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Solidification and PrecipitationChapter 22 & 23 (Kinetics of Materials)

Rommel Paulo B. ViloanMSE 233

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Solidification

• we restrict ourselves to two cases: when the liquid/solid interface is stable and plane-front solidification is achieved, and when the interface is unstable and cellular or dendritic growth occurs.

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22.1 Plane-Front Solidification in One Dimension22.1.1 Scheil Equation

In Fig. 22.1a one-dimensional solidification is depicted; a liquid binary alloy initially of uniform composition co is placed in a bar-shaped crucible of length L. The bar is progressively cooled from one end, so it solidifies from one end to the other with a stable and planar liquid/solid interface.

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22.1.1 Scheil Equation

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22.1.1 Scheil EquationAt a relatively short time after the establishment of the quasi-steady-state concentration spike, the flux relative to an origin at the interface moving at velocity v is,

The diffusion equation in the liquid is,

General solution

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22.1.1 Scheil Equation

Substituting CL(x) and using

The effective partition ratio, k’, is defined as the ratio of the concentration in the solid being formed over the concentration in the bulk liquid. Therefore,

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22.1.1 Scheil EquationIf the concentration spike moves by dx, the amount of solute that must berejected into the liquid is

The slowly changing concentration in the bulk liquid is represented by

The change in concentration in the liquid is then

(Scheil Equation)

Distribution of solute after the solidification

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22.1.1 Scheil EquationTwo limiting forms of the Scheil Equation,

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•Lack of convection•High solidification rate•Slow rate of diffusion in the liquid

cSL = co

the composition of the solid being formed and the composition of the bulk liquid are the same

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22.1.1 Scheil EquationTwo limiting forms of the Scheil Equation,

2.

There is then rapid mixing in the liquid, the diffusion barrier is nonexistent, and there can be a large difference between the compositions of the solid being formed and the bulk liquid, depending on the factor k.

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22.1.1 Scheil EquationTwo limiting forms of the Scheil Equation,

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22.1.2 Zone Melting and Zone Leveling

Heating from one end

A melted zone of

length l is produced

The zone is then moved along the entire specimen at a constant rate while keeping l constant.

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22.1.2 Zone Melting and Zone Leveling

The zone is generally much longer than the width of the liquid boundary layer (i.e., 1 >> 6).

When the zone moves a distance dx, the amount of solute gained by the zone is (cO - csL) dx, and therefore

cL is the concentration in the liquid in the zone.

Because the initial composition of the liquid in the zone is co.

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22.1.2 Zone Melting and Zone Leveling

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22.1.2 Zone Melting and Zone Leveling

When the zone length is relatively short, k’ is large, and when the numberof passes is small, the bulk of the specimen solidifies at very nearly a uniform composition corresponding to CO. Zone solidification can be used in this manner to produce compositional uniformity, a technique known as zone leveling.

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22.2 Cellular and Dendritic Solidification22.2.1 Formation of cells and dendrites

When the liquid/solid interface is unstable (undercooling) a cellular or dendritic structure is developed. When the degree of instability is relatively low, an array of protuberances develops on the interface. These protuberances, called cells, advance perpendicular to the interface.

For <100> liquid/solidinterfaces in cubic metals.

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22.2.1 Formation of cells and dendrites

However, for a <110> interface, the cells take on a corrugated configuration of long hills and furrows. When the degree of instability is increased by increasing the rate of solidification, fully formed dendrites develop.

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22.2.1 Formation of cells and dendrites

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22.2.2 Solute Segregation during Dendritic Solidification

During dendritic growth, extensive solute segregation occurs in the interdendritic spaces; this phenomenon is a serious problem in the casting of alloys. The segregation occurs because of the tendency of the solidifying solid to reject excess solute into the remaining liquid and can be understood using the model developed to analyze plane-front solidification

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22.3 Structure of Castings and IngotsCastings are typically produced by pouring liquid into a relatively cold mold and allowing solidification to take place. Heat is removed from the solidifying material by conduction out through the mold.

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Precipitation

• Precipitation occurs when a new phase forms discontinuously within a homogeneous metastable phase to form a two-phase mixture of lower energy. The process occurs by the nucleation and growth of particles (precipitates) of the new phase embedded in the original phase.

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23.1 General Features of Precipitation

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23.2 Nucleus Morphology and EnergySimplest case: α and β-phase crystals have different compositions but still

match almost exactly in all three dimensions.

The critical nucleus can then form with a coherent interface and is therefore of relatively low energy.

When there is no near lattice matching between precipitate and matrix structures in any dimension, the interfacial energy will be relatively high. In such cases, homogeneous nucleation will be slow to occur and the nucleation will be inhomogeneous.

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23.2 Nucleus Morphology and Energy

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23.3 Loss of Precipitate Coherency During Growth

Just after nucleation, a small coherent precipitate generally possesses both interfacial energy and strain energy. Its strain energy increases linearly with its size. On the other hand, its total interfacial energy increases linearly with its area.

interfacial energy dominates at small

sizes where the interface-to-volume

ratio is large.

However, the strain energy becomes dominant as the size increases. Nuclei and small-sized precipitates

therefore tend to be coherent because this

minimizes the interfacial energy (and the total

energy).

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