Modeling the concurrent growth of inter- and intragranular Si precipitates during slow cooling of the alloy AA6016

Abstract

This work combines two established models for precipitate growth in the bulk and at the grain boundary to investigate the growth of pure Si precipitates in the alloy AA6016 during slow cooling after hot rolling. Despite the simplicity of the approach, the predicted phase fractions and the predicted width of the precipitate-free zone are in agreement with the measured values. The model predictions suggest that ­raising the Si content and increasing the density of primary Fe-based particles reduces the fraction of grain boundary precipitates. The model correctly reproduces the technologically important effect of the cooling rate on intra- and intergranular growth. The model predictions can be used to optimize hot rolling schedules and the alloy design.

Keywords:

• Al-Mg-Si alloy
• Si phase
• collector plate model
• hot rolling

1. Introduction

The Al-Mg-Si alloy AA6016 is widely used for car body parts because it offers a balanced mix of strength, formability, corrosion resistance, and weldability (Miller et al., 2000; Prillhofer et al., 2014). It differs from other common automotive sheet alloys of the 6xxx series because of its high Si content ranging from 1 to 1.5 wt.%. and a high Si/Mg ratio, typically above three. As a consequence, pure Si becomes the dominant equilibrium phase in AA6016 (Chakrabarti and Laughlin, 2004) instead of Mg₂Si. The pure Si phase appears during over-aging (Meyruey et al., 2018), slow quenching following solution annealing (Castany et al., 2013; Fan et al., 2018) or after slow heating from the supersaturated state (Lang et al., 2014). It can also be found in as-cast material (Engler et al., 2023) and after hot rolling or intermediate annealing treatments (Engler and Hirsch, 2002). The technological significance of Si precipitates formed during the production process of AA6016 sheets is twofold. On the one hand, they act as potential sites for particle-stimulated nucleation, which reduces the crystallographic texture intensity in the final sheet (Bollmann et al., 2010; Wang et al., 2022). On the other hand, the Si precipitates must be dissolved during solution annealing to exploit the full hardening potential of the alloy (Zhang et al., 2016). Solute Si accelerates the precipitation hardening kinetics (Gupta et al., 2001) and enhances the work-hardening and strength (Zhang et al., 2023), while non-dissolved precipitates may decrease the formability (Garrett et al., 2005). In spite of this technological relevance, studies of the formation of the Si phase in wrought alloys are scarce. Schumacher et al. (2015) studied the formation of pure Si precipitates in a hot-rolled binary Al-Si alloy, but similar investigations for industrial AA6016 do not exist.

In this work, we investigate the growth of pure Si precipitates in AA6016 during coil cooling after hot rolling, i.e. prior to cold rolling. An essential aspect of the material at this processing step is the simultaneous occurrence of inter- and intragranular Si-precipitates together with Mg₂Si precipitates. The kinetics of grain boundary precipitates have already been studied in other alloys such as Al-Cu (Huang et al., 2010), Al-Mg (Yi et al., 2014; Zhang et al., 2017), and Al-Mg-Zn alloys (Kamp et al., 2006; Liu et al., 2021) but not in the case of pure Si precipitates. Following Kamp et al. (2006), we employ a growth model based on the collector plate mechanism (Aaron and. Aaronson, 1968) in combination with a classical growth model for intragranular precipitates. This simple physical approach proves especially useful for the present purpose because it allows the simulation of the effect of varying Si-content, precipitate density, grain size, and cooling rate. Additionally, it can be easily applied to thermal histories of industrial sheet production, thus providing a fast and useful tool for alloy development and process optimization.

To calibrate the model, the fraction and arrangement of Si precipitates in an industrial sample are experimentally determined with scanning electron microscopy (SEM). Standard SEM sample preparation methods of 6xxx series alloys (Wang et al., 2022) are not able to identify the Si-phase unambiguously in the presence of Mg₂Si. Etching or electro-polishing changes the composition of Mg₂Si, which also reacts with water during mechanical polishing (Österreicher et al., 2016). We overcome this problem by using argon milling to avoid any chemical or mechanical contamination of the sample surface. Thus, we are able to distinguish between the phases and quantify the fraction of inter- and intragranular Si.

2. Material and electron microscopic analysis

The investigated alloy was AA6016 for automotive applications with the main alloying elements as measured with optical emission spectroscopy (OES) given in Table 1. The alloy additionally contains trace elements, such as Ti, which are not relevant to the present analysis. A thermodynamic equilibrium calculation with the commercial software FactSage (Bale et al., 2002) predicts a solvus temperature of 535 °C and 490 °C for pure Si and Mg₂Si, respectively. The material was directly chill cast, homogenized, and hot rolled to a thickness of 4 mm. The hot rolled coil was solution annealed in a batch oven at 540°C and air cooled to room temperature. At this point, a sample was extracted from the coil. To avoid any chemical or mechanical contamination, the sample was prepared with an Ilion argon milling system. The analysis was carried out on a Zeiss Ultra 55 scanning electron microscope (SEM) operated at 6 kV in conjunction with an energy-dispersive X-ray spectroscopy system (EDS) by Ametek. Figure 1a shows the microstructure as measured by an inlens secondary electron detector (SE), and Figures 1b–1d show element scans for Si, Mg, and Fe as measured with the EDS system. The sample contains coarse primary particles which consume the Fe and Mn content of the alloy (Engler et al., 2023) and appear white in the SE detector. Cu is mostly consumed by small metastable Mg-Si phases. Si and Mg₂Si precipitates appear black and dark gray, respectively. Both phases are distinguishable from the matrix and from each other. Large, elongated Si-precipitates are found along the grain boundaries. Adjacent to the grain boundaries, there is a precipitate-free zone (PFZ) with a width of several microns. In the grain interior, Si and Mg₂ Si precipitates coexist. Both phases are often found at primary Fe-based particles, which act as preferred nucleation sites. A considerable part of the Mg₂Si precipitates was found at Si-precipitates, indicating that the latter can also act as a nucleation site for Mg₂Si. The quantitative analysis of Si and Mg₂Si precipitates was carried out with the ImageJ software (Schneider et al., 2012) for two micrographs at different positions (A and B) of the sample. The micrographs are shown in Figures 2 and 3, respectively. The first step was to identify the two types of precipitates using gray-scale correlation. The second step was to distinguish between precipitates located at the grain boundary and in the bulk of the grain. This was done by identifying the grain boundaries based on the channeling contrast in the SE image and manually separating the grain boundaries from the grain interior. The resulting intergranular phase fraction was assumed to consist completely of Si-precipitates. The resulting intragranular phase fraction contains a non-negligible amount of Mg₂Si and, therefore, was corrected. The correction factor was determined from additional micrographs at higher resolution, allowing a better distinction between dark gray Mg₂Si and black Si-precipitates. The phase fractions for intergranular Si precipitates thus obtained for the two positions A and B are ${0.0052}_{-0.0022}^{+0.0026}$ and ${0.0058}_{-0.0024}^{+0.0028}$, respectively. The uncertainty originates from the gray-scale correlation because the interface between the precipitate and matrix on the micrograph can be blurred. The upper and lower uncertainties have been calculated by adding and subtracting the size of one pixel of 0.25 µm to both the minor and the major principal axis of each precipitate. The phase fractions for intragranular Si precipitates for the two positions A and B are ${0.0013}_{-0.0007}^{+0.001}$ and ${0.0028}_{-0.0013}^{+0.0014}$, respectively. The mean diameters of intragranular Si precipitates at the two investigated positions A and B are 1.62 ± 0.37 µm and 1.77 ± 0.62 µm, respectively. Here, the sample standard deviation of all detected intragranular precipitates was chosen as an error estimate because it exceeds the uncertainty associated with the gray-scale correlation, which is equal to the pixel size of 0.25 µm in this case. The PFZ was determined from micrographs at higher resolution by measuring the distance between the grain boundary and the closest intragranular precipitate for 20 grain boundary segments. The measured values for the two positions A and B are 9.1 ± 1.5 µm and 10.6 ± 2.7 µm, respectively, where again, the sample standard deviation is given. The total grain boundary length at positions A and B was 6.3 mm and 5.4 mm, and the length per area was 38.8 mm⁻¹ and 31.4 mm⁻¹, respectively. All values are summarized in Table 2. The differences between the phase fractions at the two measurement positions lie within the uncertainty of the gray-scale correlation, and the values of the width of the PFZ and the diameter of the intragranular precipitates are within the standard deviation. Hence, the investigated area appears to be representative of the sample.

Figure 1. Microstructure of the alloy AA6016 containing primary Al-Fe-Si particles (white), Si (black), and Mg₂Si (dark-gray) precipitates. The width of the images is 100 µm. (a) Image from inlens secondary electron (SE) detector. (b–d) Energy dispersive X-ray spectrometry scans of the main alloying elements Si, Mg, and Fe.

Figure 2. Microstructure of the AA6016 as measured by the inlens secondary electron detector at Position A. The width of the image is 500 µm.

Figure 3. Microstructure of the AA6016 as measured by the inlens secondary electron detector at Position B. The width of the image is 500 µm.

Table 1. Main alloying elements of AA6016 in wt.%.

Mg	Si	Fe	Mn	Cu
0.35	1.15	0.15	0.10	0.05

Table 2. Microstructure after coil cooling.

Quantity	Position A	Position B	Model prediction
Intergranular Si phase fraction	${0.0052}_{-0.0022}^{+0.0026}$	${0.0058}_{-0.0024}^{+0.0028}$	0.0048
Effective intragranular Si phase fraction	${0.0013}_{-0.0007}^{+0.001}$	${0.0028}_{-0.0013}^{+0.0014}$	0.0012
Mean diameter of intragranular Si	1.62 ± 0.37 µm	1.77 ± 0.62 µm	2.29 µm
Width of PFZ	9.1 ± 1.5 µm	10.6 ± 2.7 µm	6.6 µm

3. Modeling

3.1. Growth of intergranular precipitates

The so-called collector-plate mechanism for the growth of grain boundary precipitates proposed by Aaron and Aaronson (1968) considers a single growing precipitate at a flat grain boundary. Solute atoms migrate from the grain interior to the grain boundary and along the grain boundary to the precipitate. Assuming that the grain boundary diffusion occurs much faster than volume diffusion inside the grain, only the latter is considered by the model. The original work contains the exact isothermal solution of Fick’s second law for the collector plate model with constant bulk concentration. Here, the shape of the concentration profile of the exact isothermal solution is taken as an approximation for the case of general processing histories with varying temperature, solute content, and grain boundary area. The approximation of the concentration profile c(u) along the coordinate u perpendicular to the grain boundary is defined as: (1) $$c(u)=c_b+\left(c_i-c_b\right)\text{erf}\left(\frac{u}{2u_D}\right)$$(1) where c_b and c_i are the solute concentrations at the grain boundary and in the grain interior, respectively. u_D represents a length measure that determines the slope of the profile, and A is the size of the collector plate. All solutes which migrate to the grain boundary are assumed to be consumed by precipitate growth, which leads to the following continuity equation of the solute flux at the grain boundary $J^{\star }$: (2) $$A{J}^{\star }=-\dot{V}\left(c_{\theta }-c_b\right)$$(2)

Here, $\dot{V}=\frac{\mathrm{d}V}{\mathrm{d}t}$ is the time derivative of the precipitate half-volume V, and $c_{\theta }$ is the concentration of solute in the precipitate. The flux at the grain boundary is also determined by Fick’s first law as: (3) $$J^{\star }={D\overline{)\frac{\partial c\left(u\right)}{\partial u}}}_{u=0}=-\frac{D}{\sqrt{\pi }}\frac{(c_i-c_b)}{u_D}$$(3) with the volume diffusion coefficient D. The solutes migrating from the grain interior to the precipitate fulfil the mass balance: (4) $$V(c_{\theta }-c_b)=A(c_i-c_b){\int }_0^{u_0}[1-\text{erf}\left(\frac{u}{2u_D}\right)]du\approx A(c_i-c_b)\frac{2u_D}{\sqrt{\pi }}$$(4)

The approximation of the integral applies as long as $u_0>2u_D$, where u₀ represents the distance from the grain boundary to the grain center. Inserting the grain boundary flux given by Fick’s first law in equation (3)(3) $$J^{\star }={D\overline{)\frac{\partial c\left(u\right)}{\partial u}}}_{u=0}=-\frac{D}{\sqrt{\pi }}\frac{(c_i-c_b)}{u_D}$$(3) into the continuity equation (2)(2) $$A{J}^{\star }=-\dot{V}\left(c_{\theta }-c_b\right)$$(2) and using the mass balance in equation (4)(4) $$V(c_{\theta }-c_b)=A(c_i-c_b){\int }_0^{u_0}[1-\text{erf}\left(\frac{u}{2u_D}\right)]du\approx A(c_i-c_b)\frac{2u_D}{\sqrt{\pi }}$$(4) to eliminate u_D leads to the following evolution equation for the precipitate volume: (5) $$\dot{V}=\frac{2D}{\pi }\frac{A^2}{V}\frac{{\left(c_i-c_b\right)}^2}{{\left(c_{\theta }-c_b\right)}^2}$$(5)

This growth equation reproduces the original isothermal result by Aaron and Aaronson when only the volume V is allowed to vary in time while holding constant the temperature, concentrations, and boundary area. For the combined use with the intragranular growth model, it is convenient to replace the precipitate half-volume and the collector plate area with the phase fraction $f_b=V/A{u}_0$ and the density of grain boundaries $S={(2u_0)}^{-1}$  , respectively, leading to (6) $${\dot{f}}_b=\frac{8}{\pi }\frac{S^2}{f_b}D\frac{{\left(c_i-c_b\right)}^2}{{\left(c_{\theta }-c_b\right)}^2}.$$(6) for the growth equation.

3.2. Simultaneous growth of intragranular precipitates

The growth of intragranular precipitates is modeled with the classical Zener equation (Zener, 1949) for spherical particles at low supersaturations: (7) $$\dot{d}=\frac{4D}{d}\frac{\left(c_i-c_{\mathit{\text{eq}}}\right)}{\left(c_{\theta }-c_{\mathit{\text{eq}}}\right)}$$(7)

Here, d is the diameter of the spherical precipitate. C_eq is the equilibrium concentration of the solute in the Al matrix. In analogy to equation (6)(6) $${\dot{f}}_b=\frac{8}{\pi }\frac{S^2}{f_b}D\frac{{\left(c_i-c_b\right)}^2}{{\left(c_{\theta }-c_b\right)}^2}.$$(6) , the above expression can be rewritten in terms of the volume fraction of intragranular precipitates $f_i\equiv N\pi d^3/6$ with the number density of intragranular precipitates N, leading to: (8) $$\stackrel{.}{f_i}=3{\left(\frac{4\pi N}{3}\right)}^{\frac{2}{3}}{f}_i^{\frac{1}{3}}D\frac{\left(c_i-c_{\mathit{\text{eq}}}\right)}{\left(c_{\theta }-c_{\mathit{\text{eq}}}\right)}$$(8)

The evolution equations for inter- and intragranular growth, equation (6)(6) $${\dot{f}}_b=\frac{8}{\pi }\frac{S^2}{f_b}D\frac{{\left(c_i-c_b\right)}^2}{{\left(c_{\theta }-c_b\right)}^2}.$$(6) and equation (8)(8) $$\stackrel{.}{f_i}=3{\left(\frac{4\pi N}{3}\right)}^{\frac{2}{3}}{f}_i^{\frac{1}{3}}D\frac{\left(c_i-c_{\mathit{\text{eq}}}\right)}{\left(c_{\theta }-c_{\mathit{\text{eq}}}\right)}$$(8) are solved simultaneously. The equations are coupled via the solute concentration in the matrix which is reduced by the fraction of intragranular precipitates: (9) $$c_i=c_i^0-c_{\theta }{f}_i$$(9) where $c_i^0$ denotes the initial concentration of the solute. The growth equations (8)(8) $$\stackrel{.}{f_i}=3{\left(\frac{4\pi N}{3}\right)}^{\frac{2}{3}}{f}_i^{\frac{1}{3}}D\frac{\left(c_i-c_{\mathit{\text{eq}}}\right)}{\left(c_{\theta }-c_{\mathit{\text{eq}}}\right)}$$(8) and (6)(6) $${\dot{f}}_b=\frac{8}{\pi }\frac{S^2}{f_b}D\frac{{\left(c_i-c_b\right)}^2}{{\left(c_{\theta }-c_b\right)}^2}.$$(6) are controlled by different diffusion fields but also by different driving forces. Only the bulk mobility of solutes is the same for both mechanisms. The coupling of both growth equations is expected to lead to a nontrivial interplay depending on the thermal history, the Si-content, and the relative amount of grain boundaries and intragranular nucleation sites.

The effective fraction of intragranular precipitates observed in the microstructure $\overline{f}$ is lower than the local fraction f_i because of the PFZ. There are different ways to account for the PFZ based on the assumption of a simplified grain morphology. We choose to adhere to the planar collector plate concept, and therefore, we apply the following linear correction (10) $${\overline{f}}_i=f_i\left(1-\mathit{\text{Sw}}\right)$$(10) where w is the width of the PFZ. Based on a literature suggestion (Kamp et al., 2006), we estimate $w\approx u_D$, leading to (11) $$w=\frac{\sqrt{\pi }{f}_b}{4S}\frac{\left(c_{\theta }-c_b\right)}{\left(c_i-c_b\right)}$$(11) via the mass balance of the grain boundary precipitates given by equation (4)(4) $$V(c_{\theta }-c_b)=A(c_i-c_b){\int }_0^{u_0}[1-\text{erf}\left(\frac{u}{2u_D}\right)]du\approx A(c_i-c_b)\frac{2u_D}{\sqrt{\pi }}$$(4) . We use both expressions only for the evaluation of $\overline{f}$ and w at room temperature after cooling. The estimate is not generally valid for the present model because as long as $c_i-c_b$ is significantly higher than $c_i-c_{\mathit{\text{eq}}}$, the driving force for intragranular precipitate growth cannot be neglected in the vicinity of the grain boundary. Combining the mass balances in the bulk of the grain and at the grain boundary given by equations (4)(4) $$V(c_{\theta }-c_b)=A(c_i-c_b){\int }_0^{u_0}[1-\text{erf}\left(\frac{u}{2u_D}\right)]du\approx A(c_i-c_b)\frac{2u_D}{\sqrt{\pi }}$$(4) and (9)(9) $$c_i=c_i^0-c_{\theta }{f}_i$$(9) and accounting for the PFZ with equation (10)(10) $${\overline{f}}_i=f_i\left(1-\mathit{\text{Sw}}\right)$$(10) provides the following total mass balance: (12) $$c_i^0=\left(c_i+c_{\theta }{f}^i\right)\left(1-\mathit{\text{Sw}}\right)+c_b\mathit{\text{Sw}}+\left(c_{\theta }-c_b\right)f_b$$(12)

Because of the PFZ, the total phase fraction is the sum of the effective phase fraction ${\overline{f}}_i$ and the intergranular phase fraction f_b: (13) $$f_{\mathit{\text{total}}}={\overline{f}}_i+f_b$$(13)

3.3. Model parameters

For the pure Si phase, $c_{\theta }=1$ and the diffusion of Si in the aluminium matrix is given by $D=202\times \hspace{0.17em}\text{exp}\hspace{0.17em}(-\frac{136 \text{kJ}{\text{mol}}^{-1}}{\mathit{\text{RT}}})$ mm² s⁻¹ (Fujikawa et al., 1978) with the gas constant R and the temperature T. The equilibrium concentration of Si solutes in the matrix can be expressed as $c_i=33\times \hspace{0.17em}\text{exp}\hspace{0.17em}(-\frac{54 \text{kJ}{\text{mol}}^{-1}}{\mathit{\text{RT}}})$ (Starink, 1996). The formation of Si precipitates starts at a temperature below the solvus temperature T_s, characterized by a critical undercooling $T=T_s-\Delta T$. Differential scanning calorimetry (DSC) (Schumacher et al., 2015) shows the start of substantial precipitate growth below 450 °C at moderate cooling rates for a binary Al-Si containing 0.69 at wt.% Si with a corresponding solvus temperature of 495 °C. We apply the critical undercooling of 60 K, as indicated by this data also, to the present alloy.

The grain boundaries act as traps for Si atoms, and therefore, the equilibrium concentration at the grain boundary c_b is higher than the equilibrium concentration in the matrix. The grain boundary concentration can be calculated with the help of a classical equation attributed to McLean and Oriani (McLean, 1957; Oriani, 1970; Svoboda et al., 2015), which for dilute solute concentrations reads as: (14) $$\frac{c_i}{1-c_i}=\frac{c_b}{1-c_b}\mathrm{\text{exp}}\hspace{0.17em}\left(-\frac{\Delta G}{\mathit{\text{RT}}}\right)$$(14)

Direct experimental measurements of the trapping energy $\Delta G$ of Si solutes at grain boundaries in aluminium do not exist. There are, however, numerical estimates from ab-initio simulations predicting values in the range of 0–15 kJmol⁻¹ depending on the type of grain boundary (Wang et al., 2015). Within this range, we choose a value of 4.0 kJmol⁻¹ because this value leads to incipient growth at the desired critical undercooling. A similar value of 5.0 kJmol⁻¹ was used for the trapping energy of Mg in Al-Mg to model solute drag (Buken and Kozeschnik, 2020), while slightly lower values of approximately 2.0 kJmol⁻¹ for the trapping energies of Zn, Mg, and Cu were used to describe grain boundary segregation of these elements (Liu et al., 2021). The effective initial Si concentration in the matrix is lower than the nominal Si content because primary Al-Fe(Mn)-Si particles consume a part of the Si concentration (Myhr et al., 2001). With regard to the low Fe and Mn content, we assume an approximate value, $c_i^0=0.011$. The number of available nuclei N required in the growth equation (8)(8) $$\stackrel{.}{f_i}=3{\left(\frac{4\pi N}{3}\right)}^{\frac{2}{3}}{f}_i^{\frac{1}{3}}D\frac{\left(c_i-c_{\mathit{\text{eq}}}\right)}{\left(c_{\theta }-c_{\mathit{\text{eq}}}\right)}$$(8) is used as a fitting parameter. Table 3 shows a summary of the model parameters used in this work.

Table 3. Summary of model parameters for Si in AA6016.

Parameter	Description	Value
D	Diffusion coefficient	$202\times \hspace{0.17em}\text{exp}\hspace{0.17em}(-\frac{136 kJ{\text{mol}}^{-1}}{\mathit{\text{RT}}})$ mm^–2 s^–1 [Fujikawa et al., 1978]
ceq	Equilibrium concentration	$33\times \hspace{0.17em}\text{exp}\hspace{0.17em}(-\frac{54 kJ{\text{mol}}^{-1}}{\mathit{\text{RT}}})$ [Starink, 1996]
$c_i^0$	Initial matrix concentration	0.011
$\Delta G$	Trapping energy	4.0 kJmol^−1
$\Delta T$	Critical undercooling	60 K
S	Grain boundary length per area	38.8 mm^–1
N	Intragranular precipitate density	$5\times {10}^4$ mm^−3

4. Results and discussion

The evolution equations of the combined model are evaluated for the temperature-time curve of slow coil cooling shown in Figure 4, starting from the fully solutionized state. The resulting growth of inter- and intragranular Si precipitates is shown in Figure 5a. The corresponding concentrations of solute Si in the matrix and the grain boundary are shown in Figure 5b, together with the matrix equilibrium concentration. Growth starts at the prescribed critical undercooling at 475 °C. At this temperature, the matrix concentration surpasses the boundary concentration, thus initiating the migration of Si atoms to the grain boundary and promoting intergranular precipitate growth. The rate of the latter exceeds the rate of intragranular growth. The reason for this is not the driving force, which is actually higher for intragranular growth, but the size and geometry of the diffusion field. Only limited growth occurs below 350 °C because of the decreasing mobility of Si atoms. At lower temperatures, grain boundary concentration and matrix equilibrium concentration do not converge, but their difference vanishes in comparison with the matrix concentration. At room temperature, the calculated phase fractions are 0.0048 and 0.0025 for inter- and intragranular precipitates, respectively. A mean diameter of 2.29 µm was determined from the intragranular phase fraction and the constant intragranular precipitate density. The width of the PFZ, calculated with equation (11)(11) $$w=\frac{\sqrt{\pi }{f}_b}{4S}\frac{\left(c_{\theta }-c_b\right)}{\left(c_i-c_b\right)}$$(11) , is 6.6 µm, and the effective intragranular phase fraction, calculated with equation (10)(10) $${\overline{f}}_i=f_i\left(1-\mathit{\text{Sw}}\right)$$(10) , is 0.0012. These calculated microstructural quantities are compared with the measured values in Table 2. Given the simplifying assumptions, such as the perfectly flat grain boundary, the simple representation of the diffusion fields and the mean-field modeling approach, the agreement between the calculated and the measured values is reasonable. Only the intragranular precipitate density N has been used to fit the results. The largest discrepancy between measured and predicted values is found for the width of the PFZ. This is due to the fact that the precipitation in the depleted zone adjacent to the grain boundaries has not been explicitly calculated. Instead, the PFZ was deduced from the phase fraction of intergranular precipitates after coil cooling at room temperature via Equation (11)(11) $$w=\frac{\sqrt{\pi }{f}_b}{4S}\frac{\left(c_{\theta }-c_b\right)}{\left(c_i-c_b\right)}$$(11) , where the continuous concentration profile at the grain boundary was approximated by a simple step function.

Figure 4. Temperature curve (black line) and cooling rate (gray line) used for the coil cooling simulation.

Figure 5. Model results for coil cooling: (a) Growth of inter- and intragranular Si precipitates. (b) Evolution of the Si concentration in the grain boundary and in the matrix in comparison with the matrix equilibrium concentration.

The following three subsections present the effect of a variation of the Si content, the intragranular precipitate density, the grain boundary area, and the cooling rate. The variations are applied while all other parameters are held constants.

4.1. Effect of Si-content

Considerable amounts of Si are found in Al-scraps, especially from cast automotive components such as engines and transmissions or, to a lesser degree, from wrought alloys in outer skin parts (Paraskevas et al., 2019). Therefore, increased use of scrap may increase the Si content in AA6016. The effect of Si content in the present case of slow coil cooling, as predicted by the model, is shown in Figures 6a and 6b. A higher Si-content increases the solvus temperature of the Si-phase and consequently—based on the assumption of constant critical undercooling—also the start temperature for precipitate growth of both types. The total, as well as the intragranular fractions, are proportional to the Si content. The intergranular fraction, however, exhibits a maximum at approximately 1.1 at.-% and decreases at higher contents. The model predicts that the intergranular phase fraction is larger than the intragranular fraction up to a Si-content of approximately 1.2 at.-%. At higher Si-contents, the relation is reversed, and intragranular precipitate growth becomes the dominant process. This result is a consequence of the coupling of the two growth mechanisms. Intragranular growth reduces the matrix concentration of Si and, thus, the driving force for intergranular growth. Conversely, intergranular growth promotes the formation of a PFZ but does not reduce the driving force for intragranular growth. Taken individually, the growth models would predict a phase fraction directly proportional to the Si-content for both intra- and intergranular precipitates. Figure 6b shows the effect of Si-content on the width of the PFZ and the mean intragranular precipitate diameter. They follow the same trends as the phase fraction, with the difference that the maximum width of the PFZ is reached at a higher Si-content of 1.15 at.% compared to the maximum intergranular phase content at 1.1 at at.-%. The decrease in the width of the PFZ follows from the reduced intergranular phase fraction at elevated Si-content according to equation (11)(11) $$w=\frac{\sqrt{\pi }{f}_b}{4S}\frac{\left(c_{\theta }-c_b\right)}{\left(c_i-c_b\right)}$$(11) .

Figure 6. Effect of a variation of the Si content on (a) the total, inter-, and intragranular phase fractions, and (b) the mean intragranular precipitate diameter (grey dashed line) and the width of the PFZ (black line) after coil cooling.

4.2. Effect of precipitate density and grain boundary area

The number density of intragranular precipitates depends on the number density of potential nucleation sites. In the case of industrial Al-Mg-Si alloys, primary Fe-rich particles serve as preferred sites for heterogeneous nucleation (Milkereit et al., 2012; Falkinger et al., 2022). Also, in this study, a considerable part of intragranular precipitates is found adjacent to coarse Fe-rich particles, as shown in Figure 7. The phase fraction of coarse Fe-rich particles primarily correlates with the Fe content. The number density of coarse Fe-rich particles depends on the phase fraction but also on the amount of fragmentation which occurs during hot rolling. Therefore, apart from the Fe content, the hot rolling parameters and the Mn content may also affect the number density, as Mn promotes the formation of Fe-rich particles with a different fragmentation behavior (Kuijpers et al., 2005). Figures 8a and 8b show the effect of a variation of the number density of intragranular precipitates on the precipitation kinetics in the case of slow coil cooling. The intragranular phase fraction is proportional to the number density, but the correlation is not linear. The mean diameter of intragranular precipitates decreases only slightly because the increase in number density is mostly compensated by the increased phase fraction. With increasing number density, intragranular precipitates consume more Si at the expense of the intergranular phase fraction, and accordingly, the width of the PFZ decreases. Since the total phase fraction remains fairly constant, the number density of intragranular precipitates merely changes the ratio of intra- and intergranular phase fractions. The predicted changes are substantial. At the lowest number density of $2.0\times {10}^4$ mm⁻³, the intergranular fraction is three times higher than the intragranular fraction, while at the highest value of $2.0\times {10}^5$ mm⁻³, this relation is reversed. On a technological level, the results indicate that the number density of intragranular nucleation sites, i.e. the number density of primary Fe-based particles, is an essential microscopic feature controlling the proportions of inter- and intragranular precipitate growth.

Figure 7. Microstructure of the Al-Mg-Si alloy as shown in Figure 1(a). Dark intragranular precipitates adjacent to bright primary Fe-based particles are highlighted with circles to emphasize their role as heterogeneous nucleation sites.

Figure 8. Effect of a variation of the intragranular precipitate number density on (a) the total, inter-, and intragranular phase fraction, and (b) the mean intragranular precipitate diameter (gray dashed line) and the width of the PFZ (black line) after coil cooling.

Figure 9 shows the effect of variations in the grain boundary density. In the present model, intragranular growth beyond the PFZ is assumed to be independent of intergranular growth, and therefore, the intragranular phase fraction and the diameter of intragranular precipitates are not affected by variations of the grain boundary density. This assumption appears justified by recent investigations on an Al-Zn-Mg-Cu alloy by other authors (Zhao et al., 2018), who found that intragranular precipitates adjacent to the PFZ and in the bulk of the grain do not show any differences. Variations of the grain boundary density affect only the phase fraction of the intergranular precipitates. This known effect of the grain boundary density (Kozeschnik et al., 2009; Mucsi, 2014) is highly relevant in the production process, where the grain boundary area may vary considerably because of rolling deformation and recrystallization. Only those Si solutes that are located close to a grain boundary contribute to intergranular growth because their diffusion path is short enough to reach the boundary. Increasing the grain boundary area means that more Si solutes are within a critical distance of the grain boundary, resulting in increased intergranular growth. The linear dependence between grain boundary area and intergranular phase fraction reflects the simplifying model assumption of planar grain boundaries. This is an approximation of the real material, where the network of grain boundaries assumes irregular shapes, but the trend predicted by the model is plausible. The model additionally assumes that there is no overlap of diffusion fields towards different grain boundaries. Again, this assumption represents an approximation because overlaps are inevitable at high values of the grain boundary area. As a consequence of this assumption, a variation of the grain boundary density does not affect the width of the precipitate zone. Formally, the ratio $\frac{f_b}{S}$, which defines the width of the PFZ in equation (11)(11) $$w=\frac{\sqrt{\pi }{f}_b}{4S}\frac{\left(c_{\theta }-c_b\right)}{\left(c_i-c_b\right)}$$(11) , remains constant upon a variation of S because f_b depends linearly on S. In the present analysis, the model predictions are limited to cases where the inverse of the grain boundary density exceeds four times the width of the precipitate-free zone. Within this limit, the model approximation of equation (4)(4) $$V(c_{\theta }-c_b)=A(c_i-c_b){\int }_0^{u_0}[1-\text{erf}\left(\frac{u}{2u_D}\right)]du\approx A(c_i-c_b)\frac{2u_D}{\sqrt{\pi }}$$(4) applies. Si solutes which are not located in the vicinity of a grain boundary are not affected by the grain boundary density, and consequently, the intragranular growth remains constant. The total phase fraction, which is given by the sum of the intergranular and the effective intragranular phase fraction given by equation (10)(10) $${\overline{f}}_i=f_i\left(1-\mathit{\text{Sw}}\right)$$(10) , increases linearly as well, but the slope is lower than that of the intergranular fraction.

Figure 9. Total, inter-, and intragranular phase fraction after coil cooling dependent on the grain boundary density. The intergranular and the total phase fraction both increase linearly with the grain boundary density but with different slopes.

4.3. Effect of cooling rate

Figure 10 shows the effect of the linear cooling rate on the inter- and intragranular phase fraction at room temperature. For the sake of simplicity, the cooling rate is also varied at constant precipitate density and constant critical undercooling. The predicted shape of the intragranular curve agrees well qualitatively with the measured curves in (Schumacher et al., 2015) for the binary Al-Si wrought alloys with Si contents of 0.69 and 0.25 at wt.%. Towards high cooling rates, the intergranular curves exhibits a long tail in correspondence with experimental and simulated results for grain boundary precipitates, e.g. in a Al-Mg-Zn alloy (Kamp et al., 2006). With increasing intragranular phase fraction, the intergranular fraction decreases, showing a maximum at a cooling rate of approximately 0.01 K s⁻¹. The model predictions at lower rates could not be compared with literature results because previous studies investigated either intragranular or intergranular growth but not the coupled effect of both mechanisms occurring simultaneously. Only the work of Kamp et al. (Kamp et al., 2006) analyzes concurrent precipitation at varying cooling rates but not during slow cooling rates lower than 1.0 K s⁻¹. The present approach predicts that at these low rates, the Si matrix concentration is consumed by intragranular precipitates, thereby reducing the driving force for intergranular growth. This promotes the intragranular at the expense of the intergranular phase fraction while the total phase fraction given by equations (10)(10) $${\overline{f}}_i=f_i\left(1-\mathit{\text{Sw}}\right)$$(10) and (13)(13) $$f_{\mathit{\text{total}}}={\overline{f}}_i+f_b$$(13) increases continuously towards lower cooling rates. The fact that the total fraction is lower than the local intragranular fraction at low cooling rates is a consequence of the PFZ. The present calculation of the width of the PFZ, however, relies on approximations which have not been validated at very low cooling rates.

Figure 10. Effect of the cooling rate on the total, inter-, and intragranular phase fractions after continuous cooling.

5. Conclusion and outlook

We draw the following conclusions from the present results:

• Sample preparation using argon milling in combination with SEM and the inlens SE-detector enables an unambiguous quantification of coarse Si-precipitates in AA6016.

• Based on simple assumptions, the classical collector-plate model for intergranular precipitate growth can be generalized to include transient thermal histories and solute segregation at the grain boundary.

• The coupling of two established growth models for inter- and intragranular precipitates leads to reasonable predictions of the inter- and intragranular phase fractions and the width of the PFZ in the case of AA6016 after slow cooling.

• While the predicted effects of grain boundary area and high cooling rate on intergranular growth are well-known, the combined model suggests a hitherto unknown effect: An increase of Si-content and primary Fe-based particles, which are both consequences of increased scrap content (Raabe et al., 2022), may suppress intergranular growth and minimize the width of the PFZ after hot rolling.

The present results highlight the importance of grain size and cooling rate during the process to control the evolution of the precipitate state. Further work will, therefore, be dedicated to coupling the present precipitate models with a microstructure model, including recrystallization (Sherstnev et al., 2012), and applying them to industrial production schedules.

Funding

R.K. gratefully acknowledges the financial support under the scope of the COMET program within the K2 Center “Integrated Computational Material, Process and Product Engineering (IC-MPPE).” This program is supported by the Austrian Federal Ministries for Climate Action, Environment, Energy, Mobility, Innovation and Technology (BMK) and for Labour and Economy (BMAW), represented by the Austrian Research Promotion Agency (FFG), and the federal states of Styria, Upper Austria, and Tyrol.

Disclosure statement

The authors report there are no competing interests to declare.