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2.4 Theoretical description 15
For s =+1, the energetically favorable solution to Eq. 2.2 for α > 0 is
ψ+ = ±π/2 (pure bend configuration), and ψ+ = 0,π for α < 0 (pure splay),
Fig. 2.11. For s = −1, the director field around the defect is given in implicit
form by (Landau et al., 1995)
φ[ψ−,k−] ≡ k−
ψ−
0
1 + α cos2x
1 + αk2 1/2 dx, (2.3)
− cos2x
where the integration constant k− is determined by φ[−4π,k−]=2π, which
is the topological condition for the director field of a −1 defect in polar coordinates,
ψ[φ + 2π] =ψ[φ] − 4π, for φ = 0. Since a negative defect is a
mixture of splay and bend configurations, for α = 0, the defect is deformed
with respect the corresponding isotropic one, minimizing the splay or bend
domain depending on the sign of the anisotropy, Fig. 2.12.
(a) (b) 0
(c)
ψ
-1
-2
-3
-4
-5
-6
0 0.5 1 1.5 2 2.5 3
Fig. 2.12. Deformation of a −1 defect. (a) Negative defect for isotropic elastic constants. (b)
The director angle ψ as a function of the polar angle φ, for isotropic elastic constants (dashed
line) and for an anisotropy α = 0.9 (solid line). (c) Negative defect for an anisotropy α = 0.9.
2.4.3 Defect motion
During defect motion, the director field is distorted. We have shown that back
flows are negligible in our system and consequently, in a quasistatic approximation,
we can assume that Eq. 2.3 holds during motion and describes the
instantaneous director field during rectilinear defect motion at velocity v.
The dissipation rate per unit length associate to rotations of the director field
is (Kleman and Laverntovich, 2003)
Σ = γ
dS
φ
2 ∂Ψ
, (2.4)
∂t