Orientational order parameter
In physics,
tensor is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase.[1] The
tensor is a second-order, traceless, symmetric tensor and is defined by[2][3][4]
![{\displaystyle \mathbf {Q} =S\left(\mathbf {n} \otimes \mathbf {n} -{\tfrac {1}{3}}\mathbf {I} \right)+R\left(\mathbf {m} \otimes \mathbf {m} -{\tfrac {1}{3}}\mathbf {I} \right)}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/af52326983ec731ece438257ef6ba1970b0afd08)
where
and
are scalar order parameters,
are the two directors of the nematic phase and
is the temperature; in uniaxial liquid crystals,
. The components of the tensor are
![{\displaystyle Q_{ij}=S\left(n_{i}n_{j}-{\tfrac {1}{3}}\delta _{ij}\right)+R\left(m_{i}m_{j}-{\tfrac {1}{3}}\delta _{ij}\right)}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/12033923e4de53661e9c20f38018dd450f029bad)
The states with directors
and
are physically equivalent and similarly the states with directors
and
are physically equivalent.
The
tensor can always be diagonalized,
![{\displaystyle \mathbf {Q} ={\frac {1}{3}}{\begin{bmatrix}2S-R&0&0\\0&2R-S&0\\0&0&-S-R\\\end{bmatrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/3f13eff626a62e938198ecf62c623784d0e5ff82)
The following are the two invariants of the
tensor,
![{\displaystyle \mathrm {tr} \,\mathbf {Q} ^{2}=Q_{ij}Q_{ji}={\frac {2}{3}}(S^{2}-SR+R^{2}),\quad \mathrm {tr} \,\mathbf {Q} ^{3}=Q_{ij}Q_{jk}Q_{ki}={\frac {1}{9}}[2(S^{3}+R^{3})-3SR(S+R)];}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/8e77c27d0c9a52bbfd374ee2ff47bdf009f85094)
the first-order invariant
is trivial here. It can be shown that
The measure of biaxiality of the liquid crystal is commonly measured through the parameter
![{\displaystyle \beta =1-6{\frac {(\mathrm {tr} \,\mathbf {Q} ^{3})^{2}}{(\mathrm {tr} \,\mathbf {Q} ^{2})^{3}}}={\frac {27S^{2}R^{2}(S-R)^{2}}{4(S^{2}-SR+R^{2})^{3}}}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/e4e8f383ae4142bb2d3ee4032990da9d3925f785)
In uniaxial nematic liquid crystals,
and therefore the
tensor reduces to
![{\displaystyle \mathbf {Q} =S\left(\mathbf {n} \mathbf {n} -{\frac {1}{3}}\mathbf {I} \right).}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/ee3c907c9b391356a14a462af5c2052339e6a42a)
The scalar order parameter is defined as follows. If
represents the angle between the axis of a nematic molecular and the director axis
, then[2]
![{\displaystyle S=\langle P_{2}(\cos \theta _{\mathrm {mol} })\rangle ={\frac {1}{2}}\langle 3\cos ^{2}\theta _{\mathrm {mol} }-1\rangle ={\frac {1}{2}}\int (3\cos ^{2}\theta _{\mathrm {mol} }-1)f(\theta _{\mathrm {mol} })d\Omega }](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/e61b6a1e354315d029b74a89c9616bdd1c69ba52)
where
denotes the ensemble average of the orientational angles calculated with respect to the distribution function
and
is the solid angle. The distribution function must necessarily satisfy the condition
since the directors
and
are physically equivalent.
The range for
is given by
, with
representing the perfect alignment of all molecules along the director and
representing the complete random alignment (isotropic) of all molecules with respect to the director; the
case indicates that all molecules are aligned perpendicular to the director axis although such nematics are rare or hard to synthesize.
- ^ De Gennes, P. G. (1969). Phenomenology of short-range-order effects in the isotropic phase of nematic materials. Physics Letters A, 30 (8), 454-455.
- ^ a b De Gennes, P. G., & Prost, J. (1993). The physics of liquid crystals (No. 83). Oxford university press.
- ^ Mottram, N. J., & Newton, C. J. (2014). Introduction to Q-tensor theory. arXiv preprint arXiv:1409.3542.
- ^ Kleman, M., & Lavrentovich, O. D. (Eds.). (2003). Soft matter physics: an introduction. New York, NY: Springer New York.