An Age-Hardening Model for Al-Mg-Si Alloys Considering Needle-Shaped Precipitates

Precipitation is a process in which the initial supersaturated alloy is decomposed into matrix and a new phase, generally containing a higher concentration of solute atoms. The precipitation is traditionally categorized into three stages: nucleation, growth, and coarsening. In this model, simultaneous nucleation and growth is assumed to take place as long as the equilibrium volume fraction has not been reached. After reaching the equilibrium value, coarsening starts. In the microstructure model, a single type of precipitate is considered. These assumptions already were successfully applied to modeling the precipitation hardening in Al-Mg-Si alloys.[3‐5] For a complete model description, there are additional assumptions, as follows.

Assuming that precipitates nucleate with a cylindrical morphology, the change in the Gibbs free energy of the system due to nucleation can be written aswhere ΔG _ν is the driving force per unit volume to form a precipitate from supersaturated solid solution and γ is the interfacial energy between the precipitate and the matrix. Using the definition of the aspect ratio (A = h/r), the critical radius of precipitates, r _cr , corresponding to the maximum of ΔG _nucl, can be expressed as

Inserting the expression of ΔG _ν given in Reference 23 into Eq. [2] yieldswhere C _m is the mean concentration of Mg in the matrix, \( C_{e}^{\alpha \beta } \) is the equilibrium concentration of Mg in the matrix at the aging temperature, and V _m is the molar volume of precipitate. Provided that the incubation time can be neglected, i.e., assuming steady-state nucleation, the nucleation rate J is expressed as[7]where J ₀ is a pre-exponential term, and Q _d is the activation energy for bulk diffusion of Mg. \( \Updelta G^{ * } , \) the activation energy for nucleation, can be obtained from Eqs. [1] and [3] and is given as

Myhr et al.[16] proposed an approximate expression to calculate the energy barrier\( \Updelta G^{ * } , \) given bywhere A ₀ is treated as a fitting parameter. Peripheral and longitudinal growth rates of a needlelike precipitate of radius r and half-length h are determined by the composition gradient outside the precipitate; the bulk diffusivity of Mg, D; and the aspect ratio of precipitates. Extending the approach of Ferrante and Doherty[22] and Liu et al.,[23] the diffusion-controlled thickening and lengthening of precipitates approximately obeywhere \( C^{\beta \alpha } \) is the concentration of Mg inside the precipitate, and \( C_{r}^{\alpha \beta } \) is the equilibrium interface concentration around the precipitates, taking into account the Gibbs–Thomson effect,[22] given as

Since both C _m and \( C_{r}^{\alpha \beta } \) are a function of the degree of precipitation, the aging time during underaging is divided into a series of small time intervals (Δt). At each time-step, a new population of JΔt precipitates nucleates, having a radius equal to r _cr (Eq. [3]). In the same time interval, other previously formed precipitates grow. Thus, a precipitate size distribution evolves. If Δt is chosen to be very small, one can assume that the change in supersaturation during the time period Δt is negligible. Therefore, the increment in the radius of precipitates group j, having the radius r _j, is calculated asbeing the time derivative of Eq. [7]. Consequently, the radius of precipitates in group j in the ith time-step is written as

By keeping track of the growth of each group of precipitates formed during aging, one can calculate the mean radius of precipitates, r _m . At the end of each time-step, the mean concentration of alloying element in the matrix is updated using the mass balance. The aging time increases step-by-step until the calculated mean concentration of Mg in the matrix becomes equal to the equilibrium interface concentration of precipitates of smallest size. The highest equilibrium interfacial concentration in the matrix occurs at the smallest precipitates (due to the Gibbs–Thomson effect). This group of precipitates, therefore, is the first group reaching the criterion \( C_{m} < C_{{r_{\text{j}} }}^{\alpha \beta } , \) when the growth stops (Eq. [10]). This is a numerically imposed criterion. From the moment that \( C_{m} < C_{{r_{\text{j}} }}^{\alpha \beta } , \) the nucleation/growth stage is terminated and coarsening starts. According to the Gibbs–Thomson equation, during coarsening, the stability of precipitates increases by increasing their size, meaning that fine precipitates dissolve and bigger precipitates grow to reduce the free energy of the system. Therefore, the driving force for coarsening is provided by the difference between the size-dependent interfacial concentrations of alloying element and the average concentration of alloying element in the matrix. Assuming that during coarsening precipitates of the average size are in equilibrium with the matrix (\( C_{{r^{m} }}= C_{m} \)), the supersaturation during coarsening for a precipitate of size r can be written as[22]where \( C_{{r^{m} }} \) is the equilibrium interfacial concentration around the precipitate of mean radius r _m . The value of \( C_{{r^{m} }} \) can be obtained from Eq. [8]. Inserting the values of \( C_{r}^{\alpha \beta } \) and \( C_{{r_{m} }} \) into Eq. [11] yields

This equation clearly shows that for precipitates larger than r _m , the supersaturation is positive, meaning that during coarsening, when r is bigger than r _m , precipitates grow. From Eq. [7], the coarsening rate can be written as

Inserting the value of supersaturation from Eq. [12] into the coarsening rate equation yields

Using the approximation that the mean radius changes at the same rate as the maximum thickening rate, when r = 2r _m ,[22] the coarsening rate can be written as

Integrating this equation gives an analytical equation for precipitates radius during coarsening as follows: with r _PA being the mean radius at peak age and t _PA being the time to peak age, which corresponds to the time when coarsening starts.

The strength model is a framework in which the overall strength of the artificially aged alloy can be obtained by the addition of the intrinsic strength of aluminum, the solid solution strength contribution, and the precipitate strength contribution.[4] Assuming that different strengthening contributions to the overall strength can be added linearly, the yield strength of Al-Mg-Si alloys can be expressed aswhere σ _i is the yield strength of pure aluminum chosen as 10 MPa,[3] and σ _ss is the solid solution strengthening term, given by[3]where k _j is a constant with a specific value for element j. The effect of precipitates on the strength is given by σ _ppt. The precipitation-strengthening model correlates the size and volume fraction of precipitates with strength. Two main mechanisms exist for precipitation hardening. When a dislocation encounters a precipitate, it will either cut through it, a mechanism known as shearing, or bypass the precipitate by looping around it. Shearing is more common in coherent precipitates, which have an orientation relationship with the matrix, whereas in the case of larger precipitates, coherency usually breaks down and looping occurs instead. Initial precipitation hardening involves strengthening of the alloy due to the formation of a high density of small coherent precipitates. These precipitates are sheared during deformation by moving dislocations. By increasing the aging time, the precipitates become larger and stronger. Precipitates larger than a transition radius cannot be sheared anymore and a dislocation can only bypass by looping around them. This leaves an Orowan loop around the precipitates, which enhances the strength of the alloy. Let F and l denote mean precipitate strength and effective mean interprecipitate distance along the dislocation line, respectively. Then, precipitate strengthening is given by[3]where M is the Taylor factor and b is the length of the Burgers vector. There are two influencing parameters in precipitation strengthening: (1) mean precipitates strength, F, and (2) effective precipitates spacing, l. The mean precipitate strength is defined as the interaction force between the precipitate and the dislocation. Experimental observation of dislocation-precipitate interaction in Al-Mg-Si alloys reveals that precipitates are still shearable at the peak-age condition,[2] and a part of them remains shearable even after a long-time overaging.[24] Based on this observation, Esmaeili et al.[2] divided the aging process into three parts: (1) underaging up to peak age when precipitates are shearable; (2) the stage between peak age to the transition point where precipitates are still shearable; and (3) after the transition point, where precipitates behave as nonshearable particles. The transition radius, r _trans, is defined as the radius where the strengthening mechanism is changing from shearing to bypassing. The mean precipitate strengths at these three stages are given as[2]where β is a constant equal to 0.5, and m is 0.6. The effective precipitate spacing is dependent on the precipitate strength.[24] Precipitate spacing can be estimated based on the planar center-to-center distance between precipitates. Since the most important precipitate species, β″, is elongated in 〈100〉_Al direction, it is very important to consider the effect of its needle-like morphology and its orientation on the effective obstacle spacing.

Considering the orientation relationship between the needle-shape precipitates along the 〈100〉_Al direction and the {111}_Al slip planes in aluminum, the effective obstacle spacing between precipitates is calculated as[25,26]where Γ is the dislocation line tension and f is the volume fraction of precipitates. Knowing the values of effective precipitate spacing for shearable and nonshearable precipitates as well as the values of mean obstacle strength (Eq. [21]), the contribution of precipitates to the overall yield strength can be calculated aswhere f _PA is the volume fraction of precipitates at peak age.

In this model, it is assumed that all precipitates are cylindrical with a constant aspect ratio, which is a parameter for which different choices can be considered. The alloy used to check the validity of the model is AA6061 (1.12 wt pct Mg, 0.57 wt pct Si, 0.25 wt pct Cu). Assuming that all Si atoms take part in precipitation, the chemical composition of the alloy is balanced. The equilibrium interfacial concentration of Mg for a flat interface, C _e , is given by[3]where \( Q_{s} \) is the apparent solvus boundary enthalpy. In the strength model, there are two important radii: r _PA and r _trans. The value r _PA is dependent on the aging temperature and chosen aspect ratio. The microstructure model is developed in such a way that peak age correlates to the time when the precipitation reaction is changing from the nucleation-growth regime to the coarsening regime. Consequently, at peak age, the precipitate density, N _PA , is maximum. r _PA can be approximately determined, assuming that the amount of solute elements left in the matrix is negligible compared to the amount used to form the precipitates by

As is seen, r _PA is a function of aspect ratio; the maximum number density of precipitates, which is obtained from the simulation; the initial concentration of alloying elements, C ₀; and the concentration in the precipitate. Another important parameter in the strength model is the transition radius, r _trans, at which the strengthening mechanism changes from shearing to bypassing. Cheng et al.[27] proposed that for the alloy AA6111, the shearable to nonshearable transition occurs when the yield strength is equal to 0.8 σ _PA (σ _PA is the yield strength at peak age). Assuming that this is also the case in alloy AA6061 and knowing that the peak strength of alloy AA6061 is in the range of 250 to 270 MPa, σ _ppt at the transition point, \( \sigma_{\text{ppt}}^{\text{trans}} , \) assuming that the contribution from solid solution hardening is negligible, is given by

Using Eq. [23], r _trans can be calculated as

Assuming that the volume per atom is constant, the final volume fraction of precipitates at the peak age can, by the lever rule of phase equilibria, be approximated as

Depending on the chosen values, r _trans varies between 3.0 and 4.0 nm. In the developed model, the approximate value of 3.5 nm is used for all conditions. Table I shows the input data used in the microstructure and strength model.

Figure 1 shows the simulated length (2h) of precipitates in alloy AA6061 during aging at 463 K (190 °C) together with experimental data. The aspect ratio is adjusted to obtain the correct value for the length of precipitates at peak age (≈10 ks). The aspect ratio, which gives the correct prediction of the length at peak age, is A = 10. In the underage regime, the model slightly overestimates the length of precipitates, showing the highest overestimation in the beginning of aging. By increasing the aging time, the difference between the modeling results and experimental data becomes smaller. In the overage regime, the modeling results are in good agreement with the experimental data.

Figure 2 shows the prediction of the volume fraction of precipitates in alloy AA6061 aged at 463 K (190 °C) together with experimental data. As is seen, the model shows a faster aging kinetics during underaging compared to the real aging kinetics. This is possibly due to the assumption of the incubation time being zero. Besides, the maximum precipitate volume fraction predicted by the model is slightly higher than the experimental maximum value. This is due to the fact that it is assumed that all Si atoms are partitioned to the precipitates during precipitation. In reality, a part of Si content of the alloy is used by Fe- and Mn-containing particles.

Figure 3 shows the reproduction of the yield strength of alloy AA6061 at different temperatures, using A = 10, compared with experimental data. Except for aging temperature 443 K (170 °C), in which there is reasonably good agreement between model and experiment, the model overestimates the yield strength of alloy AA6061 in both underage and overage regimes. The overestimation, however, is more pronounced in the underage regime.

Figure 4 shows the comparison between the modeling results for A = 1 with the case when A = 10. Obviously, for A = 1, the model yields a better fit for the yield strength of the alloy in the underage regime. However, the peak strength is predicted more accurately when A = 10. Also, it is interesting to note that the yield strength at the early stage of coarsening is predicted very well when A = 10, while when A = 1, the model underestimates the yield strength in the beginning of coarsening. This changes when the overaging time is increased: the model with A = 1 becomes more accurate, while the model with A = 10 gradually loses its accuracy.

Figure 5 shows the effects of aspect ratio on the evolution of number density, mean radius, volume fraction, and precipitate effective spacing during aging at 463 K (190 °C). The model shows that precipitates with A = 1 have maximum values of both number density and mean radius compared to precipitates of A = 20 and 50. Figure 5(c) shows that even though both the number density and thickness of precipitates in case A = 1 are higher compared to those of precipitates with A = 20 and 50, the volume fraction of precipitates in the underage regime (when A = 1) shows the lowest magnitude. This is due to the fact that longer precipitates have larger volumes (and, therefore, lower number density). This also has an effect on the precipitate effective spacing, as seen in Figure 5(d), where initially the mean precipitate distance is largest for precipitates of A = 1. The difference between the mean precipitate distance for precipitates of different aspect ratios decreases up to the peak age. At peak age, the mean precipitate distance values for different aspect ratios are similar.

Figure 6 shows the effects of aspect ratio on the predicted yield strength of alloy AA6061. Clearly, the higher the aspect ratio, the faster the hardening kinetics. Also, the yield strength increases over the entire time range with increasing aspect ratio. This is more pronounced in the underage regime.

The precipitation sequence in Al-Mg-Si alloys is very complex with a series of precipitates having different morphologies. β″ phase, the most important precipitate in terms of strengthening, is a needlelike precipitate. Apart from that, other precipitates including pre-β″, β′, and Q′ have elongated cylindrical morphologies. Therefore, a model that is able to predict the evolution of precipitate radius and length is more realistic than an aging model based on the assumption of spherical morphology for precipitates. An age-hardening model was developed considering a cylindrical morphology for precipitates. The model is fitted for alloy AA6061 in such a way that it gives a correct prediction of the precipitate mean length at peak age at 463 K (190 °C). With the fitted value of aspect ratio equal to A = 10, the model yields a satisfactory prediction of precipitate mean length in the overage regime (Figure 1). However, it overestimates the mean length in the underage regime.

The reason for this overestimation is that not all precipitates formed in the beginning of aging have a needlelike morphology. Especially, a large fraction of GP zones have spherical morphology. In addition, the aspect ratio of β″ phase changes during the aging process in such a way that it initially increases to a maximum value and thereafter decreases,[23] meaning that in the beginning of aging and in the very end of overage regime, the value A = 10 is too high for the aspect ratio of precipitates. For this reason, the yield strength in the beginning of aging and in the very end of overaging is predicted more accurately when A = 1 (Figure 4). The aspect ratio of precipitates is influenced not only by aging time but also by aging temperature. At higher aging temperatures, the aspect ratio becomes lower due to a smaller lengthening driving force and the readily occurring loss of coherency or semicoherency of peripheral plane.[23] This would result in a stronger deviation of predicted yield strength from the experimental values in the case of A = 10, when the aging temperature increases (Figure 3).

The shape of precipitates influences the strength evolution both through its effects on precipitation kinetics (size and density) and by altering the precipitate-dislocation interaction. In order to separate these two effects, we will assume that the number density and volume fraction of precipitates are not dependent on the chosen value of the aspect ratio; i.e., at any time, the volume fraction and number density of precipitates for the case A = 1 are supposed to be the same as for cases A = 20 and 50. The only difference is the difference between the radius and length of precipitates, i.e., shape difference. Figure 7 shows the evolution of σ _ppt in alloy AA6061 for the different aspect ratio and similar number density and volume fraction. It is seen that in the absence of shape effect on precipitation kinetics, the evolution of strength up to peak age is hardly dependent on aspect ratio. This implies that in the underage regime, the effects of precipitate thickness and number of precipitate/slip-plane intersections cancel. When precipitates become more elongated, they also become thinner, and therefore weaker (due to lower F). However, as they also become longer, the number of intersections with slip planes increases and consequently l decreases. So, in the underage regime, when precipitates are shearable, the morphology by itself does not have a significant effect on the yield strength. The difference in the yield strength of the alloy at the underage regime, as seen in Figure 6, is a kinetics effect (Figure 5). Longer precipitates, due to the faster growth kinetics in the underage regime (Figure 5(c)), have more intersections with slip planes (smaller effective precipitate distances, as shown in Figure 5(d)), making their contributions to yield strength higher. By entering the overaging regime, the strength becomes independent of F. It is also obvious that the aging kinetics does not have any influence on the yield strength, since both number density and volume fraction are the same, independent of aspect ratio (Figures 5(a) and (c)). This implies that the only influencing parameter in the overage regime is the shape of the precipitates. The longer the precipitate, the higher the density of precipitate/slip-plane intersections, leading to higher strength.