{{Short description|Orientational order parameter}} In physics, '''<math>\mathbf Q</math>-tensor''' is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase.<ref>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.</ref> The <math>\mathbf Q</math> tensor is a second-order, traceless, symmetric tensor and is defined by<ref name="degennes">De Gennes, P. G., & Prost, J. (1993). The physics of liquid crystals (No. 83). Oxford university press.</ref><ref>Mottram, N. J., & Newton, C. J. (2014). Introduction to Q-tensor theory. arXiv preprint arXiv:1409.3542.</ref><ref>Kleman, M., & Lavrentovich, O. D. (Eds.). (2003). Soft matter physics: an introduction. New York, NY: Springer New York.</ref>
:<math>\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) </math>
where <math>S=S(T)</math> and <math>R=R(T)</math> are scalar order parameters, <math>(\mathbf n,\mathbf m)</math> are the two directors of the nematic phase and <math>T</math> is the temperature; in uniaxial liquid crystals, <math>R=0</math>. The components of the tensor are
:<math>Q_{ij} = S\left(n_in_j - \tfrac{1}{3}\delta_{ij}\right) + R\left(m_im_j - \tfrac{1}{3}\delta_{ij}\right)</math>
The states with directors <math>\mathbf n</math> and <math>-\mathbf n</math> are physically equivalent and similarly the states with directors <math>\mathbf m</math> and <math>-\mathbf m</math> are physically equivalent.
The <math>\mathbf Q</math>-tensor can always be diagonalized,
:<math> \mathbf Q= \frac{1}{3}\begin{bmatrix} 2S-R & 0 &0 \\ 0 & 2R-S & 0 \\ 0 & 0& -S-R\\ \end{bmatrix} </math>
The following are the two invariants of the <math>\mathbf Q</math> tensor,
:<math>\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)];</math>
the first-order invariant <math>\mathrm{tr}\,\mathbf Q=Q_{ii}=0</math> is trivial here. It can be shown that <math>(\mathrm{tr}\, \mathbf Q^2)^3\geq 6(\mathrm{tr}\, \mathbf Q^3)^2.</math> The measure of biaxiality of the liquid crystal is commonly measured through the parameter
:<math>\beta = 1 - 6\frac{(\mathrm{tr}\, \mathbf Q^3)^2}{(\mathrm{tr}\, \mathbf Q^2)^3}= \frac{27 S^2 R^2 (S-R)^2}{4(S^2-SR+R^2)^3}.</math>
==Uniaxial nematics==
In uniaxial nematic liquid crystals, <math>R=0</math> and therefore the <math>\mathbf Q</math>-tensor reduces to
:<math>\mathbf{Q} = S\left(\mathbf n\mathbf n - \frac{1}{3}\mathbf I\right).</math>
The scalar order parameter is defined as follows. If <math>\theta_{\mathrm{mol}}</math> represents the angle between the axis of a nematic molecular and the director axis <math>\mathbf n</math>, then{{r|degennes}}
:<math>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</math>
where <math>\langle\cdot\rangle</math> denotes the ensemble average of the orientational angles calculated with respect to the distribution function <math>f(\theta_{\mathrm{mol}})</math> and <math>d\Omega = \sin \theta_{\mathrm{mol}}d\theta_{\mathrm{mol}}d\phi_{\mathrm{mol}}</math> is the solid angle. The distribution function must necessarily satisfy the condition <math>f(\theta_{\mathrm{mol}}+\pi) = f(\theta_{\mathrm{mol}})</math> since the directors <math>\mathbf n</math> and <math>-\mathbf n</math> are physically equivalent.
The range for <math>S</math> is given by <math>-1/2\leq S\leq 1</math>, with <math>S=1</math> representing the perfect alignment of all molecules along the director and <math>S=0</math> representing the complete random alignment (isotropic) of all molecules with respect to the director; the <math>S=-1/2</math> 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== {{reflist|30em}}
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Category:Soft matter Category:Phase transitions Category:Liquid crystals