Thermal Evaporation

Thermal evaporation of metals: vapour pressure from the Alcock et al. fits, Hertz–Knudsen flux, growth rate and thickness profile from a small cosine source, mean free path and structure zone.

Model: Textbook approximation

EvaporantSource temperature (°C)Source–substrate distance h (cm)Source (melt surface) diameter (cm)Substrate radius (cm)Evaporation coefficient αChamber pressure (Torr)Substrate temperature (°C)

Aluminium: M = 26.982 g/mol, ρ = 2.70 g/cm³, T_m = 933.47K. Metals only: oxides and fluorides evaporate by dissociation or sublimation, which these fits don't cover.

Vapour pressure

0.817 Pa

6.13e-3 Torr

Growth rate at the centre

0.20 nm/s

1.99 Å/s; film at bulk density, sticking 1

Thickness at the substrate edge

94.7 %

of the centre, at 5.0 cm (flat substrate)

Evaporation flux (α p √(M/2πRT))

4.84e-4 kg m⁻² s⁻¹

Mean free path (residual air)

50.0 m

Kn = λ/h = 1.7e+2: straight-line flight

Mean speed of the vapour

1075 m/s

√(8RT/πM) at the source temperature

Homologous temperature T_s/T_m

0.40

Movchan–Demchishin zone 2: dense columnar grains

Model

log₁₀(p/atm) = A + B/T + C log₁₀T (Alcock, Itkin & Horrigan 1984; solid below T_m, liquid above)

Γ = α p √(M/(2πRT)) (Hertz–Knudsen–Langmuir mass flux)

R = Γ π r_s² / (π ρ h²) at the centre; d/d₀ = 1/(1 + (l/h)²)² across a flat substrate (cosine source)

λ = 5×10⁻³ cm·Torr / P (air at room temperature); zones at T_s/T_m = 0.3 and 0.45 (metals)

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Alcock fits (±5 %). Free-evaporation flux (α ≤ 1) from a small cosine source onto a flat, parallel substrate; sticking 1, bulk density. No vapour cloud, gas scattering or source depletion. Metal zones.

Alcock C. B., Itkin V. P., Horrigan M. K. (1984). Vapour pressure equations for the metallic elements: 298–2500 K. Can. Metall. Q. 23, 309. doi:10.1179/cmq.1984.23.3.309
Ohring M. (2002). Materials Science of Thin Films, 2nd ed., ch. 3 (Hertz–Knudsen flux, surface-source distribution, thickness uniformity). Academic Press. doi:10.1016/b978-012524975-1/50006-9
Ohring M. (2002). Materials Science of Thin Films, 2nd ed., ch. 2 (mean free path of air). Academic Press. doi:10.1016/b978-012524975-1/50005-7
Thornton J. A. (1977). High rate thick film growth (Movchan–Demchishin zones). Annu. Rev. Mater. Sci. 7, 239. doi:10.1146/annurev.ms.07.080177.001323

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