10. Phase Transformations in Solids Aug 2011/Chapter10_Phase... · order-disorder and amorphization...

Light Water Reactor Materials © Donald Olander and Arthur Motta 7/20/2011

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10. Phase Transformations in Solids 10.1 Introduction............................................................................................................... 1 10.2. Thermodynamics of Phase Changes....................................................................... 1

10.2.1 Driving Force for Phase Transformations....................................................... 1 10.2.2. Phase Diagrams and their relation to Free energy curves ............................ 3 10.2.3. Examples of Important phase diagrams ......................................................... 8

10.3. Phase Nucleation ...................................................................................................... 8 10.3.1. Homogeneous nucleation.................................................................................. 9 10.3.2. Heterogeneous Nucleation.............................................................................. 11

10.4. Phase Transformation Kinetics ............................................................................ 13 10.4.1 Nucleation and growth: Johnson-Mehl-Avrami Kinetics ............................ 13 10.4.2. Diffusion controlled growth ........................................................................... 16 10.4.3. Transformation-Time-Temperature (TTT) diagrams ................................ 18 10.3.4. Phase Transformation Kinetics TTT diagrams ........................................... 18

Spinodal decomposition ............................................................................................ 19 10.4.4. Precipitate loss of coherency .......................................................................... 19 10.4.5. Precipitate coarsening-Ostwald Ripening .................................................... 20

10.5. Diffusionless Phase Transformations ................................................................... 21 10.5.1 Martensitic transformation............................................................................. 22

10.7. Order-Disorder Transformations......................................................................... 22 10.8 Amorphization......................................................................................................... 26 Problems .......................................................................................................................... 28 References ........................................................................................................................ 29


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10.1 Introduction A solid system of a given chemical composition can exist in a variety of chemical and crystallographic arrangements or phases. A phase is defined as a homogeneous region in a liquid or a solid, with a characteristic atomic arrangement and chemical composition1. Depending on the thermodynamic variables (temperature, pressure, alloy composition, etc.) different phases or mixtures of phases are stable (the equilibrium state). Phase stability and phase transformations are crucial to alloy design, behavior, and performance. This chapter deals with the criteria for phase stability and with the processes of phase transformations in condensed matter. Phase transformations follow the principle of minimization of Gibbs free energy. This means the equilibrium product is always the combination of phases and compositions that minimizes the overall Gibbs free energy of the system for a particular set of thermodynamic variables. This final result is only guaranteed if one can wait an infinite amount of time, and in practice the heat treatments followed by real materials stop short of reducing the Gibbs free energy to its absolute possible minimum value. However, the overall trends of phase transformations follow the tendencies established by thermodynamics. For example, the most stable state of a crystalline solid would be a single crystal with no defects. This is a very difficult state to attain and most solids contain a certain proportion of lattice defects, including dislocations. Although heat treatments will not eliminate dislocations, their density will be reduced, so the heat treatment can be calibrated to produce the desired dislocation density. Similar control is possible with other microstructural features such as grain size, precipitate size and density, etc. Metallurgical engineers have been using this fact to fabricate materials with a variety of microstructures that can generate a range of the macroscopic properties of interest. Since the mechanisms of phase transformations affect the performance of nuclear materials, both in the as-fabricated state and during irradiation, they are of interest to us and will be reviewed in this chapter. The chapter is organized such that the thermodynamics of phase transformations are reviewed first. This is followed by a discussion of the processes by which the transformations take place, from nucleation and growth to spinodal phase transitions and the use of such knowledge to produce microstructures with favorable macroscopic properties through thermo-mechanical processing. The chapter ends with a discussion of order-disorder and amorphization phase transitions. . 10.2. Thermodynamics of Phase Changes2 10.2.1 Driving Force for Phase Transformations

1 Only condensed phases (liquids and solids) are considered. A gas phase is always present, but is ignored when dealing with condensed-phase equilibria 2 See also section 2.7


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As mentioned above and described in Chapter 2, the driving force for phase transformations is the minimization of the overall Gibbs free energy of the system. The Gibbs free energy G is given by

G H TS= − (10.1)

where H is the system enthalpy, S is the entropy and T is temperature. Phase transformations occur in the direction of decreasing Gibbs free energy, such that if a lower free energy state is available, a driving force exists to transform the system to this lower free energy state. As a consequence, the equilibrium state is achieved when the derivative of the Gibbs free energy with respect to all possible changes in parameter X is zero (equation 2.17).

0dGdX

⎛ ⎞ =⎜ ⎟⎝ ⎠

(10.2)

Temperature is the most common variable that can change the relative phase stability. Figure 10.1 shows schematically the variation of free energy for the solid and liquid phases near the melting point (Tm). Figure 10 1: Schematic plot of the free energy of the liquid (Liq) and two solid phases (Sol1 and Sol2) in a single component system. At high temperature the free energy of the liquid phase Liq (red curve) is lower than that of the solid phase Sol1 (blue curve) which means the liquid is the most stable phase. As the temperature is decreased, the free energy difference between the two phases decreases, such that at the melting temperature the two have the same Gibbs free energy. Lowering the temperature below Tm causes the solid to become the most stable phase and

G

T Tm

Sol1

Liq

Sol2


3

creates a driving force for solidification to occur. Note that the Sol1 phase is simply the most stable (lowest free energy) solid phase. In principle there are an infinite number of solid phases possible, (e.g. the Sol2 phase shown in Fig.10.1), but they have higher free energies than the stable solid and so do not form. The variation of G with temperature occurs because of increasing entropy. In fact (from equation 2.15)

p

G ST

∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠ (10.3)

Thus, as the temperature increases, phases with larger molar entropy tend to be favored over lower entropy phases. This is one of the reasons why liquids are more stable than solids at high temperature. As indicated in Eq (10.3), total pressure p is a thermodynamic variable. However, the free energies of condensed phases are largely insensitive to pressure because their molar volumes are very small, and from Eq (1.15):

T

G Vp

⎛ ⎞∂=⎜ ⎟∂⎝ ⎠

(10.4)

10.2.2. Phase Diagrams and their relation to Free energy curves Phase diagrams are maps (for solids, most often of temperature vs. composition) indicating which phases are present at equilibrium for the given conditions. In principle, phase diagrams can be deduced if the free energy-composition curves at various temperatures for all the possible phases in the system are known. If the thermodynamic quantities that are used to calculate the free energy are known experimentally, the phase diagrams can be calculated by one of many methods [1]. In practice, phase diagrams often originate rather from empirical data, obtained by producing mixtures of the components of the system at various temperatures and compositions and determining which phases are present, normally by x-ray diffraction and chemical analysis. Although most alloys contain more than two components, for simplicity only binary phase diagrams will be studied in this chapter. However, understanding of ternary and higher order phase diagrams can help guide the design of more complex systems. Figure 10.2 shows a plot of free energy versus composition for a binary system A-B showing free energy curves for two phases: alpha and beta. The free energy of the alpha phase shows a minimum in the A-rich region and the beta phase shows a minimum in the B-rich region. The free energy curve for the liquid GL is higher than either solid phase.


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According to Figure 10.2, in the A-rich region, the stable phase is pure alpha while in the B-rich region the stable phase is pure beta. In the in-between region, for example at the point x’ the lowest free energy curve is that of the alpha phase. This would indicate that pure alpha with composition x’ would be observed. However α(x’) is unstable with respect to a combination of alpha and beta at slightly different compositions. If it is possible to nucleate and grow the beta phase, and form it with by using B atoms from the alpha phase, then the B content of the alpha phase will decrease to form the beta phase and it is possible to reach compositions x1 and x2. In that case the overall free energy can be written:

'12 1 2 iniG M G M G Gα β= + < (10.5)

where Mα, Mβ are the mole fractions of alpha and beta, and G1 and G2 the molar free energy of alpha at composition x1 and beta at composition x2, respectively. Since

1M Mα β+ = (10.6) it is clear that the free energy of the alpha-beta mixture will be a linear combination of the free energies of the two phases and therefore will lie on the line G1 - G2. Using the lever rule (or phase rule section 2.5)

1 22 1 12

2 1 2 1

( ' ) ( ')( ) ( )

x x x xG G G Gx x x x

− −= + =

− − (10.7)

Thus since G12 is lower than G’ini, the pure alpha phase at composition x’ is unstable with respect to the mixture of alpha at composition 1 and beta at composition 2. This creates a driving force for phase beta to nucleate and for B atoms to get transferred from phase alpha to phase beta. This process can continue until the compositions xA and xB are reached. A mixture of alpha at xA and beta at xB provides the lowest overall free energy G’ that can be achieved for the binary alloy shown in Figure 10.2 at composition x’.


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Figure 10 2: Schematic free energy versus composition for binary alloy A-B. The points xA and xB are found by taking the common tangent of the two free energy curves. For compositions in between the two, it is favorable for phase separation to occur. for x < xA, or x > xB single phase alpha and beta respectively are seen. We should note also that at equilibrium, the chemical potential of A atoms in the alpha and beta phases is the same. That is (see chapter 2),

AA

dGdx

αα μ= ; AA

dGdx

β βμ= and A Aα βμ μ= (10.8)

That means that there is no driving force for an A atom in either phase to move to the other, (since the chemical potential of A is identical in the alpha and beta phases) and the system is at equilibrium. The net result is that at a given temperature the equilibrium phases are those that would be predicted by the lower envelope of the possible free energy curves, if it is considered that such envelope contains the common tangents such as xA -xB shown in Figure 10.2. The above derivation allows us to relate the free energy variation with composition to the free energy-composition phase diagram. In Figure 10.2, which was sketched for a fixed temperature, the single alpha stability domain ranges from 0 to xA, the two-phase domain from xA to xB and single beta from xB to 1.

α β

G

Pure B

Gini’

xA xB x'x1 x2

G1 G2

G12

G’

GL

Pure A


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Figure 10 3: Schematic phase diagram of an eutectic A-B system Figure 10.3 shows how this is translated into a phase diagram. At temperature T1, the two phase domain is shown to extend from xA to xB, with single phase alpha and beta on the sides. As the temperature increases, the free energy of the liquid decreases relative to the alpha and beta phases and at temperature T2 the three curves exhibit a common tangent.

Pure A Pure B

T1

T2

T3

T4

T5

α β

L

α+β

L+β α+L

a b

c d e

f g h i

j k

l

xA xB

1

2

xB2


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Figure 10 4: Schematic Free energy curves corresponding to the temperatures shown in Figure 10.3. At temperature T2 the three phases are in equilibrium. The lowest temperature and composition (in this case T2 and point d) at which the liquid is stable are called the eutectic temperature and composition. As the temperature increases further there are now two two-phase regions in addition to the three single phase domains. In addition to the pure α and pure β regions at the edges there is now a pure liquid phase near equiatomic composition. In between these phases there are two regions of two-phase equilibria, α + L and β+L (dotted lines at temperature T3 in Figure 10.4). Finally at temperature T4 the L free energy curve is lower than free energy curve of the α−phase throughout the entire compositional range. This temperature then corresponds to the melting temperature of the alpha phase containing 100% A. At temperature T5 the same occurs to the beta phase. Order of Phase transformations Finally phase transformations can be classified by their order. A phase transformation is first order if during the phase transformation in the change in free energy is zero but the first derivative of the free energy is discontinuous.

0,

≠⎟⎠⎞

⎜⎝⎛

∂∂

ΔPTE

TG (10.9)

T2 T3

T4 T5

c d e g

f

h i

k l

j

m


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For a second order phase transformation the discontinuity is on the second derivative of the free energy with temperature. This implies that the first order phase transitions have a latent heat of transformation, i.e. energy needs to be supplied at constant temperature for one phase to transform into another. Thus the phases coexist once the transformation is under way. In a second order phase transition, one phase changes into the order continuously. For example, the phase transformation illustrated in Figure 10.1 is first-order because the free energy is continuous across the transformation, but the first derivative is not. 10.2.3. Examples of Important phase diagrams In analyzing the materials for the core of a nuclear reactor, some phase diagrams will be particularly useful, notably those dealing with the alloys used for fuel cladding and structural materials and those related to the fuel.

Figure 10.5: Examples of Binary phase diagrams of importance to the nuclear industry: Zr-Cr, Zr-Fe, Zr-Nb and Fe-C 10.3. Phase Nucleation


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The first part of this chapter dealt with the driving forces for phase transformation, which govern the direction of a phase transformation. Once this is established it is necessary to address how the transformation will take place and at what speed. Phase transformations may require compositional changes and large scale transfer of matter. In most instances such transfer is accomplished by atomic diffusion. When no change in composition the transformation can occur in a displacive manner, in which the atoms move a small distance (within the unit cell) in coordinated fashion. For phase transformations in which the two phases coexist, it is necessary to create a nucleus of the new phase (or nucleate the new phase). This process is described in the following section. 10.3.1. Homogeneous nucleation In homogeneous phase transformations, the new phase appears without the aid of any catalysts or defects. In heterogeneous phase transformations, such defects, such as grain boundaries, grain junctions, dislocations, or free surfaces can aid the formation of a new phase. This is because in replacive phase transformations the formation of a new phase normally entails an energy cost associated with the formation of a new surface between the nucleus of the new phase and the matrix it is embedded on. This cost has to be balanced by the change in free energy associated with the formation of the new (more stable) phase. From this idea we can derive the theory of homogeneous nucleation. Consider a precipitation reaction such as should occur when we move from point 1 to point 2 in Figure 10.4. The most stable combination of phases in point 1 is a solid solution of element A in the B-rich beta phase. When the temperature is lowered from T1 to T2, the most stable combination of phases is a solid solution of A in beta at composition xB2 and B in A at composition xA2. Since composition xA2 is much richer in A than xB2, a nucleus of the A-rich alpha phase has to be formed from atoms previously dissolved in the beta phase. Such a nucleus can be formed by statistical thermal fluctuations, that is, random clustering of A atoms during thermal motion. If such a nucleus appears, the consequent free energy change ΔG is v elastG V G A V GγΔ = − Δ + + Δ (10.10) where ΔGv is the free energy change per unit volume caused by the phase transformation V is the volume of the nucleus, A is its surface area, γ the free energy change per unit area of this nucleus and ΔGs is the misfit elastic strain energy caused by the difference in volume of the two phases. Because the creation of a new surface is energetically costly, the nucleus is often spherical. Other considerations (such as the variation of elastic properties, surface energies and epitaxy with crystallographic orientation) can cause nuclei of different shapes and morphologies to form. The free energy change caused by the formation of a spherical nucleus is


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γππ 23

4)(3

4 rGGrG elastv +Δ−Δ−=Δ (10.11)

where r is the radius of the nucleus of the new phase. Equation (10.11) is plotted schematically in figure 10.6.

Figure 10 6: Free energy as a function of nucleus size The expression in equation (10.11) is positive for small values of r and eventually becomes negative as r increases. The point at which further increases in r cause a decrease in free energy is the point where the nucleus of the new phase becomes stable. This is called the critical radius rcrit and is found by setting

0=Δdr

Gd (10.12)

and solving for r, yielding

)(2

elastvcrit GG

rΔ−Δ

=γ (10.13)

The critical radius can then be inserted into equation (10.11) to find the critical value of the free energy change necessary for nucleation of a critical radius:

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

0.00 0.50 1.00

ΔG

r rc


11

2

3

)(316

elastvcrit GG

GΔ−Δ

=Δπγ (10.14)

As shown in Fig.10.1, the free energy difference (or the driving force for the transformation) between the old and new phase is zero at the transformation temperature and increases as we depart from that temperature (superheating or undercooling). Thus, as the driving force increases, the critical radius decreases. 10.3.2. Heterogeneous Nucleation In practice homogeneous nucleation hardly ever occurs. This is because real crystals have defects and impurities. It can be shown that nucleation at such imperfections (called heterogeneous nucleation) can lower the nucleation free energy and provide an easier path for phase transformation. For example, it is often observed that bubble nucleation when boiling water in a pot occurs at the bottom surface; by analogy, when casting metals, the mould can provide nucleation sites for the solid to form from the liquid phase. This can be modeled by imagining a spherical cap of the new phase forming on a wall, as shown in Figure 10.7.

Figure 10 7: Heterogeneous nucleation at a surface (after [2], as cited by [3].) The free energy change associated with forming a nucleus of the beta phase from the alpha in this geometry is given by

het S S S SG G V A A Aαβ β αβ αβ β β α αγ γ γΔ = Δ + + − (10.15) where the γij are the surface free energies per unit area for the three types of interfaces (αβ, βS and αS) and the Aij are the corresponding interfacial areas. The first term corresponds to the free energy decrease in transforming from alpha to beta, the second and third terms are the free energy of formation of the interfaces between the alpha and beta phases and between beta and the wall surface S. The last term is a reduction in the

θ β

α γ

γSα γSβ

S


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surface free energy resulting from the elimination of the α−S interface. We can write an equilibrium of forces at the edge of the nucleus, as indicated in figure 10.7:

cosS Sα β αβγ γ γ θ= + (10.16) Substituting equation (10.16) into (10.15) we obtain

( cos )het S S SG G V A Aαβ β αβ αβ β β αβγ γ γ γ θΔ = Δ + + − − (10.17) From geometry (i) the volume of the spherical cap is Vβ is 3 2(2 cos )(1 cos ) / 3rπ θ θ+ − , (ii) the area of the cap is 2πr2(1-cosθ) and (iii) the area of the wall surface-beta interface is πr2sin2θ, and we obtain

3 22 2 2(2 cos )(1 cos )

2 (1 cos ) sin cos3het

r GG r rαβ

αβ αβ

π θ θγ π θ γ π θ θ

Δ + −Δ = + − + (10.18)

and

324

4 ( )3het

r GG r fαβ

αβ

ππ γ θ

⎡ ⎤ΔΔ = +⎢ ⎥

⎢ ⎥⎣ ⎦ (10.19)

with

3(2 3cos cos )( )4

f θ θθ − += (10.20)

Because the function above lies between 0 and 1, heterogeneous nucleation diminishes the free energy barrier, that is, according to this model, het homG GΔ < Δ . It can be seen from figure 10.8 this function goes to 1 when theta goes to 180. This is reasonable since in that case we have a sphere immersed in a liquid without touching the surface and we should then recover the value of the homogeneous nucleation free energy. When theta goes to zero, the beta phase “wets” the wall surface and only a very small amount of energy is needed for nucleation, related to the edges of the new phase, which are neglected above.


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0

0.2

0.4

0.6

0.8

1

0 45 90 135 180

θ

f( θ)

Figure 10 8: Value of the function f(θ) as a function of θ The differentiation of equation (10.19) shows that the critical radius is given also by equation (10.13) that is het hom

crit critr r= . The free energy change for the formation of a critical radius is

3

2

16 ( )3( )

hetcrit

v elast

G fG G

πγ θΔ =Δ − Δ

(10.21)

Other surfaces or defects can also serve as nucleation sites. In particular grain boundaries and especially tripe joints can provide additional surface energy credit that can be used to lower the free energy required for phase nucleation. 10.4. Phase Transformation Kinetics 10.4.1 Nucleation and growth: Johnson-Mehl-Avrami Kinetics Once the new phase is nucleated it is of interest to study how the percent volume transformed will evolve with time. In a situation of nucleation and growth, the nuclei once formed can grow by absorbing a flux of atoms from the matrix. Since the growth of the interface depends on the sequential processes of atomic migration by solid state diffusion in the matrix followed by interfacial reaction, the solute


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concentrations profile may assume different shapes depending on which process is faster. Figure 10.2 illustrates such a process, in which a nucleus of phase β advances into phase α with velocity r . The velocity depends on a skin of surface dr and volume 4πr2 dr being formed on the surface of the spherical nucleus. This new beta volume has both the structure and composition of the beta phase, which is ( B BC Cβ α− ) richer in B than the alpha phase. Thus there needs to be transport of B atoms from the alpha phase to the beta phase for the layer to advance; such transport occurs by atomic diffusion. In the case where the atomic diffusion is fast compared to interfacial reaction, the flux can remain high without a significant solute gradient we end up with the flat concentrations profile (a) in Figure 10.9. In contrast when the interfacial reaction is fast, all atoms near the interface jump quickly and thus a solute depleted region appears in the matrix near the advancing interface and we obtain profile (b). Figure 10 9: Growth of one phase into another controlled by interfacial reaction or atom migration by diffusion. The triangle approximation to the concentrations profile used below is also shown If it is assumed that the conditions near the interface do not change significantly during the layer growth, it is possible to assume that the velocity r is constant. If we also assume that the nucleation rate N is constant, then beta phase nuclei will appear throughout the transformation time and the total volume transformed will be the sum of all the volumes of the nuclei transformed.

nf V= ∑ (10.22) where the volume for a nucleus formed at time t’ is

3 3 334 4 ( ( ')) 4 ( )( ) ( ')

3 3 3r r t t rV t t tπ π π−

= = = − (10.23)

β α

r

BC β

BCα

(b) BCαβ

r L


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Since the nucleation rate is constant, in a time interval dt’ a number of nuclei Ndt’ is formed and thus the fraction of volume fraction transformed (per unit total volume) df between t and t+dt’ is

334 ( ) ( ') '

3r Ndf t t dtπ

= − (10.24)

but this is only strictly true if the volume fraction is small so that the nuclei do not interfere with each other. As the transformed fraction increases, so does the probability that this interference will occur and thus the probability of finding untransformed material (1-f) must be factored into the equation:

334 ( )(1 ) ( ') '

3r Ndf f t t dtπ

= − − (10.25)

Integrating equation (10.25) we obtain

33

0

4 ( ) ( ') '(1 ) 3

tdf r N t t dtf

π= −

−∫ ∫ (10.26)

From this we obtain

3 41 exp3

f Nr tπ⎡ ⎤= − −⎢ ⎥⎣ ⎦ (10.27)

This is the Johnson-Mehl-Avrami equation, also often associated with Kolmogorov. Notice that no explicit assumption has been made of the growth mechanism except that the nucleation rate and growth rate are constant. For short times this equation reduces to

3 4

3f Nr tπ

= (10.28)

Other assumptions about the nucleation rate and the mobility lead to other forms of the more general equation

1 exp( )nf kt= − − (10.29)

with n a constant depending on the nucleation mechanism and k a factor that depends strongly on the temperature [4]. For example, for diffusion limited growth of spherical particles, n=3/2 [3]


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Equation (10.27) predicts transformation kinetics that are increasingly faster in the beginning but gradually slow down as the transformation proceeds, and thus show a characteristic S shape. Both the mobility r and the nucleation rate N are strongly dependent on temperature. The nucleation rate increases as the undercooling increases, since the driving force is greater, as shown in Figure 10.1. Conversely the mobility decreases as the temperature decreases, because of slower atomic diffusion, thus slowing down the phase transformation. This allows metallurgists to control the amount of phase transformed by changing the cooling rates, as explained in the next section.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 100 200 300 400 500

time (s)

f(t)

10 A/s

2 A/s

1 A/s

0.5 A/s

Figure 10 10: Fraction of phase transformed for various interfacial velocities in Å/s. The nucleation rate is taken to be 1018 nuclei/cm2/s. 10.4.2. Diffusion controlled growth If the advancement of the beta layer is controlled by the change in solute content between the alpha and beta phases and the necessity of supplying such an atomic flux, then we can write

( )B BCdr C C dr Jdtβ αΔ = − = (10.30) and so the rate of advancement of the layer is

( )B B

dr Jdt C Cβ α=

− (10.31)


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The flux of atoms is given by Fick’s law. If we make the approximation that ( ) ( )B B B BC C C Cβ α α αβ− − and approximating the diffusion profile by a triangle of sides ( )B BC Cα αβ− and L then

( )B BC CJ DL

α αβ− (10.32)

From conservation of mass, the amount of solute depleted at time t, has to be equal to the size of the layer, that is

( )( )2

B BB B

L C CC C rα αβ

β α −− = (10.33)

the layer advancement rate is then

2( )2 ( )( )

B B

B B B B

D C Cdrdt L C C C C r

α αβ

β αβ β α

−=

− − (10.34)

Integrating

( )2 ( )( )

B BB

B B B B

C Cr Dt C DtC C C C

α αβ

β αβ α αβ

−= = Δ

− − (10.35)

and finally

( )( )( )

B BB

B B B B

C Cdr D Dr Cdt t tC C C C

α αβ

β αβ β α

−= = = Δ

− − (10.36)

Thus for diffusion controlled processes, the layer thickness goes as Dt and the layer advancement rate decreases with time. We can now re-evaluate the fraction transformed using the same methodology as above assuming now that all nuclei are present at time zero

3 1.53 4 ( ) ( )4 ( )3 3

BC Dtrf ππ Δ= = (10.37)

The differential growth in time dt is

3 1.5 0.5(1 )2 ( ) ( )Bdf f N C D t dtπ= − Δ (10.38)


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Integrating

1.51 exp( )f tκ= − − (10.39) with

3 1.543BC NDπκ Δ

= (10.40)

10.4.3. Transformation-Time-Temperature (TTT) diagrams A convenient way to illustrate the fraction transformed in terms of the time elapsed at a certain temperature is given by the transformation-time-temperature diagrams. The diagrams are obtained by obtaining successive isothermal transformation curves and measuring the fraction transformed after an elapsed time t.

3 1.51.541 exp( )

3BC NDf tπΔ

= − − (10.41)

The argument of the exponential is proportional to 1.5ND . Once the temperature decreases below the transformation temperature Tab, the transformation can occur. At this temperature the diffusion coefficient D is the highest, however the nucleation rate is small, since the undercooling is also small. As the temperature decreases further the driving force for the transformation (and hence the nucleation rate) increases leading to a decrease in the time required to achieve a certain percent of volume transformed. 10.3.4. Phase Transformation Kinetics TTT diagrams In general for replacive transformations, the rate of growth of the new phase is controlled by a thermally activated process. This is because they depend for example on the motion of atoms that add to the nucleus of the new phase for the nucleus to grow. In general, the replacive phase transformations, depend on having enough time at temperature to realize the transition from a higher free energy state to a lower free energy state and the amount of phase transformed is proportional to exp(-ΔG/RT), where ΔG is the activation barrier for the transformation. This process is normally depicted in a TTT (time, temperature, transformation) diagram, as shown below. As the temperature is decreased from the melt the driving force for the transformation increases, but the mobility decreases, causing the nose observed in the figure. In principle, what is needed is a sum of heat treatments for a time t and temperature T such that the sum [time x exp(−ΔG/RT)] is enough to achieve a given transformation.


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Spinodal decomposition In the phase transformations described above, there is a nucleation barrier, and phase formation occurs by nucleation and growth. This mechanism is valid when the second derivative of the free energy with respect to composition is positive. When the second derivative of the free energy is negative, phase formation occurs by spinodal decomposition

2

2 0d GdX

< (10.42)

One such situation occurs when the free energy curve for a given phase exhibits a miscibility gap, such as shown in Figure 10.11 Figure 10 11: Schematic spinodal decomposition In the region between X1 and X2, it is energetically favorable for the alloy to separate into two regions with compositions X1 and X2. This transformation occurs by nucleation and growth in the region between X1 and '

1X , and X1 and '2X , where the second derivative is

positive. However between '1X and '

2X the fact that the second derivative is negative causes the alloy to be inherently unstable with respect to small variations in composition. As a result, the alloy decomposes until the final compositions of X1 and X2 are arrived at. 10.4.4. Precipitate loss of coherency

X1 X2 '1X '

2X


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As the precipitate of the new phase forms it normally starts out coherent with the matrix, i.e., there is an orientation relationship between the planes of the precipitate and the planes of the matrix atoms. This minimizes surface energy, but increases elastic strain energy. As the precipitate grows bigger it eventually becomes incoherent with the matrix, thus minimizing elastic strain energy and increasing surface energy from the creation of the new interface. 10.4.5. Precipitate coarsening-Ostwald Ripening Once a precipitate size distribution is present in a system. it is found that energetic considerations related to their smaller surface to volume ratio causes large precipitates to grow at the expense of smaller ones. This is the so-called Ostwald ripening process. This process leads to a cubic coarsening rate law, as is shown in the following derivation [5]. Once precipitation ends, the precipitate volume fraction remains mostly constant in equilibrium with an average solute content in the matrix C. In this scenario only the precipitate size distribution changes, causing the average precipitate size to increase. In that case

0i

i

dndt

=∑ (10.43)

where ni is the number of atoms in precipitate i and the sum is performed over all precipitates. If the atomic volume is Ω then for spherical precipitates

2

1,..

41 0i i i

i N

dV r drdt dt

π=

= =Ω Ω∑ (10.44)

where ri is the particle radius and N is the number of particles. The growth of the precipitate can be controlled by either solute diffusion or surface reaction. Since the latter is normally fast compared to the former, it is assumed that precipitate growth is diffusion controlled. In that case the diffusion controlled atom flux leaving the particle is given by

24 ( )isurf

dCJ r Ddr

π ⎛ ⎞= − ⎜ ⎟⎝ ⎠

(10.45)

where D is the diffusion coefficient, r is the radial coordinate and C is the solute concentration, and the derivative is calculated at the precipitate surface. The concentration at the curved surface is enhanced relative to a flat surface because as predicted by the Gibbs-Thomson equation [4]

21surfi B

C Crk T

γ⎡ ⎤Ω= +⎢ ⎥

⎣ ⎦ (10.46)

where Ω is the surface energy. The radial solute concentration gradient at the surface of a sphere of radius ri is


21

s o

surf i

C CdCdr r

−⎛ ⎞ =⎜ ⎟⎝ ⎠

(10.47)

where Co is the concentration far from the precipitate. Combining (10.47) and (10.45)and setting that equal to equation (10.44) written for a single particle

( )i S o

i

dr D C Cdt r

−Ω −= (10.48)

Substituting (10.48) into (10.44)

2( )S o ii B

C C r NCk T

γΩ− =∑ (10.49)

Noting that i

ir

rN

=∑

2 1 1( )o SB i

C C Ck T r r

γ ⎛ ⎞Ω− = −⎜ ⎟

⎝ ⎠ (10.50)

Substituting equation (10.50) into (10.48)

2

2

2 1i i

B i

dr rDCdt k Tr r

γΩ ⎡ ⎤= −⎢ ⎥⎣ ⎦ (10.51)

Note that this equation indicates precipitate growth when ir r> and shrinkage when it is smaller. Integrating and solving for the maximum radius

3 3max

26oB

CDr r tk TγΩ

− = (10.52)

Wagner and Lifshitz using a more sophisticated theory were able to derive the evolution of the mean particle size as [6, 7]

2

3 3 89o

B

CDr r tk TγΩ

− = (10.53)

Equation (10.53) shows that once a precipitate has become incoherent it will coarsen according to cubic kinetics. 10.5. Diffusionless Phase Transformations


22

At the other extreme of phase transformation from diffusion-controlled transformations are diffusionless phase transformations. In contrast with the diffusion controlled (or replacive) transformations, diffusionless transformations (also called displacive) require no long range transport of chemical species and thus can occur relatively quickly. Such transformations are also called martensitic transformations in honor of A. Martens who first described them in steels. Some of these transformations are technologically important, such as the martensitic transformation in steels and the monoclinic-to-tetragonal transformation in oxide layers formed during zirconium alloy corrosion. 10.5.1 Martensitic transformation It has been observed throughout the ages that quenching steel from the austenitic temperature domain to the ferritic temperature range, causes the steel to harden. This has been shown to occur by the formation of a metastable phase in steel called martensite. Martensite is a supersaturated sold solution of carbon in a-Fe (bcc-Fe or ferrite). The high rate of cooling involved in quenching does not allow enough time for the atoms to come out of solution, so that the material can decompose into ferrite and cementite (Fe3C). Thus martensite is a metastable phase and which does not appear in the Fe-C equilibrium phase diagram. Because carbon is an austenite stabilizing element, some austenite is retained upon quenching when the concentration of carbon is solid solution is high. The effect of martensite formation with its high carbon supersaturation is to increase the hardness of the material and to make it more brittle. As a result, tempering heat treatments are normally conducted to allow the carbon to partially come out of solution and for some cementite to form. As this occurs, the material becomes less hard and more ductile. By varying the specific heat treatment, different degrees of transformation and thus different levels of hardening and ductility can be achieved. The martensitic transformation has been the object of much study which is still ongoing and it is beyond the scope of this text to provide the detailed transformation mechanisms which have been discussed in other specialized books [3, 8]. It is clear however that this transformation occurs without change in chemical composition and by small atomic displacements within the unit cell. These atomic motions occur in a coordinated fashion so that the ensemble of atoms moves together in lock step. Because these transformations involve such regimented atomic rearrangements they have also been called military transformations, in contrast with the relatively disordered thermally activated nucleation and growth processes also called civilian transformations. Finally because no diffusion occurs, such movements have to take place by twinning or by dislocations motion 10.7. Order-Disorder Transformations Phase transformations in alloys can further be classified by whether they are clustering or ordering, that is, whether it is more favorable to form A-B bonds than A-A and B-B bonds. This is measured by the ordering energy:


23

2

AA BBAB

V Vw V += − (10.54)

where the Vs are bond energies. When w >0, clustering is favored, and when w<0 ordering is favored. An ordered compound has at least two sub-lattices, occupied by different chemical species. The degree of order in such materials can be measured by the two different parameters defined in Chapter 9, i.e. the long range order parameter S and the short range order parameter σ. The long range order parameter S is given by

1aa a

a

P xSx

−=

− (10.55)

where Paa is the probability of finding an atom of type a on an “a” sublattice site, and xa is the overall concentration of a in the alloy. In a perfectly ordered compound the probability of finding an “a” atom in an “a” site is 100% so Paa=1 and S=1. In a perfectly disordered compound the probability of finding an “a” atom at an “a” site is purely random (i.e. the atoms are evenly mixed in the lattice) and thus Paa = xa, so that S=0. The short range order parameter is given by

dis

ab abord disab ab

N NN N

σ −=

− (10.56)

where Nab is the number of a-b pairs in the compound, dis

abN is the number of a-b pairs in the perfectly disordered compound, and ord

abN is the number of a-b pairs in the perfectly ordered compound, resulting again in an order parameter of σ=1 for a perfectly ordered material and σ=0 for perfectly disordered material. Both parameters describe the state of chemical order of the material, but in general the long range order parameter is more accessible experimentally. The classical means of detection of long range order is by the presence of superlattice reflections, which are extinguished when the long-range order parameter goes to zero.


24

Figure 10.12. The structure of Cu3Au. The Cu and Au atoms together constitute a fcc lattice. The Au atoms occupy a simple cubic lattice on the corners with the Cu atoms on each face. When the compound is fully ordered, only Cu atoms are found on Cu sites, and only Au atoms on gold sites (see Figure 10.12). When the compound is fully disordered the probability of finding atoms in given lattice positions is given the compound stoichiometry. Thus the crystal structure changes symmetry from simple cubic with a basis of 4 (3Cu and 1 Au) to fcc with an average atom in each site. The particular atomic arrangement of the fcc lattice causes diffraction from some sets of planes to interfere destructively, and so some reflections are not allowed. For example, when the 100 planes are at the exact Bragg condition for reinforcement from successive planes, there is a plane in between with the exact same atom density and which by virtue of being half way in between successive 100 planes interferes destructively and causes a diffraction extinction[9]. This does not occur in the simple cubic structure, and thus as the material goes from disordered to ordered, many new (superlattice) reflections appear which can be used to monitor the degree of order. We should note that following the evolution of S from superlattice reflections is only possible in materials that change symmetry upon disordering as do the Cu3Au (simple cubic to face centered cubic) or CsCl structures (simple cubic to body centered cubic). To detect the changes in short range order is a more complicated task. Furthermore, it is possible to have almost no long range order, but still a great deal of short range order as described in the box. Box: Relationship between S and σ For a two sub-lattice ordered crystal it is not difficult to calculate the short range order parameter corresponding to a given long range order parameter. The number of a-b atom pairs in a crystal per sub-lattice site is given by


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ab aa bb ba abN P P Z P P Z= + (10.57) where Z is the number of nearest neighbors and the Pxy are the probabilities of finding atom x on sublattice y. From equation (10.55), these are given by

)1( aaaa xSxP −+= 1ba aaP P= − (10.58) );1( bbbb xSxP −+= bbab PP −= 1

substituting equations (10.58) into (10.57) and rearranging yields:

2

2 (2 1)2 (1 ) (1 )2

aab a a a a

xN Z x x S S x x⎡ ⎤−

= − + + −⎢ ⎥⎣ ⎦

(10.59)

For two sublattices with xa=0.5, this reduces to

21 124 4abN Z S⎡ ⎤= +⎢ ⎥⎣ ⎦

(10.60)

The extreme cases are when S=1 and S=0. According to the formula above, for S=1 (perfect order) the number of pairs per lattice site is Z, as expected. For S=0, the number of a-b pairs per lattice site is Z/2, also as expected since in the case of random distribution of atoms in the sublattices in an equiatomic compound there is an equal chance of finding either a or b atoms and thus the number of pairs is reduced by half. Returning to equation (10.59), we can now write ord

abN and disabN and it is easy to verify

that when S=1, the number of ab pairs is Z. ord

abN Z= (10.61) For S=0 2 (1 )dis

ab a aN Z x x= − (10.62) Substituting equations (10.59),(10.61) and (10.62) into (10.56) the short range order parameter is in the general case for a two sublattice compound:

2 22 (1 ) (2 1) 2 (1 ) 2( (1 ))

1 2( (1 ))a a a a a a a

a a

Z x x S x S x x x x Z

x x Zσ

⎡ ⎤− + − + − − −⎣ ⎦=− −

(10.63)


26

Equation (10.63) gives the lower bound of the degree of short range order that would be expected from a given degree of long range order.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Shor

t Ran

ge O

rder

Par

amet

er σ

Long Range Order parameter S

Figure 10.13: Relationship between the long range order parameter and the short range order parameters. *****************Close box 10.8 Amorphization In a liquid, although long-range order is absent, there is some degree of short range order. Normally, a liquid is stable at higher temperatures where the entropic contribution to the Gibbs free energy increases the stability of structures with greater amount of disorder. Although the Gibbs free energy of an ordered crystalline solid is intrinsically lower than that of an amorphous solid (no topological long range order) or than that of a chemically disordered solid (no chemical long range order), it is possible to observed metastable disordered or amorphous solids in situations where the stable crystalline configuration is kinetically inaccessible. One common example is the formation of an amorphous solid upon a very fast cool from the melt. If the cooling rate is fast enough, short-range order may be established during cooling, with no topological long-range order, which creates an amorphous solid. This is why amorphous solids are often referred by the oxymoronic term “frozen liquid”. In an ideal infinite crystal at 0 K, the probability of finding an atom at one of the lattice positions is 1.00. In a real crystal, lattice imperfections, such as point defects, dislocations, and grain boundaries destroy the perfect correlation that exists between


27

lattice positions in a perfect crystal. In addition, because of the atomic vibrations around the equilibrium position associated with thermal motion, at any given moment the atoms are not found at the exact equilibrium positions, but with slight deviations, which average out to the equilibrium lattice position. As a result, the interatomic distances vary, and these variations increase with distance such that the interatomic correlations decrease with distance. Quantitatively, the degree of order of crystalline solids can be described by the radial distribution function, normally denoted g(r). This function describes the probability of finding another atom at a given distance nearby. Figure 11.14 shows an example of a g(r) calculation performed in a crystal and in a liquid. In a liquid, as in an amorphous solid, although a well-defined relationship to their first nearest neighbors exists, (as shown by the large first peak), this correlation decreases with distance such that there is only a wide peak in the second nearest neighbor distance, decreasing to a random probability henceforth. In the crystal, the peak heights decrease with distance because of the factors discussed above, but the correlation distance is much greater than in the amorphous solid.

Figure 10.14: The radial distribution functions for a solid (top) and a liquid (bottom) (courtesy of C. Iacovelli and S.Glotzer, University of Michigan) Besides fast quenching a metastable amorphous phase can form during other non-equilibrium processes, such as during intense plastic deformation from ball milling [10], and in diffusion couples between elements with large negative heats of mixing [11]. Irradiation can also make crystalline phases amorphous and cause chemically ordered phases to become disordered.


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Problems 1. Describe qualitatively what would occur to a melt of Zr-10%Cr as it is slowly cooled from 1750 °C down to 700 °C, in quasi-equilibrium. Describe what phases would appear, their compositions and how the volume fractions and compositions of these phases would change as the temperature is steadily decreased. Do a neat sketch with illustrations at important points in the process to help convey the physical processes taking place taking special attention to the regions where phase transformations take place. 2. Using the Zr-Cr phase diagram shown in the Figure 10.5 answer the questions a. For a solid containing 22 atom %Cr, and at equilibrium at T=1330 °C, identify the phases present, their relative amounts (how much of each phase is present) and their composition (Cr atom % for each). b. The maximum solubility of Cr (atom %Cr) in the beta phase of Zr and the temperature at which it occurs. At 1100 °C, what occurs when the solubility limit of Cr in the beta phase is reached? c. Construct free energy curves for T = 400 °C , 1100 °C, and 1650 °C that are consistent with the Zr-Cr phase diagram given in the notes. Indicate the phases associated with each curve, in each case. d. For a 60 at% Cr, 40 at % Zr mixture at 1100 °C at equilibrium, what phases are present, in what composition and in which proportions? e. Calculate the maximum possible equilibrium vacancy concentration in pure beta Zr, if the vacancy formation energy is 1.6 eV (5). 3. For the 2D figure below, calculate the long range S and short range order parameter σ. Use Z=4 and consider only the atoms which have all 4 nearest neighbors


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b) How many toms would have to be moved for perfect order (both S and σ) to be established? 4. During cooling from high temperature a second phase “beta” precipitates from a solid solution in the “alpha” phase. a) If the precipitate is spherical radius r, and the surface energy of the alpha-beta interface is 1 J/m2 and the change in free energy from beta to alpha is 23 KJ/mol calculate the critical radius at the transformation temperature, if the material density is 7 g/cm3. What would likely occur to the critical radius if the temperature were further decreased below the precipitation temperature? Why? b) Assume now that the precipitate is plate-like with a thickness and that the surface energy of the plate surface is xΩ, where x is a number between 0 and 1. Estimate the plate dimensions as a function of x. Assume that the two in-plane directions are equivalent and that elastic forces play no role in the transformation. References [1] P. J. Spencer, "A Brief History of CALPHAD," Calphad-Computer Coupling Of

Phase Diagrams And Thermochemistry 32 (2008) 1-8. [2] M. Volmer, Z. Elektrochem. 35 (1929) 55. [3] J. W. Christian, The Theory of Transformations in Metals and Alloys, part I:

Equilibrium and General Kinetic Theory, 2nd edition. Oxford: Pergamon Press, 1975.

[4] D. A. Porter and K. E. Easterling, Phase Transformations in Metal and Alloys, 2nd ed. London: Chapman and Hall, 1993.

[5] J. Verhoeven, Fundamentals of Physical Metallurgy. New York: Wiley and sons, 1975.

[6] J. M. Lifshiz and V. V. Slyozov, Journal of Physical Chemistry of Solids 19 (1961) 35.

[7] C. Wagner, Zeitshrift fur Elektrochem. 65 (1961) 581. [8] H. K. D. H. Bhadeshia, "Martensitic Transformation," in Encyclopedia of

Materials Science: Science and Technology, R. W. C. K. Buschow, M. C. Flemings, B. Iischner, E. J. Kramer and S. Mahajan, Ed.: Pergamon Press, Elsevier Science, 2001, pp. 5203-5206.

[9] B. D. Cullity, Elements of X-ray Diffraction. Reading, MA: Addison-Wesley, 1978.

[10] R. B. Schwarz and W. L. Johnson, "Formation of an Amorphous Alloy by Solid-State Reaction of the Pure Polycrystalline Metals," Physical Review Letters 51 (1983) 415-418.

[11] U. Gosele and K. N. Tu, Journal of Applied Physics 53 (1982) 3252.

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