Misplaced Pages

Q tensor

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Orientational order parameter

In physics, Q {\displaystyle \mathbf {Q} } tensor is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase. The Q {\displaystyle \mathbf {Q} } tensor is a second-order, traceless, symmetric tensor and is defined by

Q = S ( n n − 1 3 I ) + P ( m m − 1 3 I ) {\displaystyle \mathbf {Q} =S\left(\mathbf {n} \mathbf {n} -{\frac {1}{3}}\mathbf {I} \right)+P\left(\mathbf {m} \mathbf {m} -{\frac {1}{3}}\mathbf {I} \right)}

where S = S ( T ) {\displaystyle S=S(T)} and P = P ( T ) {\displaystyle P=P(T)} are scalar order parameters, ( n , m ) {\displaystyle (\mathbf {n} ,\mathbf {m} )} are the two directors of the nematic phase and T {\displaystyle T} is the temperature; in uniaxial liquid crystals, P = 0 {\displaystyle P=0} . The components of the tensor are

Q i j = S ( n i n j − 1 3 δ i j ) + P ( m i m j − 1 3 δ i j ) {\displaystyle Q_{ij}=S\left(n_{i}n_{j}-{\frac {1}{3}}\delta _{ij}\right)+P\left(m_{i}m_{j}-{\frac {1}{3}}\delta _{ij}\right)}

The states with directors n {\displaystyle \mathbf {n} } and − n {\displaystyle -\mathbf {n} } are physically equivalent and similarly the states with directors m {\displaystyle \mathbf {m} } and − m {\displaystyle -\mathbf {m} } are physically equivalent.

The Q {\displaystyle \mathbf {Q} } tensor can always be diagonalized,

Q = − 1 2 ( S + P 0 0 0 S − P 0 0 0 − 2 S ) {\displaystyle \mathbf {Q} =-{\frac {1}{2}}{\begin{pmatrix}S+P&0&0\\0&S-P&0\\0&0&-2S\\\end{pmatrix}}}

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

t r Q 2 = Q i j Q j i = 1 2 ( 3 S 2 + P 2 ) , t r Q 3 = Q i j Q j k Q k i = 3 4 S ( S 2 − P 2 ) ; {\displaystyle \mathrm {tr} \,\mathbf {Q} ^{2}=Q_{ij}Q_{ji}={\frac {1}{2}}(3S^{2}+P^{2}),\quad \mathrm {tr} \,\mathbf {Q} ^{3}=Q_{ij}Q_{jk}Q_{ki}={\frac {3}{4}}S(S^{2}-P^{2});}

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

β = 1 − 6 ( t r Q 3 ) 2 ( t r Q 2 ) 3 = 1 − 27 S 2 ( S 2 − P 2 ) 3 ( 3 S 2 + P 2 ) 3 . {\displaystyle \beta =1-6{\frac {(\mathrm {tr} \,\mathbf {Q} ^{3})^{2}}{(\mathrm {tr} \,\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 {\displaystyle P=0} and therefore the Q {\displaystyle \mathbf {Q} } tensor reduces to

Q = S ( n n − 1 3 I ) . {\displaystyle \mathbf {Q} =S\left(\mathbf {n} \mathbf {n} -{\frac {1}{3}}\mathbf {I} \right).}

The scalar order parameter is defined as follows. If θ m o l {\displaystyle \theta _{\mathrm {mol} }} represents the angle between the axis of a nematic molecular and the director axis n {\displaystyle \mathbf {n} } , then

S = ⟨ P 2 ( cos ⁡ θ m o l ) ⟩ = 1 2 ⟨ 3 cos 2 ⁡ θ m o l − 1 ⟩ = 1 2 ∫ ( 3 cos 2 ⁡ θ m o l − 1 ) f ( θ m o l ) d Ω {\displaystyle S=\langle P_{2}(\cos \theta _{\mathrm {mol} })\rangle ={\frac {1}{2}}\langle 3\cos ^{2}\theta _{\mathrm {mol} }-1\rangle ={\frac {1}{2}}\int (3\cos ^{2}\theta _{\mathrm {mol} }-1)f(\theta _{\mathrm {mol} })d\Omega }

where ⟨ ⋅ ⟩ {\displaystyle \langle \cdot \rangle } denotes the ensemble average of the orientational angles calculated with respect to the distribution function f ( θ m o l ) {\displaystyle f(\theta _{\mathrm {mol} })} and d Ω = sin ⁡ θ m o l d θ m o l d ϕ m o l {\displaystyle d\Omega =\sin \theta _{\mathrm {mol} }d\theta _{\mathrm {mol} }d\phi _{\mathrm {mol} }} is the solid angle. The distribution function must necessarily satisfy the condition f ( θ m o l + π ) = f ( θ m o l ) {\displaystyle f(\theta _{\mathrm {mol} }+\pi )=f(\theta _{\mathrm {mol} })} since the directors n {\displaystyle \mathbf {n} } and − n {\displaystyle -\mathbf {n} } are physically equivalent.

The range for S {\displaystyle S} is given by − 1 / 2 ≤ S ≤ 1 {\displaystyle -1/2\leq S\leq 1} , with S = 1 {\displaystyle S=1} representing the perfect alignment of all molecules along the director and S = 0 {\displaystyle S=0} representing the complete random alignment (isotropic) of all molecules with respect to the director; the S = − 1 / 2 {\displaystyle 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

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.
Categories: