Coating Stress & Curvature

Stoney equation: relates thin-film stress to substrate curvature. Valid for thin films (film thickness ≪ substrate thickness).

Model: Textbook approximation

σ_film (MPa, tensile > 0)d_film per layer (nm)Number of layersE_substrate (GPa)ν_substratet_substrate (mm)

Curvature, 1 layer

7.114e-3 m⁻¹

R = 140.6 m, film side concave

Curvature, 10 layers

7.114e-2 m⁻¹

R = 14.06 m, film side concave

Curvature vs Number of Layers

Curvature vs Layer Thickness (10 layers)

Model

κ = 6 σ_f d_f N (1 − ν_s) / (E_s t_s²)

Stoney's equation with the substrate's biaxial modulus E_s/(1 − ν_s) (Janssen et al., Thin Solid Films 517, 1858, 2009). The forces σ_fd_f of identical layers add. Textbook approximation: valid for a total coating thickness much smaller than the substrate, deflections small compared with t_s, uniform stress and an isotropic substrate.

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

Stoney's equation with the substrate's biaxial modulus, for a coating much thinner than the substrate, small deflections, uniform stress and an isotropic substrate. The forces of identical layers add.

Stoney G. G. (1909). The tension of metallic films deposited by electrolysis. Proc. R. Soc. Lond. A 82, 172–175. doi:10.1098/rspa.1909.0021
Janssen G. C. A. M. et al. (2009). Celebrating the 100th anniversary of the Stoney equation for film stress: Developments from polycrystalline steel strips to single crystal silicon wafers. Thin Solid Films 517, 1858–1867. doi:10.1016/j.tsf.2008.07.014

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