An analytical model for secondary phase dissolution kinetics

Abstract

An analytical model for dissolution kinetics of secondary phase particles upon isothermal annealing has been proposed. Considering the interactions of solute diffusion fields in front of the secondary phase/matrix interface upon dissolution, a Johnson–Mehl–Avrami type equation, subjected to necessary modification, was derived, in combination with a classic dissolution model for single-particle system. Compared with the semiempirical dissolution models, which are used to fit the experimental results and phase-field method simulation, the current model follows an analogous form, but with the time-dependent kinetic parameters. Distinct from the model fitting work published recently, the current model is derived from the diffusion-controlled transformation theory, while the modeling quality is guaranteed by the physically realistic model parameters. On this basis, the current model calculation leads to a clear relationship between the secondary phase volume fraction and the time. Accordingly, model predictions for isothermal θ′ dissolution in Al–3.0wt%–Cu alloy and silicon dissolution in Al–0.8wt%–Si alloy were performed; good agreement with the published experimental data has been achieved.

Appendix

Analogous to a mathematical treatment in Ref. [47], the extended transformed fraction is reformulated proceeding as follows.

For the two parts x _e1 and x _e2 in Eq. (34), two corresponding parameters b ₁ and b ₂ can be chosen in such a way that x _e is due to only the first part,

$$ x_{{{\text{e}}1{\prime }}} = \frac{{b_{1} }}{{R_{0}^{3} }} \cdot 3R_{0} \cdot \left( {\sqrt {\frac{k}{\pi }} \cdot \left( {R_{1} - \sqrt {\frac{{kDt_{m} }}{m}} } \right) + \sqrt {R_{1}^{2} - \frac{{kDt_{m} }}{m}} } \right) \cdot \left( {\frac{{\sqrt {kDt_{m} } }}{{\sqrt m \cdot R_{1} + \sqrt {mR_{1}^{2} - kDt_{m} } }} + \sqrt {\frac{k}{\pi }} } \right) \cdot \left( {\frac{{kDt_{m} }}{m}} \right)^{\frac{1}{2}} $$

(A.1)

Or that x _e is due to only the second part,

$$ x_{{{\text{e}}2{\prime }}} = \frac{{b_{2} }}{{R_{0}^{3} }} \cdot \left( {\frac{{\sqrt {kDt_{m} } }}{{\sqrt m \cdot R_{1} + \sqrt {mR_{1}^{2} - kDt_{m} } }} + \sqrt {\frac{k}{\pi }} } \right)^{3} \cdot \left( {\frac{{kDt_{m} }}{m}} \right)^{\frac{3}{2}} $$

(A.2)

With x _e1′ = x _e2′ = x _e. Then, the total extended transformed fraction can be written as,

$$ x_{\text{e}} = \frac{1}{{a_{1} + a_{2} }}\left( {a_{1} \cdot x_{e1'} + a_{2} \cdot x_{{{\text{e}}2{\prime }}} } \right) $$

(A.3)

And

$$ \left\{ \begin{aligned} b_{1} & = & 1 + \frac{{a_{2} }}{{a_{1} }} \\ b_{2} & = & 1 + \left( {\frac{{a_{2} }}{{a_{1} }}} \right)^{ - 1} \\ \end{aligned} \right. $$

(A.4)

During dissolution, it can be seen that both x _e1 and x _e2 are positive and smaller than x _e. Always two integers, a ₁ and a ₂, can be found to satisfy Eq. (35). Moreover, for integers, a ₁ and a ₂, it holds that $a_{1} x_{{{\text{e}}1'}} = \sum\nolimits_{i = 1}^{{a_{1} }} {x_{{{\text{e}}1}} (i)}$ if x _e1(1) = x _e1(2) = …··· = x _e1(a ₁) = x _e1′; and $a_{2} x_{{{\text{e}}2{\prime }}} = \sum\nolimits_{i = 1}^{{a_{2} }} {x_{{{\text{e}}2}} (i)}$ if x _e2(1) = x _e2(2) = …··· = x _e2(a ₂) = x _e2′. Then, taking both x _e1(i) and x _e2(i) equal to x _e, Eq. (A.3) can be rewritten as,

$$ x_{\text{e}} = \frac{1}{{a_{1} + a_{2} }}\left( {\sum\nolimits_{i = 1}^{{a_{1} }} {x_{{{\text{e}}1}} (i)} + \sum\nolimits_{i = 1}^{{a_{2} }} {x_{{{\text{e}}2}} (i)} } \right) $$

(A.5)

Note that all fraction terms in Eq. (A.5) are equal. Substituting all x _e1(i) and x _e2(i) combined with Eqs. (A.1), (A.2) and (A.4), Eq. (A.5) becomes,

$$ x_{\text{e}} = \left\{ \begin{gathered} \frac{1}{{R_{0}^{3} }} \cdot \left( {\frac{{kDt_{m} }}{m}} \right)^{{\frac{1}{2} + \frac{1}{{1 + \left( {\frac{{a_{2} }}{{a_{1} }}} \right)^{ - 1} }}}} \cdot \left( {\frac{{\sqrt {kDt_{m} } }}{{\sqrt m \cdot R_{1} + \sqrt {mR_{1}^{2} - kDt_{m} } }} + \sqrt {\frac{k}{\pi }} } \right)^{{1 + \frac{2}{{1 + \left( {\frac{{a_{2} }}{{a_{1} }}} \right)^{ - 1} }}}} \cdot \left( {1 + \left( {\frac{{a_{2} }}{{a_{1} }}} \right)^{ - 1} } \right)^{{\frac{1}{{1 + \left( {\frac{{a_{2} }}{{a_{1} }}} \right)^{ - 1} }}}} \hfill \\ \left[ {3R_{0} \cdot \left( {\sqrt {\frac{k}{\pi }} \cdot \left( {R_{1} - \sqrt {\frac{{kDt_{m} }}{m}} } \right) + \sqrt {R_{1}^{2} - \frac{{kDt_{m} }}{m}} } \right) \cdot \left( {1 + \frac{{a_{2} }}{{a_{1} }}} \right)} \right]^{{\frac{1}{{1 + \frac{{a_{2} }}{{a_{1} }}}}}} \hfill \\ \end{gathered} \right\} $$

(A.6)