Abstract
In the electrochemical deposition of thin films, the measurement of the current-time curve does not allow for a direct determination of the nucleus growth law, electrode surface coverage, and mean film thickness. In this work, we present a theoretical approach suitable to gain insight into these quantities from the knowledge of nucleation density, solution parameters, and current-time behavior. The model applies to both isotropic and anisotropic growth rates of nuclei and a study on the effect of nucleus shape and aspect ratio on the kinetics is presented. Computer simulations and experimental results from literature are also discussed in the framework of the present approach.
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The authors are indebted with Dr. E. Tamburri for the helpful discussions and the critical reading of the manuscript.
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Appendix
Appendix
Let Q be a point at height h from the electrode surface and O its projection on the electrode surface. The area of the capture zone of point Q for nuclei born at time t′ < t is equal to πr(t, t′)2 where \( r\left(t,{t}^{\prime}\right)\equiv \overline{ON} \) and the point N, on the electrode surface, is such that a nucleus born at N will reach point Q at time t (Fig. 10). Since the points on the nucleus surface satisfy the equation, \( \frac{x^2+{y}^2}{A^2\left(t-{t}^{\prime}\right)}+\frac{z^2}{B^2\left(t-{t}^{\prime}\right)}=1 \), we get
namely
where \( \alpha =\frac{B}{A} \). Since r(t, t′)2 > 0 and \( {t}^{\prime }<{t}_{\mathrm{mx}}^{\prime }=t-\frac{h^2}{A^2{\alpha}^2}=t-\frac{h^2}{B^2} \), ξc(h, t) is eventually computed by summing over the population of nuclei,
For site-saturation nucleation Eq. 23 gives Eq. 14:
In what follows, we illustrate the two-rate model of [7]. In this method, growth rates pr and ph are defined for “on-plane” and “normal“directions, respectively. These quantities provide the time dependence of the coordinates, r and h, of a point lying on the nucleus surface at time t:
In these equations, t′ is the birth time of the nucleus, and w > t′ is the time of arrival of the nucleus at height h, i.e., the semi-axis of the ellipsoid at time w equals h: B2(w − t′) = h2. Equations 25 and 26 provide h = h(t′, w) that can be inverted to get w = w(t′, h). Next, by considering the 2D phase transition on the layer at height h, we obtain (Fig. 10): \( {\xi}_{\mathrm{c}}\left(h,t\right)=H\left({t}_{\mathrm{mx}}^{\prime}\right)\pi {\int}_0^{t_{\mathrm{mx}}^{\prime }}r{\left(t,w\left({t}^{\prime },h\right)\right)}^2\dot{N}\left({t}^{\prime}\right)\mathrm{d}{t}^{\prime } \) with \( w\left({t}_{\mathrm{mx}}^{\prime },h\right)=t \), which becomes, using Eq. 25,
that is the expression derived in [6]. Next, we apply Eq. 27 to the ellipsoidal and cone-shaped nuclei in diffusional growth. To this purpose, the pr and ph functions have to be determined.
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Ellipsoidal nuclei. In this case, pr and ph (Eqs. 25 and 26) should satisfy Eq. 21. It is possible to show that \( {p}_{\mathrm{r}}(t)=\frac{A}{2\sqrt{t-w}} \) and \( {p}_{\mathrm{h}}(t)=\frac{B}{2\sqrt{t-{t}^{\prime }}} \) are the appropriate functions. Use of these rates in Eq. 27 provides
Moreover, Eq. 26 gives \( h\left({t}^{\prime },w\right)=B\sqrt{w-{t}^{\prime }} \) which implies \( w\left({t}^{\prime },h\right)=\frac{h^2}{B^2}+{t}^{\prime } \). Performing the integral in Eq. 28, we eventually obtain
For site-saturation nucleation, Eq. 29 gives
that leads to Eq. 24.
Cone-shaped nuclei. In the case of right circular cones \( {p}_{\mathrm{r}}=\frac{\mu_1}{\sqrt{t}} \), \( {p}_{\mathrm{h}}=\frac{\mu_2}{\sqrt{t}} \) (with constant μi) and \( w\left({t}^{\prime },h\right)={\left(\frac{h}{2{\mu}_2}+\sqrt{t^{\prime }}\right)}^2 \). For site saturation nucleation, Eq. 27 becomes (\( {t}_{\mathrm{mx}}^{\prime }=\sqrt{t}-\frac{h}{2{\mu}_2} \))
The volume of the deposit is equal to
The current density is obtained from the time derivative of Eq. 32, \( J(t)= zF\frac{\rho }{M}\frac{\mathrm{d}V(t)}{\mathrm{d}t} \), as
that is Eq. 19.
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Politi, S., Tomellini, M. Kinetics of island growth in the framework of “planar diffusion zones” and “3D nucleation and growth” models for electrodeposition. J Solid State Electrochem 22, 3085–3098 (2018). https://doi.org/10.1007/s10008-018-4011-2
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DOI: https://doi.org/10.1007/s10008-018-4011-2