Liquid Crystals. Iam-Choon Khoo

Liquid Crystals

href="#fb3_img_img_66981a42-c73d-5b41-95bb-1d06c7b10637.png" alt="ModifyingAbove upper M With right harpoon with barb up"/>_, the magnetic induction ModifyingAbove upper B With right harpoon with barb up

ModifyingAbove upper B With right harpoon with barb up

, and the magnetic strength ModifyingAbove upper H With right harpoon with barb up

(3.20) ModifyingAbove upper M With right harpoon with barb up equals StartFraction ModifyingAbove upper B With right harpoon with barb up Over mu 0 EndFraction minus ModifyingAbove upper H With right harpoon with barb up equals ModifyingAbove Above ModifyingAbove chi With right harpoon with barb up With right harpoon with barb up colon ModifyingAbove upper H With right harpoon with barb up

and

(3.21) ModifyingAbove upper B With right harpoon with barb up equals mu 0 left-parenthesis 1 plus ModifyingAbove Above ModifyingAbove chi With right harpoon with barb up With right harpoon with barb up Subscript m Baseline right-parenthesis colon ModifyingAbove upper H With right harpoon with barb up period

The magnetic susceptibility tensor ModifyingAbove Above ModifyingAbove chi With right harpoon with barb up With right harpoon with barb up Subscript m is anisotropic. For a uniaxial material such as a nematic, the magnetic susceptibility takes the form

(3.22)

Note that this is similar to the dielectric constant ModifyingAbove Above ModifyingAbove epsilon With right harpoon with barb up With right harpoon with barb up .

Nematic liquid crystals, in fact, liquid crystals in general, are diamagnetic. Therefore, chi Subscript up-tack Superscript m and chi Subscript parallel-to Superscript m are negative of vanishingly small magnitude. As a result of the smallness of these magnetic susceptibilities, the magnetic interactions among the molecules comprising the liquid crystal are small (in comparison with their interaction with the externally applied field). Consequently, the local field acting on the molecules differs very little from the external field, and in general, magnetic measurements are the preferred method to study liquid crystal order parameters and other physical processes.

3.3.2. Free Energy and Torques by Electric and Magnetic Fields

In this section, we consider the interactions of nematic liquid crystals with applied fields (electric or magnetic); we will limit our discussion to only dielectric and diamagnetic interactions.

For a generally applied (dc, low frequency, or optical) electric field ModifyingAbove upper E With right harpoon with barb up , the displacement ModifyingAbove upper D With right harpoon with barb up may be written in the form

(3.23) ModifyingAbove upper D With right harpoon with barb up equals epsilon Subscript up-tack Baseline ModifyingAbove upper E With right harpoon with barb up plus left-parenthesis epsilon Subscript parallel-to Baseline minus epsilon Subscript up-tack Baseline right-parenthesis left-parenthesis n dot ModifyingAbove upper E With right harpoon with barb up right-parenthesis n period

The electric interaction energy density is therefore

(3.24)

Note that the first term on the right‐hand side of Eq. (3.24) is independent of the orientation of the director axis. It can therefore be neglected in the director axis deformation energy. Accordingly, the free‐energy density term associated with the application of an electric field is given by

(3.25) upper F Subscript upper E Baseline equals minus StartFraction normal upper Delta epsilon Over 2 EndFraction left-parenthesis n dot ModifyingAbove upper E With right harpoon with barb up right-parenthesis squared

in SI units (in cgs units, upper F Subscript upper E Baseline equals minus left-parenthesis normal upper Delta epsilon slash 8 pi right-parenthesis left-parenthesis ModifyingAbove n With ampersand c period circ semicolon dot ModifyingAbove upper E With right harpoon with barb up right-parenthesis squared ). The molecular torque produced by the electric field is given by

(3.26)

Similar considerations for the magnetic field yield a magnetic energy density term U_m given by

(3.27)

a magnetic free‐energy density (associated with director axis reorientation) F_m given by

(3.28) upper F Subscript m Baseline equals StartFraction 1 Over 2 mu 0 EndFraction normal upper Delta chi Superscript m Baseline left-parenthesis n dot ModifyingAbove upper B With right harpoon with barb up right-parenthesis squared comma

and a magnetic torque density

(3.29)Скачать книгу