Q tensor

Template:Short description In physics, $𝐐$ tensor is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase. The $𝐐$ tensor is a second-order, traceless, symmetric tensor and is defined by

$𝐐=S(𝐧\otimes 𝐧-\frac{1}{3}𝐈)+R(𝐦\otimes 𝐦-\frac{1}{3}𝐈)$

where $S=S(T)$ and $R=R(T)$ are scalar order parameters, $(𝐧,𝐦)$ 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

$Q_{ij}=S(n_i{n}_j-\frac{1}{3}{\delta }_{ij})+R(m_i{m}_j-\frac{1}{3}{\delta }_{ij})$

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

The $𝐐$ tensor can always be diagonalized,

$𝐐=\frac{1}{3}\left[\begin{array}{ccc}2S-R& 0& 0\\ 0& 2R-S& 0\\ 0& 0& -S-R\end{array}\right]$

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

$\mathrm{t}\mathrm{r}{𝐐}^2=Q_{ij}{Q}_{ji}=\frac{2}{3}(S^2-SR+R^2),\mathrm{t}\mathrm{r}{𝐐}^3=Q_{ij}{Q}_{jk}{Q}_{ki}=\frac{1}{9}[2(S^3+R^3)-3SR(S+R)];$

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

$\beta =1-6\frac{(\mathrm{t}\mathrm{r}{𝐐}^3{)}^2}{(\mathrm{t}\mathrm{r}{𝐐}^2{)}^3}=\frac{27S^2{R}^2(S-R{)}^2}{4(S^2-SR+R^2{)}^3}.$

Uniaxial nematics

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

$𝐐=S(𝐧𝐧-\frac{1}{3}𝐈).$

The scalar order parameter is defined as follows. If ${\theta }_{\mathrm{m}\mathrm{o}\mathrm{l}}$ represents the angle between the axis of a nematic molecular and the director axis $𝐧$, thenTemplate:R

$S=⟨P_2(\cos {\theta }_{\mathrm{m}\mathrm{o}\mathrm{l}})⟩=\frac{1}{2}⟨3{\cos }^2{\theta }_{\mathrm{m}\mathrm{o}\mathrm{l}}-1⟩=\frac{1}{2}\int (3{\cos }^2{\theta }_{\mathrm{m}\mathrm{o}\mathrm{l}}-1)f({\theta }_{\mathrm{m}\mathrm{o}\mathrm{l}})d\mathrm{\Omega }$

where $⟨\cdot ⟩$ denotes the ensemble average of the orientational angles calculated with respect to the distribution function $f({\theta }_{\mathrm{m}\mathrm{o}\mathrm{l}})$ and $d\mathrm{\Omega }=\sin {\theta }_{\mathrm{m}\mathrm{o}\mathrm{l}}d{\theta }_{\mathrm{m}\mathrm{o}\mathrm{l}}d{\varphi }_{\mathrm{m}\mathrm{o}\mathrm{l}}$ is the solid angle. The distribution function must necessarily satisfy the condition $f({\theta }_{\mathrm{m}\mathrm{o}\mathrm{l}}+\pi )=f({\theta }_{\mathrm{m}\mathrm{o}\mathrm{l}})$ since the directors $𝐧$ and $-𝐧$ are physically equivalent.

The range for $S$ is given by $-1/2\le S\le 1$, 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