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Q tensor

From Wikipedia, the free encyclopedia

In physics, tensor is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase.[1] The tensor is a second-order, traceless, symmetric tensor and is defined by[2][3][4]

where and are scalar order parameters, are the two directors of the nematic phase and is the temperature; in uniaxial liquid crystals, . The components of the tensor are

The states with directors and are physically equivalent and similarly the states with directors and are physically equivalent.

The tensor can always be diagonalized,

The following are the invariants of the tensor

the first-order invariant is trivial here. It can be shown that

Uniaxial nematics

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In uniaxial nematic liquid crystals, and therefore the tensor reduces to

The scalar order parameter is defined as follows. If represents the angle between the axis of a nematic molecular and the director axis , then[2]

where denotes the ensemble average of the orientational angles calculated with respect to the distribution function and is the solid angle. The distribution function must necessarily satisfy the condition since the directors and are physically equivalent.

The range for is given by , with representing the perfect alignment of all molecules along the director and representing the complete random alignment (isotropic) of all molecules with respect to the director; the case indicates that all molecules are aligned perpendicular to the director axis although such nematics are rare or hard to synthesize.

See also

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References

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  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. ^ a b 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.