In physics,
tensor is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase.
tensor is a second-order, traceless, symmetric tensor and is defined by[2][3][4] where
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
are physically equivalent and similarly the states with directors
are physically equivalent.
tensor can always be diagonalized, The following are the two invariants of the
tensor, the first-order invariant
{\displaystyle \mathrm {tr} \,\mathbf {Q} =Q_{ii}=0}
The measure of biaxiality of the liquid crystal is commonly measured through the parameter In uniaxial nematic liquid crystals,
tensor reduces to The scalar order parameter is defined as follows.
θ
{\displaystyle \theta _{\mathrm {mol} }}
represents the angle between the axis of a nematic molecular and the director axis
denotes the ensemble average of the orientational angles calculated with respect to the distribution function
θ
{\displaystyle f(\theta _{\mathrm {mol} })}
is the solid angle.
The distribution function must necessarily satisfy the condition
θ
are physically equivalent.
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.