baseline 2nd Row minus one half sigma-summation Underscript alpha comma beta Endscripts upper Q Subscript italic alpha beta Baseline upper H Subscript alpha Baseline upper H Subscript beta Baseline period EndLayout

      Without solving the problem explicitly, we can infer from the magnetic interaction term that a lower energy state corresponds to some alignment of the molecules in the direction of the magnetic field (for Δχ m > 0).

      Using a similar approach, we can also deduce that the electric interaction contribution to the free energy is given by (in inks units)

      (2.31)upper F Subscript i n t Superscript upper E Baseline equals minus one half integral Subscript 0 Superscript upper E Baseline bold upper D dot bold upper E equals minus one half epsilon Subscript up-tack Baseline upper E squared minus one half normal upper Delta epsilon left-parenthesis ModifyingAbove n With ampersand c period circ semicolon dot bold upper E right-parenthesis squared period

      (2.32)upper F Superscript upper E Baseline equals minus one half normal upper Delta epsilon left-parenthesis ModifyingAbove n With ampersand c period circ semicolon dot bold upper E right-parenthesis squared equals minus one half sigma-summation Underscript alpha comma beta Endscripts upper Q Subscript italic alpha beta Baseline upper E Subscript alpha Baseline upper E Subscript beta Baseline comma

      where Qαβ is defined in Eq. (2.4).

      In Chapter 8, we present a detailed discussion of isotropic phase molecular orientations by an applied optical field from a short intense laser pulse. It is shown that both the response time and the induced order Q depend on the temperature vicinity (T – Tc) in a critical way; they both vary as (T – Tc)−1, which becomes very large near Tc. This near‐Tc critical slowing down behavior of the order parameter Q of the isotropic phase is similar to the slowing down behavior of the order parameter S of the nematic phase discussed in the previous section. Besides the nematic ↔ isotropic phase transition, which is the most prominent order ↔ disorder transition exhibited by liquid crystals, there are other equally interesting phase transition processes among the various mesophases [13], such as smectic‐A ↔ smectic‐C*, which will be discussed in Chapter 4.

      1 1. deGennes, P. G. 1974. The Physics of Liquid Crystals. Oxford: Clarendon Press.

      2 2. deGennes, P.G. 1971. Mol. Cryst. Liq. Cryst. 12: 193.

      3 3. Landau, L. D. 1965. Collected Papers. D. Ter Haar (ed.). New York: Gordon & Breach.

      4 4. Litster, J. D. 1971. Critical Phenomena. R. E. Mills (ed.). New York: McGraw‐Hill.

      5 5. Khoo, I.C. and S. T. Wu. 1993. Optics and Nonlinear Optics of Liquid Crystals. Singapore: World Scientific.

      6 6. See, for example, Blinov, L. M. 1983. Electro‐optical and Magneto‐optical Properties of Liquid Crystals. Chichester: Wiley.

      7 7. Maier, W. and A. Saupe. 1959. Z. Naturforsch. 14A: 882; for a concise account of the theory, see Khoo and Wu [5].

      8 8. Humphries, R.L., and O. R. Lukhurst. 1972. Chem. Phys. Lett. 17: 514; Luckhurst, G. R., C. Zannoni, P. L. Nordio, et al. 1975. Mol. Phys. 30:1345; Freiser, M. J. 1971. Mol. Cryst. Liq. Cryst. 14: 165.

      9 9. Blinov, L.M., V. A. Kizel, V. G. Rumyantsev, et al. 1975. J. Phys. (Paris) Colloq. 36: C1–C69; see also Blinov [6].

      10 10. DeJeu, W.H. 1980. Physical Properties of Liquid Crystalline Materials. New York: Gordon and Breach.

      11 11. Khoo, I. C., R. G. Lindquist, R. R. Michael, et al. 1991. J. Appl. Phys. 69: 3853.

      12 12. Khoo, I.C., and R. Normandin. 1985. IEEE J. Quantum Electron. QE21: 329.

      13 13. Chandrasekhar, S. 1992. Liquid Crystals. 2nd ed. Cambridge: Cambridge University Press; see also deGennes [1].

      3.1. INTRODUCTION

      Nematics best exemplify the dual nature of liquid crystals – fluidity and crystalline structure. To describe their liquid‐like properties, one needs to invoke hydrodynamics. On the other hand, their crystalline properties necessitate theoretical formalisms pertaining to solids or crystals. To study their optical properties, it is also necessary that we invoke individual molecular electronic structures and collective crystalline properties. In this chapter, we discuss all three aspects of nematogen theory: solid‐state continuum theory, hydrodynamics, and electro‐optical properties, in that order.

      3.2.1. The Vector Field: Director Axis

      In elastic continuum theory, introduced and refined over the last several decades by several workers [1–3], nematics is basically viewed as crystalline in form. An aligned sample may thus be regarded as a single crystal in which the molecules are, on average, aligned along the direction defined by the director axis ModifyingAbove n With ampersand c period circ semicolon left-parenthesis ModifyingAbove r With right harpoon with barb up right-parenthesis.

      The crystal is uniaxial and is characterized by a tensorial order parameter:

      (3.1)upper S Subscript italic alpha beta Baseline equals upper S left-parenthesis normal upper T right-parenthesis left-parenthesis n Subscript alpha Baseline n Subscript beta Baseline minus one third delta Subscript italic alpha beta Baseline right-parenthesis period

Скачать книгу