8. Principles of Solidification

Transcription

1 CBE4010 Introduction to Materials Science for Chemical Engineers 8. Principles of Solidification The Driving Force a Phase Change We expect a material to solidify when the liquid cools to just below its freezing (melting) temperature, because the energy associated with the crystalline structure of the solid is less than the energy of the liquid. This energy difference between the liquid and the solid is the free energy per volume ( G ) and is the driving force for solidification. v At the equilibrium temperature, both phases coexist because they have the same free energy. The expression for the change in free energy for the liquid to solid (L S) transformation has the form: G H T S L S L S L S (1) L S G L S H L S S : The free energy change : The enthalpy change : The entropy change 1

2 At the equilibrium temperature, T=T E, the solid and liquid have the same free energy (ΔG =0). Substituting this condition into the eq. (1) and solving for at T E yields: L S L S S ( H / T E ) () Assuming the heat capacities of the two phases are about the same, L S L S L S G H T( H / T E ) (3) Rearranging Eq. (3) and substituting T TE T L S ( L G H S / T ) T E (4) The solidification of a pure substance is an exothermic transformation, so that L S S L H H H 0 Eq. (4) shows that the liquid-to-solid phase transformation is thermodynamically favored for all T<T E. This equation also shows that the thermodynamic driving force for the phase transformation increases as T increases.

3 Nucleation Nucleation refers to the formation of the first nano-sized crystallites from molten material. In a broad sense, the initial stage of formation of one phase from another phase. - Homogeneous nucleation: the new phase nucleates randomly in the parent phase. - Heterogeneous nucleation: the new phase nucleates preferentially at specific sites in the parent phase. Specific sites - Structural discontinuities in the parent phase (e.g., grain boundaries or dislocations) or foreign objects (mold wall), Impurity particle trapped in the liquid. The distinction between these two types of nucleation events is important, and specific terms have been developed to identity the different mechanisms. Homogenous Nucleation of Phase A simplified but helpful view of a liquid is to visualize the atoms comprising the liquid as a random distribution of hard spheres. The interatomic distances in the liquid are similar to those of the crystalline phase, but each atom on average has fewer nearest neighbors in the liquid than in the crystal. The structure is therefore more open and allows greater atomic mobility than does that of the solid state. Distributed throughout the liquid are small closely packed atomic clusters having a packing arrangement similar to that of the solid. Because of the open structure, these clusters form and disperse quickly. The relationship between the size and the stability of the clusters depends on T. For the purpose of conducting a simplified calculation relating size to stability, we make two assumptions. 1. The interfacial energy can be regarded as isotropic, that is, Independent of the specific crystal plane that forms the interface (transforming phase is spherical).. The interfacial energy per unit surface area is independent of the size of the solid phase. 3

4 Two specific components are associated with the free energy change of the liquid-solid transformation. The first is the change in energy associated with the creation of the liquid-solid interface. The second is the difference in bulk free energies of the liquid and solid phases. The total interfacial energy associated with a spherical particle is the product of the area of the interface (4πr ) and the interfacial energy per unit area. - The nd energy term is the bulk free energy. It is the difference in free energy of the solid and liquid phases multiplied by the volume of the particle. When T<T E, The bulk free energy of solid is smaller than that of liquid, L S free energy change is negative. the product of change in free energy per unit volume (ΔGv) and the volume of the growing phase, (4/3)πr 3 is negative. Figure: The dependence of the various energy terms associated with nucleation as a function of temperature: (a) The relationship between cluster radius and surface energy of growing spherical solid phase in a liquid, (b) The relationship between the cluster radius and (c) The sum of the previous two curves 4

5 Note that the two energy terms act in opposite directions. Initially the r area term dominates, and as the particle becomes larger, there is an increase in ther net free energy. However, once a critical radius(r*) is reached, V(r 3 ) term begins to dominate, and further increases in the particle size result in the particle size result in a reduction in the free energy of the system. Thus, r* represents the size of a stable nucleus smaller particles redissolve into the liquid phase while larger particles will continue to grow. The total change in free energy as a function of r : 3 G( r) (4 r ) (4/ 3) r ( ) (5) SL G v - r* can be determined by taking the derivative of Eq.(5) with respect to r and setting the resultant expression for the slope of the energy curve in Figure(c) equal to zero: d[ G( r)]/ dr 0 8 r 4 r ( ) (6) SL G v - Solving this expression for r, which is r*, gives: r* ( ) / (7) SL G v - Substituting the expression for Gv given in Eq. (4) into the previous equation yields: r* ( ) T /( H T ) (8) SL E v - The critical radius(r*) decreases as the degree of undercooling increases, that is, r* (1/ T). - The expression for G(r) given in Eq. (5) to obtain the change in free energy necessary to form stable nuclei. S / L E v T 3 G* [16 ( r ) T ]/[3( H ) ] (1/ ) (9a) 5

6 G * (1/ T ) (9b) - G* can be thought of as the size of the energy barrier to the nucleation process. That is, as the undercooling increases, the energy barrier to nucleation decreases. With larger undercoolings, both r* and G * decrease, suggesting that simply lowering the temperature of the system, nucleation to occur ever more readily. However, there are practical kinetic limits to this effect. The random fluctuation in the local arrangements of atoms is the process that provides the clusters. The formation of the clusters depends on atomic mobility. With decreasing temperature, there is a corresponding reduction in atomic mobility and subsequently reduction in the rate of clustering. 6

7 Heterogeneous Nucleation of a Phase - Many different microstructural features can serve as preferential nucleation sites. 1. The wall of the mold that contains a liquid.. A small piece of a ceramic crucible that may have broken off during the pouring of the molten metal into the mold. 3. Small crystals of a higher melting temperature material may be intentionally added to the liquid to increase the nucleation rate. How can one predict whether a specific material will act as a heterogeneous nucleation site for a specific solidification event? A major factor is a characteristic referred to as wetting. If a droplet of the molten metal does not wet a candidate nucleating agent, then that agent will not be effective innoculants. In contrast, a material that is wet by the molten metal is likely to be an effective nucleating agent. 7

8 Consider the case of heterogeneous nucleation of a solid on the wall of a mold containing a liquid. (Fig.5) LM cos (10a) MS LS cos ( ) / (10b) LM MS LS θ: The contact angle LM : the energy of the liquid/mold interface : the energy of the liquid/solid interface LS : the energy of the mold/solid interface MS The homogeneous nucleation barrier (ΔG* hom ) can be modified by taking into account the effectiveness of a particular feature for heterogeneously nucleating a new solid phase. The barrier to heterogeneous nucleation has the form: * * G het ( G ) f ( ) (11) hom ( cos )(1 cos ) Where f (θ) =. The value of f (θ) is related to the contact 4 angle θ. Since f ( ) has a value between 0 and 1, the energy barrier for heterogeneous nucleation is always lower than that for homogeneous nucleation. 8