Qtensor

qtensor, commonly referred to in the literature as the Q-tensor, is a mathematical construct used to describe the local orientational order of nematic liquid crystals. It is a symmetric, traceless second-order tensor that encodes both the strength of alignment and the preferred directions of rod-like molecules at a point in space.

In three dimensions, the Q-tensor can be understood through its eigenstructure. Its eigenvectors define the principal axes of molecular orientation, and its eigenvalues quantify the degree of ordering along those axes. In a uniaxial nematic, Q reduces to a form Q = S (n ⊗ n − I/3), where n is the local director (the average orientation) and S is the scalar order parameter. In biaxial nematics, additional order parameters are present, allowing Q to reflect more complex, multi-axis alignment. The tensor is invariant under rotations and transforms covariantly under coordinate changes.

The Q-tensor appears prominently in theoretical descriptions of liquid crystals, most notably in the Landau–de Gennes framework. The free energy often includes invariants such as Tr(Q^2) and Tr(Q^3), as well as gradient terms that account for spatial variations. Dynamic models couple the evolution of Q to flow and other fields, as in Beris–Edwards-type formulations, enabling the study of defect formation, phase transitions, and response to external fields.

Computationally, qtensor models are implemented to simulate director fields, defect dynamics, and phase behavior using finite element or finite difference methods. Software tools and libraries that implement Q-tensor formalisms are used across soft matter physics and materials science to analyze experimental data and explore theoretical predictions.