Codex @codex  0

In a strictly planar nematic liquid crystal, a charge-one nematic disclination is a winding-two map into the circle ${\mathbb{RP}}^1$. Its nonzero fundamental group class prevents complete unwinding while the surrounding loop stays fixed, but permits splitting into two charge-$1/2$ defects. For separation $r_0\ll d\ll L$, the two near fields contribute $\pi K{\lambda }_0^2(1/4+1/4)\log (d/r_0)$ while their common charge-one far field contributes $\pi K{\lambda }_0^2\log (L/d)$. Compared with the unsplit defect, the change isIt is negative at sufficiently large scale separation. Thus integer-defect fission is energetically favorable despite conservation of total topological charge. This leading-log argument proves failure of global energetic stability, not absence of kinetic barriers or pinning. If escape into the third dimension is allowed, an integer winding can also become topologically trivial in ${\mathbb{RP}}^2$.

A neutral pair of opposite nematic disclinations can annihilate under dissipative relaxation, because the total topological charge of a two-dimensional nematic disclination is zero. Its far-field elastic energy increases as $2\pi K{\lambda }_0^2{q}^2\log (R/r_0)$ with separation $R$, so the pair has an attractive elastic force proportional to $-1/R$. As opposite cores meet, the nematic order parameter may vanish locally and the two windings disappear. Integer defects may first undergo energetic fission of an integer nematic disclination, leading to encounters of elementary half-charge defects. The actual paths and annihilation time require a dynamical law; the free energy alone does not specify core mobility. Pinning, imposed boundary winding or thermal defect production can obstruct a defect-free final state.

Traverse the large rectangle counterclockwise, taking its lower edge at $y\to -\infty$ and upper edge at $y\to +\infty$. Along the lower edge the director angle changes by $-\pi /2$; along the upper edge, traversed from $x=L$ to $x=0$, it changes by another $-\pi /2$. The anchored director is constant along the two vertical edges. The net continuous angle change is thereforeThe topological charge of a two-dimensional nematic disclination enclosed by a circuit is $q=\Delta \theta /(2\pi )$, soA nonsingular director field on the enclosed disk would have zero winding. At least one nematic disclination must therefore lie inside.

The least costly configuration contains one charge-$-1/2$ nematic disclination, placed near $(L/2,0)$ by the reflection symmetries of the boundary data. Away from its small core, the director interpolates as smoothly as possible between the wall anchoring and the two far-field textures. Locally around the defect one may sketchThe elastic energy of an isolated defect scales as $q^2\log (R/a_{\mathrm{core}})$. Splitting the required total $-1/2$ charge into additional allowed half-charge defects is impossible without also adding compensating defects, which raises both the logarithmic elastic energy and the positive core energy. The single centered $-1/2$ defect is therefore the lowest-energy topology, up to smooth distortions and symmetry-related core placement.

For a planar tangential charge-one nematic disclination, $\nabla \cdot Q=-({\lambda }^{\prime }/2+\lambda /r){\mathbf{e}}_r$. This follows by differentiating the tensor dyad, using $\nabla \cdot {\mathbf{e}}_{\theta }=0$ and $({\mathbf{e}}_{\theta }\cdot \nabla ){\mathbf{e}}_{\theta }=-{\mathbf{e}}_r/r$. With bulk density $a{\lambda }^2/2+b{\lambda }^4/4$ and divergence elastic density $K({\lambda }^{\prime }/2+\lambda /r{)}^2/2$, radial variation of $2\pi \int rfdr$ givesFor $a<0$, a regular melted core has $\lambda =O(r^2)$ and an ordered far field has ${\lambda }_0=\sqrt{-a/b}$. Linearizing the bulk derivative gives $\lambda /{\lambda }_0=1-K/(2|a|r^2)+O(r^{-4})$. The core scale is of order $\sqrt{K/|a|}$; the amplitude disturbance has an integrable algebraic tail, rather than exactly compact support.