Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 344 / 3

A nematic director is an unoriented axis: $\mathbf{n}$ and $-\mathbf{n}$ describe the same molecular alignment. A vector average cannot represent this head-tail symmetry. The orientational second moment $⟨n_i{n}_j⟩$ is a symmetric second-rank tensor, and removing its isotropic part gives the traceless second-rank tensor $Q_{ij}\propto ⟨n_i{n}_j-{\delta }_{ij}/2⟩$. It vanishes for an isotropic angular distribution and transforms correctly under rotation matrices.

Every nonzero real symmetric traceless $2\times 2$ matrix has eigenvalues $\lambda /2,-\lambda /2$ and can be written $Q=\lambda (\mathbf{n}{\mathbf{n}}^{\mathsf{T}}-I/2)$ by choosing the eigenvector of the positive eigenvalue as director and $\lambda \ge 0$. Consequently $\mathrm{Tr}{Q}^2={\lambda }^2/2$. The bulk Landau-de Gennes free energy becomesFor $a<0$, its minimum occurs at ${\lambda }^2=-a/b$. The elastic density $K|\nabla \cdot Q{|}^2/2$ is nonnegative and vanishes for a uniform field. Thus, with no boundary condition imposing a texture, the pointwise lower bound is attained byThe negative-amplitude representation is equivalent in two dimensions to rotating the director by $\pi /2$, so the positive choice loses no tensor minima. This is minimization of the specified coarse-grained functional, not a claim of true thermodynamic long-range nematic order at every temperature in an infinite two-dimensional system. The PDF explicitly uses the divergence elastic term; the TeX's $|\nabla Q{|}^2$ loses the dot and would give different elastic coefficients.

Put $\mathbf{t}=\mathbf{\theta }$. The product rule applied to $Q_{ij}=\lambda (t_i{t}_j-{\delta }_{ij}/2)$ givesSince $\mathrm{Tr}{Q}^2={\lambda }^2/2$, substituting this vector into the divergence elastic term proves the requested form of the local density. For the radial amplitude and tangential director, the polar coordinates identities areHence the tangential integer nematic defect core hasThe $1/r$ director-curvature term survives when the amplitude is constant. Replacing the divergence norm by a full tensor-gradient norm would not give this expression; it is the PDF's divergence convention that is being evaluated.

Outside a core of radius $r_0$, set $\lambda \simeq {\lambda }_0$ to leading order. The uniform-state bulk energy cancels on subtraction, while the divergence elastic density is $K{\lambda }_0^2/(2r^2)$. Its two-dimensional area integral isThus the logarithmic energy of a planar nematic disclination givesThe core suppresses the nematic amplitude to regularize the singular director texture. Balancing lost bulk ordering energy against elasticity gives $r_0$ of order $\sqrt{K/|a|}$. The exact core radius and additive core energy depend on the amplitude profile.

The “local” amplitude disturbance need not be literally compactly supported. The radial Euler-Lagrange equation following from the PDF density isIts far-field expansion is $\lambda ={\lambda }_0[1-K/(2|a|r^2)+O(r^{-4})]$. These algebraic tails change only finite contributions, not the logarithm. A regular core has $\lambda =O(r^2)$ for this integer winding.

The large cutoff $L$ is set by the sample size for an isolated defect with corresponding imposed far-field winding, or by the distance to compensating defects or other texture-screening boundaries. An isolated nonzero winding in an infinite ordered plane has a logarithmically divergent excess energy; in a neutral pair, far-field distortions cancel beyond the separation. If $L$ is not much larger than $r_0$, the leading logarithm is not a dominant approximation.

For a strictly planar nematic director, the order-parameter manifold at fixed nonzero amplitude is ${\mathbb{RP}}^1$, homeomorphic to a circle. Its fundamental group is $\mathbb{Z}$. The topological charge of a two-dimensional nematic disclination is $q=\Delta \chi /(2\pi )$ for director angle $\chi$, defined modulo $\pi$; the integer winding of the physical doubled angle $2\chi$ is $m=2q$. Thus a $q=+1$ texture has $m=2\ne 0$ and cannot be continuously replaced by a uniform nonvanishing planar texture while the surrounding loop is held fixed:It can nevertheless split into two $q=+1/2$ defects, because their charges add to one. Such splitting preserves the total winding and is not topological unwinding. Allowing escape into the third dimension changes the manifold to ${\mathbb{RP}}^2$, with fundamental group $\mathbb{Z}/2\mathbb{Z}$; the doubled winding of this integer defect is then trivial. That escape is excluded by the stated two-dimensional director model.

Energetically, the defect is not the preferred isolated charge-one configuration at large separation of scales. At constant amplitude, the planar nematic divergence-elasticity identity gives $F_{\mathrm{el}}=(K{\lambda }_0^2/2)\int |\nabla \chi {|}^2$, so a charge $q$ has self-energy $\pi K{\lambda }_0^2{q}^2\log (L/r_0)$. Separating two $+1/2$ defects by $d$, with $r_0\ll d\ll L$, makes the far field still charge one but reduces the near-field logarithm:The logarithmic gain eventually dominates the finite core cost. HenceThis is energetic fission of an integer nematic disclination. It proves failure of global energetic stability, not the absence of every microscopic kinetic barrier or a full linear stability spectrum of the radial core. Core structure, boundaries or pinning can make a charge-one texture metastable over a finite time or length scale.

The total charge of the widely separated $+1,-1$ pair is zero, so a uniform planar nematic director is compatible with the far boundary if it imposes no additional winding. Each integer defect is susceptible to energetic fission of an integer nematic disclination: the positive defect can split into two $+1/2$ defects and the negative defect into two $-1/2$ defects. Like-sign elementary defects repel, whereas opposite signs attract through their elastic distortion. This commonly produces subsequent encounters and annihilations of opposite half-integer defects.

The attraction can also be seen before fission. In the fixed-amplitude outer theory, a neutral pair of charges $q,-q$ separated by $R$ hasIts energy therefore falls as the separation decreases. In nematic defect-antidefect annihilation, opposite defects meet, the amplitude can vanish in their merging cores, and the local winding disappears; the remaining smooth director distortion relaxes towards the uniform energy minimum.

Direct approach and annihilation of the unsplit integer pair is also topologically allowed. Whether fission happens first, the paths of the four elementary defects, and the annihilation time require a specified dynamical law and core mobilities; the free energy alone does not fix them. Pinning, imposed boundary textures, or sustained thermal production of defect pairs can prevent a strictly defect-free final state. The stated final state presumes dissipative relaxation with no such obstruction.