Q tensor

In physics, ${\displaystyle \mathbf {Q} }$ tensor is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase. ^[1] The ${\displaystyle \mathbf {Q} }$ tensor is a second-order, traceless, symmetric tensor and is defined by ^{[2] [3] [4]}

${\displaystyle \mathbf {Q} =S\left(\mathbf {n} \mathbf {n} -{\frac {1}{3}}\mathbf {I} \right)+P\left(\mathbf {m} \mathbf {m} -{\frac {1}{3}}\mathbf {I} \right)}$

where ${\displaystyle S=S(T)}$ and ${\displaystyle P=P(T)}$ are scalar order parameters, ${\displaystyle (\mathbf {n} ,\mathbf {m} )}$ are the two directors of the nematic phase and ${\displaystyle T}$ is the temperature; in uniaxial liquid crystals, ${\displaystyle P=0}$. The components of the tensor are

${\displaystyle Q_{ij}=S\left(n_{i}n_{j}-{\frac {1}{3}}\delta _{ij}\right)+P\left(m_{i}m_{j}-{\frac {1}{3}}\delta _{ij}\right)}$

The states with directors ${\displaystyle \mathbf {n} }$ and ${\displaystyle -\mathbf {n} }$ are physically equivalent and similarly the states with directors ${\displaystyle \mathbf {m} }$ and ${\displaystyle -\mathbf {m} }$ are physically equivalent.

The ${\displaystyle \mathbf {Q} }$ tensor can always be diagonalized,

${\displaystyle \mathbf {Q} =-{\frac {1}{2}}{\begin{pmatrix}S+P&0&0\\0&S-P&0\\0&0&-2S\\\end{pmatrix}}}$

The following are the invariants of the ${\displaystyle \mathbf {Q} }$ tensor

${\displaystyle \mathrm {tr} \,\mathbf {Q} ^{2}=Q_{ij}Q_{ji}={\frac {1}{2}}(3S^{2}+P^{2}),\quad \mathrm {tr} \,\mathbf {Q} ^{3}=Q_{ij}Q_{jk}Q_{ki}={\frac {3}{4}}S(S^{2}-P^{2});}$

the first-order invariant ${\displaystyle \mathrm {tr} \,\mathbf {Q} =Q_{ii}=0}$ is trivial here. It can be shown that ${\displaystyle (\mathrm {tr} \,\mathbf {Q} ^{2})^{3}\geq 6(\mathrm {tr} \,\mathbf {Q} ^{3})^{2}.}$ The measure of biaxiality of the liquid crystal is commonly measured through the parameter

${\displaystyle \beta =1-6{\frac {(\mathrm {tr} \,\mathbf {Q} ^{3})^{2}}{(\mathrm {tr} \,\mathbf {Q} ^{2})^{3}}}=1-{\frac {27S^{2}(S^{2}-P^{2})^{3}}{(3S^{2}+P^{2})^{3}}}.}$

Uniaxial nematics

In uniaxial nematic liquid crystals, ${\displaystyle P=0}$ and therefore the ${\displaystyle \mathbf {Q} }$ tensor reduces to

${\displaystyle \mathbf {Q} =S\left(\mathbf {n} \mathbf {n} -{\frac {1}{3}}\mathbf {I} \right).}$

The scalar order parameter is defined as follows. If ${\displaystyle \theta _{\mathrm {mol} }}$ represents the angle between the axis of a nematic molecular and the director axis ${\displaystyle \mathbf {n} }$, then ^[2]

${\displaystyle S=\langle P_{2}(\cos \theta _{\mathrm {mol} })\rangle ={\frac {1}{2}}\langle 3\cos ^{2}\theta _{\mathrm {mol} }-1\rangle ={\frac {1}{2}}\int (3\cos ^{2}\theta _{\mathrm {mol} }-1)f(\theta _{\mathrm {mol} })d\Omega }$

where ${\displaystyle \langle \cdot \rangle }$ denotes the ensemble average of the orientational angles calculated with respect to the distribution function ${\displaystyle f(\theta _{\mathrm {mol} })}$ and ${\displaystyle d\Omega =\sin \theta _{\mathrm {mol} }d\theta _{\mathrm {mol} }d\phi _{\mathrm {mol} }}$ is the solid angle. The distribution function must necessarily satisfy the condition ${\displaystyle f(\theta _{\mathrm {mol} }+\pi )=f(\theta _{\mathrm {mol} })}$ since the directors ${\displaystyle \mathbf {n} }$ and ${\displaystyle -\mathbf {n} }$ are physically equivalent.

The range for ${\displaystyle S}$ is given by ${\displaystyle -1/2\leq S\leq 1}$, with ${\displaystyle S=1}$ representing the perfect alignment of all molecules along the director and ${\displaystyle S=0}$ representing the complete random alignment (isotropic) of all molecules with respect to the director; the ${\displaystyle S=-1/2}$ case indicates that all molecules are aligned perpendicular to the director axis although such nematics are rare or hard to synthesize.

See also

• Landau–de Gennes theory

References

1. 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.
2. De Gennes, P. G., & Prost, J. (1993). The physics of liquid crystals (No. 83). Oxford university press.
3. Mottram, N. J., & Newton, C. J. (2014). Introduction to Q-tensor theory. arXiv preprint arXiv:1409.3542.
4. Kleman, M., & Lavrentovich, O. D. (Eds.). (2003). Soft matter physics: an introduction. New York, NY: Springer New York.