1、Phase TransformationZheng JiangChongqing UniversityAprail-20-2016Outline1 Objectives2 State Equations3 Boundary and Initial ConditionsOutline1 Objectives2 State Equations3 Boundary and Initial Conditions1 ObjectivesnReview the concepts of phase diagrams and the equilibrium conditions that apply at a
2、n interface between two phases.nIntroduce the different scales of a process and the local transformation path concept.nDerive the average continuity equations for two phases.The following the all problems involving phase transformations:nIron in a blast furnacenAluminum in an electrolysis cell start
3、ing from their respective mineralsnThe decarburizing of steels, the solidification of metal alloys and their solid state transformationsnThe production by a flux technique of high melting point single crystalsnThe precipitation of ceramic powders starting from aqueous solutionsnThe growth of a film
4、at the interface between two materials (diffusion couple), nCorrosion or oxidation reactions on a surface, etc. Outline1 Objectives2 State Equations3 Boundary and Initial Conditions2.1 Phase diagramsThe equilibrium conditions for Np phases vi (i = 1, Np), with Nc chemical constituents Aj (j = 1, Nc)
5、， the given by thermodynamics. For example, in the aluminum-silicon system, for which the phase diagram is shown in figure 5.1, we will have, v1 = l, the liquid phase of the mixture (Al + Si), v2 = ，the cubic face centered phase (cfc) of aluminum, and v3 = ，the diamond structure of silicon. There ar
6、e two chemical constituents in this case: A1 = Al and A2 = Si. The thermodynamic equilibrium conditions stipulate that:In other words, the pressure, pvi , the temperature, TVi ,as well as the chemical potentials, Ajvi , of each of the elements Aj must be equal for all the phases, vi , present.When N
7、p phases are present, the number of variables describing the system is equal to (Np(Nc -1) + 2， that is, the (Nc -1) concentrations in each of the Np phases, the temperature and the pressure. However, equations (5.1c) introduce (Np -1 ) constraints for each of the chemical species Aj present in the
8、system, thus, Nc(Np -1) equations. The number of degrees of freedom of the system is then equal to (Nc + 2 - Np), including the pressure (or to (Nc + 1 - Np) if p is fixed). This is Gibbs phase rule.2.2 Out-of-equilibrium phenomenaAs material transformation processes take place over a finite length
9、of time，the phases present the rarely in equilibrium! Therefore, two types of deviation from the equilibrium given by a phase diagram are distinguished: In the first case, the chemical species and/or the heat do not have sufficient time to diffuse and thus to even out the gradients of solute and tem
10、perature that the inevitably created in a phase transformation. The interfaces themselves can depart from the equilibrium given by the phase diagram for three reasons. The interface surface energy ?, between the phases, modifies the energy balance, and thus the equilibrium, when the characteristic s
11、ize of the phases is small.where the Gibbs-Thomson coefficient, Lf the specific latent heat of fusion and = 2R-1 the mean curvature of the interface.The kinetics of attachment of atoms or molecules is associated with the rearrangement of the atoms or molecules as they pass from the structure of one
12、phase to the other. The kinetic undercooling will then be greater the more the structures present differ (and thus have elevated transformation entropies). Solute trapping is linked to the redistribution of the solute elements across the moving interface of an alloy. 2.3 Simple models of phase trans
13、formations Two extreme cases can be presented for the occurrence of a phase transformation; they are illustrated in figure 5.4. In the first case (fig. 5.4(a), the interface between the liquid and solid phases of a pure material that solidifies in a Bridgman furnace is relatively smooth, that is, it
14、s geometric characteristics are on the scale of the size of the process.Figure 5.4(b) illustrates the situation of a transformation governed by the nucleation of the second phase in the volume of the primary phase (transformation ). the volume fraction f of the transformed phase is given by the well
15、 known Avrami equation:where dn is the number of grains formed per unit volume between times and + ，and R( ,t) is the radius of the grains that nucleate at instant and are observed at instant t.where R( , ) is the initial radius of the seeds at the moment of their nucleation.where b and q are adjust
16、able parameters. General equation2.4 Interface conservation equationsClass finished !where n is the outgoing normal of the domain . The first integral and the integral of the source term are zero as the thickness, , of the chosen volume element goes to zero. Similarly, in this limit, the contributio
17、ns of the surface integrals are only from the surfaces parallel to the interface.where the notation A = A - A has been adopted to represent jump of a value A across the interface between and . Conservation of mass Conservation of soluteTo signify that the concentrations are those at the interface, w
18、e have added the index *.In the case where the solute transport associated with the velocities vn and vn can be neglected (vn =vn = 0), equation (5.12) establishes that the velocity of the interface is given by the solute flux jump divided by the concentration jump, with a minus sign: Conservation o
19、f energywhere K is the thermal conductivity.For the case where vn =vn = 0, the velocity of the interface is given by the difference of the heat flux in the two phases divided by the volumetric enthalpy jump. Conservation of momentumIn the case where the densities of the two phases are equal, the ter
20、m to the left is zero and the equation is reduced to the continuity of the normal stresses at the interface: n = n. For the two tangential components, equation (5.9) gives: v n - v n/ v n - v n - v n/ v n = n - n2.5 Domain Composed of Two PhasesInside each of the sub-domains， and , characterizing th
21、e phases and , the heat equation can be writtenFirst, the equilibrium condition (if equilibrium is assumed):In the case of a binary alloy with two phases present, the problem is similar: the heat equation and the solute conservation equation are to be solved in each phase with the heat and solute co
22、nservation equations at the interface / described in the previous section, and with the equilibrium conditions at the interface given by the equality of the temperatures and of the chemical potentials.where T/ is the equilibrium temperature between the two phases (for example the melting point in th
23、e case of crystallization).2.6 Representative Volume to Take Average ValuesThe diffusion of solute is considerably slower than the diffusion of heat in many materials: the non dimensional Lewis number defined as the ratio of the thermal and chemical diffusion coefficients (Le = ID with = K(/(cp) is
24、on the order of 104 for a liquid metal (107 for a solid metal).It would be unfortunately not realistic for calculation of the deformation of a bridge (tens of meters) with a resolution at the scale of aggregates (centimeters or even millimeters). Because of this, when very different scales are prese
25、nt in a heterogeneous material, averaged continuity equations are used (one also speaks of homogenization theory in mechanics of heterogeneous materials). The characteristic dimension of the microstructure, d at the scale of the interface is much smaller than the characteristic size of the domain, d
26、, in which one hopes to solve the continuity equations. The size of the domain V chosen for the continuity balances is situated between the two extremes, d V d. Now suppose that we take the average inside V of a specific value such as the specific mass, the solute concentration or the enthalpy per u
27、nit volume. We denote this value :When the size of the domain increases, this average value tends to stabilize as statistically sufficient fractions of the two phases are taken into account (shaded zone in figure 5.11). One then speaks of a representative volume element.We always place ourselves in
28、the case of a representative volume neither too small so that the fluctuations at the scale of the microstructure are averaged, nor too large” with respect to the macroscopic variations of the value at the scale of the domain.We define the following values: The volume fraction, f, of a phase inside
29、V:Where X is the distribution function of the phase :For the two phase system chosen here, the relation V= V + V, or f + f =1, is of course satisfied.The average values, and , of a value weighted over the sub-domains V and V :With the definition (5.21) of , we obtain again the complementary relation
30、:The intrinsic average values, and , specified in each sub-domain V or V : Temporal Derivative of The distribution function of the phase is such that its temporal derivative is zero everywhere except at the interface /. More precisely:The temporal derivative of the average of the value is equal to t
31、he average of its temporal derivative plus an average on the interface of the product of *, the value taken at the interface in the phase ， and the normal component of the interface velocity vn/. Spatial Derivative of It is thus equal to the average of the gradient less an average on the interface o
32、f the product of * and the normal vector at the interface, n/.2.7 Average Continuity Equations Mass Conservation EquationA continuity equation in physics is an equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities ar
33、e conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.Note that the signs appearing in front of the averages on the interface change according to the convention for the direction of the normal n/ at the interface (outgo
34、ing from the domain entering into ).Using =f Supposing that the two phases, and , do not have the same mass density but that both are constant (=cnst and =cnst, ), the two phase media is no longer incompressible in the sense that its average mass density () is no longer constant. Therefore movements
35、 of material are necessary (v0) to accommodate the variations of mass density related to phase transformations.The first term represents the shrinkage upon solidification: as s l for most metals and alloys, this term is positive if the solid fraction fs increases (solidification). In this case, the
36、liquid must move to feed the growth front, in the opposite direction of the front velocity: this liquid flux is represented by the second term of(negative divergence). Solute Conservation Equationwhere the internal chemical reaction (source) term has been omitted for simplicity.The diffusion of chem
37、ical species at this scale is very slow (typically 105-108 seconds on the scale of a centimeter!). It thus appears reasonable to neglect this term at the macroscopic scale. As for the transport term, it is only important if a liquid phase is present or in the case of large deformations. If the diffu
38、sion at the scale of the small volume element V is sufficiently rapid in the two phases, the average intrinsic local concentrations are then equal to those at the interface (v=cv* =cv, v=, ) , the latter being imposed by the phase diagram. With this assumption, the well known lever rule is obtained
39、for phase transformations:where co is the nominal composition of the alloy. In the case of a very simple binary phase diagram, the equilibrium concentrations c and c are given as functions of the temperature by the corresponding equilibrium lines.For c = co, the fraction f is zero while f = 1 for c
40、= co/k/. Another situation often encountered in phase transformations is that of a phase in which the diffusion is rapid at the scale d (D = ) and a phase for which there is practically no diffusion (D= 0). This approximation, named Scheil or Scheil-Gulliver, describes relatively well the solidifica
41、tion path of the majority of alloys. In this case, the concentration in phase , here, the liquid, is always such that: = c *= c , while the average concentration of phase is given by (fig. 5.15): Energy Conservation Equation Momentum Conservation EquationThis average equation is only interesting if
42、one of the two phases is in motion relative to the other.3 Initial and Boundary Conditions3.1 GeneralitiesThe thermal boundary conditions for a domain could be of three types: Dirichlet condition (or essential condition): the surface temperature of the domain is imposed, T(x, t) = for . Note that th
43、is imposed temperature can depend on time and position. Neumann condition (or natural condition): the heat flux, qT, at the surface of the domain is imposed: Cauchy condition (or mixed natural condition): the heat flux at the surface of the domain is imposed via a thermal transfer coefficient, h:3.2 Transfer by Forced Convection3.3 Transfer by Natural Convection3.4 Transfer by Boiling/condensation Thanks !
