Atoms evaporated onto a cold

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Non-equilibrium statistical mechanics and stochastic thermodynamics

Soft Matter at the Thermal Scale, Ch Nucleation and the lifetime of a metastable phase

Problem

Atoms evaporated onto a cold, flat substrate form a two-dimensional adsorbed layer (an adlayer) that is supersaturated with respect to a crystalline island phase growing on the surface. At the substrate temperature T=400 KT = 400\,\mathrm{K} the adlayer's chemical potential exceeds that of the crystalline islands by Δμ=0.060 eV\Delta\mu = 0.060\,\mathrm{eV} per atom. The islands are flat, one atom thick and roughly circular in outline, with an edge line tension Λ=0.15 eV nm−1\Lambda = 0.15\,\mathrm{eV\,nm^{-1}} and an area per atom of a0=0.20 nm2a_0 = 0.20\,\mathrm{nm^2} in the crystalline phase. For a flat, disc-shaped nucleus of radius rr, the free energy of formation is ΔF(r)=2πrΛ−πr2Δμa0,\Delta F(r) = 2\pi r\Lambda - \pi r^2\frac{\Delta\mu}{a_0}, the first term the cost of the disc's edge, the second the two-dimensional version of the bulk term, released as an area πr2\pi r^2 of the new phase forms. ((a)) By the same maximisation used for a three-dimensional nucleus, derive r∗r^* and ΔF∗\Delta F^* in terms of Λ\Lambda, a0a_0 and Δμ\Delta\mu. Evaluate r∗r^* in nanometres and ΔF∗\Delta F^* in units of kBTk_BT at 400 K400\,\mathrm{K}. ((b)) Islands nucleate at a constant rate II per unit area per unit time, and each grows with its radius increasing at a constant speed GG until it meets a neighbour. Build the extended area fraction Xext(t)X_{\rm ext}(t), and use the same random-overlap counting that turned the extended volume into X(t)X(t) in three dimensions to find the transformed area fraction X(t)X(t). ((c)) A second substrate is instead pre-seeded with n0n_0 islands per unit area, all present at t=0t = 0, with no further nucleation. Find X(t)X(t) for this case. ((d)) A plot of ln⁡[−ln⁡(1−X)]\ln[-\ln(1-X)] against ln⁡t\ln t for a real film is a straight line of slope 22. Which of the two nucleation histories does it indicate?

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