3. A surface tension and viscosity relation model for multicomponent alloys

Based on the formula originally proposed by Egry [34], a new formula is derived for the relation between surface tension and viscosity for multicomponent alloys, that is,

$$\frac{\sigma\_d}{\eta\_a} = \frac{15}{16} \sqrt{N\_{av} \ k \ T} \cdot \frac{1}{\sqrt{\sum\_{i=1}^n \varkappa\_i \cdot M\_i}} \tag{12}$$

where �<sup>1</sup> <sup>σ</sup><sup>a</sup> is the alloy surface tension in <sup>ð</sup>N:<sup>m</sup> <sup>Þ</sup>, <sup>η</sup><sup>a</sup> is the alloy viscosity in <sup>ð</sup>Pa:sÞ, xi is the molar fraction of the i alloy component, n is the total number of components of the alloy, and Mi is the molar weight of the alloy component i.

Substituting Eq. (12) into Eq. (6), we get,

Dependence of Surface Tension and Viscosity on Temperature in Multicomponent Alloys DOI: http://dx.doi.org/10.5772/intechopen.82307

$$\sigma\_a(T, \mathbf{x}\_i, \mathbf{x}\_j, \dots, \mathbf{x}\_n) = \frac{15}{16} \cdot \frac{h \cdot N\_{Av}}{\sum\_i \mathbf{x}\_i \cdot V\_i + \Delta V^E} \cdot \sqrt{\frac{\overline{R} \cdot T}{\sum\_{i=1}^n \mathbf{x}\_i \cdot M\_i}} \cdot \exp\left[\frac{\sum\_i \mathbf{x}\_i \cdot \Delta G\_i^\* - a \cdot \Delta H}{\overline{R} \cdot T}\right] \tag{13}$$

#### 4. Results and discussion

Figure 1 presents the viscosity of pure aluminum as a function of temperature, simulated by Kaptay [7] theoretical model compared to experimental data [37]. The model fits the experimental scatter well, but a correction in the melting temperature for pure aluminum was carried out according to the model description provided in [7], although, for the case of Al, no correction was applied by the author [7]. Figure 2 shows the evolution of viscosity of molten Cu as a function of temperature, where the theoretical model is compared to experimental results [37], where a deviation is noticed for temperatures close to the melting temperature.

In Figure 3, the viscosity of pure molten silicon is plotted against temperature. As mentioned, the melting temperature of Si has been assumed as Tm corr,Si ffi 596:85 K [22], which is less than the correction of melting temperature of pure Si proposed by Kaptay [7], that is, Tm corr,Si ffi 870 K. This correction was carried out by the author for some pure molten metals, so that the model could also fit the experimental scatter of these elements. As can be observed, a correction in the melting temperature was needed not only for Si, Ge, Sb, and Bi but also for pure molten Al, based on the same set of experimental data.

In Figure 4, it can be seen that Kaptay's model [7] well the experimental data well for the viscosity of molten magnesium as a function of temperature [38–40].

In Figures 5–8, the surface tension for pure Al, Cu, Si, and Mg is depicted as a function of temperature, calculated from surface tension-viscosity relations according to Egry [34] and Kaptay [7] formulations. For all cases, except for Si in Figure 7, the surface tension exhibits a trend to decrease as the temperature increases. It also can be noticed that for all cases, they diverge close to the melting point. For high temperatures, both models yield very close results. The best agreement observed between the two models is that of pure Mg, as shown in Figure 8.

Figures 9–12 show the evolution of viscosity as a function of temperature for all examined alloys, where it can be seen that the two models generally exhibit similar

Figure 1. Calculated viscosities of the pure molten Al as a function of temperature compared to experimental data [37].

Figure 2. Calculated viscosities of the pure molten Cu as a function of temperature compared to experimental data [37].

Calculated viscosities of the pure molten Si as a function of temperature compared to the experimental data [37].

Figure 4. Calculated viscosities of the pure molten Mg as a function of temperature compared to the experimental data.

Dependence of Surface Tension and Viscosity on Temperature in Multicomponent Alloys DOI: http://dx.doi.org/10.5772/intechopen.82307

Figure 5.

Comparison between surface tension of pure molten Al as a function of temperature provided by both Egry's and Kaptay's surface tension-viscosity relation models.

Figure 6.

Comparison between surface tension of pure molten Cu as a function of temperature provided by both Egry's and Kaptay's surface tension-viscosity relation models.

#### Figure 7.

Comparison between surface tension of pure molten Si as a function of temperature provided by both Egry's and Kaptay's surface tension-viscosity relation models.

#### Figure 8.

Comparison between surface tension of pure molten Mg as a function of temperature provided by both Egry's and Kaptay's surface tension-viscosity relation models.

#### Figure 9.

Simulations of viscosity of Al-6wt%Cu-1wt%Si as function of temperature: Arrhenius-type equation and Kaptay model.

#### Figure 10.

Simulations of viscosity of Al-6wt%Cu-3wt%Si as function of temperature: Arrhenius-type equation and Kaptay model.

Dependence of Surface Tension and Viscosity on Temperature in Multicomponent Alloys DOI: http://dx.doi.org/10.5772/intechopen.82307

#### Figure 11.

Simulations of viscosity of Al-6wt%Si-3wt%Cu as function of temperature: Arrhenius-type equation and Kaptay model.

Simulations of viscosity of Al-6wt%Si-3wt%Cu as function of temperature: Arrhenius-type equation and Kaptay model.

#### Figure 13.

Application of derived surface tension-viscosity relation for Al-Cu-Si ternary alloys and quaternary Al-Cu-Si-Mg alloys as function of temperature.

results. The greatest deviation between the models can be observed for the Al-6wt% Cu-3wt%Si alloy, Figure 10; but even in this case, a relatively good agreement can be considered. This may be related to the non-ideal term of viscosity of the Redlich-Kister equation, whose coefficients do not depend on the alloy composition but depend on the temperature [36].

Figure 13 shows the application of the derived surface tension relation equation for alloys, Eq. (13), to ternary Al-Cu-Si and quaternary Al-Cu-Si-Mg alloys. As can be noticed, the increase in the alloy Si content decreases the surface tension. The lowest surface tension profile is associated with the alloy having the highest Si content, that is, for the Al-12wt%Si-1wt%Cu-1wt%Mg quaternary alloy.
