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2.4 Theoretical description 13
the effect of anisotropic boundary conditions can be neglected with respect to
inner elasticity in the regimes where pure bend textures are obtained (Ignés-
Mullol et al., 2005). More generally, Eq. 2.1 should include contributions
from a density order parameter so as to account for the finite compressibility
of this system (Hinshaw et al., 1988; Tabe et al., 1999). Nevertheless, at a
fixed lateral pressure, explicit density contributions can be renormalized into
effective elastic constants and Eq. 2.1 is of general applicability, keeping in
mind that the value of the elastic constants will depend on the applied pressure.
a b
v−
Fig. 2.9. The motion of a negative defect along the curved boundary (a) can be mapped into
that of a defect in rectilinear motion (b). This mapping is justified because the director field
around ±1/2 boundary defects rotates when defects move along the curved boundary. In (a),
we see a sketch of a −1/2 defect with a virtual −1/2 part outside the boundary (in dashed
lines) moving at a velocity v− and its director field is rotated accordingly to its motion (the
director normal to the boundary is preserved as the defect moves as indicated with the lines
pointing to the center of the droplet). In addition, the effect of the curvature of the droplet on
the velocities is negligible in our experimental conditions (droplet radii considered in the range
30-60 µm ). In (b), a −1 defect is decomposed in a −1/2 defect and a virtual −1/2 defect in
dashed lines. This corresponds to having a −1/2 defect in the boundary and consequently, the
results of rectilinear ±1 defect motion can be extrapolated to that of semi-integer boundary
defects.
In the experimental section, we observed topological defects of charges
±1 in the bulk and ±1/2 on the boundary, Fig. 2.8. Fractional defects in a
nematic order parameter can not exist on the bulk by topological constraints
and must be confined on the boundary. Fig. 2.8A shows that the director field
around ±1/2 defects rotates when defects move along the curved boundary,
that is, the angle between the ±1/2 director field and the tangent to the
boundary is kept constant as the defect moves, Fig. 2.9a. Therefore, the director
field around a defect moving along the boundary can be mapped into that
of a defect in rectilinear motion. On the other hand, each boundary defect can
be regarded as a ±1 with half its spatial extend being virtual. This allows to