Q Tensor

In physics, $\mathbf Q$-tensor is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase. The $\mathbf Q$ tensor is a second-order, traceless, symmetric tensor and is defined byDe Gennes, P. G., & Prost, J. (1993). The physics of liquid crystals (No. 83). Oxford university press.Kleman, M., & Lavrentovich, O. D. (Eds.). (2003). Soft matter physics: an introduction. New York, NY: Springer New York.

:$\mathbf = S\left(\mathbf n\otimes\mathbf n - \tfrac\mathbf I\right) + R\left(\mathbf m\otimes\mathbf m - \tfrac\mathbf I\right)$

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

:$Q_ = S\left(n_in_j - \tfrac\delta_\right) + R\left(m_im_j - \tfrac\delta_\right)$

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

The $\mathbf Q$-tensor can always be diagonalized,

:$\mathbf Q= \frac\begin 2S-R & 0 &0 \\ 0 & 2R-S & 0 \\ 0 & 0& -S-R\\ \end$

The following are the two invariants of the $\mathbf Q$ tensor,

:\mathrm\, \mathbf Q^2= Q_Q_ = \frac(S^2-SR+R^2), \quad \mathrm\,\mathbf Q^3 = Q_Q_Q_ = \frac (S^3+R^3)-3SR(S+R)

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

:$\beta = 1 - 6\frac= \frac.$

Uniaxial nematics

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

:$\mathbf = S\left(\mathbf n\mathbf n - \frac\mathbf I\right).$

The scalar order parameter is defined as follows. If $\theta_$ represents the angle between the axis of a nematic molecular and the director axis $\mathbf n$, then

:$S = \langle P_2(\cos \theta_)\rangle = \frac\langle 3 \cos^2 \theta_-1 \rangle = \frac\int (3 \cos^2 \theta_-1)f(\theta_) d\Omega$

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

The range for $S$ is given by $-1/2\leq S\leq 1$, with $S=1$ representing the perfect alignment of all molecules along the director and $S=0$ representing the complete random alignment (isotropic) of all molecules with respect to the director; the $S=-1/2$ 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

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Soft matter
Phase transitions
Liquid crystals