Stress

A coating pulls on the part it is deposited on. If the substrate is thin enough the part bends; if the film stores enough energy it cracks, or lets go of what is under it. Stress predicts all three from the constants on your materials and four numbers about the run.

No stress model is as good as the optical one. Film stress depends on the deposition process, not only on the material, so the constants are yours to measure and the numbers here compare one design against another rather than predicting a part in absolute terms.

What it needs

Every constant comes off the Mechanical tab of the Material Editor, one set per material. The window substitutes nothing: a constant nobody entered leaves the rows that need it blank, and the notice badge names the material and the field.

Quantity	Needs
Film stress	the film’s intrinsic stress; with its modulus, Poisson’s ratio, expansion coefficient and reference temperature, and the substrate’s expansion coefficient, the thermal terms too
Strain energy	the film’s stress, modulus and Poisson’s ratio
Delamination factor	the strain energies, and the surface energy of the two materials meeting at the interface
Cracking parameter	every film’s strain energy and surface energy
Edge shear	the modulus and Poisson’s ratio of every film and of the substrate, and the substrate thickness
Radius of curvature	the substrate’s modulus, Poisson’s ratio and thickness
Centre deflection	the radius, and the substrate diameter

The four inputs

T is the temperature the part is at when you read the stress off. T deposition is the temperature of the substrate while the films grow, which is what your thermocouple or pyrometer on the planet reads. It is not the temperature of the evaporant: the vapour arrives hot, but each atom gives up that energy to the growing film in nanoseconds, and what fixes a film’s strain-free lateral size is the substrate lattice it bonds to.

Leave both empty and every film simply carries its intrinsic stress, which is the honest answer for a design that describes no process. Set either one and both apply, the other reading 20 °C.

Substrate d is the thickness, the same value the Design Editor carries; editing it here edits that. The bow goes as one over its square, so it is the strongest lever in the window. Substrate ⌀ is the diameter, needed only for the centre deflection.

All four are saved with the design, so the same file prints the same table on any machine.

How a film’s stress is built

The film condenses strain free at its own deposition temperature and then takes the substrate’s temperature without changing its lateral size, which is the intrinsic strain. Its reference stress is that stress for a film deposited at the reference temperature on its record. From there to the evaluation temperature the film follows the substrate, and the expansion mismatch adds to the strain. Both strains become biaxial stress through E/(1 − ν):

σ = σ_ref + E/(1 − ν) · [ α_f (T_ref − T_dep) + (α_s − α_f) (T − T_dep) ]

Tensile is positive throughout. A film that expands more than its substrate goes tensile on cooling.

Reading the result

The chart is a bar per film in its material’s colour, tensile up and compressive down, so the layers carrying the force are visible without reading a number. Films are numbered 1 at the substrate, as everywhere else in TFStudio.

Whole part holds what the part does. Film force is Σ σ·d, the quantity Stoney’s equation turns into a curvature and the same number the STR merit operand targets, so a stack optimized to zero force reads zero here. Radius of curvature and centre deflection are both positive for the convex face a compressive coating leaves. Cracking parameter is the sum over films of U/2γ: at 1 the stack has the energy to pay for the two faces of a crack.

Per film holds the rest. Delamination is the strain energy of that film and everything outside it against the surface energy of the two materials meeting below it; at 1 there is the energy to let go there, and the number usually peaks at the substrate interface, which is where a coating does let go. Edge shear is the peak shear at the rim of the coating, k·|F|. Under a uniform biaxial stress the shear is zero everywhere inside the coating and rises only at its edge, which is why coatings fail from the rim inward.

Which coatings count follows the design’s evaluation mode, as in every analysis window: the active side alone with “ignore the other side” on, both otherwise. With coatings on both faces the two bending moments oppose, so the force is F_front − F_back; a mirrored back coating cancels the bow exactly and the window says so.

Nulling the bow

A target of zero film force is the zero-deflection condition. You do not have to reach it by hand: put an STR row in the Merit Function Editor with a target of 0 and the optimizer balances the stack for you. See Operands for how to weight it.

References

• E. Klokholm, “Intrinsic stress in evaporated metal films,” IBM J. Res. Dev. 31, 585 (1987): the strain-energy criteria for cracking and delamination.
• E. Suhir, “Predicted thermally induced stresses in, and the bow of, a circular substrate/thin-film structure,” J. Appl. Phys. 88, 2363 (2000).
• C. A. Klein, “How accurate are Stoney’s equation and recent modifications,” J. Appl. Phys. 88, 5487 (2000).
• C. A. Klein, “Normal and interfacial stresses in thin-film coated optics: the case of diamond-coated zinc sulfide windows,” Opt. Eng. 40, 1115 (2001): the interface forces, the curvature and the zero-deflection condition.