1-Transformación de fases

Ingeniería de Materiales Cerámicos I

Capitulo 1

Phase Transformations, Glass Formation, and Glass-Ceramics


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3.2 Kinetic Processes In Ceramics And Glasses3.2.1 Nucleation and Growth

• Nucleation and growth kinetics are composed of two separate processes.

• Nucleation is the initial formation of small particles of the product phase from the parent phase.

• The resulting nuclei are often composed of just a few molecules.

• The growth step involves the increase in size of the nucleated particles.

• Many natural phenomena follow nucleation and growth kinetics, such as ice formation.

Edwin J. Urday U.


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• Unlike martensitic transformations, both spinodal decomposition and nucleation and growth processes involve diffusion, but there are some subtle differences.

• Figure 3.11 shows a comparison based on composition and spatial extent between a spinodal transformation and a nucleation and growth transformation.

• Spinodal transformations are large in spatial extent but involve relatively small concentration differences throughout the sample, whereas the nucleation process involves the formation of small domains of composition very different from the parent phase, which then grow in spatial extent via concentration-gradient-driven diffusion.

Edwin J. Urday U.


4Edwin J. Urday U.

Figure 3.11 Schematic comparison of dimensional changes that occur in (a) spinodal and(b) nucleation and growth transformation processes.



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• Let us first consider the liquid-solid phase transformation.

• At the melting point (or more appropriately, fusion point for a solidification process), liquid and solid are in equilibrium with each other.

• At equilibrium, we know that the free energy change for the liquid-solid transition must be zero.

• The equation for this situation is

Edwin J. Urday U.


(3.31)


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• where – ΔGv is the free energy change per unit volume for fusion,

– ΔHf is the enthalpy of fusion and is related to the strength of interatomic or intermolecular forces of the solid,

– ΔSf is the entropy of fusion, representing the degree of disorder in the solid, and

– Tm is the melting point.

• We can solve Eq. (3.31) for the entropy change of fusion:

Edwin J. Urday U.


(3.32)


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• If we assume that the entropy and enthalpy are relatively independent of temperature, we can drop the subscript “m” on temperature in Eq. (3.32), thereby obtaining a more general relationship between the entropy and enthalpy of fusion as a function of temperature.

• Substitution of this modified form of Eq. (3.32) into Eq. (3.31) gives the following relation for the free energy:

• This expression will be very useful in the discussion of nucleation and growth kinetics.

Edwin J. Urday U.

(3.33)



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• In the nucleation step, there must be sites upon which the crystals can form.

• This is similar to “seeding” the clouds to cause water to precipitate (rain).

• There are two sources for these nucleating particles: homogeneous and heterogeneous agents.

Edwin J. Urday U.



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• In homogeneous nucleation, critical nuclei (nuclei with enough molecules to initiate growth) form without the aid of a foreign agent.

• This happens very rarely in practice, but is useful for theoretical discussions.

• Consider the formation of a small, solid, spherical particle from the parent (liquid) phase.

• There is a free energy change, ΔG, associated with this process.

Edwin J. Urday U.

3.2.1 Nucleation and Growth3.2.1.1 Homogeneous Nucleation


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• Much like the potential energy function between two atoms, there are two competing forces that contribute to the overall free energy change:

(1) the free energy change due to a volume change (a negative contribution) and

(2) the free energy change due to formation of an interphase (a positive contribution).

• These forces are quantified in Eq. (3.34):

Edwin J. Urday U.


(3.34)


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• where

– ΔGv is the free energy change per unit volume for the phase change (which must be negative);

– r is the nucleated particle radius;

– γ is the interfacial energy (often called “surface tension” for a liquid-gas interface), which has units of energy/area (erg/cm2) or force/length (dyne/cm).

• The first term in Eq. (3.34) is the free energy change associated with the volume change, and the second term is the free energy change associated with the interphase formation.

Edwin J. Urday U.



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• As with any activated process, there is a free energy barrier, ΔG∗, or “critical free energy” that must be surmounted in order to nucleate a particle.

• This is illustrated in Figure 3.12 as a plot of free energy versus the particle size, r.

• Once the overall free energy change decreases past ΔG∗, the nucleation process becomes spontaneous.

• This occurs as enough molecules gather to form a nucleus with critical radius, r∗.

• We can it equal to zero (the slope of ΔG vs. r in Figure 3.12 is zero at ΔG = ΔG∗), and replacing r by r∗ to get

Edwin J. Urday U.



13Edwin J. Urday U.


Figure 3.12 Representation of activation energy barrier for nucleation.


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• The height of the activation energy barrier, ΔG∗, can now be found by evaluating Eq. (3.34) at r by r∗ with the aid of Eq. (3.35)

• To summarize the nucleation stage, then, particles that achieve a critical radius r > r∗ may enter the growth stage of this phase transformation process.

Edwin J. Urday U.


(3.35)

(3.36)


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Cooperative Learning Exercise 3.3

Work with a neighbor. Consider Eq. (3.34), which describes the free energy change associated with the homogeneous nucleation of a solid, spherical particle of radius, r, from a parent liquid phase:

Recall that this equation could be minimized with respect to particle radius to determine the critical particle size, r , as given by Eq. (3.35). This critical radius could ∗then be used to determine the height of the free energy activation energy barrier, ΔG∗, as given by Eq. (3.36). A similar derivation can be performed for a cubic particle with edge length, a.

Person 1: What is the equivalent expression for the first term in Eq. (3.34) for the cube? Recall that ΔGv is the free energy change per unit volume.

Edwin J. Urday U.



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Cooperative Learning Exercise 3.3

Person 2: What is the equivalent expression for the second term in Eq. (3.34) for the cube? Recall that γ is the interfacial energy for the liquid-solid interface.

Combine your results to arrive at an equivalent expression for Eq. (3.34) for the cube. Minimize this equation to determine the critical cube length, a . Substitute this value ∗back into your expression to determine the critical free energy barrier for the cube, ΔGv

∗. Why is ΔGv∗ higher for a cube than for a sphere? What does this tell you about

the likelihood of cubes nucleating rather than spheres?

• Answer: a3Gv; 6a2γ ; a∗ = −4γ/Gv; Gv∗ = 32γ3/Gv

2. The barrier is lower for spherical particles so their formation is more likely.

Edwin J. Urday U.



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• In heterogeneous nucleation, the critical nuclei form with the assistance of nucleating sites.

• These sites can be impurities, container walls, cracks, or pores and may either be naturally present or intentionally added to promote nucleation.

• The effect of nucleating agents is to decrease ΔG∗.

• Unlike homogeneous nucleation, heterogeneous nucleation involves an interface between two compositionally different materials, so we must account for the interaction of the parent phase with the nucleating particle.

Edwin J. Urday U.

3.2.1 Nucleation and Growth3.2.1.2 Heterogeneous Nucleation


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• This is accomplished through the introduction of the contact angle, θ, at the three-phase interphase of the parent liquid, its solid phase, and the nucleating agent (see Figure 3.13).

• Without derivation, we present the free energy barrier for heterogeneous nucleation, , as

• Where ΔG ∗ is the free energy barrier for homogeneous nucleation and the subscript “s” onindicates heterogeneous or “surface” nucleation.

Edwin J. Urday U.


(3.37)


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Figure 3.13 Schematic diagram of contact angle formed at three phase intersection during heterogeneous nucleation.


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• We can now define an overall nucleation rate:

• where

• is the number of critical nuclei that form per unit volume per unit time,

• ns is the number of molecules in contact with critical nucleus,

• n∗ is the number of critical size clusters per unit volume, and

• ν is the collision frequency of single molecules with the nuclei (see Figure 3.14).

Edwin J. Urday U.

3.2.1 Nucleation and Growth3.2.1.3 Nucleation Rate

(3.38)


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Figure 3.14 Schematic illustration of a nucleation site formation.



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• The number of critical nuclei, n∗ can be related to the free energy barrier, ΔG∗, through an Arrhenius-type expression:

• where n0 is the number of single molecules per unit volume, kB is Boltzmann’s constant, and T is absolute temperature.

• Similarly, the collision frequency, ν, can be expressed by an Arrhenius relation:

Edwin J. Urday U.

(3.39)

(3.40)



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• where ν0 is the molecular jump frequency, and ΔGm is the activation energy for transport across the nucleus-matrix interface, which is related to short-range diffusion.

• Putting these expressions back into Eq. (3.38) gives

• For heterogeneous nucleation, we simply use instead of ΔG∗:

Edwin J. Urday U.

(3.41)

(3.42)



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• Though Eqs. (3.41) and (3.42) are somewhat cumbersome, they describe the rate at which nuclei form as a function of temperature.

• Note that in both equations, ns and n0 have dimensions of number per unit volume, and ν0 has units of time-1.

• What would be more useful are forms of Eqs. (3.41) and (3.42) that contain some measurable, physical properties of the system.

• Free energies hardly fit this description.• The danger in doing this is that we have to make some

simplifiºcations and assumptions.

Edwin J. Urday U.



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• Let us begin by recalling that the free energy for homogeneous nucleation is given byEq. (3.36):

• Recall also that for liquid-solid transitions at temperatures close to Tm, ΔGv is given by Equation (3.33):

• where ΔHv is the heat of transformation per unit volume.

Edwin J. Urday U.

(3.36)

(3.33)



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• Substitution of Eqs. (3.33) and (3.36) into the first exponential in Eq. (3.42) gives us the temperature dependence of the nucleation rate for homogeneous nucleation:

• While this may appear even more cumbersome than Eq. (3.41), it contains some parameters that are directly measurable such as the interfacial surface energy, γ, and the heat of fusion, ΔHv, but more importantly, it contains the temperature difference (Tm - T), which is the degree of undercooling.

Edwin J. Urday U.

(3.43)



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• The two exponentials in Eq. (3.43) compete against each other. As T decreases below the melting point:– (Tm - T ) becomes larger (remember that this term is squared, so

it is always a positive value), the term inside the first exponential becomes smaller, but because it is negative, the entire exponential gets bigger and the nucleation rate increases due to this exponential - the driving force for nucleation becomes greater.

– The term in the second exponential becomes more negative, the exponential gets smaller and the nucleation rate decreases due to this exponential term - diffusion becomes more difficult as the temperature decreases and particles cannot migrate to the nucleation surface.

Edwin J. Urday U.



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• These generalizations are true for both homogeneous and heterogeneous nucleation.

• As a result, we would expect that the two competing exponentials give rise to a maximum nucleation rate at some temperature below the melting point, and this is indeed the case, as illustrated in Figure 3.15.

• It is also logical that heterogeneous nucleation has a higher absolute nucleation rate than homogeneous nucleation and that heterogeneous occurs at a higher temperature (lower degree of undercooling) than homogeneous nucleation.

Edwin J. Urday U.



29Edwin J. Urday U.


Figure 3.15 Effects of temperature and undercooling on homogeneous and heterogeneous nucleation rates.


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• The second portion of nucleation and growth is the growth process.

• Here the tiny nuclei grow into large crystals through the addition of molecules to the solid phase.

• There are two primary types of crystal growth: – thermally activated (diffusion controlled) and – diffusionless (martensitic).

• The thermally activated crystal growth will be considered here.

Edwin J. Urday U.

3.2.1 Nucleation and Growth3.2.1.4 Growth


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• The development of the proper growth rate expression is highly dependent upon the type of phase transformation - that is, crystallization from the melt, vapor phase, or dilute solution.

• We will simply use a general form of the growth rate expression which is based upon an Arrhenius-type expression:

• where is the growth rate, ΔG is the molar free energy difference between product and parent phase, and A is a pre-exponential factor.

Edwin J. Urday U.


(3.44)


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• The form of the pre-exponential factor, A, depends on the type of theory one wishes to employ.

• It can be directly related to the liquid phase viscosity, as in A = νa0, where ν is the frequency factor for transport across the interphase

• in which – η is the viscosity of the liquid phase, and

– a0 is the molecular diameter;

Edwin J. Urday U.


(3.45)


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• or A can be related to the diffusivity of atoms across the interphase, as in A = KD, where K is a constant and the diffusivity of atoms jumping across interphase, D, is given by

• These two expression are actually quite similar since D and η are interrelated.

• Other expressions exist as well for such phenomena as surface growth and screw dislocation growth.

Edwin J. Urday U.


(3.46)


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• Finally, the overall transformation rate, is given by the product of the nucleation and growth expressions [Eqs. (3.41) or (3.42), and (3.44), respectively):

• The overall transformation rate is shown qualitatively in Figure 3.16.

• Notice that the maximum nucleation rate occurs at a lower temperature than the maximum growth rate, and that the maximum transformation rate may not be at either of these two rate maxima.

Edwin J. Urday U.

3.2.1 Nucleation and Growth3.2.1.5 Overall Phase Transformation Rate

(3.46)


35Edwin J. Urday U.


Figure 3.16 Schematic representation of transformation rates involved in crystallization by nucleation and growth kinetics.


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• Note also that there is some finite transformation rate, even at very low temperatures.

• This is why some glasses can crystallize over very long periods of time.

• As long as there is some molecular motion, there is a probability of crystallization taking place.

• Finally, it should be pointed out that the breaking down of the liquid to solid phase transformation into two separate steps, nucleation and growth, is not entirely artificial.

• An interesting application of the distinct nature of the nucleation and growth processes is found in the formation of glass ceramics.

Edwin J. Urday U.



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• As shown in Figure 3.17, certain glass-forming inorganic materials can be heat-treated in very controlled ways in order to affect the structure of the crystals that form.

• For instance, a glass can be rapidly cooled below its melting point, then heated to the maximum nucleation rate, which (recall) is below both the melting point and the maximum growth rate.

• At this temperature and with sufficient time, many small nuclei will be formed.

• The glass, now replete with many tiny nuclei, is heated to the maximum growth rate temperature and held there while the crystals grow.

Edwin J. Urday U.



38Edwin J. Urday U.


Figure 3.17 Schematic time-temperature cycle for the controlled crystallization of aglass-ceramic body.


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• The crystals can only grow so large, though, since they soon run into another growing crystal from a neighboring nucleation site.

• In this way, the size of the crystals, or crystallites, in the glass can be controlled, and a glass ceramic is formed.

• Glass ceramics are unique in that they are crystalline materials, yet they are transparent in many cases.

• They also possess unique physical properties, such as low thermal expansion, due to the large amount of interphase relative to the same material made of larger crystals.

Edwin J. Urday U.



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• In the extreme, crystallites on the order of 10-9 meters in diameter can be formed, resulting in so-called nanostructured ceramics.

• These materials, like glass ceramics, have unique physical properties, which may one day lead to improved ductility and thermal properties in ceramics.

Edwin J. Urday U.


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