Microsegregation: partition coefficient and Scheil’s model

When the solidification occurs in thermodynamic equilibrium conditions, i.e.

when the rate R of the solidification front tends to zero, the solute concentrations in the liquid 𝐶_𝐿 and in the solid 𝐶_𝑆 phases are uniform and equal, at each temperature, to the equilibrium value indicated by the phase diagram. The ratio 𝑘 = 𝐶_𝑆⁄𝐶_𝐿, approximately independent from the temperature in the solidification range, is called the partition coefficient of the solute, therefore 𝑘 is a measure of the solute tendency to segregate in the solid phase during the solidification: the solute tends to dissolve more in the solid when 𝑘 > 1, instead it will be more present in the liquid until the end of the solidification if when 𝑘 < 1. However, in most of the technological processes of industrial interest, the alloys solidification is too fast and the thermodynamic equilibrium conditions cannot be maintained. In these cases, the solute concentrations corresponding to the thermodynamic equilibrium can be established only locally at the solid/liquid interface and the establishment of a certain grade of compositional microsegregation cannot be avoided because there is no time for the solute diffusion to uniform the concentration in the system.

This situation can be described through the Scheil’s model [53], that is based on the follow simplifying assumptions:

1. perfect mixing in the liquid, i.e. the solute concentration 𝐶_𝐿 is uniform in the liquid domain;

2. the solute diffusion in the solid phase does not occur or it is negligible;

3. local equilibrium at the solid/liquid interface, in other words the solute concentrations 𝐶_𝑆 and 𝐶_𝐿 respectively in the solid and in the liquid in reciprocal contact are the ones reported by the phase diagram at each temperature;

4. the difference between the molar volumes of the liquid and the solid can be neglected;

5. the partition coefficient 𝑘 is constant with the temperature;

6. the solid/liquid interface is planar or it has a curvature sufficiently low that the Gibbs-Thomson effect can be neglected.

At the beginning of the solidification, the liquid has the nominal solute concentration 𝐶₀, therefore the first solid is formed at the temperature 𝑇₁ with the solute concentration 𝐶_𝑆(𝑇₁) = 𝑘𝐶₀. In the case of 𝑘 < 1, the formed solid has a solute concentration lower than the liquid, therefore the solute must be rejected in the remaining liquid as the solidification process proceeds. Conversely, in the case 𝑘 > 1, the liquid is progressively depleted in the solute. During the temperature decreasing, the solute concentration 𝐶_𝐿(𝑇) of the liquid phase remain uniform over all the liquid volume at the corresponding value indicated by the phase diagram because of the assumptions N.1 and N.3. Conversely, in the solid phase the equilibrium solute concentration value 𝐶_𝑆(𝑇) = 𝑘𝐶_𝐿(𝑇) is established only at the solid/liquid interface, instead the solid formed previously between 𝑇₁ and the current temperature 𝑇 has a lower or higher solute concentration respect to the equilibrium value if 𝑘 is minus or major to 1 respectively. Therefore, a compositional gradient is formed in the solid that cannot be reduced by the solute diffusion because of the assumption N.2. The trend of the mean value 𝐶_{𝑆,𝑚𝑒𝑎𝑛}(𝑇) of the solute concentration in the solid phase does not follow the solidus curve of the phase diagram, as in the case of a quasi-static solidification, but it traces a curve placed below or above it in the case of 𝑘 < 1 or 𝑘 > 1, respectively, as shown in figure 1.14.

Figure 1.14. Solute profiles in the solid and liquid phase and path followed in the phase diagram by the mean concentration in the solid according to the Scheil’s model in the cases of k < 1 and k

> 1.

At each reduction 𝑑𝑇 of the system temperature, an infinitesimal liquid volume 𝑑𝑉_𝑙 solidifies. The difference between the solute quantity 𝐶_𝐿(𝑇) ∙ 𝑑𝑉_𝐿 in the infinitesimal volume before the solidification and the solute quantity 𝐶_𝑆(𝑇) ∙ 𝑑𝑉_𝑆 after the solidification corresponds to the solute rejected to (if 𝑘 < 1) or removed from (if 𝑘 > 1) the liquid. This difference is equal to:

𝐶_𝐿(𝑇)𝑑𝑉_𝐿− 𝐶_𝑆(𝑇)𝑑𝑉_𝑆 = 𝐶_𝐿(𝑇)𝑑𝑉_𝐿− 𝑘𝐶_𝐿(𝑇)𝑑𝑉_𝑆 = (1 − 𝑘)𝐶_𝐿(𝑇)𝑑𝑉_𝑆 eq. 1.1 where the last equality is justified by the assumption N.4, for which it is possible to neglect the volume variation associated to the solidification, i.e. 𝑑𝑉_𝑆 ≅ 𝑑𝑉_𝐿. The solute rejected to or removed from the liquid during the solidification of 𝑑𝑉_𝐿 leads to an infinitesimal variation of the solute concentration in the liquid equals to:

𝑉_𝐿[𝐶_𝐿(𝑇 − 𝑑𝑇) − 𝐶_𝐿(𝑇)] = 𝑉_𝐿𝑑𝐶_𝐿 eq. 1.2

By equating the last terms of equations 1.1 and 1.2 and dividing by the total volume 𝑉 of the system:

(1 − 𝑘)𝐶_𝐿(𝑇)𝑑𝑓_𝑆 = 𝑓_𝐿𝑑𝐶_𝐿 ⇒ (1 − 𝑘) ^𝑑𝑓𝑆

1−𝑓_𝑆 = ^𝑑𝐶𝐿

𝐶_𝐿(𝑇) eq. 1.3

with 𝑓_𝑆 = 𝑉_𝑆⁄𝑉 and 𝑓_𝐿 = 𝑉_𝐿⁄𝑉 the volume fractions of the solid and the liquid, respectively. By integrating equation 1.3 between the temperature 𝑇₁, at which

corresponds 𝑓_𝑆 = 0 and 𝐶_𝐿 = 𝐶₀, and a generic temperature 𝑇 in the solidification

The Scheil’s equation 1.4, also called non-equilibrium lever rule, predicts that, in the case of 𝑘 < 1, the solute concentration in the liquid diverges when 𝑓_𝐿 go to zero indipendently from the solute quantity present in the system. In other word, when 𝑘 < 1 and the solute diffusion in the solid cannot occur, the solidifcation always ends with an eutectic transformation in presence of an arbitrary low quantity of solute.

The Scheil’s model can be modified in order to include an eventual not negligible solute diffusion in the solid during the solidification, the Brody-Flemings equation 1.5 is obtained in this way [14]:

𝐶_𝑆 = 𝑘𝐶₀[1 − ^𝑓𝑆

1+^{𝐷𝑆𝑡𝑓} 𝐿2 𝑘]

𝑘−1

eq. 1.5

where 𝐷_𝑆 is the solute diffusivity in the solid, 𝑡_𝑓 is the local solidification time, i.e. the period during which both the liquid and solid phases are present, and 𝐿 is the length scale of the microsegregation, usually half the arm dendrite spacing is used as 𝐿.

In Inconel 718 alloy, there are both elements with a partition coefficient 𝑘 < 1, they are Nb, Mo and Ti, and elements with 𝑘 > 1, i.e. Fe and Cr. The partition coefficients of the alloying elements of Inconel 718 are reported in table 1.6.

According to the Scheil’s model, a certain grade of segregation is expected to occur during a fast solidifcation: the Fe and Cr contents tend to be higher in the zones where the solidification has started, i.e. the cores of the dendrites, instead Nb, Mo and Ti prevail where the last liquid solidified, i.e. the interdendritic zones. As it can be observed in table 1.6, the highest microsegregation level is expected with the niobium. At first aproximation, it is possible to describe the solidification of Inconel 718 as a binary alloy by considering only the role of the Nb solute [54] [55].

Table 1.6. Partition coefficients k of the alloying elements of Inconel 718 alloy. From: [56].

Element Ni Cr Fe Nb Mo Ti Al C

Experimental 𝑘 value 1.03 1.09 1.20 0.28 0.73 0.41 0.79 0.12