Q tensor explained

In physics,

Q tensor is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase.^[1] The Q tensor is a second-order, traceless, symmetric tensor and is defined by^{[2] [3] [4]} Q = S\left(n n - (1)/(3) I\right) + P\left(m m - (1)/(3) I\right)

where

S = S(T) and P = P(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 areQ _{ij} = S\left(n _{in} _{j} - (1)/(3) \delta _{ij} \right) + P\left(m _{im} _{j} - (1)/(3) \delta _{ij} \right)

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= - (1)/(2) \begin{pmatrix} S + P & 0 & 0 \\ 0 &S - P & 0 \\ 0 & 0 & -2 S\\ \end{pmatrix}

The following are the invariants of the

Q tensor\delta = Q _{ij} Q _{ij} = (1)/(2) ( 3 S ^{2+P} ^{2),} \Delta = Q _{ij} Q _{jk} Q _{ki} = (3)/(4) S(S ^{2-P} ^{2);}

the first-order invariant

Q _{ii} =0 is trivial here. It can be shown that \delta ^{3\geq} 6 \Delta ^{2.}

Uniaxial nematics

In uniaxial nematic liquid crystals,

P =0 and therefore the Q tensor reduces toQ = S\left(n n - (1)/(3) I\right).

The scalar order parameter is defined as follows. If

\theta _{mol} represents the angle between the axis of a nematic molecular and the director axis n , thenS = \langle P _{2(\cos} \theta _{mol} )\rangle = (1)/(2) \langle 3 \cos ^{2} \theta _{mol} -1 \rangle = (1)/(2) \int ( 3 \cos ^{2} \theta _{mol} -1 )f(\theta _{mol} ) d\Omega

where

\langle ⋅ \rangle denotes the ensemble average of the orientational angles calculated with respect to the distribution function f(\theta _{mol} ) and d\Omega = \sin \theta _{mol} d\theta _{mol} d\phi _{mol} is the solid angle. The distribution function must necessarily satisfy the condition f(\theta _{mol} + \pi) = f(\theta _{mol} ) since the directors n and - 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

Notes and References

1. De Gennes, P. G. (1969). Phenomenology of short-range-order effects in the isotropic phase of nematic materials. Physics Letters A, 30 (8), 454-455.
2. De Gennes, P. G., & Prost, J. (1993). The physics of liquid crystals (No. 83). Oxford university press.
3. Mottram, N. J., & Newton, C. J. (2014). Introduction to Q-tensor theory. arXiv preprint arXiv:1409.3542.
4. Kleman, M., & Lavrentovich, O. D. (Eds.). (2003). Soft matter physics: an introduction. New York, NY: Springer New York.