Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 2 / b

Table of contents
- Solution b

Solution

The bulk potential gives fluctuations in

λ

a nonzero restoring force, whereas slowly rotating the nematic director costs only gradients. At lengths much larger than the amplitude correlation length, it is therefore the leading gradient expansion approximation to set

λ = λ_{0}

while retaining

n (r)

.

Differentiating

Q_{ij} = 2 λ_{0} (n_{i} n_{j} - \frac{1}{2} δ_{ij})

gives

\nabla_{i} Q_{ij} = 2 λ_{0} [(\nabla \cdot n) n_{j} + (n \cdot \nabla) n_{j}] .

Substitution into the elastic term yields

F_{elastic} = 2 K λ_{0}^{2} ∣ (\nabla \cdot n) n + (n \cdot \nabla) n ∣^{2} .

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