Thesis (pdf) - Espci

2.4 Theoretical description 17
To prove this statement more rigorously, we consider the angular part
of the dissipation, 2π
0 dφ (∂Ψ/∂φ)2 . Recalling the definition of the defect
charge,
2π
0
dφ ∂Ψ
∂φ
= Ψ(2π) −Ψ(0) ≡ 2πs, (2.7)
we show that the overall absolute value of the rotation of the director field
Ψ in both positive and negative defects is the same. In the case of a positive
defect, the derivative of the director field with respect to the angular direction
is equal to 1 independently of the anisotropy, whereas in the negative case,
it is a function of the angular direction and the anisotropy (Eq. 2.3). As a
consequence,
2π
0
dφ
2 ∂Ψ+
=
∂φ
1
2π
dφ
2π 0
∂Ψ±
2 2π
≤ dφ
∂φ 0
2 ∂Ψ−
, (2.8)
∂φ
where in the last step we have used a general inequality of functional analysis
(the continuous version of the discrete triangular inequality). The immediate
implication of this result is that, due to elastic anisotropy, isolated positive
defects are always faster than negative ones (independently of the anisotropy
sign).
2.4.5 Ratio of defect velocities
The power supplied per unit length needed to maintain the defect moving at
a fixed velocity v is P = fv, where f is the drag force, and must be equal to
the dissipation rate Σ. This leads to the expression
f− 1
2 γv−
⎡
R 2π
log dφ ⎣1+ 1
1+αk2 ⎤2
− cos2ψ− ⎦ ,
rc
0
k−
rc
1+α cos2ψ−
R
f+ πγv+ log , (2.9)
for the negative and positive defect respectively. Although the attractive force
between defects is of elastic origin and can be in general a complicated function
of the position, the anisotropy α, and the viscosity of the material can
vary as a function of the lateral pressure, to compare the ratio between velocities
we only need to know that both the defects feel the same force f
in opposite directions (Newton’s third law), and the same material viscosity.
Using the expression for the drag force in Eq. 2.9 for the +1 and −1 defects,