11.5: The Stoney Equation - the Thin Coating Limit
- Page ID
- 7854
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Origin of the Stoney Equation
The Stoney equation is still in widespread use. It relates the curvature of a substrate with a thin coating to the stress level within the coating (for an equal biaxial case). It was proposed in 1909 - a long time before the relationship described in earlier pages (for the general case in which the coating thickness is not negligible compared to that of the substrate) was established. However, it is easy to show that the Stoney equation can be derived from that relationship, by imposing the h << H condition, which allows all of the denominator terms except the last one to be discarded and (h + H) = H to be assumed, so that
\[\kappa=\frac{6 E_{\mathrm{d}}^{\prime} h \Delta \epsilon}{E_{\mathrm{s}}^{\prime} H^{2}}\]
with the biaxial moduli now being used. Furthermore, the \(h \ll H\) condition allows the assumption to be made that all of the misfit strain is accommodated in the coating (deposit), so only the coating is under stress. In addition, the Stoney equation is based on an equal biaxial stress state, so that the biaxial versions of these Young’s moduli should be used and the misfit strain can be expressed as
\[\Delta \epsilon=\frac{-\sigma_{d}}{E_{\mathrm{d}}^{\prime}}\]
recognizing that, with the convention we are using for \(\Delta \epsilon\), a positive value will generate a negative value for \(\sigma_{d}\) (ie a compressive stress). Substitution of this then leads to the Stoney equation:
\[\kappa=\frac{-6 E_{\mathrm{d}}^{\prime} h \sigma_{\mathrm{d}}}{E_{\mathrm{s}}^{\prime} H^{2} E_{\mathrm{d}}^{\prime}}=\frac{-6 h \sigma_{\mathrm{d}}}{E_{\mathrm{s}}^{\prime} H^{2}} \quad \therefore \sigma_{\mathrm{d}}=\frac{-E_{\mathrm{s}}^{\prime} H^{2}}{6 h\left(1-\nu_{\mathrm{s}}\right)} \kappa\]
(The minus sign is not always included, but the curvature should have a sign and, using the convention that a convex upper surface corresponds to positive curvature, this implies a negative deposit stress.) This equation allows a coating stress to be obtained from a (measured) curvature. Only E and ν values for the substrate are needed - this is convenient, since they are often known (whereas those of the coating may not be). A single stress value is obtained - if the coating is thin, then any through-thickness variation in its value is likely to be small. It is really the misfit strain that is the more fundamental measure of the characteristics of the system, but a stress value is often regarded as more easily interpreted.
Approach to Stoney Conditions
The Stoney equation is easy to use and, indeed, is widely used. However, it does have the limitation of being accurate only in a regime in which the curvatures tend to be relatively small. In some applications – such as with semiconductor wafers – surfaces are very smooth, so that highly accurate optical methods of curvature measurement are feasible and this is not such a problem. However, when curvatures are high (or need to be high for reliable measurement), the Stoney equation should not be used. The simulation below allows exploration of the conditions under which the Stoney equation gives a good or poor approximation to the actual behaviour. It should be noted that, while the stresses are scale-independent, the curvatures are not - changing the substrate thickness thus affects the plots on the left, but not those on the right.