Scheil equation

The Scheil equation (often also: Scheil-Gulliver equation ) describes in metallurgy the distribution of an alloying agent in an alloy during solidification . A solidification described in this way is also often referred to as Scheil solidification or directional solidification . It was first described by GH Gulliver in 1913 in phenomenological and by Erich Scheil in 1942 in mathematical form.

description

The properties of an alloy depend heavily on its composition. To describe the properties, it is helpful to know the concentration of the alloying elements in the material. As a rule, after solidification, the alloying elements are not evenly distributed over the workpiece and the mathematical description of this distribution is difficult in general.

Solutions can be found if certain assumptions and simplifications are made.

The Scheil approach requires a local equilibrium between the solid and the melt on the advancing solidification front. This enables the use of equilibrium phase diagrams in the solidification analysis .

In contrast to equilibrium solidification, it is also assumed that no diffusion takes place in the solidified solid. This is e.g. B. the case when the characteristic diffusion length is much smaller than the workpiece length. There is also no diffusion of the alloying element from the melt into the solid.

Complete mixing and thus homogeneous distribution of the alloying element is required for the melt. This can be achieved either by diffusion, convection or by mechanical stirring of the melt.

Derivation

Under these assumptions, the Scheil equation can be derived, which describes the composition of the workpiece and the melt as a function of the solidified volume fraction during solidification.

As was assumed, the diffusion coefficient is in the solid because no diffusion should take place and in the melt because the mixing should be complete. indicates the concentration of the alloying element in the melt and in the solid. indicates the volume fraction of the melt and that of the solid. is the so-called distribution coefficient, which results from the ratio of the equilibrium concentrations in the solid and the melt and can be determined from the equilibrium phase diagram: ${\displaystyle D_{S}=0}$ ${\displaystyle D_{L}=\infty }$ ${\displaystyle C_{L}}$ ${\displaystyle C_{S}}$ ${\displaystyle f_{L}}$ ${\displaystyle f_{S}}$ ${\displaystyle k}$

${\displaystyle (1)\quad k={\frac {C_{S}}{C_{L}}}}$

When a first amount of solid is produced, it has the alloying agent concentration . Since it is necessary to preserve the total amount of alloying agent: ${\displaystyle df_{S}}$ ${\displaystyle kC_{0}}$

${\displaystyle (2)\quad (C_{L}-C_{S})\ df_{S}=f_{L}\ dC_{L}}$

The derivation of Eq. can be understood from Figure 1 . The dashed areas in the picture represent the amount of alloying element in the solid and liquid phase at the phase boundary. ${\displaystyle (2)}$

Figure 1 : Schematic representation of the state of local equilibrium at the liquid-solid phase boundary for the derivation of Eq. . ${\displaystyle (2)}$

The total mass must also be retained:

${\displaystyle (3)\quad f_{S}+f_{L}=1}$

With and we can write as: ${\displaystyle (1)}$ ${\displaystyle (3)}$ ${\displaystyle (2)}$

${\displaystyle (4)\quad C_{L}(1-k)\ df_{s}=(1-f_{S})\ dC_{L}}$

The boundary condition at the beginning of the solidification at enables an integration of the differential equation : ${\displaystyle C_{L}=C_{0}}$ ${\displaystyle f_{S}=0}$ ${\displaystyle (4)}$

${\displaystyle (5)\quad \displaystyle \int _{0}^{f_{S}}{\frac {df_{S}}{1-f_{S}}}={\frac {1}{1-k}}\displaystyle \int _{C_{o}}^{C_{L}}{\frac {dC_{L}}{C_{L}}}}$

with which the concentration curve in the melt

${\displaystyle (6)\quad C_{L}=C_{0}\ (1-f_{S})^{k-1}}$

and the concentration profile in the solidified workpiece

${\displaystyle (7)\quad C_{S}=kC_{0}\ (1-f_{S})^{k-1}}$

can be specified as a function of the progress of solidification (volume fraction of the solid ). Figure 2 shows this process. ${\displaystyle f_{S}}$

Figure 2 : Course of the concentration C of the alloying element B with progressive solidification or the volume fraction of the solidified phase. is the concentration at the beginning of the solidification. ${\displaystyle f_{S}}$ ${\displaystyle C_{0}}$

The Scheil equation predicts a concentration profile in which the concentration at the end of the workpiece can become infinitely large, i.e. H. In a binary system AB, the melt will consist of 100% B at the end of the solidification and thus solidify as pure B. An alternative case is a eutectic system AB in which the melt at the point where the eutectic composition is reached also solidifies in the state, so that a typical eutectic structure is present at the end of the workpiece.

swell

• University of Cambridge - Teaching and Learning: The Scheil Equation
• Gulliver, GH, J. Inst. Met. , 9: 120, 1913.
• Kou, S., Welding Metallurgy , 2nd edition, Wiley-Interscience, 2003.
• Porter, DA, and Easterling, KE, Phase Transformations in Metals and Alloys (2nd edition), Chapman & Hall, 1992.
• Scheil, E., Z. Metallk. , 34:70, 1942.