# 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 by
    $\mathbf {Q} =S\left(\mathbf {n} \otimes \mathbf {n} -{\tfrac {1}{3}}\mathbf {I} \right)+R\left(\mathbf {m} \otimes \mathbf {m} -{\tfrac {1}{3}}\mathbf {I} \right)$
where $S=S(T)$ 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, $P=0$. The components of the tensor are
    $Q_{ij}=S\left(n_{i}n_{j}-{\tfrac {1}{3}}\delta _{ij}\right)+R\left(m_{i}m_{j}-{\tfrac {1}{3}}\delta _{ij}\right)$
The states with directors $\mathbf {n}$ 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,
    $\mathbf {Q} ={\frac {1}{3}}{\begin{bmatrix}2S-R&0&0\\0&2R-S&0\\0&0&-S-R\\\end{bmatrix}}$
The following are the two invariants of the $\mathbf {Q}$ tensor,
    $\mathrm {tr} \,\mathbf {Q} ^{2}=Q_{ij}Q_{ji}={\frac {2}{3}}(S^{2}-SR+R^{2}),\quad \mathrm {tr} \,\mathbf {Q} ^{3}=Q_{ij}Q_{jk}Q_{ki}={\frac {1}{9}}[2(S^{3}+R^{3})-3SR(S+R)];$
the first-order invariant $\mathrm {tr} \,\mathbf {Q} =Q_{ii}=0$ is trivial here. It can be shown that $(\mathrm {tr} \,\mathbf {Q} ^{2})^{3}\geq 6(\mathrm {tr} \,\mathbf {Q} ^{3})^{2}.$ The measure of biaxiality of the liquid crystal is commonly measured through the parameter
    $\beta =1-6{\frac {(\mathrm {tr} \,\mathbf {Q} ^{3})^{2}}{(\mathrm {tr} \,\mathbf {Q} ^{2})^{3}}}={\frac {27S^{2}R^{2}(S-R)^{2}}{4(S^{2}-SR+R^{2})^{3}}}.$

## Uniaxial nematics

In uniaxial nematic liquid crystals, $P=0$ and therefore the $\mathbf {Q}$ tensor reduces to
    $\mathbf {Q} =S\left(\mathbf {n} \mathbf {n} -{\frac {1}{3}}\mathbf {I} \right).$
The scalar order parameter is defined as follows. If $\theta _{\mathrm {mol} }$ represents the angle between the axis of a nematic molecular and the director axis $\mathbf {n}$, then
    $S=\langle P_{2}(\cos \theta _{\mathrm {mol} })\rangle ={\frac {1}{2}}\langle 3\cos ^{2}\theta _{\mathrm {mol} }-1\rangle ={\frac {1}{2}}\int (3\cos ^{2}\theta _{\mathrm {mol} }-1)f(\theta _{\mathrm {mol} })d\Omega$
where $\langle \cdot \rangle$ denotes the ensemble average of the orientational angles calculated with respect to the distribution function $f(\theta _{\mathrm {mol} })$ and $d\Omega =\sin \theta _{\mathrm {mol} }d\theta _{\mathrm {mol} }d\phi _{\mathrm {mol} }$ is the solid angle. The distribution function must necessarily satisfy the condition $f(\theta _{\mathrm {mol} }+\pi )=f(\theta _{\mathrm {mol} })$ 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.