Distortion free energy density

The Distortion free energy density is a quantity that describes the distortion of a liquid crystal from its preferred state in which all of the liquid crystal molecules are aligned parallel to one common axis. It also commonly goes by the name Frank free energy density named after Frederick Charles Frank.

1 Nematic Liquid Crystal
2 Chiral Liquid Crystal
3 Electric and Magnetic Field Contributions
4 Notes
5 References

Nematic Liquid Crystal

The Distortion free energy density is a measure of the Helmholtz free energy per unit volume of a liquid crystal. For a non-chiral nematic liquid crystals it typically is taken to consist of three terms and is given by:

$\mathcal{F}_{d}=\frac{1}{2}K_1(\nabla\cdot\mathbf{\hat{n}})^2%2B\frac{1}{2}K_2(\mathbf{\hat{n}}\cdot\nabla\times\mathbf{\hat{n}})^2%2B\frac{1}{2}K_3(\mathbf{\hat{n}}\times\nabla\times\mathbf{\hat{n}})^2$

The unit vector $\mathbf{\hat{n}}$ is the normalized director of the molecules $(|\mathbf{\hat{n}}|=1)$ , which describes the nature of the distortion. The three constants $K_i$ are known as the Frank constants and are dependent on the particular liquid crystal being described. They are usually of the order of $10^{-6}$ dyn.^[1] Each of the three terms represent a type of distortion of a nematic. The first term represents pure splay, the second term pure twist, and the third term pure bend. A combination of these terms can be used to represent an arbitrary deformation in a liquid crystal. It is often the case that all three Frank constants are of the same order of magnitude and so its commonly approximated that $K_1=K_2=K_3=K$ .^[2] This approximation is commonly referred to as the one-constant approximation and is used predominantly because the free energy simplifies when used to the much more computationally compact form:

$\mathcal{F}_{d}=\frac{1}{2}K((\nabla\cdot\mathbf{\hat{n}})^2%2B(\nabla\times\mathbf{\hat{n}})^2)=\frac{1}{2}K\partial_{\alpha}n_{\beta}\partial_{\alpha}n_{\beta}$

A fourth term is also commonly added to the Frank free energy density called the saddle-splay energy that describes the surface interaction. It is often ignored when calculating director field configurations since the energies in the bulk of the liquid crystal are often greater than those due to the surface. It is given by:

$\frac{1}{2}K_{24}\nabla\cdot((\mathbf{\hat{n}}\cdot\nabla)\mathbf{\hat{n}}-\mathbf{\mathbf{\hat{n}}}(\nabla\cdot \mathbf{\hat{n}}))$

If inclusions are added to a liquid crystal the free energy density associated with their presence is often given by the Rapini approximation:

$\mathcal{F}_{s}=-\oint\frac{1}{2}W(\mathbf{\hat{n}}\cdot\mathbf{\hat{\nu}})^2\mathrm{d}S$

The anchoring energy is given by $W$ and the unit vector $\mathbf{\hat{\nu}}$ is normal to the particles surface.^[3]

Chiral Liquid Crystal

For the case when the liquid crystal consists of chiral molecules an additional term to the distortion free energy density is added. The term changes sign when the axes are inverted and is given by:

$k_2(\mathbf{\hat{n}}\cdot\nabla\times\mathbf{\hat{n}})$

And so for the case of a chiral liquid crystal the distortion free energy density is given by:

$\mathcal{F}_{d}=\frac{1}{2}K_1(\nabla\cdot\mathbf{\hat{n}})^2%2B\frac{1}{2}K_2(\mathbf{\hat{n}}\cdot\nabla\times\mathbf{\hat{n}}%2Bq_0)^2%2B\frac{1}{2}K_3(\mathbf{\hat{n}}\times\nabla\times\mathbf{\hat{n}})^2$

The quantity $2 \pi /q_0$ is the cholesteric pitch.

Electric and Magnetic Field Contributions

As a result of liquid crystal mesogens' anisotropic diamagnetic properties and electrical polarizability, electric and magnetic fields can induce alignments in liquid crystals. By applying a field one is effectively lowering the free energy of the liquid crystal.^[4]

To understand the effect a magnetic field produces on the distortion free energy density, a small region of local nematic order $\mathbf{\hat{n}}$ is often considered in which $\chi_\perp$ and $\chi_\parallel$ is the magnetic susceptibility perpendicular and parallel to $\mathbf{\hat{n}}$ . The value $\Delta\chi\equiv\chi_\parallel-\chi_\perp=N<P_2(\cos{\theta})>$ , where N is the number of mesogens per unit volume. The work per unit volume done by the field is then given by:

$W_{magnetic}=\int_{0}^{H}(-M_\perp\sin{\theta}-M_\parallel\cos{\theta})\, dH=-\frac{H^2}{2}(\chi_\perp%2B\Delta\chi\cos{\theta}^2)$

where:

$M_\parallel=H\chi_\parallel\cos{\theta}$

$M_\perp=H\chi_\perp\sin{\theta}$

Since the $-\frac{H^2\chi_\perp}{2}$ term is spatially invariant it can be ignored and so the magnetic contribution to the distortion free energy density becomes:

$-\frac{\Delta\chi}{2}[\mathbf{H}\cdot\mathbf{\hat{n}}]^2$

From similar arguments the electric field's contribution to the distortion free energy can be found and is given by:

$-\frac{\Delta\epsilon}{8\pi}[\mathbf{E}\cdot\mathbf{\hat{n}}]$

The quantity $\Delta\epsilon\equiv\epsilon_\parallel-\epsilon_\perp$ is the difference between the local dielectric constants perpendicular and parallel to $\mathbf{\hat{n}}$ .

Notes

^ de Gennes & Prost 1995, p. 103
^ Chandrasekhar 1992, p. 118
^ Kuksenok et al. 1996, p. 5199
^ Priestley, Wojtowicz & Sheng 1975, pp. 107–110

References

Chandrasekhar, Sivaramakrishna (1992). Liquid Crystals (2nd ed.). Cambridge University Press. ISBN 0521417473.
de Gennes, Pierre-Gilles; Prost, J. (10 August 1995). The Physics of Liquid Crystals (2nd ed.). Oxford University Press. ISBN 0198517858.
Kamien, Randall D.; Selinger, Jonathan V. (22 January 2001). "Order and frustration in chiral liquid crystals". Journal of Physics: Condensed Matter 13 (3). arXiv:cond-mat/0009094. Bibcode 2001JPCM...13R...1K. doi:10.1088/0953-8984/13/3/201.
Kuksenok, O. V.; Ruhwandl, R. W.; Shiyanovskii, S. V.; Terentjev, E. M. (November 1996). "Director structure around a colloid particle suspended in a nematic liquid crystal". Physical Review E 54 (5): 5198–5203. Bibcode 1996PhRvE..54.5198K. doi:10.1103/PhysRevE.54.5198.
Priestley, E. B.; Wojtowicz, Peter J.; Sheng, Ping (1975). Introduction to Liquid Crystals. Plenum Press. ISBN 0306308584.