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Q tensor - Wikipedia

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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} \otimes \mathbf {n} -{\tfrac {1}{3}}\mathbf {I} \right)+R\left(\mathbf {m} \otimes \mathbf {m} -{\tfrac {1}{3}}\mathbf {I} \right)}

where {\displaystyle S=S(T)} and {\displaystyle R=R(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}-{\tfrac {1}{3}}\delta _{ij}\right)+R\left(m_{i}m_{j}-{\tfrac {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}{3}}{\begin{bmatrix}2S-R&0&0\\0&2R-S&0\\0&0&-S-R\\\end{bmatrix}}}

The following are the two invariants of the {\displaystyle \mathbf {Q} } 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)];}

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}}}={\frac {27S^{2}R^{2}(S-R)^{2}}{4(S^{2}-SR+R^{2})^{3}}}.}

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.

  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. ^ a b 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.