Modeling of precipitation hardening in Mg-based alloys

Abstract

This work deals with the development of Mg-based alloys with enhanced properties at elevated temperatures. This is achieved by precipitation of binary phases such as MgZn₂ and Mg₂Sn during the aging of these alloys. The aim of the present work is to develop and calibrate a model for precipitation hardening in Mg-based alloys, as different types of precipitates form simultaneously. The modified Langer-Schwartz approach, while taking into account nucleation, growth and coarsening of the new phase precipitations, was used for the analysis of precipitates’ evolution and precipitation hardening during aging of Mg-based alloys. Two strengthening mechanisms associated with particle-dislocation interaction (shearing and bypassing) were considered to be operating simultaneously due to particle size-distribution. Parameters of the model, R _N i and k _{σ i}, were found by fitting of calculated densities and average sizes of precipitates with ones estimated from experiments. The effective diffusion coefficients of phase formation processes, which determine the strengthening kinetics, were estimated from the hardness maximum positions on the aging curves.

Introduction

Extensive experimental research work has been carried out over the last several years to develop an Mg-based alloy with enhanced properties at elevated temperatures. Various studies were made in order to find additional elements to improve the castability and creep resistance at elevated temperatures. Previous research on Mg–Zn–Sn and Mg–Zn–Sn–Ca alloys has shown that Zn improves the castability and has several stable intermetallics with Mg, Sn forms the stable Mg₂Sn compound [1], Ca behaves as a grain refiner [2], and creates the stable intermetallic Mg₂Ca [3]. Theoretical description of the precipitation and hardening processes in these alloys may assist in finding their optimal compositions and heat treatments for achieving the required mechanical properties. Models for precipitation kinetics were developed and calibrated for Cu–Ti [4, 5] and Al-based alloys [6, 7]. The microstructure evolution of an Mg–Zn–Sn alloy recently studied by combining microhardness experiments [8], X-ray diffraction, transmission electron microscopy (TEM) and small angle scattering (SAXS) [9, Rashkova et al. (submitted, 2006)] provided determination of precipitated phases, their volume fraction and size distribution. The aim of the present work is to develop and calibrate a model for precipitation hardening in Mg-based alloys, as different types of precipitates form simultaneously.

The model

The model used in the present work is based on the classical theory proposed by Langer and Schwartz [4] and modified by Kampmann and Wagner [5]. This model takes into account three mechanisms involved in the solid-state decomposition of a supersaturated solid solution taking place during aging: nucleation, growth, and coarsening of precipitates.

It should be noted that in this analytical model droplets of a new phase are considered to be spheres, and no elastic energy or anisotropy of interfacial energy effects are taken into account. Precipitates in the α-Mg-matrix are often of needle-like shape, as in the case of MgZn₂, for example. For such a case, some corrections should be introduced in the model, in particular in the law of steady state growth of precipitates and in the Gibbs–Thomson equation. The coarsening process of the needle-like precipitates may differ from that of equiaxed ones, and is driven by the needles’ widths. However, if the average width of needles, w, increases proportionally to their average length, l, an equivalent radius of the droplet, R _p = (3w ²l/16)^1/3 is proportional to w, and a conception of equivalent radius can be used in the model calculations. Baring in mind this simplification, the following model can be developed.

The rate of homogeneous nucleation of precipitates is determined by the following equation:

$$ J_{\rm p}=\hbox{ZN}_0 \beta\hbox{exp} (-\Delta {G}_{\rm p}^\ast /{kT})\hbox{exp}(-{t}_{\rm w}/{t}), $$

(1)

where ΔG ^*_p = (4π/3)(R ^*)²γ is the formation energy of critical nucleus with radius R*, as R* represents the limit above which a particle becomes stable:

$$ {R}^{\ast}=\frac{2\gamma {V}_{\rm m}}{X_\beta R_{\rm g} {T}}\cdot \frac{1}{\hbox{ln} (X_\alpha /X_{\rm e})} $$

(2)

β = D _α X _α/a ², D _α is the diffusion coefficient of the species responsible for new phase nucleation, a is the atomic jump distance, X _α and X _e are the mean and equilibrium concentrations of the species in the matrix, X _β is the concentration in the precipitate, γ is the surface energy of the nuclei, t _w is the incubation period, Z is the Zeldovich factor, V _m is molar volume of the nucleated phase, k is the Bolzman’s constant, R _g is the gas constant, and T is the absolute temperature.

The growth rate of the precipitates is approximated by the steady state solution of the second Fick’s law:

$$ \frac{{\hbox{d}}R_{\rm p}}{{\hbox{d}}t}=\nu _{\rm p} =\frac{D_\alpha }{\varphi \cdot R_{\rm p}}\cdot \frac{X_\alpha-X_{\alpha /\beta }^{R_{\rm p} } }{X_\beta - X_{\alpha /\beta }^{R_{\rm p}}} $$

(3)

where R _p is the equivalent radius of the precipitate, φ = 1 for spherical particles and φ = ln(d/w) = (1/3)ln(π l/4fw) for cylindrical (needle-like) droplets of width w and length l, d being the average distance between needles and f is their volume fraction, ${X_{\alpha /\beta }^{R_{\rm p}}}$ is the concentration at the matrix/precipitate interface, as the latter depends on R _p according to Gibbs–Thomson equation:

$$ X_{\alpha /\beta }^{R_{\rm p}} =X_{\rm e}\cdot \hbox{exp} \left\{{\frac{2\gamma V_{\rm m}}{R_{\rm g}T}\cdot \frac{1}{\theta \cdot R_{\rm p}}} \right\}, $$

(4)

where θ is the shape dependent parameter; for a sphere shape θ = 1, for needles θ = w/2R _p < 1.

It is assumed in the Langer-Schwartz model that the system contains N _p droplets per unit volume of uniform size R _p > R ^*; the number of particles is given by integration of equation:

$$ \frac{\hbox{d}N_{\rm p}}{\hbox{d}t}={J}_{\rm p}-{f}_{\rm a}(R^{\ast},t)\cdot \frac{\hbox{d}R^{\ast}}{\hbox{d}t}, $$

(5)

where f _a(R ^*,t) is the apparent density of particles:

$$ f_{\rm a} (R^{\ast},t)=N_{\rm p} \frac{b_{\rm LSW} }{R_{\rm p} -R^{\ast}} $$

(6)

b _LSW = 0.317 is chosen to force the average particle size to tend toward the long time LSW coarsening rate.

The time evolution of the droplet average size, R _p, is described by the following equation:

$$ \frac{{\hbox{d}}\bar{R}_{\rm p}}{{\hbox{d}}t}=\nu _{\rm p} (\bar{{R}}_{\rm p})+(\bar{{R}}_{\rm p} -R^{\ast})\cdot \frac{f_{\rm a} (R^{\ast},t)}{N_{\rm p}}\cdot \frac{{\hbox{d}}R^{\ast}}{{\hbox{d}}t}+\frac{1}{N_{\rm p}}J_{\rm p} (R^{\ast}(t))\cdot (R^{\ast}+\delta R^{\ast}-\bar{{R}}_{\rm p}) $$

(7)

The third term in Eq. 7 is concerned with the nucleation of particles slightly larger than those of a critical size, R = R ^* + δR ^*, and δR ^* ≪ R ^*. Here δR ^* is a parameter of the model. Equations 5 and 7 should be coupled with the solute conservation equation:

$$ \frac{X_{0\alpha } -\bar{{X}}_\alpha }{X_\beta -\bar{{X}}_\alpha }=\frac{4\pi }{3}\bar{{R}}_{\rm p}^3 \cdot N_{\rm p}^3 $$

(8)

In the case of multi-component alloys, different types of precipitates, with different kinetic and thermodynamic parameters, may develop simultaneously. As a result, Eqs. 5–8 should be written for each type of precipitate. It should be noted that the model in the present form may be applied only to the nucleation and growth of binary phases. In the case of Mg–Zn–Sn alloys, two types of binary precipitates, MgZn₂ and Mg₂Sn, are assumed to form independently, and the model can describe simultaneous evolution of them both.

Using dimensionless parameters

$$ \begin{array}{lll} \rho_{i}\,=\,\bar{R}_{i}/{R}_{\rm Ni},&\quad \xi _{i}\,=\,\bar{X}_{\alpha {\rm i}}/{X}_{\rm ei},&\, n_{i}\,=\,{N}_{\rm pi}\cdot \frac{4\pi}{3}\cdot{R}_{\rm Ni}^3,\\ \tau \,=\,{D}_1{t}/{R}_{{\rm N}1}^2,&{\bar{J}_{i}}\,=\,{J}_{\rm pi}\frac{\hbox{d}n_{i}}{\hbox{d}N_{\rm pi}}\cdot \frac{\hbox{d}t}{\hbox{d}\tau}, &{k_{\gamma {i}}\,=\,\gamma_{\rm i}/\gamma_0}, \,{i\,=\,1,2} \\ \end{array} $$

(9)

where R _Ni = 2V _mi γ₀/(X _βi R _g T) and γ₀ are the scaling length and interphase surface energy, V _mi is the molar volume of the new phase i, one can obtain a system of four differential equations:

$$ \left\{{\begin{aligned} &\frac{{\hbox{d}}\rho _1 }{{\hbox{d}}\tau }+\frac{\rho _1 }{3}\left\{ {\frac{1}{\xi _{01} -\xi _1 }+\frac{b\cdot k_{\sigma 1} }{\rho _1 \cdot \hbox{ln } \xi _1 }\cdot \frac{1}{\xi _1 \cdot \hbox{ln } \xi _1 }} \right\}\cdot \frac{{\hbox{d}}\xi _1 }{{\hbox{d}}\tau }=-\bar{{J}}_1 \cdot \frac{\rho _1^4 }{3}\cdot \frac{\xi _{p1}-1}{\xi _0 -\xi _1}\\ &\frac{{\hbox{d}}\rho _1 }{{\hbox{d}}\tau }+\frac{b\cdot k_{\sigma 1} }{\xi _1 (\hbox{ln } \xi _1 )^{2}}\cdot \frac{{\hbox{d}}\xi _1 }{{\hbox{d}}\tau }=\frac{1}{\xi _{p1} -1}\cdot \frac{1}{\rho _1 }\cdot \left\{ {\xi _1 -\hbox{exp} \left( {\frac{k_{\sigma 1} }{\rho _1 }} \right)} \right\}+\rho _1^3 \cdot \frac{\xi _{p1} -1}{\xi _{01} -\xi _1 }\cdot \bar{{J}}_1 \cdot \left[ {\frac{k_{\sigma 1} }{\hbox{ln } \xi _1 }+\delta \rho ^{\ast}-\rho _1 } \right]\\ &\frac{{\hbox{d}}\rho _2 }{{\hbox{d}}\tau }+\frac{\rho _2 }{3}\left\{ {\frac{1}{\xi _{02} -\xi _2 }+\frac{b\cdot k_{\sigma 2} }{\rho _2 \cdot \hbox{ln } \xi _2 }\cdot \frac{1}{\xi _2 \cdot \hbox{ln } \xi _2 }} \right\}\cdot \frac{{\hbox{d}}\xi _2 }{{\hbox{d}}\tau }=-\bar{{J}}_2 \cdot \frac{D_2 }{D_1 }\cdot \frac{\rho _2^4 }{3}\cdot \frac{\xi _{p2} -1}{\xi _{02} -\xi _2}\\ &\frac{{\hbox{d}}\rho _2 }{{\hbox{d}}\tau }+\frac{b\cdot k_{\sigma 2} }{\xi _2 (\hbox{ln } \xi _2 )^{2}}\cdot \frac{{\hbox{d}}\xi _2 }{{\hbox{d}}\tau }=\frac{1}{\xi _{p2} -1}\cdot \frac{1}{\rho _2 }\cdot \left\{ {\xi _2 -\hbox{exp} \left( {\frac{k_{\sigma 2} }{\rho _2 }} \right)} \right\}+\rho _2^3 \cdot \frac{\xi _{p2} -1}{\xi _{02} -\xi _2 }\cdot \bar{{J}}_2 \cdot \frac{D_2 }{D_1 }\cdot \left[ {\frac{k_{\sigma 2} }{\hbox{ln } \xi _2 }+\delta \rho ^{\ast}-\rho _2 } \right]\\ \end{aligned}}\right. $$

(10)

The conservation conditions in a dimensionless form complete this system of equations:

$$ \begin{array}{ll} n_i =\frac{1}{\rho _i^3 }\cdot \frac{\xi _{0i} -\xi _i }{\xi _{pi} -1},&{i=1,2} \end{array} $$

(11)

The set of Eq. 10 were solved numerically by using a fourth-order Runge-Kutta method, using Eq. 11 to compute the dimensionless density of precipitates. Different parameters of k _γ1 and k _γ2, the ratio of diffusivities D ₂/D ₁ and different initial supersaturations ξ_i were used in the calculations. Incubation periods in the scaled version were estimated as

$$ \begin{array}{l} \tau _{{\rm w}1}\,=\,\frac{1}{2}(\rho \ast)^{2}c_{{\rm w}1}^2 \frac{\xi _{{\rm p}1} -1}{\xi _{01} -1}\,\equiv \,\tau _{{\rm w}1,{\rm min}} c_{{\rm w}1}^2 ,\\ \tau _{{\rm w}2}\,=\,\frac{1}{2}({\rho^\ast})^{2}c_{{\rm w}2}^2 \frac{\xi _{{\rm p}2} -1}{\xi _{02} -1}\frac{D_1}{D_2}\,\equiv \,\tau_{{\rm w}2,{\rm min}}c_{{\rm w}2}^2\\ \end{array} $$

(12)

derived from the assumption that the minimum time, τ_w,min, for a particle to reach the critical size ρ* is given by Eq. 3 with ${X_{\alpha /\beta }^{R_{\rm p}} =X_{\rm e}}$ . The latter equation usually overestimates the growth rate of subcritical nuclei (since ${X_{\alpha /\beta }^{R_{\rm p}} > X_{\rm e})}$ , and it is counterbalanced by the parameters c _wi, which are determined by fitting of results with experimental data. Typical variations of precipitates’ dimensionless densities and average radius as a function of a dimensionless time are shown in Figs. 1 and 2. The values used for the calculations are given in the figure captions. As can be seen, the height and width of the density peak are very sensitive to the dimensionless surface energy k _γ (Fig. 1a). Its location depends on the incubation time (Fig. 1b), which, in turn, depends on corresponding initial supersaturation and interface energy. Shape dependent parameters, φ and θ, may change the effective diffusion coefficients and surface energies: D′ = D/φ and γ′ = γ/θ, that should be taken into account in the analysis of calculation results.

Fig. 1

Calculated dimensionless densities of precipitates as a function of a dimensionless ageing time of supersaturated alloy; curves 1,2 – n₁ for supersaturation ξ₀₁ = 1.6 and interfacial energies k _γ1 = 0.06 (1) and k _γ1 = 0.08 (2); curves 3,4 – n ₂ for k _γ2 = 0.5, D ₂/D ₁ = 0.084 and supersaturations ξ₀₂ = 9 (3) and ξ₀₂ = 5 (4)

Fig. 2

Calculated average normalized radii of two kinds of precipitates as a function of the dimensionless time; 1−ρ₁ = R ₁/R _N1 for ξ₀₁ = 1.6, k _γ1 = 0.08; 2−ρ₂ = R ₂/R _N2 for ξ₀₂ = 9, k _γ2 = 0.5, D ₂/D ₁ = 0.084

Precipitation hardening due to particle-dislocation interaction

Hardening during aging of supersaturated alloys is determined by precipitates-dislocation interaction. Coherent precipitates, depending on their size and spacing, can be sheared or bypassed by dislocations [10]. When the precipitates are sheared, the stress at which the particles are overcome is proportional to their average size and inversely proportional to the effective distance between particles along a dislocation line, λ:

$$ \sigma_{\rm sh} =\Gamma _1 \frac{\bar{R}_{\rm p}}{\lambda}, $$

(13)

where Γ₁ depends on the nature of the interaction between precipitates and dislocations, as well as on the type of dislocation, Γ₁ = 2γ_a/b, γ_a represents the force per unit length opposing the motion of dislocation as it penetrates the particle, whereas b is the Burgers vector. In the following consideration Γ₁ was assumed as a parameter averaged over all types of dislocations interacted with precipitates. Taking into account a relationship between λ and the square lattice spacing, λ_s, between obstacles in the glide plane [11]: λ = λ_s/β ^1/2_c , where ${\beta _{\rm c}=\Gamma _1 {b}\bar{R}_{\rm p} /2\Gamma }$ , one can obtain:

$$ \sigma_{\rm shear}=\tilde{\Gamma }\frac{\bar{R}_{\rm p}^{3/2} }{\lambda_{\rm s}} $$

(13a)

where ${\tilde {\Gamma }=\Gamma _1^{3/2} (b/2\Gamma )^{1/2}}$ , Γ is the dislocation line tension. When the precipitates increase in size, and the distance between precipitates also increases, it becomes easier for the dislocation to bypass them. In this case the strengthening depends only on the distance between precipitates in the glide plane λ_s and on the dislocation line tension, according to Orowan equation [12, 13]:

$$ \sigma _{\rm by-pass}=0.81\frac{2\Gamma}{b\lambda_{\rm s}} $$

(14)

This prevails for precipitates having a radius higher than a critical value, ${\bar{R}_{\rm C}}$ given by the equality of two stresses σ_shear = σ_by-pass:

$$ \bar{R}_{\rm C}=1.74\frac{\Gamma }{b\Gamma_1}=0.87\frac{\Gamma }{\gamma _{\rm a}} $$

(15)

Two strengthening mechanisms associated with particle-dislocation interaction (shearing and bypassing) should be considered as operating simultaneously due to particle size-distribution, ${f(\bar{R}_{\rm p})}$. The contribution of precipitates sheared by dislocations to hardness can be written as follows [6]:

$$ \begin{array}{ll} \sigma_{i}^{\rm shear}=\frac{\tilde{\Gamma}_{\rm i}}{\lambda_{\rm si} }\int\limits_0^{\bar{R}_{\rm Cii}}{R_{\rm p}^{3/2}}\cdot f_{\rm i}({R}_{\rm p})\hbox{d}R_{\rm p}&{i=1,2} \end{array} $$

(16)

and the contribution of by-passed precipitates as:

$$ \begin{array}{ll} \sigma _{\rm i}^{\rm by-pass}=\frac{1.64\Gamma\cdot \bar{R}_{\rm Ci} }{b\lambda_{\rm si} }\cdot \int\limits_{\bar{R}_{\rm Ci}}^\infty {f(R_{\rm p})\hbox{d}R_{\rm p}}&i=1,2 \end{array} $$

(17)

In the case of two kinds of precipitated phases, the total contribution of four sets of discrete obstacles to precipitation hardening can be evaluated according to different rules mentioned in the literature (e.g., [11, 14]). Computer simulation [15] and theoretical investigations [16] have shown that the so-called Pythagorean addition rule is the best approximation for the case of distinct obstacles of two or more strengths. In the present consideration the precipitation hardening can be estimated as:

$$ \Delta \sigma _{\rm prec}=\sqrt{(\sigma _1^{\rm shear} )^{2}+(\sigma _2^{\rm shear})^{2}+(\sigma _1^{\rm by-pass} )^{2}+(\sigma _2^{\rm by-pass} )^{2}} $$

(18)

For the size distribution, f(R _p), as suggested by Deschams and Brechet [6], the Gaussian law was used:

$$ f(R_{\rm p})=\frac{2}{\Delta \sqrt{\pi }}\cdot \frac{1}{1+erf(\bar{{R}}_{\rm p} /\Delta )}\cdot \hbox{exp} \left[ {-\frac{(R_{\rm p} -\bar{{R}}_{\rm p} )^{2}}{\Delta ^{2}}} \right] $$

(19)

${\bar{R}_{\rm Ci} ,\tilde{\Gamma}_{i}}$ and Δ will serve as the fitting parameters of the hardening model. For particle spacing, λ_s, the expression of Kocks, obtained on the basis of computer simulations [14] was used:

$$ \lambda_{\rm s}=1.15\sqrt{\frac{2\pi }{3\cdot f_{\rm v}}}\cdot \bar{R} $$

(20)

where f _v is the particle volume fraction and ${\bar{R}}$ is the overall average radius of precipitates.

The strengthening of the alloy is usually considered as the result of a mixture of different contributions [14]:

$$ \sigma _{\rm tot} =\sigma _0 +\sigma _{\rm sol} +\Delta \sigma _{\rm prec} $$

(21)

where σ₀ is the friction stress of the matrix, σ_sol is the solid solution contribution; it is usually calculated as:

$$ \sigma _{\rm sol} =K_{\rm i} X_{\rm i}^{2/3} , $$

(22)

where X _i are the concentrations of impurities in the matrix. In our calculations, σ₀ was considered constant. The contribution of the solid solution strengthening can be evaluated using the difference in hardness between the alloyed and pure magnesium, σ ⁰_tot  − σ ⁰_tot (pure) = σ ⁰_sol = K _i X _0i ^2/3. Assuming that the microhardness of the alloy is proportional to the total stress, σ_tot, the relative change in the microhardness during precipitation can be found as:

$$ \frac{H_{\rm tot} -H_0}{H_0}=\frac{\Delta\sigma_{\rm prec}-{K}_{\rm i} (X_{0{\rm i}}^{2/3} -X_{\rm i}^{2/3} )}{\sigma _0 +K_{\rm i} {X}_{0{\rm i}}^{2/3}} $$

(23)

Experimental results and discussion

Mg–Zn–Sn alloy with 4.53 wt.% Zn and 4.75 wt.%Sn was solution treated at 465 °C for 96 h and then water quenched. Aging at 175, 200, 225, and 250 °C up to 96 h has led to the precipitation of the binary phases MgZn₂ and Mg₂Sn. The formation of these phases was studied experimentally. The two maxima during aging were found to be related to the formation of two types of precipitates, which were detected by XRD and SEM investigations: the first is MgZn₂ and the second is Mg₂Sn [8].

TEM investigations revealed that Mg₂Sn particles are plate-like and mainly equiaxed in the α-Mg-matrix while precipitates of MgZn₂ have a needle-like shape and are semi-coherent during almost all stages of growth and coarsening [9], so internal stress due to misfit strains may influence their kinetics of coarsening. Such influence can be substantial if the ratio of elastic and interfacial energies, measured by the dimensionless parameter ${\hat{S}=\varepsilon^{2}LC_{44}/\gamma}$ , is more than 4.5 [17, 18], where ɛ is the particle-matrix misfit, L is a characteristic length, e.g. an equivalent radius of a particle, C ₄₄ is an elastic constant and γ is the interfacial energy. Evaluation of this parameter for MgZn₂-precipitates embedded in the Mg-matrix in such a way that the ${\{11\overline 2 0\}_{\,{\rm MgZn}2}}$ lattice planes are parallel to the (0002)_α-Mg planes (which is the case realized [9]), for ɛ = 0.04 [Rashkova et al. (submitted, 2006)], C ₄₄ = 16.4 GPa [19], assuming L = 100 nm and γ = 10 mJ/m² yields ${\hat{S}\approx 0.26}$ . Thus, in this case, a coarsening process can be considered as stress-free for MgZn₂ as well as for Mg₂Sn-precipitates which are incoherent in the α-Mg-matrix.

The variation in the measured hardness with time is given in Fig. 3. The solid curves correspond to the calculated values. Precipitation hardening gives the main contribution to the results. The variation in solid solution strengthening was evaluated by comparison with the hardness of pure magnesium that undergone the same thermal treatment. It was found that this variation is small and can be neglected in the present analysis. The simulation parameters k _γ1 and k _γ2 were chosen by fitting the shapes of the calculated relative microhardness peaks, Eqs. 17–23, with experimental microhardness maxima corresponding to the evolution of given precipitates [8, 9] (Fig. 3); the location of peaks on the dimensionless time axis is determined by parameters c _w1, c _w2 and D ₂/D ₁. Additional temperature dependence of results is connected with variations in the limit solubilities, hence in the initial supersaturations ξ_0i (see Table 1). Experimental alloy’s concentrations, X _0,Zn = 1.7 at.% and X _0,Sn = 1.0 at.%, and the limit solubilities of Zn and Sn in Mg taken from the Mg–Zn and Mg–Sn phase diagrams were used to obtain the initial supersaturations, ξ₀₁ = X _0,Zn/X _e,Zn and ξ₀₂ = X _0,Sn/X _e,Sn. The chosen values c _w1 = c _w2 = 1.4 correspond to incubation times τ_i ≈ 2τ_i,min needed to first particles of MgZn₂ and Mg₂Sn to become ‘observaible’. The parameters found are given in Table 1.

Fig. 3

Precipitation hardening of water quenched Mg–Zn–Sn alloys as a function of the dimensionless (a) and dimensional (b) ageing time at different temperatures. Parameters used in calculations are present in the Table 1, scaling parameters t/τ—in Table 2

Table 1 Parameters used in calculations of the relative microhardness change.

Critical values ${\rho _{\rm Ci}=\bar{R}_{\rm Ci}/R_{\rm Ni}}$ , determining the transition from shearing to bypassing strengthening mechanism were chosen as 0.20 and 0.22 for i = 1 and i = 2 respectively that gives the best fitting with the initial growth of the hardness peaks (${\sigma \sim f^{1/2}\bar{R}_{\rm p}^{1/2}}$) followed by a behavior that corresponds with the bypassing mechanism (${\sigma \sim {f}^{1/2}/\bar{R}_{\rm p}\sim 1/\lambda _{\rm s}}$). So, the latter mechanism prevails during most of the growth period of precipitates. Unfortunately, direct experimental examination of mechanisms responsible for different stages of precipitation hardening in the Mg-based alloys was not done, and it requires a special TEM investigation which is out of the scope of the present work.

The calculated density and average size of precipitates correspond to experimental values according to Eq. (9). Using the available experimental data on volume fractions of MgZn₂-precipitates, found by SAXS (Fig. 4) and comparing it with the calculated volume fraction f ₁ = n ₁ρ ³₁ , one can find the scaling ratio (t/τ) = R _N ²₁ /D ₁. The average needle’s size evolution (Fig. 5, [Rashkova et al. (submitted, 2006)]), allows finding the scaling lenghts R _{N i} which were evaluated as follows: R _{N 1} ≈ (6.2/T)μm and R _{N 2} = (V _m2 X _β1/V _m1 X _β2)R _{N_1} ≈ (27.4/T)μm (Table 2). On the other hand, by using the values of molar volumes V _m1 = 2.1·10⁻⁵m³, V _m2 = 4.65·10⁻⁵m³ for MgZn₂ and Mg₂Sn respectively, the value of the scaling surface energy, γ₀, can be evaluated as: γ₀ = (R _Ni X _βi R _gT)/2V _mi ≈ 0.82 J/m². The corresponding surface energies of the nuclei are as follows: γ₁ =$\gamma_{MgZn_{2}}$= k _γ1 γ₀ ≈ 0.065 J/m² and γ₂ =$\gamma_{Mg_{2}Sn}$= k _γ2 γ₀ ≈ 0.41 J/m² (Table 2). The low value of interface surface energy, γ₁, is in line with the coherent interface between MgZn₂ precipitates and Mg-matrix. A rather large value of γ₂ is indicative of incoherent interfaces between Mg₂Sn-particles and Mg.

Fig. 4

Variation in the volume fraction of MgZn₂-precipitates during aging at 175 °C; parameters used in calculations are present in Tables 1, 2. Comparison of experimental results [Rashkova et al. (submitted, 2006)] with calculated ones (solid line) allows finding the ratio t/τ = R ²_N1 /D ₁ = 1.5 h

Fig. 5

Variation in the effective radius of MgZn₂(R ₁) and Mg₂Sn (R ₂) precipitates during aging at 175 °C of the quenched alloy Mg–Zn–Sn [9, Rashkova et al. (submitted, 2006)] and calculated values R ₁ = ρ₁ R _N1 and R ₂ = ρ₂ R _N2 with R _N1 = 14 nm, R _N2 = 62 nm, R ^*₁ = R _N1/ln ξ₀₁ = 2.4 nm and R ^*₂ = R _N2 /ln ξ₀₂ = 13.2 nm

Fig. 6

Calculated effective diffusion coefficients D ₁ and D ₂ responsible for evolution kinetics of MgZn₂ and Mg₂Sn precipitates; 1— D _Zn = 1.05exp(−125.8 kJ/RT)cm²/s [20] and 2−D _Sn = 4.27exp(−149.6 kJ/RT)cm²/s [21]

Table 2 Physical parameters evaluated from calculations compared with experimental data

The experimental times, $t_{\rm ml}=\tau_{\rm ml}R^2_{{N_{1}}/D_{1}}$, elapsed before the first maximum strength is achieved, and the ratios D ₂/D ₁ were used to determine the effective diffusion coefficients D ₁ and D ₂ responsible for the precipitates formation process (see Table 2). Comparison of these values with diffusion coefficients of Zn and Sn in Mg, D _Zn = 1.05exp(−125.8 kJ/RT)cm²/s [20] and D _Sn = 4.27exp(−149.6 kJ/RT)cm²/s [21] extrapolated from high temperatures (Fig. 5) shows that D ₁ is about ${3\div 4}$  times smaller than D _Zn. The decreasing parameter φ introduced in Eq. 3 can be estimated as (${2.0\div 2.4}$) for measured volume fractions of the needles f = (${0.4\div 0.8}$)% and the aspect ratio l/w found for the MgZn₂-needles as ${4\div 8}$ [9, Rashkova et al. (submitted, 2006)]. Decrease in the effective diffusion coefficient may be related to the absorption of Zn-atoms, mainly by incoherent needle tips that reduces the diffusion flux by the factor l/w. It can be also connected with compressive coherent stresses near the tips and tensile stresses along the needles so that small Zn-atoms are directed to the needle tips. It should be noted that the formation of the MgZn₂-phase in the Mg-matrix occurs with substantial decrease in the average atomic volume: ${\bar{v}_a ({\hbox{MgZn}}_2 )=1.16\cdot 10^{-29}\,{\hbox{m}}^{3}}$ and ${\bar{{v}}_a ({\hbox{Mg}})=2.32\cdot 10^{-29}\,{\hbox{m}}^{3}}$ and should be accompanied by the formation of vacancies. The coefficient D ₂ is close to D _Sn yet slightly higher. It is possibly caused by increased vacancy concentration mentioned before. With this in mind one can conclude that the first maximum is actually related to the precipitation of MgZn₂, and the second is related to Mg₂Sn precipitation.

Using scaling lengths R _{N i}, one can evaluate critical radii ${\bar{{R}}_{C_i}: \bar{R}_{C1} \approx 2.8\,{\hbox{nm}}}$ and ${\bar{{R}}_{C2} \approx 13.6\,{\hbox{nm}}}$ and the corresponding surface energies ${\gamma_a =1.87\cdot \Gamma /\bar{R}_{\rm C} }$ . Assuming Γ ≈ Gb ²/2, G = C ₄₄ = 16.4 GPa, b = (1/3)a _Mg = 0.1 nm, one can find γ_a1 ≈ 55 mJ/m² and γ_a2 ≈ 12 mJ/m² (Table 2). These values may correspond to energies of antiphase boundaries formed by dislocations penetrating the precipitated ordered phases, or to a difference in the stacking fault energies between the matrix and precipitates. Further detailed investigation of the hardening mechanisms in Mg-based alloys is required to resolve this question.

Thus, a reasonable agreement between calculated and experimental precipitation hardening was found for the temperatures 175 and 200 °C. For higher temperatures, 225 and 250 °C, the first maximum was not detected, apparently because of very short aging times when it appears at such temperatures. The measured values of hardness at the second maximum are lower (especially for 250 °C) than the calculated ones. One can assume that these deviations are connected with a recrystallization process occurring at such high temperatures, which are above 0.5T _melt, where T _melt is the alloy melting temperature.

Conclusion

The model for the precipitation and hardening processes in Mg-based alloys was developed on the basis of modified Langer-Schwartz approach. Analysis of the evolution and precipitation hardening of precipitates during the aging of Mg–Zn–Sn alloy was carried out while taking into account the simultaneous formation of the binary MgZn₂ and Mg₂Sn phases. Two strengthening mechanisms associated with particle-dislocation interaction (shearing and bypassing) were considered as operating simultaneously due to particle size-distribution. A reasonable agreement between the calculations and observations was found.

The scaling parameters of the model, R _Ni and γ₀, were found by fitting the calculated densities and average size of precipitates with the measured ones. Interface surface energies, γ₁ and γ₂, and effective diffusion coefficients responsible for the strengthening kinetics, D ₁ and D ₂, were estimated based on the microhardness experiments. Two maxima of hardness during aging were found to be related to the formation and coarsening of two types of precipitates, MgZn₂ and Mg₂Sn. The values of γ₁ and γ₂ are in line with the nucleation of coherent MgZn₂-particles and incoherent Mg₂Sn-particles. Effective diffusion coefficients correspond to the solute diffusion of Zn and Sn in Mg but can be affected by the features of the new phase particles: in the case of MgZn₂-needles, D ₁ is reduced by the factor l/w in comparison with the usual bulk diffusion coefficient D _Zn.

The transition from shearing to bypassing strengthening mechanism was found to occur at rather early stages of the particle growth. The bypassing was found to be the prevailing strengthening mechanism in the investigated alloys. The model allows the description of competitive hardening mechanisms taking into account the distribution of precipitate size and strength as well as the transition from one mechanism to another. However, additional investigations are required to identify these mechanisms.