A CALPHAD-Based Investigation of the Sludge Factor

Abstract

The formation of $\alpha \text {-AlFeSi}$ sludge in AlSi10Mg has been studied by computational thermodynamics based on the CALPHAD method. Both the amount of sludge and the sludge onset temperature $T_{\text{Sludge}}$ have been investigated. We find that the amount of sludge increases linearly with the empirical sludge factor for the studied composition interval, which agrees well with published experimental data. We also find a notable difference between the total amount of $\alpha \text {-AlFeSi}$ phase and the amount of pre-eutectic sludge. The $T_{\text{Sludge}}$ follows a similar linear relationship but only when there is no Cr present in the material. However, we propose a modified sludge factor like expression for the temperature onset with a significantly improved predictability compared to previous empirical expressions. With Cr present, a $\alpha \text {-AlMSi}$ phase, rich in Cr but poor in Mn, forms at a significantly higher temperature which leads to higher amounts of sludge suggesting Cr to be more detrimental for sludge formation than previously thought. Finally, we also suggest ranges of validity for the linearized sludge factor expression outside which full thermodynamic calculations accounting for all multicomponent effects must be used.

Introduction

Most aluminium foundry alloys contain silicon in concentrations typically 5–12 wt.%. Silicon contributes to higher strength and stiffness and improves castability by reducing the liquidus temperature, increasing fluidity and reducing porosity.³ At the same time, these alloys contain iron and other transition metals such as manganese or chromium. Iron can be present as an impurity, particularly if the alloy is produced from secondary metal, or added deliberately to reduce metal-mould reactions during die-casting. During solidification of aluminium-silicon alloys, iron will form ternary or higher intermetallic phases, notably $\alpha$ ($Al_{15}M_{3}Si_{2}$), $\beta$ (${Al_{5}FeSi}$) and $\pi$ ($Al_{8}Mg_{3}FeSi_{6}$). Here, M indicates Fe, Mn or Cr. These phases are not entirely stoichiometric and other proposed formulae exist. Here, the ${Al_{15}M_{3}Si_{2}}$ and ${Al_{5}FeSi}$ phases will be denoted $\alpha \text {-AlFeSi}$ and $\beta \text {-AlFeSi},$  respectively. The precipitation behaviour of iron-rich intermetallics in Al-Si alloys has been described in detail by Bäckerud.² The balance between $\alpha \text {-AlFeSi}$ and $\beta \text {-AlFeSi}$ formation during solidification is given by the Si/Fe ratio and will also be influenced by Mn additions, which favours $\alpha \text {-AlFeSi}$ formation. This is utilized in order to prevent formation of $\beta \text {-AlFeSi}$, which forms flake-like brittle precipitates, reducing the ductility of the alloy.

At moderate iron contents, $\alpha \text {-AlFeSi}$ and $\beta \text {-AlFeSi}$ form in binary- or higher eutectics together with Al and Si. At high contents these phases, particularly $\alpha \text {-AlFeSi}$, can precipitate in the melt during solidification forming “sludge”. Sludge particles can form in holding furnaces, in shot cylinders or in casting moulds. They are large, bulky and brittle; they will affect the properties and machinability of the casting negatively and must therefore be avoided.

The likelihood for sludge formation has historically been estimated from the empirical sludge factor (SF)^4,7,8

$$\begin{aligned} \textrm{SF} = 1 \times (\% \textrm{Fe}) + 2 \times (\% \textrm{Mn}) + 3 \times (\% \textrm{Cr}) \end{aligned}$$

(1)

which relates the alloy composition in wt.% to the likelihood for sludge formation. Its applicability spans over a large composition range. Work has been published relating the SF to the amount of sludge.^6,9,10,11 Perhaps the most important and frequent usage of the SF is found in the estimation of the sludge onset temperature $T_{\text{Sludge}}$.^4,5,7,8,9,11 The early works^4,7,8 as well as the later works^5,11 all describe the $T_{\text{Sludge}}$ as function of the SF. However, an often cited relation is that by Shabestari⁹ which relates the $T_{\text{Sludge}}$ to the square of the Fe content.

The present work uses computational thermodynamics based on the CALPHAD method to calculate the sludge fraction and sludge onset temperature of AlSi10Mg  (EN AC-43500), an alloy which has been used extensively and for a long time in diecasting. It also belongs to a select group of aluminium alloys which is considered printable in additive manufacturing.

The article starts with an investigation of how well the sludge factor can describe the amount of sludge formed, and assesses how the calculated results agree with published experimental data. The report focuses on three of the most important properties; the fraction $\alpha \text {-AlFeSi}$ ($F_{\alpha \text {-AlFeSi}}$), the fraction $\beta \text {-AlFeSi}$ ($F_{\beta \text {-AlFeSi}}$), and the $T_{\text{Sludge}}$. It will be shown that there is an important difference between $F_{\alpha \text {-AlFeSi}}$ and the amount of sludge ($F_{\text{Sludge}}$), where $F_{\alpha \text {-AlFeSi}}$ is the total amount of $\alpha \text {-AlFeSi}$ while $F_{\text{Sludge}}$ is the amount of primary sludge formed before the onset temperature of $\alpha$-Al. Along the way a set of constitutive equations relating the chemical composition of the alloy with the said properties will be developed.

Computational Setup

Thermodynamic calculations according to the method were performed using the Thermo-Calc software,¹ version 2021a and the thermodynamic database TCS Al-based Alloy Database Version 7. Calculations were performed for a total of more than 6000 compositions. The compositional range investigated is found in Table 1, with Al as balance. Throughout the investigation, Cr levels were kept at 0%, except for Section “Sludge Fraction Versus Sludge Factor”, where also the Cr levels were varied up to 0.30%, and the section relating to Figure 7.

Table 1 The Alloy Composition Range Investigated, in wt.%. Al as Balance

Results and Discussion

In the presentation below, an alloy with the composition 10.0% Si, 0.35% Mg 0.80% Mn and 0.30% Fe serves as an illustration for qualitative discussions of the phase equilibria and solidification behaviour. This alloy is within the investigated composition range of EN AC-43500 and can be considered a prototype alloy. Starting from this composition different amounts of the constituent elements are varied. Although the results and figures shown here are limited to this specific prototype composition the general trends and ideas hold also for the other compositions within the interval listed in Table 1, except for when Cr is added. The effect of Cr will be discussed separately.

Solidification

Figure 1 shows the fraction of the different constituent phases in the solid prototype alloy. The upper panel of Figure 1 shows the volume fractions of the different phases at different temperatures under equilibrium conditions and the lower shows the mass fractions under Scheil solidification conditions. At higher temperatures and low fractions solid phase, the results from the two calculations agree well and will therefore be discussed together. Note that all Fe containing intermetallic phases are written using only the Greek letter rather than the full designation, e.g. $\alpha \text {-AlFeSi}$ in order to save space in the figure.

When the temperature is decreased the first solid phase to solidify is the Fe-rich intermetallic phase $\alpha \text {-AlFeSi}$. This occurs at the sludge onset temperature, $T_{\text{Sludge}}$, of about 610$^\circ$C for the present composition. It is an important fact that $\alpha \text {-AlFeSi}$ can solidify at a higher temperature than the $\alpha$-Al, which is the next phase to solidify at an onset temperature $T_{\text{Al}}$ of about 590$^\circ$C. At about 570$^\circ$C we find the Al-Si eutectic temperature $T_E$ at which Si starts to solidify.

Figure 1

Calculated fraction of constituent phases under (top) equilibrium conditions and (bottom) during Scheil solidification for the example alloy containing 10.0% Si, 0.35% Mg, 0.80% Mn and 0.30% Fe.

Other phases which are stable under equilibrium conditions are $\pi$ below 483 °C, $\beta \text {-AlFeSi}$ below 361 °C and ${Mg_{2}Si}$. For the present prototype composition, ${Mg_{2}Si}$ is not thermodynamically stable until below 182 °C, where $\pi$-phase is no longer thermodynamically stable. This temperature is thus the upper limit for heat treatment intended at precipitating ${Mg_{2}Si}$. 182 °C is also an upper threshold for $\beta \text {-AlFeSi}$, which is stable in appreciably smaller amounts above this temperature. However, $\beta \text {-AlFeSi}$ still forms under normal solidification conditions due to segregation. To understand this one has to simulate solidification conditions.

Under normal cooling conditions there is not enough time for the system to equilibrate. Instead, we here assume Scheil solidification conditions to prevail. Under Scheil conditions, no diffusion of elements in the solid phase is allowed, but perfect mixing of the liquid phase is assumed. This will cause the liquid to become enriched in solute elements.

The total fraction of the different constituent solid phases under Scheil solidification conditions is shown in the lower panel of Figure 1. The equilibrium result, corresponding to the results shown in the upper panel, is indicated with a dashed line to facilitate a comparison. As long as the fraction of solid phase is low, the differences between the two are small.

Up until the Al-Si eutectic temperature $T_E$ no considerable segregation occurs and the fractions of the different phases calculated under Scheil conditions are comparable to those obtained from equilibrium calculations. When the fraction of solid phase exceeds 0.6 the differences become more apparent. The remaining liquid is now enriched in alloying elements allowing the formation of phases, such as $\pi$-Fe and ${Mg_{2}Si}$, but perhaps most importantly, $\beta \text {-AlFeSi}$. This identification of the order of the reaction steps is supported by the observations in References 5,11.

The inset in the lower panel of Figure 1 shows the evolution of the fraction solidified $\alpha \text {-AlFeSi}$ during the Scheil solidification. Similar curves have been obtained for all constituent phases. From these curves, the amount of the phase at various important phase transition temperatures can be extracted.

The solidification of $\alpha \text {-AlFeSi}$ can be divided into three intervals. For the current composition these are approximately equal and one third of the total amount of $\alpha \text {-AlFeSi}$ is formed in each interval. In the first interval, after the sludge onset temperature, $T_{\text{Sludge}}$, and before the onset temperature of primary $\alpha$-Al, $T_{Al}$, about one third of the total amount of $\alpha \text {-AlFeSi}$ is formed as sludge. This is marked with a blue line. The second third is formed eutectically together with primary aluminium $\alpha$-Al, before the Al-Si eutectic temperature, $T_E$. This is marked green. The last and third interval represents a complex eutectic with many phases and is formed after $T_E$ ending up in a fraction $\alpha \text {-AlFeSi}$ of just below 3%.

The fraction $\alpha \text {-AlFeSi}$ formed in each interval is important. In this work the word sludge (or sometimes primary sludge) is used for the $\alpha \text {-AlFeSi}$ formed in the interval between $T_{\text{Sludge}}$ and $T_{\text{Al}}$. Primary sludge will form larger precipitates⁶ and is more detrimental to the fluidity of the melt and the final mechanical properties than the smaller secondary precipitates formed later in the solidification process. We use the notation $F_{\alpha \text {-AlFeSi}}$ for the total fraction of $\alpha \text {-AlFeSi}$ and $F_{\text{Sludge}}$ for the fraction sludge. While $F_{\alpha \text {-AlFeSi}}$ is almost 3% in the example composition in Figure 1$F_{\text{Sludge}}$ amounts to less than 1%.

Sludge Fraction Versus Sludge Factor

Figure 2

The total fraction $\alpha \text {-AlFeSi}$ ($F_{\alpha \text {-AlFeSi}}$) plotted against the sludge factor (SF) for Cr/Fe ratios less than or equal to one. Each dot represents a composition within the composition range defined in Table 1. It is clear that the total $F_{\alpha \text {-AlFeSi}}$ is proportional to the SF.

Figure 2 shows the total amount of $\alpha \text {-AlFeSi}$ formed in a Scheil solidification simulation plotted against the sludge factor (SF). Each dot represents a composition within the composition range defined in Table 1.

The range of validity of Eqn. 1 is an important notion which is seldom mentioned. Compositions with a Cr/Fe ratio above 1 lead to high SF values but not to a corresponding increase in the amount of $\alpha \text {-AlFeSi}$. Instead, the Cr-rich ${Al_{13}Cr_4Si_4}$ phase is formed. Similarly, compositions with Mn/Fe ratios above one form rather high amounts of $\beta \text {-AlFeSi}$ in addition to $\alpha \text {-AlFeSi}$. These compositions are therefore deemed to lie outside of the validity range of Eqn. 1 and are omitted in Figure 2.

There is a clear linear trend in Figure 2 between SF and $F_{\alpha \text {-AlFeSi}}$ which can be described by the equation

$$\begin{aligned} F_{\alpha \text {-AlFeSi}}= 1.65 \times (\text {SF}) - 0.27. \end{aligned}$$

(2)

Note that despite the very limited solubility of Fe, Cr or Mn in primary Al, an SF of 0.16 is possible before any sludge is formed. At higher SF the fraction $\alpha \text {-AlFeSi}$ increases linearly with the SF. The quality of the fit, marked with a dashed line, is good with the $R^2$ score 0.989.

Previous works^8,11 have proposed other equations for the sludge factor with different weights to the content of Fe, Mn and Cr. The attentive reader may also note that there is also a slight shift to smaller slope with increasing Cr content in Figure 2. This indicates that the factor for Cr in Figure 2 is too high. By fitting the coefficients using linear regression, a new sludge factor can be constructed

$$\begin{aligned} \text {SF}^{\prime } = 1.5 \times Fe + 2.0 \times Mn + 2.5 \times Cr \end{aligned}$$

(3)

which lead to an equation $F_{\alpha }^\prime = 1.72 \times (\text {SF}^\prime ) - 0.53$ with $R^2$ values of 0.997. We note that this equation is similar but not identical to the alternative versions mentioned in References 8 and 11.

With regards to the $\alpha \text {-AlFeSi}$ the impact of the amount of Cr on the SF is only important implicitly through its contribution to the SF. However, this is not always the case. For the sludge onset temperature, the behaviour is markedly different when the alloy contains Cr. This will be discussed in greater detail in Section “Sludge Onset Temperature”.

Comparison with Published Experimental Results

Figure 3

The fraction primary sludge plotted against the sludge factor. Primary sludge is the fraction $\alpha \text {-AlFeSi}$ formed before the formation of primary aluminium. The figure also includes data from $^a$Reference 9, $^b$Reference 6,$^c$Reference 10,$^d$Reference 11, where the symbols represent the measurements and the lines the fits obtained and reported on within the respective article.

Figure 3 shows a comparison between calculated fraction primary sludge and measured values found in the literature.^6,9,10,11 Contrary to above, Figure 3 only shows the calculated fraction of primary sludge, i.e. the fraction $\alpha \text {-AlFeSi}$ before the formation of primary aluminium.

There is a linear relationship between the SF and the $F_{\text{Sludge}}$  which can be described by the equation

$$\begin{aligned} F_{\text{Sludge}}\ = 1.75 \times (SF) - 2.37. \end{aligned}$$

(4)

The slope is essentially the same as that in Figure 2, Eqn. (2), but the intercept is markedly different. Below a SF of 1.35 (but above 0.16, cf. above) $\alpha \text {-AlFeSi}$ will form but only after primary aluminium has started to form. This form of eutectic $\alpha \text {-AlFeSi}$ is less harmful than the primary sludge which can form already in the melt.

Overall, the agreement between the measured $F_{\text{Sludge}}$ and the calculated data is good, with similar values for the intersection with the x-axis for all fitted lines. The slopes on the other hand are somewhat different. An important observation is that the experiments have been performed on different alloys. Reference 9 studied the variation of both Fe, Mn, and Cr in a near eutectic Al-Si alloy, with about 12.7% Si and only small traces of impurity elements. The studies from the Timelli group (References 6,10,11) are performed on an ${AlSi_{9}Cu_{3}}$ alloy, with significant content of both Mg and Zn. In that perspective, it is an interesting fact that there is more spread between the measurements on the same alloy than between the different alloys.

Reference 11 investigated the $F_{\text{Sludge}}$ at different cooling rates. In Figure 3, only the data from the 20$^\circ$C/min measurements are included. The fit to the data at lower cooling rates are essentially parallel with different intersects. At lower cooling rates more sludge is allowed to form. The extreme is the rapid quench in Reference 9. The cooling rates in References 6 and 10 are not explicitly given.

The Beta Phase

Figure 4

(left) The total fraction $\beta \text {-AlFeSi}$ plotted against weight percentage Fe, colour coded by the amount of Mn. There is a clear linear trend with a well-defined slope and intersect for each concentration of Mn. The dashed line is fitted to an Mn level of 0.8%. (right) The total fraction $\beta \text {-AlFeSi}$ (log scale) against the percentage Mn, colour coded by the amount of Fe. The slope of the log $\beta \text {-AlFeSi}$ as a function of Mn is not constant within the investigated interval, with slopes varying between 1.2 down to 0.3.

The left panel of Figure 4 shows the total fraction $\beta \text {-AlFeSi}$ as function of Fe content. Different Mn content is indicated with different colours, from 0.1 to 0.8 weight percent Mn. The dashed line is fitted to an Mn level of 0.8%. Similar to the relations for $\alpha \text {-AlFeSi}$ above, the fraction $\beta \text {-AlFeSi}$ increases linearly with increasing Fe content. From the colour coding, it is already evident that increasing Mn content will reduce the amount of $\beta \text {-AlFeSi}$  although the relationship is more complicated.

One important observation is that at high amounts of Mn, above about 0.4%, the intersection between the fitted dashed line and the x-axis is found at an Fe concentration of about 0.10%. Only for Mn concentrations below 0.4% does one see appreciable amounts of $\beta \text {-AlFeSi}$ at an Fe concentration of 0.10%. Hence, below 0.10% Fe a sufficiently high amount of Mn will eliminate $\beta \text {-AlFeSi}$. It will not be possible to completely suppress $\beta \text {-AlFeSi}$ formation at higher concentrations. At a constant Mn concentration of, e.g. 0.8%, an increase in Fe from 0.15% to 0.20% will double the amount of $\beta \text {-AlFeSi}$, which will again increase by the same amount for each additional 0.05% Fe added.

The right panel of Figure 4 shows the logarithm of the total fraction $\beta \text {-AlFeSi}$ as a function of Mn content. Contrary to above there is no simple linear trend, even for the logarithm. The exponent varies from about 1.2 at low concentration levels of Mn to 0.3 at higher Mn amounts.

Sludge Onset Temperature

Figure 5

Sludge onset temperature, $T_{\text{Sludge}}$, for $\alpha \text {-AlFeSi}$ marked with circles and $\beta \text {-AlFeSi}$ marked with diamonds. The eutectic temperature, $T_{E}$, is marked by one shaded line for each configuration. Since the eutectic temperature is approximately equal for each configuration, they form a grey area together. In both panels, the $T_{\text{Sludge}}$ for $\alpha \text {-AlFeSi}$ is approximately linear as a function of Mn and Fe, respectively, while the $T_{\text{Sludge}}$ for $\beta \text {-AlFeSi}$ has a more complicated relationship, in particular in the left panel.

Figure 5 shows the calculated sludge onset temperature, $T_{\text{Sludge}}$, for both $\alpha \text {-AlFeSi}$ and $\beta \text {-AlFeSi}$, marked with circles and diamonds, respectively. For $\alpha \text {-AlFeSi}$ the trend is once again nearly linear with a well-defined slope and intersect, both as a function of Mn and of Fe, through the onset temperature for primary Al, which is around 590$^\circ$C depending on the exact composition, down to the Al-Si eutectic temperature of about 574$^\circ$C. When the content of Mn is decreased to below 0.4% (not shown) the $T_{\text{Sludge}}$ is below $T_{E}$ and a simple trend can no longer be observed.

For $\beta \text {-AlFeSi}$ the relationship is more complicated. This is shown in the right panel of Figure 5. The onset temperature at a certain Fe content shows only a weak but linear dependence on the Mn content. A similar trend is not seen as a function of Fe in the left panel, either on a linear or a logarithmic scale. However, the onset temperature for $\beta \text {-AlFeSi}$ is always below $T_{E}$ in the investigated composition interval.

Figure 6

(top) Scheil solidification curve showing the fraction $\alpha \text {-AlMSi}$ with decreasing temperature and (bottom) the chemical composition of the $\alpha \text {-AlMSi}$ phase during solidification of the composition AlSi10Mg0.35Mn0.1Fe0.3Cr0.1 with a SF of 1.0. Note how the first $\alpha \text {-AlMSi}$ to solidify has a high Cr content and a very low Mn content, while the $\alpha \text {-AlMSi}$ formed during the Al-Si main eutectic solidification has a markedly different chemical composition. After the primary $\alpha \text {-AlMSi}$ formation all Cr is consumed.

Figure 7

Calculated phase diagram sliced along the direction of varying Fe-Al content of the alloy used in Reference 9, which contained 12.70% Si, 0.30% Mn and 0.10% Cr. The figure also contains the measurements reported therein, where filled (open) symbols represent presence (absence) of sludge.

The result for the onset temperature for $\alpha \text {-AlFeSi}$ is markedly different from the measurements in Reference 9. This discrepancy can be explained by the presence of Cr, which forms an $\alpha \text {-AlMSi}$-phase of a different chemical composition, with a dramatically higher onset temperature.

Figure 6 shows the results from solidification simulations of an alloy containing 0.1 wt.% Mn, 0.3 wt.% Fe, and 0.1 wt.% Cr. The upper panel shows how the mass fraction $\alpha \text {-AlMSi}$ changes with temperature, and the lower panel the composition. Two different regions are clearly seen with two distinctly different compositions. At temperatures above $T_{\text{Al}}$ the formed $\alpha \text {-AlMSi}$-phase contains significant amounts of Cr but practically no Mn. After the binary Al-Si liquidus $\alpha \text {-AlMSi}$ ceases to form, until the complex many phase eutectic where the $\alpha \text {-AlMSi}$ has a more typical $\alpha \text {-AlFeSi}$-like composition. To the knowledge of the authors, this compositional variation of the $\alpha \text {-AlMSi}$-phase has not yet been detected experimentally.

Figure 7 shows the calculated phase diagram together with the measurements from Reference 9, where filled (open) symbols represent presence (absence) of sludge. The agreement between experimental and calculated data at the same composition in Figure 7 instills confidence in the results based on the CALPHAD calculations which predict a Cr-rich $\alpha \text {-AlMSi}$ at higher temperatures. It also explains the cause for the comparatively low onset temperatures of the Cr free material presented in Figure 5.

Figure 8

Sludge onset temperature plotted against the sludge factor (SF). Since the sludge fraction is very nearly a linear function of the SF, the conversion from sludge fraction to SF through the relation in Eqn. 2 is marked in the top axis. The different configurations are marked with dots coloured by the fraction primary sludge. Only configurations with $T_{\text{Sludge}}> T_{E}$ are shown. The figure includes the data from the 20$^\circ$/min experiment in Reference 11.

Figure 8 shows the sludge onset temperature plotted against the sludge factor. Once again, the alloy contains no Cr. Since the sludge fraction is close to proportional to the SF, this is included as a second axis at the top.

There is an overall trend that an increase in SF is correlated with an increase in sludge onset temperature. Thus, a high SF not only causes a higher sludge fraction, but also causes sludge to form at a higher temperature. The figure includes the data from the 20$^\circ$/min in Reference 11. Just as in Figure 3, the calculated and measured data are in good agreement. There is some spread in the calculated data depending on the exact composition, which indicates that the relationship is not of a trivial kind.

One complication is that the cited experiments in both Figs. 7 and 8 are performed on alloys containing 0.1% Cr. Both works contain alloys with the same amount of Cr, Fe and Mn but differ in the amount of Si (12.5% and 8.5%, respectively) and Cu (0.0% and 2.4%). Considering that both alloys contain the same amount of Cr it is noteworthy that while the data from Reference 9 can only be explained by adding Cr to the calculation, the data from Reference 11 agree well with the calculated Cr free compositions.

Figure 9

The sludge onset temperature calculated using Thermo-Calc versus predicted values from the linear regression fit in Eqns. 5 and 6. When the predicted value agrees perfectly with the calculated the dots end up on the diagonal dashed line.

Although the relationship between the alloy composition and $T_{\text{Sludge}}$ is seemingly complicated, a sludge factor like expression can be constructed from the calculated data through a linear regression. From the obtained expression for the temperature factor

$$\begin{aligned} \textrm{TF} = 1\times (\% \textrm{Fe}) + 3 \times (\% \textrm{Mn}) - 0.1 \times (\% \textrm{Si}) \end{aligned}$$

(5)

the onset temperature can be obtained as

$$\begin{aligned} T_{\text{Sludge}}= 570 + 28.5 \times (\textrm{TF}), \end{aligned}$$

(6)

an equation which is valid provided that no Cr is present.

Figure 9 shows the $T_{\text{Sludge}}$ obtained from calculations on the x-axis versus that obtained using Eqns. 5 and 6 on the y-axis. For a perfect prediction, the points should lie on the diagonal dashed line. At low temperatures, the TF prediction differs from the calculations. This should not come as a surprise because the $T_{\text{Sludge}}$ has now fallen below the $T_{E}$.

When Cr is present, a Cr-rich $\alpha \text {-AlMSi}$-phase with a different chemical composition will form at a temperature well above that for Cr-free alloys, as discussed above. Attempting to fit a relationship similar to Eqns. 5 and 6 including Cr would lead to an equation which would at best be linear in Mn, but less so in Fe, and certainly not in Cr.

The composition of Si has very little influence on the total $F_{\alpha \text {-AlFeSi}}$ and $F_{\beta \text {-AlFeSi}}$ (not shown). For the $F_{\text{Sludge}}$, however, which is formed before the formation of primary Al, there is a clear dependence on the composition of Si. This stems from the shift in the onset temperature for primary aluminium, $T_{\text{Al}}$. When the Si concentration increases, the liquidus temperature decreases, leaving more room for sludge to form.

Conclusions and Outlook

We have used computational thermodynamics based on CALPHAD to calculate the amount of sludge formed in AlSi10Mg and investigated the applicability of the sludge factor SF. We show that there is an important distinction to be made between total amount of $\alpha \text {-AlFeSi}$ and the amount of sludge $F_{\text{Sludge}}$. We conclude that the sludge factor can be used to estimate both of the total amount of $\alpha \text {-AlFeSi}$ and $F_{\text{Sludge}}$. Both $\alpha \text {-AlFeSi}$ and $F_{\text{Sludge}}$ follow similar linearly increasing trends with SF, with similar slopes but with different intersects. While the first $\alpha \text {-AlFeSi}$ forms already at a SF of 0.16, a full 1.35 can be accommodated before sludge is formed.

By linear regression an adjustment to the sludge factor can be obtained. However, although the adjusted sludge factor, here denoted $\text {SF}^{\prime }$, leads to a slightly improved predictability for $\alpha \text {-AlFeSi}$ it leads to a significantly reduced predictability for $F_{\text{Sludge}}$.

The SF is not universally applicable and the range of applicability must be respected. Only for alloys with compositions satisfying $\mathrm {Cr/Fe<1}$ and $\mathrm {Mn/Fe>1}$ can the SF be relied on. For the prediction of $F_{\text{Sludge}}$ an additional restriction is that $\textrm{Mn} \ge 0.6$ wt.% if $\textrm{Cr} \ge 0.1$. The reason seems to be that when Cr is present sludge starts to form at a significantly higher temperature and with a different chemical composition, with a high Cr content and a very low Mn content. Although the total amount of $\alpha \text {-AlFeSi}$ can be rather low, as indicated by the SF, most of it is sludge which forms at temperatures above the onset temperature of primary aluminium. We would encourage more experimental work to investigate the Cr dependence on the composition of sludge and the sludge onset temperature.

Also the sludge onset temperature, $T_{\text{Sludge}}$, increases almost linearly with increasing SF, as long as the alloy does not contain Cr. However, the predictive power of the SF is lower. By linear regression, a SF-like expression can be constructed which predicts the $T_{\text{Sludge}}$ with a higher precision. This leads to an almost perfect correlation between the predicted $T_{\text{Sludge}}$ and the valued calculated using CALPHAD. Similar expressions for $\mathrm {Cr > 0}$ cannot be made with an acceptable predictability at linear order.