“Linear diffusion domain” approach for modeling the kinetics of electrodeposition: a two-dimensional study

Abstract

A study is presented on diffusion-controlled growth of multi-nuclei systems in 2D space. The effect of correlation among diffusion fields on growth law of nuclei is investigated by means of numerical computations. The outputs of these simulations are employed to test the validity of the analytical approach based on the concept of linear diffusion domain. Both growth law of single nucleus and the total rate of film growth have been investigated by taking into account the finite-size effect of nuclei on the size probability density of the Voronoi cells. It is shown that the linear-diffusion-domain approach provides a satisfactory description of such an involved process. The application of the Voronoi cell approach is extended to a 3D system and makes it possible to determine the kinetics of nucleus growth as a function of cell size.

Appendices

Appendix 1

In order to test the numerical method, we consider diffusional growth of an isolated nucleus. For this system, an analytical solution can be obtained solving Fick’s law into a circular crown. Using polar coordinates, the boundary and initial conditions are the following: c(R₀, t) = 0, c(R₁, t) = c₀, and c(r, 0) = c₀ for R₀ ≤ r ≤ R₁, where the R₁ value has to be chosen in order to avoid border effects and to ensure that at the maximum time considered in the computation ${\left.{\partial}_rc\left(r,t\right)\ \right|}_{r={R}_1}\cong 0$. Using reduced variables, $\chi =\frac{c}{c_0}$, $\tau =\frac{Dt}{R_0^2}$, $\xi =\frac{r}{R_0}$ and $\beta =\frac{R_1}{R_0}$, 2D diffusion equation in polar coordinates becomes

$$ \frac{\mathrm{\partial \upchi }}{\partial \tau }=\frac{\partial^2\upchi}{\partial {\upxi}^2}+\frac{1}{\upxi}\frac{\mathrm{\partial \upchi }}{\mathrm{\partial \upxi }}. $$

(17)

The solution of Eq. 17, with boundary and initial conditions χ(1, t) = 0, χ(ξ, 0) = χ(β, t) = 1 (for 1 ≤ ξ ≤ β), is

$$ \chi \left(\xi, \tau \right)=\frac{\ln \xi }{\ln \beta }+\pi \sum \limits_{n=1}^{\infty}\frac{J_0\left({y}_n\right)U\left(\xi, {y}_n\right)}{J_0\left({y}_n\right)+{J}_0\left({\beta y}_n\right)}{e}^{-{y}_n^2\tau }-\pi \sum \limits_{n=1}^{\infty}\frac{J_0\left({y}_n\right)U\left(\xi, {y}_n\right)}{J_0\left({y}_n\right)-{J}_0\left({\beta y}_n\right)}{e}^{-{y}_n^2\tau }, $$

(18)

where J_m and Y_m are the first and second order Bessel functions of order m, respectively, the y_n‘s are the roots of the equation U(y) = 0, with U(y) = J₀(y)Y₀(βy) − J₀(βy)Y₀(y) and U(ξ, y_n) = J₀(ξy_n)Y₀(βy_n) − J₀(βy_n)Y₀(ξy_n) [26].

The computation of the growth rate is done through Fick’s first law which requires the determination of the gradient of χ:

$$ \frac{\partial \chi \left(\xi, \tau \right)}{\partial \xi }=\frac{1}{\xi \ln \beta }+\pi \sum \limits_{n=1}^{\infty}\frac{J_0\left({y}_n\right){U}_{\xi}\left(\xi, {y}_n\right)}{J_0\left({y}_n\right)+{J}_0\left({\beta y}_n\right)}{e}^{-{y}_n^2\tau }-\pi \sum \limits_{n=1}^{\infty}\frac{J_0\left({y}_n\right){U}_{\xi}\left(\xi, {y}_n\right)}{J_0\left({y}_n\right)-{J}_0\left({\beta y}_n\right)}{e}^{-{y}_n^2\tau}\kern0.5em , $$

(19)

where

$$ {U}_{\xi}\left(\xi, {y}_n\right)={y}_n{J}_0\left(\beta {y}_n\right){Y}_1\left(\xi {y}_n\right)-{y}_n{Y}_0\left(\beta {y}_n\right){J}_1\left(\xi {y}_n\right). $$

The growth of the nucleus (in dimensionless units) is computed from Eq. 3 by exploiting the circular symmetry of the diffusion problem and using $I\left(\tau \right)=\pi {R}_0^2{c}_0{\left.\frac{\partial \chi \left(\xi, \tau \right)}{\partial \xi}\right|}_{\xi =1}$:

$$ \overset{\sim }{R}\left(\tau \right)=\frac{R\left(\tau \right)}{{\alpha R}_0}={\left[{\int}_0^{\tau }{\left.\frac{\partial \chi \left(\xi, \tau \prime \right)}{\partial \xi}\right|}_{\xi =1} d\tau \prime \right]}^{1/2}, $$

(20)

where $\alpha =\sqrt{2{c}_0/\rho }$. In Fig. 11 of the Appendix, a comparison between numerical and analytical growth laws of an isolated nucleus is given, confirming the validity of the method here proposed.

Fig. 11

Comparison between numerical (solid symbols) and analytical (open symbols, Eqs. 19 and 20) growth law of isolated nucleus. The good agreement between this data set validates the proposed numerical method. In the inset, the numerical results and the approximate analytical solution (Eq. 21) are displayed as solid and open symbols, respectively

In the inset of Fig. 11 of the Appendix, we also report the $\overset{\sim }{R}\left(\tau \right)=\frac{R\left(\tau \right)}{{\alpha R}_0}$ function obtained from the concentration profile

$$ \chi \left(\upxi, \tau \right)=1-\frac{\varPhi \left(\upxi /\sqrt{\tau}\right)}{\varPhi \left(1/\sqrt{\tau}\right)}, $$

(21)

where ξ = r/R₀ and $\varPhi (x)={\int}_x^{\infty}\frac{1}{z}{e}^{-{z}^2/4} dz$. Although this growth law is in good agreement with the other curve, we point out that Eq. 21 is not the exact solution of the diffusion equation, as it can be proved by inserting Eq. 21 in Eq. 17. Nevertheless, it can be useful to approximate the solution by means of an analytical closed-form expression, in an alternative to the more involved series expansion (Eqs. 17–20).

Appendix 2

In this appendix, we demonstrate that the iso-concentration (IC) curves for multi-nuclei are not given by the union of circumference arcs, with equal radii, centered at each nucleus. In fact, this assumption implies that (i) the flux is radial (towards any center), being parallel to the gradient vector that is normal to the IC curve and (ii) the flux has the same value at each point of the IC curve.

We consider two ICs defined by radius r and r + ∆r. The mass balance in this strip reads

$$ \frac{\partial c\left(r,t\right)}{\partial t}={\lim}_{\Delta r\to 0}\frac{-{\int}_{\varSigma \left(r+\Delta r\right)}\overrightarrow{J}d\overrightarrow{\sigma}+{\int}_{\varSigma (r)}\overrightarrow{J}d\overrightarrow{\sigma}}{\omega (r)\Delta r}\kern0.5em , $$

(22)

where r is the radius of the circumference arcs, ω(r) the total length of the arcs and the line integral is performed on the border of the strip, Σ, where $\overrightarrow{\sigma}$ is the external normal to the IC. According to hypothesis (i), Eq. 22 provides

$$ \frac{\partial c\left(r,t\right)}{\partial t}={\lim}_{\Delta r\to 0}-\frac{\left[J\left(r+\Delta r\right)\omega \left(r+\Delta r\right)-J(r)\omega (r)\right]}{\omega (r)\Delta r}=-\frac{1}{\omega (r)}\frac{\partial \left[J\left(r,t\right)\omega (r)\right]}{\partial r}. $$

The use of the flux expression $J\left(r,t\right)=-D\frac{\partial c\left(r,t\right)}{\partial r}$ eventually leads to

$$ \frac{\partial c\left(r,t\right)}{\partial t}=D\left[\frac{\partial^2c\left(r,t\right)}{\partial {r}^2}+\left(\frac{d\ \ln \omega (r)\ }{dr}\right)\frac{\partial c\left(r,t\right)}{\partial r}\right]. $$

(23)

with boundary and initial conditions: c(R₀, t) = 0, c(r, 0) = c(r, t)|_{r → ∞} = c₀.

However, under the hypothesis above, the mass balance can equally be performed by considering an infinitesimal sector of the strip centered at an arbitrary nucleus and defined by the polar angle ∆θ. By setting the origin of the coordinate system at this chosen nucleus, one obtains

$$ \frac{\partial c\left(r,t\right)}{\partial t}={\lim}_{\Delta r\to 0}-\frac{\left[J\left(r+\Delta r\right)\left(r+\Delta r\right)-J(r)r\right]\Delta \theta }{r\Delta r\Delta \theta }=-\frac{\partial J\left(r,t\right)}{\partial r}-\frac{J\left(r,t\right)}{r}. $$

The use of the J(r, t) expression provides the cylindrical diffusion equation,

$$ \frac{\partial c\left(r,t\right)}{\partial t}=D\left[\frac{\partial^2c\left(r,t\right)}{\partial {r}^2}+\frac{1}{r}\frac{\partial c\left(r,t\right)}{\partial r}\right] $$

(24)

with the same boundaries and initial conditions as above. Equation 24 differs, in general, from Eq. 23. Through the ω(r) function, Eq. 23 depends upon the arrangement and linear density of nuclei, at odd with Eq. 23 which is independent of these quantities. The two differential equations provide the same solution only for non-overlapping circumferences.² It follows that the hypotheses on which Eqs. 23 and 24 rest do not hold in the case of overlap. Interaction among diffusional fields entails the non-radial symmetry of the diffusion problem, which is not described by Eq. 23. However, the contour plots of Fig. 2 show that the circular symmetry is lost even for the IC curves surrounding only a nucleus.

Appendix 3

According to the scheme of Fig. 12 of the Appendix, we distinguish the following cases depending on the number of collisions with diffusion domains of the next neighbors: (i) the linear domain is entirely within the Voronoi cell, i.e., zero collision with the neighbors; (ii) one collision; and (iii) two collisions. The reduced length of the diffusion zone reads

$$ \mathrm{Case}\ \left(\mathrm{i}\right)\kern2em l=2{l}_d\kern1.25em \mathrm{for}\kern0.75em {l}_d<\frac{a-z}{2} $$

$$ \mathrm{Case}\ \left(\mathrm{ii}\right)\kern1.75em l=\frac{a-z}{2}+{l}_d\kern2.25em \mathrm{for}\kern1.25em \frac{a-z}{2}<{l}_d<\frac{a+z}{2} $$

$$ \mathrm{Case}\ \left(\mathrm{iii}\right)\kern1.5em l=a\kern4em \mathrm{for}\kern1.25em {l}_d>\frac{a+z}{2}. $$

Fig. 12

Graphical representation of 1D-Voronoi cell of size a. Nuclei are solid circles and the cell boundaries are marked by thick segments. The thick red line is the diffusion domain of the central nucleus (length 2l_d)

It follows that

$$ l\left(\tau \right)=2\ {l}_d\left(\tau \right)H\left(\frac{a-z}{2}-{l}_d\left(\tau \right)\right)+\left[\frac{a-z}{2}+{l}_d\left(\tau \right)\right]\chi \left(\tau \right)+ aH\left({l}_d\left(\tau \right)-\frac{a+z}{2}\right). $$

(25)

In Eq. 25, H(∙) is the Heaviside step function and $\chi \left(\tau \right)=H\left({l}_d\left(\tau \right)-\frac{a-z}{2}\right)-H\left({l}_d\left(\tau \right)-\frac{a+z}{2}\right)$ the characteristic function. Equation 25 can be rewritten as follows

$$ l\left(\tau \right)=2\ {l}_d\left(\tau \right){\chi}_{0,{\tau}_1}\left(\tau \right)+\left[\frac{a-z}{2}+{l}_d\left(\tau \right)\right]{\chi}_{\tau_1,{\tau}_2}\left(\tau \right)+a{\chi}_{\tau_2,\infty}\left(\uptau \right), $$

(26)

where τ₁ and τ₂ satisfy the equations, ${l}_d\left({\tau}_1\right)=\frac{a-z}{2}$ and ${l}_d\left({\tau}_2\right)=\frac{a+z}{2}$.

Appendix 4

To determine the cell size probability density for correlated nuclei, according to the hard rod model, we made use of the results of ref. [36]. The nearest-neighbor probability density was determined according to ${H}_p(x) dx=K{e}^{-\lambda \left(x-{x}_0\right)}H\left(x-{x}_0\right) dx$, where x₀ is the rod length, K the normalization constant, and $\lambda =\frac{2N{R}_0}{1-N{R}_0{x}_0}=\frac{2\kappa }{1-\kappa {x}_0}$, with x₀ and x normalized to R₀. H_p(x)dx, is the probability that, given a rod of length x₀, the center of the nearest nucleus lies between x and x + dx (on either left or right side). Therefore, the probability to get a cell whose shorter side is between y and y + dy is computed from the relationship ${\overset{\sim }{H}}_p(y) dy={H}_p(x) dx$ with y = x/2, namely ${\overset{\sim }{H}}_p(y)=2{H}_p(2y)$. Also, setting y = x/2, y′  = x′ /2 with a = y + y′ the cell length, we define the joint probability P(y, y′)dydy′ as the probability to find a cell with left and right side in the range y, y + dy  and y′, y′ + dy, respectively. It follows that ${\overset{\sim }{H}}_p(y) dy=\left[{\int}_y^{\infty }P\left(y,y^{\prime}\right) dy^{\prime}\right] dy=2K{e}^{-\lambda \left(2y-{x}_0\right)}H\left(2y-{x}_0\right) dy$. Let us consider $P\left(y,y^{\prime}\right)\propto {e}^{-\lambda \prime \left(2y-{x}_0\right)}{e}^{-\lambda \prime \prime \left(2y\prime -{x}_0\right)}H\left(2y-{x}_0\right)H\left(2y^{\prime }-{x}_0\right)$ as a test function. This function satisfies the equation above for λ = λ′ + λ ′ ′. Since P(y, y′) is symmetric for variable exchange, $\lambda^{\prime }=\lambda^{\prime\prime }=\frac{\lambda }{2}$. The cell size probability density is eventually obtained from the relation

$$ P(a)={\int}_0^a{\int}_0^aP\left(y,{y}^{\prime}\right)\delta \left(y+{y}^{\prime }-a\right) dyd{y}^{\prime}\kern0.5em , $$

(27)

where δ(∙) is Dirac delta distribution. It follows,

$P(a)=H\left(a-{x}_0\right)K^{\prime }{\int}_{\frac{x_0}{2}}^{a-\frac{x_0}{2}}{e}^{-\frac{\lambda }{2}\left(2y-{x}_0\right)}{e}^{-\frac{\lambda }{2}\left(2\left(a-y\right)-{x}_0\right)} dy=K^{\prime }{e}^{-\lambda \left(a-{x}_0\right)}\left(a-{x}_0\right)H\left(a-{x}_0\right)$, with K′ normalization constant. K′ = λ² is obtained. In our case x₀ = 2 and $\lambda =\frac{2\kappa }{1-2\kappa }$,

$$ P(a)=\frac{4{\kappa}^2}{{\left(1-2\kappa \right)}^2}{e}^{-\frac{2\kappa }{1-2\kappa}\left(a-2\right)}\left(a-2\right)H\left(a-2\right). $$

(28)

Equation 28 also provides $\overline{a}-2=\frac{1-2\kappa }{\kappa }$ that is $\overline{a}=\frac{1}{\kappa }$ and $\frac{a-2}{\overline{a}-2}=\frac{\kappa \left(a-2\right)}{1-2\kappa }$. In terms of $s=\frac{a}{\overline{a}}$ and.

${s}^{\prime }=\frac{a-2}{\overline{a}-2}$ variables the distribution eventually becomes

$$ F(s)=P(a)\frac{da}{ds}=\frac{4}{{\left(1-2\kappa \right)}^2}{e}^{-\frac{2}{1-2\kappa}\left(s-2\kappa \right)}\left(s-2\kappa \right)\ H\left(s-2\kappa \right) $$

(29)

$$ F\left({s}^{\prime}\right)=P(a)\frac{da}{ds^{\prime }}=4{e}^{-2{s}^{\prime }}{s}^{\prime } $$

(30)

with s^′ ≥ 0.