24
2 Active Nematics
Fig. 2.2 Left to right: Splay, bend, and twist distortion of nematic alignment (Kleman and Lavrentovich, 2003)
Thus, the standard expression for the elastic energy density of a uniaxial nematic is
F =
1
2
K 1 (div n)
2
+ K 2 (n · curl n)
2
+ K 3 (n × curl n)
2
.
(2.1)
The three terms in this expression correspond to the energies of splay, bend, and twist
distortion of nematic alignment, sketched in Fig. 2.2, which depend on the invariant
quadratic combinations of the director with its divergence and curl; the coefficients
are the respective elasticities. In 2D, only splay and bend distortions remain, and the
respective energies are expressed through the partial derivatives of the components
of the 2D director n x , n y as (div n) 2 = (∂ x n x + ∂ y n y ) 2 and (curl n) 2 = (∂ y n x − ∂ x n y ) 2 .
Notably, in this formulation there is no difference between distortions of the polar
or nematic order. Yet, what this definition of the elastic energy is lacking is that,
because n is presumed to be a unit vector, it cannot be extended to situations where
the liquid is not perfectly oriented locally. This is a serious flaw, since distorted
textures commonly include defects, where alignment is completely lost, and defects
in polar and nematic order are quite different (see Sect. 2.4).
What fully characterizes the nematic state is not a vector but a symmetric tensor
Q with zero trace. In 2D, it is expressed as
Q =
ρ
√ 2
cos 2θ sin 2θ
sin 2θ − cos 2θ
,
(2.2)
where θ is the inclination angle and ρ measures the deviation from isotropy, whence
ρ = 1 corresponds to the perfectly aligned nematic with the director having vector
components n x = cos θ, n y = sin θ. The tensorial description can also be extended in
3D to the case where the alignment is not axially symmetric, e.g., where microscopic
elements are shaped as ellipsoids with three unequal axes rather than like the spindles
in Fig. 2.1. In this formalism, elastic interactions, taking account also of the energy
of defects, are characterized by a tensor of the fourth rank, which retains far more
terms than those present in (2.1), even when this number has been reduced to the
minimum using the available symmetries.
2 Active Nematics
Fig. 2.2 Left to right: Splay, bend, and twist distortion of nematic alignment (Kleman and Lavrentovich, 2003)
Thus, the standard expression for the elastic energy density of a uniaxial nematic is
F =
1
2
K 1 (div n)
2
+ K 2 (n · curl n)
2
+ K 3 (n × curl n)
2
.
(2.1)
The three terms in this expression correspond to the energies of splay, bend, and twist
distortion of nematic alignment, sketched in Fig. 2.2, which depend on the invariant
quadratic combinations of the director with its divergence and curl; the coefficients
are the respective elasticities. In 2D, only splay and bend distortions remain, and the
respective energies are expressed through the partial derivatives of the components
of the 2D director n x , n y as (div n) 2 = (∂ x n x + ∂ y n y ) 2 and (curl n) 2 = (∂ y n x − ∂ x n y ) 2 .
Notably, in this formulation there is no difference between distortions of the polar
or nematic order. Yet, what this definition of the elastic energy is lacking is that,
because n is presumed to be a unit vector, it cannot be extended to situations where
the liquid is not perfectly oriented locally. This is a serious flaw, since distorted
textures commonly include defects, where alignment is completely lost, and defects
in polar and nematic order are quite different (see Sect. 2.4).
What fully characterizes the nematic state is not a vector but a symmetric tensor
Q with zero trace. In 2D, it is expressed as
Q =
ρ
√ 2
cos 2θ sin 2θ
sin 2θ − cos 2θ
,
(2.2)
where θ is the inclination angle and ρ measures the deviation from isotropy, whence
ρ = 1 corresponds to the perfectly aligned nematic with the director having vector
components n x = cos θ, n y = sin θ. The tensorial description can also be extended in
3D to the case where the alignment is not axially symmetric, e.g., where microscopic
elements are shaped as ellipsoids with three unequal axes rather than like the spindles
in Fig. 2.1. In this formalism, elastic interactions, taking account also of the energy
of defects, are characterized by a tensor of the fourth rank, which retains far more
terms than those present in (2.1), even when this number has been reduced to the
minimum using the available symmetries.