Particle-Size-Grouping Model of Precipitation Kinetics in Microalloyed Steels

Abstract

The formation, growth, and size distribution of precipitates greatly affects the microstructure and properties of microalloyed steels. Computational particle-size-grouping (PSG) kinetic models based on population balances are developed to simulate precipitate particle growth resulting from collision and diffusion mechanisms. First, the generalized PSG method for collision is explained clearly and verified. Then, a new PSG method is proposed to model diffusion-controlled precipitate nucleation, growth, and coarsening with complete mass conservation and no fitting parameters. Compared with the original population-balance models, this PSG method saves significant computation and preserves enough accuracy to model a realistic range of particle sizes. Finally, the new PSG method is combined with an equilibrium phase fraction model for plain carbon steels and is applied to simulate the precipitated fraction of aluminum nitride and the size distribution of niobium carbide during isothermal aging processes. Good matches are found with experimental measurements, suggesting that the new PSG method offers a promising framework for the future development of realistic models of precipitation.

Appendix: Calculation of Interfacial Energy

According to Turnbull[89] and Liu and Jonas,[90] the interfacial energy consists of two parts: a chemical part (σ _c) and a structural part (σ _st), so that

$$ \sigma = \sigma_{\text{c}} + \sigma_{\text{st}} $$

(A1)

The chemical interfacial energy is estimated from the difference between the energies of bonds broken in the separation process and of bonds made in forming the interface, with only the nearest neighbors considered. As given by Russell[36]

$$ \sigma_{c} = \frac{{\Updelta E_{0} N_{s} Z_{s} }}{{N_{A} Z_{l} }}(X_{\text{P}} - X_{\text{M}} )^{2} $$

(A2)

where ΔE ₀ is the heat of solution of precipitates in a dilute solution in the matrix, N _s is the number of atoms per unit area across the interface, Z _s is the number of bonds per atom across the interface, Z _l is the coordinate number of nearest neighbors within the precipitate crystal lattice, and X _P and X _M are the molar concentrations of the precipitate-forming element in the precipitate (P) and matrix (M) phase, respectively. ΔE ₀ is estimated to equal –ΔH, the heat of formation of the precipitate. X _P = 0.5 and X _P >> X _M.

Van Der Merwe[91] presented a calculation of structural energy for a planar interface. When the two phases have the same structure and orientation but different lattice spacing, the mismatch may be accommodated by a planar array of edge dislocations. Including the strain energy in both crystals, σ _st is given as

$$ \sigma_{st} = \frac{{\mu_{I} \bar{c}}}{{4\pi^{2} }}\left\{ {1 + \beta - (1 + \beta^{2} )^{1/2} - \beta \ln \left[ {2\beta (1 + \beta^{2} )^{1/2} - 2\beta^{2} } \right]} \right\} $$

(A3)

$$ {\text{with}}\,\frac{2}{{\bar{c}}} = \frac{1}{{c_{\text{M}}^{e} }} + \frac{1}{{c_{\text{P}}^{e} }},\,\beta = 2\pi \delta \frac{{\lambda_{ + } }}{{\mu_{I} }},\,\frac{2}{{\mu_{\text{I}} }} = \frac{1}{{\mu_{\text{M}} }} + \frac{1}{{\mu_{\text{P}} }},\,\delta = \frac{{2\left| {c_{\text{M}}^{e} - c_{\text{P}}^{e} } \right|}}{{c_{\text{M}}^{e} + c_{\text{P}}^{e} }},\,\frac{1}{{\lambda_{ + } }} = \frac{{1 - \nu_{\text{M}} }}{{\mu_{\text{M}} }} + \frac{{1 - \nu_{\text{P}} }}{{\mu_{\text{P}} }} $$

(A4)

where $c_{\text{M}}^{e}$ and $c_{\text{P}}^{e}$ are the nearest-neighbor distance across the interface, which are estimated from the lattice parameters c _M, c _P, and interface orientations; $\bar{c}$ is the spacing of a reference lattice across the matrix/precipitate interface; μ _M, μ _P, and μ _I are shear modulii in the matrix (M), precipitate (P), and interface (I), respectively; ν _M and ν _P are Poisson’s ratios; and δ is the lattice misfit across the interface.

The crystallographic relationships between the AlN (hexagonal close packed [hcp]), NbC (face centered cubic [fcc]), and steel matrix austenite phase (fcc) or ferrite phase (base-centered cubic [bcc]) are chosen as $(100)_{\text{NbC}} //(100)_{{\alpha{\text{-Fe}}}}$,[92]$(0001)_{\text{AlN}} //(111)_{{\gamma{\text{-Fe}}}}$,[93,94] and $(0001)_{\text{AlN}} //(110)_{{\alpha{\text{-Fe}}}}$.[95]

The physical properties used in the calculation are $- \Updelta H_{\text{A1N}} ({\text{KJ/mol}}) = 341.32 - 4.98 \times 10^{ - 2} T - 1.12 \times 10^{ - 6} T^{2} - 2813/T$,[96] $- \Updelta H_{\text{NbC}} ({\text{KJ/mol}}) = 157.76 - 4.54\,\, \times \,\,10^{ - 2} T - 3.84\,\, \times \,\,10^{ - 6} T^{2}$,[97] $\mu_{{\gamma{\text{-Fe}}}} ({\text{GPa}}) = 81\left[ {1 - 0.91(T - 300)/1810} \right]$,[98] $\nu_{{\gamma{\text{-Fe}}}} = 0.29$,[99] $c_{{\gamma{\text{-Fe}}}} (nm) = 0.357$,[73] $\mu_{{\alpha{\text{-Fe}}}} (GPa) = 69.2\left[ {1 - 1.31(T - 300)/1810} \right]$,[98] $\nu_{{\alpha{\text{-Fe}}}} = 0.29$,[99] $c_{{\alpha{\text{-Fe}}}} ({\text{nm}}) = 0.286$,[73] $\mu_{\text{AlN}} ({\text{GPa}}) = 127$,[100] $\nu_{\text{AlN}} = 0.23$,[100] $a_{\text{AlN}} ({\text{nm}}) = 0.311$, $c_{\text{AlN}} ({\text{nm}}) = 0.497$,[73] $\mu_{\text{NbC}} ({\text{GPa}}) = 134\left[ {1 - 0.18(T - 300)/3613} \right]$,[98] $\nu_{\text{NbC}} = 0.194$,[98] $c_{\text{NbC}} ({\text{nm}}) = 0.446$.[73]

For γ-Fe (111) plane, $Z_{\text{s}}^{{\gamma - {\text{Fe}}}} = 3$ and $N_{\text{s}}^{{\gamma - {\text{Fe}}}} = 4/(\sqrt 3 c_{{\gamma - {\text{Fe}}}}^{2} )$. For α-Fe, (110) plane $Z_{\text{s}}^{{\alpha{\text{-Fe}}}} = 4,$ $N_{\text{s}}^{{\alpha{\text{-Fe}}}} = \sqrt 2 /c_{{\alpha{\text{-Fe}}}}^{2}$, (100) plane $Z_{\text{s}}^{{\alpha{\text{-Fe}}}} = 4$, and $N_{\text{s}}^{{\alpha{\text{-Fe}}}} = 1/c_{{\alpha - {\text{Fe}}}}^{2}$. For both fcc and hcp precipitate structures, $Z_{l} = 12$. The calculated interfacial energy decreases slightly as temperature increases and also decreases for NbC (relative to AlN) because of lower heat of formation. The values used in the current simulations are

$$ \sigma_{\text{AlN}}^{{\gamma{\text{-Fe}}}} (840\,^{\text{o}} {\text{C}}) = 0.908\,\,{\text{J/m}}^{2} $$

$$ \sigma_{\text{AlN}}^{{\alpha{\text{-Fe}}}} (700\,\,^{\text{o}} {\text{C}}) = 0.997\,\,{\text{J/m}}^{2} $$

$$ \sigma_{\text{NbC}}^{{\alpha{\text{-Fe}}}} (700\,\,^{\text{o}} {\text{C}}) = 0.432\,\,{\text{J/m}}^{2} $$