Chapter 2
Active Nematics
2.1 Liquid Crystals
Anisotropic liquids display varieties of order different from polar alignment, while
retaining their fluidity. The most common kind of order is nematic, arising when
molecules are elongated and stiff. The uniaxial nematic alignment is characterized
by a director, which differs from a vector by lacking an arrow, and therefore is
invariant under rotation by π. Further kinds of ordering, sketched in Fig. 2.1, are
more restrictive. In the cholesteric phase, the prevailing orientation rotates around
some axis. In smectics, molecules are layered; within each layer, they are oriented
normally in the A phase, at a certain angle in the C phase, and at an angle rotated
between one layer and the next in the C* phase; within each layer, their positions are
disordered (de Gennes and Prost, 1995).
On the face of it, nematic, sometimes called apolar, order is less organized than
the polar (vectorial, or magnetic) order of flocks, but its description is actually more
involved. The director is not a well-defined geometric object. When it is presented as
a vector n, its paired combinations have to appear in all expressions; otherwise they
would be invariant under rotation by 2π, as vectors are, but not under rotations by π.
Fig. 2.1 Liquid-crystalline phases. Left to right: Nematic, cholesteric, smectic A, smectic C, and
smectic C ∗ (by Kebes – Own work, CC)
23
L. Pismen, Active Matter Within and Around Us, The Frontiers Collection,
https://doi.org/10.1007/978-3-030-68421-1_2
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Active Nematics
2.1 Liquid Crystals
Anisotropic liquids display varieties of order different from polar alignment, while
retaining their fluidity. The most common kind of order is nematic, arising when
molecules are elongated and stiff. The uniaxial nematic alignment is characterized
by a director, which differs from a vector by lacking an arrow, and therefore is
invariant under rotation by π. Further kinds of ordering, sketched in Fig. 2.1, are
more restrictive. In the cholesteric phase, the prevailing orientation rotates around
some axis. In smectics, molecules are layered; within each layer, they are oriented
normally in the A phase, at a certain angle in the C phase, and at an angle rotated
between one layer and the next in the C* phase; within each layer, their positions are
disordered (de Gennes and Prost, 1995).
On the face of it, nematic, sometimes called apolar, order is less organized than
the polar (vectorial, or magnetic) order of flocks, but its description is actually more
involved. The director is not a well-defined geometric object. When it is presented as
a vector n, its paired combinations have to appear in all expressions; otherwise they
would be invariant under rotation by 2π, as vectors are, but not under rotations by π.
Fig. 2.1 Liquid-crystalline phases. Left to right: Nematic, cholesteric, smectic A, smectic C, and
smectic C ∗ (by Kebes – Own work, CC)
23
L. Pismen, Active Matter Within and Around Us, The Frontiers Collection,
https://doi.org/10.1007/978-3-030-68421-1_2
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021