A New Approximate Solution for Diffusional Growth and Dissolution of Cylindrical Precipitates

Abstract

A new approximate solution has been proposed for diffusional growth and dissolution of cylindrical precipitates, for which practical exact solutions and useful approximate solutions are not available. The present approximation uses the solutions of the invariant field (steady state) approximation and the mass conservation conditions. The present approximation is compared with the numerical simulations based on the finite difference equation considering the moving boundary. It is shown that the agreement between the present approximation and the numerical simulation results is good for growth and dissolution of cylindrical precipitates.

Fig. 3 Growth rate constants as a function of the supersaturation Ω for various solutions of cylindrical precipitates.

1. Introduction

In the present paper diffusional growth and dissolution of cylindrical precipitates^1,2) are considered. The diffusion equation in the matrix phase is solved in the cylindrical coordinate system under the moving boundary condition.³⁾ For growth and dissolution of cylindrical precipitates practical exact solutions and useful approximations are not available.^1,2,4) The purpose of this study is to propose a new approximate method applicable to growth and dissolution of cylindrical precipitates.

For planar and spherical precipitates several approximate solutions are available.^5,6) For planar precipitates, the linearized gradient approximation is useful both for growth and dissolution.⁵⁾ For growth of spherical precipitates, the linearized gradient approximation is available but is not a good approximation. In this approximation a cubic equation must be solved to obtain the solution.⁷⁾ And this approximation is not applicable to dissolution.⁵⁾ The present authors proposed the modified linearized gradient approximation,⁸⁾ which showed the excellent agreement with the exact solution for growth of spherical precipitates. For dissolution of spherical precipitates, the numerical method based on the modified linearized gradient approximation was shown to be useful.⁸⁾

In this paper it is shown that the linearized gradient approximation for planar precipitates and the modified linearized gradient approximation for spherical precipitates are based on the invariant field (steady state) solutions. This fact is thought to be the reason that these two approximations work well for each precipitate shape. The invariant field approximate solution cannot satisfy the far-field condition in the planar and the cylindrical coordinate systems. Therefore, the invariant field approximate solution is applied to the region of the diffusional field, of which extent is determined by the mass conservation condition.

The new approximate method for growth and dissolution of cylindrical precipitates uses the invariant field approximate solution in the cylindrical coordinate system for the diffusional region. The extent of the diffusional region is determined by the mass conservation condition as in the case of spherical and planar precipitates.

2. Mathematical Model

To treat diffusional growth and dissolution of an isolated spherical or cylindrical precipitate in an infinite matrix the following radial diffusion equation for the concentration field in the spherical or the cylindrical coordinates must be solved.   

\begin{equation} \frac{\partial c}{\partial t} = \frac{D}{r^{d}}\frac{\partial}{\partial r}\left(r^{d}\frac{\partial c}{\partial r}\right). \end{equation} (1)

  

\begin{align*} &(\text{$d = 1$ for a cylindrical precipitate},\\ &\text{$d = 2$ for a spherical precipitate}) \end{align*}

Additionally, the case that d = 0 corresponds to a planar precipitate, where the radial coordinate r is replaced by the linear coordinate x. In eq. (1), D is the constant volume diffusion coefficient in the matrix, c = c(r, t) is the concentration field of the solute in the matrix surrounding the precipitate.

The boundary and the initial conditions for growth and dissolution of precipitates are the following.   

\begin{equation} \begin{split} & c(R,t) = c^{m}.\quad 0 < t, \\ & c(\infty,t) = c^{\infty},\quad 0 < t, \\ & c(r,0) = c^{\infty},\quad 0 < r. \end{split} \end{equation} (2)

  

\begin{equation} \begin{split} & R(t = 0) = 0,\quad \text{for growth}, \\ & R(t = 0) = R_{0} > 0.\quad \text{for dissolution}. \end{split} \end{equation} (3)

Here, R is the radius of the precipitate, r = R at the precipitate/matrix interface, c^m is the concentration in the matrix at the precipitate/matrix interface, and c^∞ is the far-field concentration. The flux balance condition at the precipitate/matrix interface shown below must be satisfied.   

\begin{equation} (c^{p} - c^{m})\frac{dR}{dt} = D \frac{\partial c}{\partial r}\bigg|_{r = R}, \end{equation} (4)

where c^p is the concentration in the precipitate. c^p and c^m are assumed to be constant during the growth and the dissolution processes. This also means that the Gibbs-Thomson effect is not considered here.

The schematic concentration fields for growth and dissolution of a cylindrical precipitate are shown in Fig. 1. The concentration fields shown in the figure are for solute-rich precipitates. In the following, solute-rich precipitates are considered but it is easy to extend the present analyses to solute-poor precipitates.

Fig. 1

Schematic illustrations of concentration fields for growth and dissolution.

The solution of eq. (1) with the boundary and the initial conditions of eqs. (2) and (3) gives the concentration field c(r, t). The interface position R(t) is calculated from eq. (4) using the calculated concentration field c(r, t).

3. Linearized Gradient Approximations for Planar and Spherical Precipitates

The linearized gradient approximation is a good approximation for planar precipitates. The concentration field for this approximation is as follows.   

\begin{equation} \begin{split} & c(x) = c^{\infty} + (c^{m} - c^{\infty})\left(1 - \frac{x - X}{L}\right)\\ &\qquad \quad \text{for $X \leq x \leq X + L$},\\ & c(x) = c^{\infty}\ \text{for $X + L < x$}. \end{split} \end{equation} (5)

Here, x is the linear coordinate and X is the position of the interface between the precipitate and the matrix. L is the extent of the diffusional field. L is determined by the mass conservation condition. Equation (5) satisfies the following boundary conditions.   

\begin{equation} c(X) = c^{m},\quad c(X + L) = \text{c}^{\infty}. \end{equation} (6)

The linearized gradient approximation for spherical precipitates is not as good as the linearized gradient approximation for planar precipitates. The present authors proposed the modified linearized gradient approximation for spherical precipitates. This approximation was derived heuristically. This was shown to be a good approximation. The concentration field for this approximation is shown below.   

\begin{equation} \begin{split} & c(r) = c^{\infty} + (c^{m} - c^{\infty})\frac{R}{r}\left(1 - \frac{r - R}{L}\right)\\ &\qquad \quad \text{for $R \leq r \leq R + L$},\\ & c(r) = c^{\infty}\ \text{for $R + L < r$} \end{split} \end{equation} (7)

The linear term of r in the conventional linearized gradient approximation is multiplied by the factor R/r. Equation (7) satisfies the following boundary conditions.   

\begin{equation} c(R) = c^{m},\quad c(R + L) = \text{c}^{\infty}. \end{equation} (8)

It should be noticed that the concentration fields eqs. (5) and (7) are the solutions of the invariant field approximation (steady state, ∂c/∂t = 0) for planar and spherical precipitates, respectively. The steady state diffusion equation for a plane precipitate is as follows.   

\begin{equation} 0 = \frac{\partial^{2}c}{\partial x^{2}}. \end{equation} (9)

The general solution of eq. (9) is c(x) = Ax + B. Here, A and B are constants, which are determined by boundary conditions. Equation (5) is the solution of eq. (9) that satisfies the boundary conditions of eq. (6). For a spherical precipitate the steady state diffusion equation is as follows.   

\begin{equation} 0 = \frac{\partial}{\partial r}\left(r^{2}\frac{\partial c}{\partial r}\right). \end{equation} (10)

The general solution of eq. (10) is c(r) = A/r + B. It can be shown that eq. (7) is the solution of eq. (10) that satisfies the boundary conditions of eq. (8).

The mass conservation condition must be satisfied for the concentration field in these approximations. It may be said that the linearized gradient approximation for planar precipitates and the modified gradient approximation for spherical precipitates are ‘the invariant field approximations satisfying the mass conservation conditions’.

It is possible to apply both the conventional and the modified linearized gradient approximations [eq. (5) and eq. (7)] to the case of the cylindrical precipitates without any mathematical limitations. x and X in eq. (5) are replaced by the radial coordinate r and the radius R, respectively. Table 1 shows the growth rate constants that are derived using the conventional and the modified linearized gradient approximations for cylindrical precipitates. The concentration fields for these two approximations are not the solutions of the invariant field approximation for cylindrical precipitates.

Table 1 Approximate solutions for the growth rate constant of a cylindrical precipitate.

4. A New Approximate Solution for Cylindrical Precipitates

A new approximation for growth and dissolution of a cylindrical precipitate, which is based on the solutions of the invariant field approximation satisfying the mass conservation conditions, is proposed here.

The steady state diffusion equation for a cylindrical precipitate is as follows.   

\begin{equation} 0 = \frac{\partial}{\partial r}\left(r\frac{\partial c}{\partial r}\right) \end{equation} (11)

The general solution of eq. (11) is c(r) = A ln r + B. The concentration field is obtained from the general solutions of eq. (11) for each coordinate interval.   

\begin{equation} \begin{split} & c(r) = c^{\infty} + (c^{m} - c^{\infty})\cfrac{\ln \biggl(\cfrac{r}{R + L}\biggr)}{\ln \biggr(\cfrac{R}{R + L}\biggr)}\\ &\qquad \quad \text{for $R\leq r \leq R + L$}, \\ & c(r) = c^{\infty}\ \text{for $R + L < r$}. \end{split} \end{equation} (12)

The concentration field above satisfies the following boundary conditions.   

\begin{equation} c(R) = c^{m},\quad c(R + L) = \text{c}^{\infty}. \end{equation} (13)

The concentration field [eq. (12)] must satisfy the mass conservation condition both for growth and dissolution. These two cases are considered separately below.

4.1 Growth

From eq. (12) and the mass conservation condition for growth (Fig. 2(a)) the following relation is obtained.   

\begin{equation} \pi (c^{p} - c^{\infty})R^{2} = 2\pi(c^{\infty} - c^{m})\int_{R}^{R + L}r\cfrac{\ln \biggl(\cfrac{r}{R + L}\biggr)}{\ln \biggl(\cfrac{R}{R + L}\biggr)}\, dr \end{equation} (14)

The integral in the right side of eq. (14) can be calculated using partial integration. This equation determines the extent of the diffusional field L. Here, L is normalized by R (y ≡ L/R). The following equation for y is obtained from eq. (14).   

\begin{equation} y^{2} + 2y - \frac{2}{\Omega}\ln (1 + y) = 0, \end{equation} (15)

where the supersaturation Ω is defined as follows.   

\begin{equation} \Omega = \frac{c^{\infty} - c^{m}}{c^{p} - c^{m}}.\quad (0 \leq \Omega \leq 1) \end{equation} (16)

Unfortunately, eq. (15) for y cannot be solved analytically and must be solved numerically using for example Newton-Raphson method. The concentration field [eq. (12)] is substituted to the flux balance equation [eq. (4)], then the following equation for R is obtained.   

\begin{equation} \frac{dR^{2}}{dt} = \frac{2\Omega D}{\ln(1 + y)}. \end{equation} (17)

As y is a function of Ω and independent of t and R, eq. (17) is integrated and R is obtained.   

\begin{equation} R = 2\sqrt{\frac{\Omega}{2\ln (1 + y)}} \cdot \sqrt{Dt}. \end{equation} (18)

From the definition shown in Table 1 and eq. (15) the growth rate constant λ_n is obtained as follows.   

\begin{equation} \lambda_{n} = \sqrt{\frac{1}{y^{2} + 2y}}, \end{equation} (19)

where y is the numerical solution of eq. (15).

Fig. 2

Mass conservation conditions for growth and dissolution of cylindrical precipitates. The condition, V^p = V^m, is shown in the figure.

4.2 Dissolution

The concentration field of the present approximation for dissolution is also described by eq. (12). The mass conservation condition for the dissolution case is shown in Fig. 2(b). The following equation is obtained.   

\begin{align} &\pi(c^{p} - c^{\infty})(R_{0}^{2} - R^{2})\\ &\quad = 2\pi(c^{m} - c^{\infty})\int_{R}^{R + L}r\cfrac{\ln \biggl(\cfrac{r}{R + L}\biggr)}{\ln \biggl(\cfrac{R}{R + L}\biggr)}\,dr. \end{align} (20)

The extent of the diffusional field L is obtained from eq. (20). L is normalized by R (y ≡ L/R) and the following equation for y is derived.   

\begin{equation} y^{2} + 2y - 2\ln (1 + y)\left(\frac{R_{0}^{2}}{R^{2}} \cdot \frac{1 + \Omega}{\Omega} - \frac{1}{\Omega}\right) = 0. \end{equation} (21)

The supersaturation Ω is given by the following.   

\begin{equation} \Omega = \frac{c^{m} - c^{\infty}}{c^{p} - c^{m}}.\quad (0 \leq \Omega) \end{equation} (22)

This is different from the supersaturation in the growth case. In eq. (21) $R^{2}/R_{0}^{2}$ is replaced by the average value $\overline{R^{2}}/R_{0}^{2}$ during the dissolution process. Then, y can be obtained solving eq. (21) numerically.

The concentration field [eq. (12)] is substituted to the flux balance equation [eq. (4)], then the following equation for R is obtained.   

\begin{equation} \frac{dR^{2}}{dt} = -\frac{2\Omega D}{\ln (1 + y)}. \end{equation} (23)

Here, t and R are normalized as follows.   

\begin{equation} \tau\equiv Dt/R_{0}^{2},\quad Z \equiv R/R_{0}. \end{equation} (24)

Equation (23) becomes the following.   

\begin{equation} \frac{dZ^{2}}{d\tau} = -\frac{2\Omega}{\ln(1 + y)}. \end{equation} (25)

y is independent of the time, because the term $R^{2}/R_{0}^{2}$ is replaced by its average. Then y is a function of Ω. Equation (25) can be expressed as follows.   

\begin{equation} \frac{dZ^{2}}{d\tau} = -\frac{1}{\tau_{dis}(\Omega)}. \end{equation} (26)

Equation (26) is integrated with the initial condition that Z = 1 for τ = 0, then Z² = 1 − τ/τ_dis. τ_dis is the dissolution time. $\overline{Z^{2}}$ ($ = \overline{R^{2}}/R_{0}^{2}$) is obtained as follows.   

\begin{equation} \overline{Z^{2}} = \frac{1}{\tau_{dis}}\int_{0}^{\tau_{dis}}(1 - \tau/\tau_{dis})d\tau = \frac{1}{2} \end{equation} (27)

Substituting the value of 1/2 to $R^{2}/R_{0}^{2}$ (= Z²) in eq. (21), the following equation for y is obtained.   

\begin{equation} y^{2} + 2y - 2\frac{1 + 2\Omega}{\Omega}\ln (1 + y) = 0. \end{equation} (28)

y is calculated for given Ω by solving eq. (28) numerically. Equation (25) can be integrated and the following is obtained using eq. (28).   

\begin{equation} Z^{2} = 1 - \frac{4(1 + 2\Omega)}{y^{2} + 2y}\tau. \end{equation} (29)

5. Results and Discussions

There is no exact solution available for growth and dissolution of cylindrical precipitates.^1,2) In order to evaluate the present approximate solution, numerical simulations of growth and dissolution of cylindrical precipitates have been performed (Appendix). The present and the other approximate solutions are compared with the numerical simulation results.

5.1 Growth

In Fig. 3, the growth rate constants for a cylindrical precipitate are plotted as a function of the supersaturation Ω using the present approximation, the conventional linearized gradient approximation and the modified linearized gradient approximation. These three approximations are compared with the numerical simulation results. The present approximation lies between the conventional and the modified linearized gradient approximations. The numerical simulation results are always larger than these three approximations for Ω > 0.4. For the small value of Ω the present approximation agrees well with the numerical results, while two other approximations deviate from the numerical results. This means that the present approximation can be applied especially to the experiments for the later stage of precipitate growth. It should be emphasized that the conventional and the modified linearized gradient approximate solutions do not satisfy the steady state equation in the cylindrical coordinate system [eq. (11)].

Fig. 3

Growth rate constants as a function of the supersaturation Ω for various solutions of cylindrical precipitates.

It is shown that the present approximation is good. In the present approximation, however, eq. (15) for y must be solved numerically, then the analytical form for the growth constant λ cannot be obtained.

To obtain the analytical form for λ the following steps are adopted. ln(1 + y) is expanded in the following infinite series.⁹⁾   

\begin{align} \ln (1 + y) &= 2\left\{\frac{y}{y + 2} + \frac{1}{3}\left(\frac{y}{y + 2}\right)^{3}{} + \frac{1}{5}\left(\frac{y}{y + 2}\right)^{5}{} + \cdots\right\},\\ &\quad 1 + y > 0 \end{align} (30)

For y → 0,   

\begin{equation} \ln(1 + y) \approx \frac{2y}{y + 2}. \end{equation} (31)

From eq. (31), eq. (15) becomes as follows.   

\begin{equation} y^{2} + 2y - \frac{4}{\Omega} \cdot \frac{y}{y + 2} = 0. \end{equation} (32)

The positive value solution of the equation above is   

\begin{equation} y = \frac{2}{\sqrt{\Omega}} - 2. \end{equation} (33)

From eqs. (19) and (33) the following expression for the growth rate constant λ_a is obtained.   

\begin{equation} \lambda_{a} = \frac{1}{2}\sqrt{\frac{\Omega}{1 - \sqrt{\Omega}}}. \end{equation} (34)

Equation (33) indicates that y → 0 for Ω → 1. This means that eq. (34) is the asymptotic form of λ for Ω → 1.

In Fig. 4 the growth rate constants λ_n and λ_a for cylindrical precipitate are plotted as a function of the supersaturation Ω. The figure shows that the agreement between λ_n and λ_a is good for Ω > 0.5. By using fitting parameter ν, λ_f defined below becomes closer to λ_n for the range of Ω > 0.0001 as shown in Fig. 4.   

\begin{equation} \lambda_{f} = \frac{1}{2}\sqrt{\frac{\Omega^{\nu}}{1 - \sqrt{\Omega}}}. \end{equation} (35)

The value of 1.124 is obtained for ν by fitting. The value of 9/8 can also be used for simple notation. In Fig. 4 λ_f is calculated using the value of 9/8 for ν. The conventional linearized gradient approximation, the modified linearized gradient approximation and the present approximation are summarized in Table 1.

Fig. 4

Growth rate constants as a function of the supersaturation Ω for cylindrical precipitates. λ_n, λ_a and λ_f by the present approximation are plotted. λ_n is calculated using y in eq. (15), which is solved numerically. λ_a is an asymptotic solution of λ_n. λ_f is a solution using the fitting parameter based on λ_a.

5.2 Dissolution

Dissolution rate constants cannot be easily defined, so the simple way to study the approximate solutions for dissolution is to show the relation between the radius of a precipitate and the time. In the following the radius and the time are normalized by R₀ and $D/R_{0}^{2}$, respectively. They are denoted by Z and τ, respectively.

Figure 5 shows the square of the normalized precipitate radius Z vs. the normalized time τ using the present approximation [eq. (29)] for dissolution of cylindrical precipitates together with the numerical simulation results for Ω = 0.05 and Ω = 0.2. y in eq. (28) is calculated using Newton-Raphson method for given Ω. The agreement between the present approximation and the numerical simulation results seems to be not good enough, but their dissolution times are close both for Ω = 0.05 and Ω = 0.2. Figure 6 shows the dissolution time vs. the supersaturation (0.01 ≤ Ω ≤ 1) for the present approximation and the numerical simulations. This result implies that the present approximation can be used to predict the dissolution time of cylindrical precipitates. Comparisons with experimental results are expected to evaluate usefulness of the present approximation.

Fig. 5

Square of normalized precipitate radius vs. normalized time for the dissolution of cylindrical precipitates by the present approximation and the numerical simulation. The results for Ω = 0.2 and Ω = 0.05 are plotted.

Fig. 6

Normalized dissolution time vs. supersaturation for the dissolution of cylindrical precipitates for the present approximation and the numerical simulation.

6. Summary

A new approximation for growth and dissolution of cylindrical precipitates has been proposed. The present approximation uses the invariant field approximation solution for the diffusional region. The mass conservation condition is imposed on the solution. The present approximation is based on the fact that the linearized gradient approximation for a planar precipitate and the modified linearized gradient approximation for a spherical precipitate use the invariant field approximate solutions satisfying the mass conservation conditions. The exact solutions for growth and dissolution of cylindrical precipitates are not available. Therefore the present approximate solutions are compared with the numerical simulation results. It is shown that the agreement between the present approximation and the numerical simulation results is good enough for growth and dissolution of cylindrical precipitate.

Acknowledgement

The authors would like to acknowledge Dr. T. Nishibata for his valuable comments.

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Appendix

Method of Numerical Simulation

Equation (1) is solved by using the finite difference method.¹⁰⁾ Equation (1) is replaced by the following finite difference equation.   

\begin{equation} \frac{c_{j}^{i + 1} - c_{j}^{i}}{\Delta t} = D\frac{r_{j + 1/2}^{d}(c_{j + 1}^{i + 1} - c_{j}^{i + 1})/(r_{j + 1} - r_{j}) - r_{j - 1/2}^{d}(c_{j}^{i + 1} - c_{j - 1}^{i + 1})/(r_{j} - r_{j - 1})}{r_{j}^{d}(r_{j + 1/2} - r_{j - 1/2})}, \end{equation} (A1)

  

\begin{equation*} (\text{$d = 1$ for a cylindrical precipitate, $d = 2$ for a spherical precipitate}) \end{equation*}

where the subscripts and the superscripts denote grid points and time steps, respectively. j ± 1/2 are the midpoints between lattice points j and j ± 1. Thus, r_j±1/2 = (r_j + r_j±1)/2. Δt is the time increment. The equation above is solved by using the Euler implicit method. The movement of the interface is calculated by using the flux balance condition [eq. (4)] at each time step.

To demonstrate the validity of the present numerical simulation the comparison with the exact solution of the growth of the spherical precipitates (d = 2) has been made. Figure A1 shows the growth rate constants calculated by using the exact solution and numerical simulations. The numerical simulation results agree very well with the exact solution. Figure A2 shows the normalized precipitate radius vs. the normalized time at two different Ω for dissolution. For dissolution of spherical precipitates no exact solution is available, therefore the Green’s function method¹¹⁾ is used for comparison. The agreement between these two solutions is excellent. Therefore the validity of the present numerical simulation method is demonstrated. It is thought that this numerical simulation method can also be applied to the case of cylindrical precipitates (d = 1).

Fig. A1

Growth rate constants as a function of the supersaturation Ω for spherical precipitates by the exact solution and the numerical simulation.

Fig. A2

Square of normalized precipitate radius vs. normalized time for the dissolution of spherical precipitates by the numerical simulation, the Green’s function method¹¹⁾ and the numerical method based on the modified linearized gradient approximation.⁸⁾ The results for Ω = 0.2 and Ω = 0.05 are plotted.