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

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

where $S=S(T)$ and $P=P(T)$ 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

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

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,

$\mathbf{Q}=-\frac{1}{2}\left(\begin{array}{ccc}S+P& 0& 0\\ 0& S-P& 0\\ 0& 0& -2S\\ \end{array}\right)$

The following are the invariants of the $\mathbf{Q}$ tensor

$\mathrm{t}\mathrm{r}{\mathbf{Q}}^2=Q_{ij}{Q}_{ji}=\frac{1}{2}(3S^2+P^2),\mathrm{t}\mathrm{r}{\mathbf{Q}}^3=Q_{ij}{Q}_{jk}{Q}_{ki}=\frac{3}{4}S(S^2-P^2);$

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

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

Uniaxial nematics

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

$\mathbf{Q}=S(\mathbf{n}\mathbf{n}-\frac{1}{3}\mathbf{I}).$

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 $\mathbf{n}$, then

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

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 $\mathbf{n}$ and $-\mathbf{n}$ 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