Fundamentals of Metallurgy PDF

Fundamentals of Metallurgy 04/12/2024  Chapter 5 Solidification and crystallization in metals and alloys  Outline Chapter Subjects 1. Introduction 2. Thermodynamics and phase Diagrams 3. Diffusion in metals...

Fundamentals of Metallurgy 04/12/2024  Chapter 5 Solidification and crystallization in metals and alloys  Outline Chapter Subjects 1. Introduction 2. Thermodynamics and phase Diagrams 3. Diffusion in metals 4. Crystal Interfaces and Microstructure 5. Solidification and crystallization in metals and alloys 6. Phase transformation 7. Transport phenomena and metals properties  Solidification Solidification is the formation of a solid from a liquid. Solidification and melting are transformations between crystallographic and noncrystallographic states of a metal or alloy. The conditions necessary for solidification Solidification generally occurs during cooling but can also occur at constant temperature by chemical change of the system.  For example, if a pure liquid metal, which is initially at a temperature above its melting point, is cooled continuously, at some temperature below its equilibrium melting point, solid will form. The temperature at which solidification actually occurs (Ti) when subtracted from the equilibrium melting temperature (Tm), is the undercooling (ΔT) necessary for the initiation of solidification.  Solidification  Similarly, if a liquid binary solution is held at constant temperature and its chemistry is changed by adding or removing one of the constituents, at some point, when the equilibrium solubility for the constituent is exceeded, solid will be precipitated. In this case, we will refer to solidification as being from a supersaturated liquid The initiation of solidification normally occurs by 1. Reducing temperature below the position of the equilibrium liquidus (from A to B) 2. Increasing the composition at constant temperature past the equilibrium liquidus (from A to C) Figure: Schematic of a solidification process in a binary solution  The Kinetics of Phase Transformations With phase transformations, normally at least one new phase is formed that has different physical/chemical characteristics and/or a different structure than the parent phase. Furthermore, most phase transformations do not occur instantaneously. Rather, they begin by the formation of numerous small particles of the new phase(s), which increase in size until the transformation has reached completion. The progress of a phase transformation may be broken down into two distinct stages:  Nucleation  Growth.  The Kinetics of Phase Transformations Nucleation involves the appearance of very small particles, or nuclei of the new phase (often consisting of only a few hundred atoms), which are capable of growing. During the growth stage, these nuclei increase in size, which results in the disappearance of some (or all) of the parent phase. The transformation reaches completion if the growth of these new-phase particles is allowed to proceed until the equilibrium fraction is attained. There are two types of nucleation: The distinction between them is made according to the site at which nucleating events occur.  Homogeneous  Heterogeneous  The Kinetics of Phase Transformations Homogeneous nucleation occurs when nuclei of the new phase form uniformly throughout the parent phase. Heterogeneous nucleation occurs when nuclei form preferentially at structural inhomogeneities, such as container surfaces, insoluble impurities, grain boundaries, dislocations, and so on. Homogeneous Nucleation: the of the theory of nucleation involves a thermodynamic parameter called free energy G. Relative to phase transformations, an important thermodynamic parameter is the change in free energy ∆G; a transformation occurs spontaneously only when ∆G has a negative value.  The Kinetics of Phase Transformations For the sake of simplicity, let us first consider the solidification of a pure material, assuming that nuclei of the solid phase form in the interior of the liquid as atoms cluster together so as to form a packing arrangement similar to that found in the solid phase Furthermore, it will be assumed that each nucleus is spherical and has a radius r. Figure: Schematic diagram showing the nucleation of a spherical solid particle in a liquid.  The Kinetics of Phase Transformations There are two contributions to the total free energy change that accompany a solidification transformation. The first is the free energy difference between the solid and liquid phases, or the volume free energy, ∆Gv. Its value is negative if the temperature is below the equilibrium solidification temperature, and the magnitude of its contribution is the product of ∆Gv and the volume of the spherical nucleus (i.e., 4/3∏r3). The second energy contribution results from the formation of the solid–liquid phase boundary during the solidification transformation. Associated with this boundary is a surface free energy, γ, which is positive; furthermore, the magnitude of this contribution is the product of γ and the surface area of the nucleus (i.e., 4 ∏r2). Finally, the total free energy change is equal to the sum of these two contributions ……..equation 1 The Kinetics of Phase Transformations a) Schematic curves for volume free energy and surface free energy contributions to the total free energy change attending the formation of a spherical embryo/nucleus during solidification. (b) Schematic plot of free energy versus embryo/nucleus radius, on which is shown the critical free energy change (G*) and the critical nucleus radius (r*).  The Kinetics of Phase Transformations As a solid particle begins to form as atoms in the liquid cluster together, its free energy first increases. If this cluster reaches a size corresponding to the critical radius r*, then growth will continue with the accompaniment of a decrease in free energy. However, a cluster of radius less than the critical value will shrink and redissolve. This subcritical particle is an embryo, and the particle of radius greater than r* is termed a nucleus. A critical free energy, G*, occurs at the critical radius and, consequently, at the maximum of the curve in above Figure b. This G* corresponds to an activation free energy, which is the free energy required for the formation of a stable nucleus. Equivalently, it may be considered an energy barrier to the nucleation process. Critical Radius(r*) : The radius of nuclei at maximum ∆G.  The Kinetics of Phase Transformations Because r* and G* appear at the maximum on the free energy-versus-radius curve of Figure b, derivation of expressions for these two parameters is a simple matter. For r*, we differentiate the G equation with respect to r, set the resulting expression equal to zero, and then solve for r ( r*). That is, ……..equation 2 which leads to the result ……..equation 3  The Kinetics of Phase Transformations Now, substitution of this expression for r* into Equation 1 yields the following expression for G*: ……..equation 4 This volume free energy change Gv is the driving force for the solidification transformation, and its magnitude is a function of temperature. At the equilibrium solidification temperature Tm , the value of Gv is zero, and with decreasing temperature its value becomes increasingly more negative.  The Kinetics of Phase Transformations It can be shown that Gv is a function of temperature as ……..equation 5 where Hf is the latent heat of fusion (i.e., the heat given up during solidification), and Tm and the temperature T are in Kelvin. Substitution of this expression for Gv into Equations 3 and 4 yields. ……..equation 6 ……..equation 7  The Kinetics of Phase Transformations Thus, from these two equations, both the critical radius r* and the activation free energy G* decrease as temperature T decreases. (The ϒ and Hf parameters in these expressions are relatively insensitive to temperature changes.) Physically, this means that with a lowering of temperature at temperatures below the equilibrium solidification temperature (Tm), nucleation occurs more readily. Figure: Schematic free energy-versus-embryo/nucleusradius curves for two different temperatures. The critical free energy change (G*) and critical nucleus radius (r*) are indicated for each temperature.  The Kinetics of Phase Transformations Furthermore, the number of stable nuclei n* (having radii greater than r*) is a function of temperature as – where the constant K1 is related to the total number of nuclei of the solid phase. For the exponential term of this expression, changes in temperature have a greater effect on the magnitude of the G* term in the numerator than the T term in the denominator. Consequently, as the temperature is lowered below Tm, the exponential term in above equation above also decreases, so that the magnitude of n* increases.  The Kinetics of Phase Transformations Heterogeneous Nucleation: smaller degree of supercooling (ΔT) is required for heterogeneous nucleation when compared with homogenous nucleation. The reason for this is that the activation energy (i.e., energy barrier) for nucleation (G* of Equation 4) is lowered when nuclei form on preexisting surfaces or interfaces, because the surface free energy (ϒ of Equation 4) is reduced. In other words, it is easier for nucleation to occur at surfaces and interfaces than at other sites. This type of nucleation is termed heterogeneous.  The Kinetics of Phase Transformations In order to understand this phenomenon, let us consider the nucleation, on a flat surface, of a solid particle from a liquid phase. It is assumed that both the liquid and solid phases “wet” this flat surface. Also noted in the figure are three interfacial energies (represented as vectors) that exist at two-phase boundaries—ϒSL, ϒSI, and ϒIL—as well as the wetting angle ϴ (the angle between the ϒSI and ϒSL vectors). Taking a interfacial energy force balance in the plane of the flat surface leads to the following expression: Figure: Heterogeneous nucleation of a solid from a liquid.  The Kinetics of Phase Transformations Now, using a somewhat involved procedure similar to the one presented for homogeneous nucleation (which we have chosen to omit), it is possible to derive equations for r* and G*; these are as follows: The S(ϴ) term of this last equation is a function only of ϴ(i.e., the shape of the nucleus), which has a numerical value between zero and unity 1. From above equation, it is important to note that the critical radius r* for heterogeneous nucleation is the same as for homogeneous nucleation, in as much as ϒSL is the same surface energy.  The Kinetics of Phase Transformations It is also evident that the activation energy barrier for heterogeneous nucleation is smaller than the homogeneous barrier by an amount corresponding to the value of this S(ϴ) function, or  This lower G* for heterogeneous nucleation means that a smaller energy must be overcome during the nucleation process (than for homogeneous nucleation), and, therefore, heterogeneous nucleation occurs more readily. Figure Schematic free energy-versus-embryo/nucleus radius plot on which are presented curves for both homogeneous and heterogeneous  The Kinetics of Phase Transformations In terms of the nucleation rate, the N -versus-T curve is shifted to higher temperatures for heterogeneous. This effect is represented in Figure below, which also shows that a much smaller degree of supercooling (T) is required for heterogeneous nucleation. Figure: Nucleation rate versus temperature for both homogeneous and heterogeneous nucleation.  The Kinetics of Phase Transformations Growth The growth step in a phase transformation begins once an embryo has exceeded the critical size, r*, and becomes a stable nucleus. – Note that nucleation will continue to occur simultaneously with growth of the new- phase particles; of course, nucleation cannot occur in regions that have already transformed into the new phase. Furthermore, the growth process will cease in any region where particles of the new phase meet because here the transformation will have reached completion. Particle growth occurs by long-range atomic diffusion, which normally involves several steps for example, diffusion through the parent phase, across a phase boundary, and then into the nucleus.  The Kinetics of Phase Transformations Consequently, the growth rate G is determined by the rate of diffusion, and its temperature dependence is the same as for the diffusion coefficient namely, Where, Q (the activation energy) C (a pre exponential) are independent of temperature. Now, at a specific temperature, the overall transformation rate is equal to some product of N and G. The third curve of Figure below, which is for the total rate, represents this combined effect. The general shape of this curve is the same as for the nucleation rate, in that it has a peak or maximum that has been shifted upward relative to the N curve.  The Kinetics of Phase Transformations Figure Schematic plot showing curves for nucleation rate (N), growth rate (G), and overall transformation rate versus temperature.