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 by²³⁴

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

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, ${\displaystyle R=0}$. 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, ${\displaystyle R=0}$ 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 ${\displaystyle \mathbf {n} }$, then²

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

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.