 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-344/1/f/solution

The interfacial tension is the excess free energy per unit interfacial area:

σ = \int_{- \infty}^{\infty} [f (ϕ) - f (ϕ_{B}) + \frac{κ ( ϕ )}{2} (ϕ^{'})^{2}] d x .

The first integral from part e equates the first two terms inside the brackets to

κ (ϕ) (ϕ^{'})^{2} /2

. Hence

σ = \int_{- \infty}^{\infty} κ (ϕ) (\partial_{x} ϕ)^{2} d x .

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