Fluctuations of topological disclination lines in nematic liquid crystals: Renormalization of the string model

Using the tensorial Landau–de Gennes theory, we study the fluctuations of a disclination line in a nematic liquid crystal. By analyzing the structure and the spectrum of the eigenmodes of a line of strength $\pm 1∕2$, we reassess the concept of line tension used in the simple string model of the disclination showing that it does not include the complete set of eigenmodes and must be renormalized. In general, the line tension considerably underestimates the thermal amplitude of a disclination and we find that it is only applicable to severely confined disclinations.

Figure 1

Tensorial core structure of the $1∕2$ disclination. The $\mathsf{Q}$ tensor eigensystem is represented by the orientation of the box, lengths of the edges are a measure of the eigenvalues.

Figure 2

Radial eigenvalue ${\lambda }_{\perp n}$ (in units of $1∕\tau$) and the norm squared (arbitrary units) of the $n$th radial eigenfunctions for two radii of confinement. For $R=200$, ${\lambda }_{\perp 1}=1.4\times {10}^{-5}$, ${\lambda }_{\perp 2}=4.5\times {10}^{-4}$, ${\lambda }_{\perp 3}=1.4\times {10}^{-3}$; note that ${\lambda }_{\perp 2}∕{\lambda }_{\perp 1}\approx 30$. Inset: the string fluctuation of the disclination line.

Figure 3

Three lowest radial eigenfunctions (not normalized) for $m=1$. For $i=0,1$ the functions $R_{i,m,n}^{\prime }$ overlap on the scale of the graph.

Figure 4

rms displacement of the disclination line (the length unit is $\xi$) vs the number of radial modes summed in the regimes of weak (upper diagram) and severe confinement; $n_{max}=1$ corresponds to the nonrenormalized string model. The maximum size of the computational domain is $R=580$, the values beyond this range ($R=1000$, $R=\mathrm{10\hspace{0.17em}000}$) are extrapolated.