Precipitation of a second phase from a solid solution has been used as a hardening—strengthening—
technique for metals for over 100 years. The aluminum frame that made possible the construction of dirigibles such as the Hindenburg was a precipitation-hardened alloy as is almost all of the metal struc- ture of modern aircraft. Here, the toughening of zirconia ceramics by precipitation is modeled. The purpose is not to compare the model to any particular experimental data but mainly to introduce some considerations involved in precipitation and their relationships to phase equilibria. In addition, the effect of boundary movement on the resulting kinetics presented in Appendix A.3 is applied in this case.
10.13.2 t
ranSFormationalt
ougheningFigure 10.25 shows part of the MgO–ZrO2 phase diagram in Figure 7.31 (Levin et al. 1964). Ceramics are often considered to be “weak” materials because they break before they bend; they have a low fracture toughness. However, partially stabilized ZrO2 can be made quite tough and crack resistant
1000 1200 1400 1600 1800 2000 2200 2400
0 5 10 15
Cubic Cubic
+ tetragonal
Tetragonal + MgO
Monoclinic + MgO
ZrO2 m/o MgO
Temperature (°C) T = 1500°C
7 m/o MgO
1.7 11.0
FIGURE 10.25 Part of the MgO–ZrO2 phase diagram showing the equilibrium compositions of the tetragonal phase, 1.7 m/o, and the cubic phase, 11.0 m/o, for a 7 m/o MgO alloy held at 1500°C to precipitate the tetragonal phase.
with strengths up to several GPa (300,000 psi). The reason for the toughening in these materials is the transformation of retained or untransformed tetragonal particles in alloys that have composi- tions lying in the cubic + tetragonal phase field of Figure 10.25. Tetragonal particles larger than 1 μm or so will transform on cooling to the monoclinic phase with a few percent increase in volume.
However, if the tetragonal particles are smaller than 1 μm, they are restrained from transforming by the surrounding cubic phase and can be retained in the tetragonal phase at room temperature. If a crack starts to propagate, some of the constraint of the cubic phase is released and the tetragonal particles transform, expand, and narrow the crack making it more difficult to propagate: increasing the toughness and strength considerably (Green et al. 1989). An ideal way to control the tetragonal particle size is by precipitation of the tetragonal phase from the cubic phase, which is in fact how it is done in industrial practice. This precipitation process is analyzed because it is the opposite of the dissolution process and reinforces the concept of the equilibrium concentration at the two-phase interface. However, in this case, the concentration versus distance profiles during precipitation are somewhat surprising.
10.13.3 t
hem
odelIn Figure 10.25, consider an alloy of 7 m/o MgO that has been densified or heat treated at a tem- perature above the (cubic + tetragonal)-cubic solvus line to form a cubic solid solution, in the neigh- borhood of 1850°C, also shown in Figure 10.26, at time equal to zero. The alloy is rapidly cooled to 1500°C and held there, while the tetragonal phase precipitates from the cubic solid solution in a nucleation and growth process. Only the growth process is considered here. From the partial phase diagram of Figure 10.25, when the precipitation process is complete—which it may not get to in practice to maximize the toughening—the microstructure will consist of tetragonal precipitate particles containing 1.7 m/o MgO in a matrix of cubic solid solution of 11.0 m/o MgO, as shown in Figure 10.26, when time goes to infinity—equilibrium. At times greater than zero and less than infin- ity, the middle concentration profile in Figure 10.26, the precipitate particle contains 1.7 m/o MgO, the matrix 7 m/o but the equilibrium concentration at the tetragonal-cubic interface is 11.0 m/o.
As a result, in order for the precipitate particle to grow, MgO must diffuse away from the particle.
In this case, the concentration versus radial distance is
C r C C a
r C
( )
=(
C− 0)
+ 0 (10.49)0 < t < ∞
t = ∞
Distance, r
C0 = 7 m/o
C0= 7 m/o CT= 1.7 m/o CC= 11 m/o
CT = 1.7 m/o CC = 11 m/o
Tetragonal precipitate particle Time
t = 0 Concentration
2a
da/dt
FIGURE 10.26 Compositions as a function of distance and time for precipitation of tetragonal ZrO2 from a 7 m/o MgO cubic solid solution at 1500°C.
10.13 Precipitation and Toughening in Zirconia 365 where CC = 11 m/o MgO, the concentration in the equilibrium cubic solid solution, and C0 = 7 m/o
MgO, the initial concentration in the cubic phase.* So dC
dr
C C
r a a
C
= −
(
−)
=
0 (10.50)
Taking into consideration the motion of the interface boundary discussed in Appendix A.3 (and has been ignored in previous models without justification), gives in this case
a D C C
CC C t
C T
2 0
=2
(
−)
(
−)
(10.51)so it does not make any difference what the units of concentration are, the units cancel so they can be left in mole %.
10.13.4 C
alCulatedh
eat-t
reatt
imeNot surprisingly, the diffusion coefficients of oxygen, magnesium, and zirconium are not known for this particular composition. However, several observations can be made and used to get an approxi- mate diffusion coefficient and heat-treatment time. First, the oxygen diffusion coefficient is much larger than that for the cations because of the large oxygen vacancy concentration produced by MgO in solid solution as was the case for CaO in solid solution, that is,
MgO ZrO MgZr OOx V
O
2
′′ + + ii.
Also, as was seen in Section 10.11 on the dissolution of NaCl, when charged species are diffusing, the slower diffusing species controls the rate. Therefore, for the growth of these tetragonal zirco- nia particles, the diffusion of magnesium and/or zirconium will be rate controlling. Cation diffu- sion data are not available in the MgO–ZrO2 system but are for the CaO–ZrO2 system (Rhodes and Carter 1966). In this system, D Ca( 2+)=0 444. ×exp( (− 419 240, J mol RT cm s/ )/ ) 2/ so at 1500°C, D=0 444. ×exp( (− 419 240 8 314 1 773, )/( . )( , ))=1 98 10. × −13cm s2/.† Assuming that this is the diffusion coefficient that controls the transport of magnesium, then the time it takes at 1500°C to get to a = 0.5 μm is
t a D
C C
C C
T C
=
(
−)
(
−)
=(
×)
−(
×)
−
− 2
0
4 2
2 13
0 5 10 1 7
2 1 98 10 11
C . (11.0 . )
. .00 7 0
104
(
−)
= × =
. t 1.47 s 4.08h
which would be a reasonable time to heat treat in practice because it is neither too short nor too long. In fact, this is entirely in the range of processing conditions used by industry in the manufacturing of transformation-toughened zirconia ceramics (Readey, M. J., pers. comm., December 2015).
* The concentrations here must be in mol/cm3 and all of the given concentrations should be converted to mol/cm3. However, the variation in density with composition is not known but could be approximated by ideal solutions, which they clearly are not, if the lattice parameters of the phases were known as a function of composition, which they also are not. In any event, the units of concentration will cancel and they can be left as mole percent.
† The oxygen diffusion coefficient in this system is about 106 times that of the cations (Rhodes and Carter 1966).
There are several assumptions implicit in Equation 10.51. The first is that the concentration of the cubic matrix phase does not change during precipitation and has remained at 7 m/o although the equilibrium value is 11.0 m/o. At equilibrium at 1500°C, the phase diagram shows that the volume fraction of tetragonal will be about (11 7 11 1 7− ) (/ − . )≅0 43. so for this assumption to be reasonably valid, perhaps no more than 4% of the tetragonal phase has precipitated, about 10% of the equilib- rium value. This is probably not a bad assumption as long as the nucleation rate is not extremely fast so that, even though the particle size is small, there are a large number of particles and the fraction of tetragonal phase is much larger. Another assumption is that the density of the solid solutions are the same regardless of the composition, which again is perhaps valid up to about 10%. The addition of MgO to the cubic phase of ZrO2 will lower its density because Mg2+ is lighter than Zr4+ and vacant oxygens are formed as well. However, not knowing how the lattice parameter changes with compo- sition the densities and molar volumes really cannot be calculated anyway. Finally, the precipitate- cubic phase boundary moves that distorts the diffusion profile. However, the assumption is made again, that because the growth rate is slow and small, this will not be a large effect and a steady-state concentration profile is assumed. If all of the necessary data are available, then the problem can be solved exactly, albeit not simply. Nevertheless, with all of the assumptions in the given model, the results of the steady-state solution are probably sufficient to quantitatively predict precipita- tion times in the MgO–ZrO2 system. The above assumptions will generate small perturbations in the results compared to what the uncertainty in the value for the diffusion coefficient will produce.*
10.13.5 g
roWthr
ateTaking the square root of both sides of Equation 10.51 gives
a C C
CC C Dt
C T
=
(
−)
(
−)
2 0
1 2
so da/dt can be written as
da dt
C C
C C
D t
C
C T
=
(
−)
(
−)
1
2 2 0
1 2
where, as before in Chapters 5 and 7, the rate of growth, da/dt, is proportional to a thermodynamic term, (CC – C0)/(CC – CT), and a kinetic term, D t/. The thermodynamic term is simply the difference between the composition of the alloy in question, C0 (7 m/o in this example), and the composition of the cubic phase in equilibrium with the tetragonal phase at a given temperature, the solvus. The solvus between the cubic and the cubic + tetragonal regions in the partial phase diagram of Figure 10.25 is essentially linear and is approximately T=2400 80 (− × m o MgO/ ). So for C0 = 7.0 m/o, the tempera- ture where this composition intersects the solvus line is about 1840°C, and at 1500°C, the composition of the cubic phase is 11.25 m/o, very close to the value of 11 m/o assumed above. So the thermody- namic term is essentially linear and increases as the temperature decreases. In contrast, the kinetic term is strongly dependent on the exponential temperature dependence of the diffusion coefficient and gets smaller rapidly as the temperature decreases because of the large activation energy. As a result, the growth rate reaches a maximum at around 1750°C and decreases with temperature as shown in Figure 10.27.
* The tetragonal particles are not actually spherical. The strain energy between the precipitate and the matrix causes the particles to take on ellipsoidal shapes. In addition, the number density of precipitate particles is rather high, so it is possible that their concentration profiles interact or overlap. Nevertheless, most importantly, the model, with all of its simplifying assumptionsis consistent with experimental results (Green, et al. 1989) and provides the engineer an excellent starting point for developing a technically robust heat-treatment schedule (Readey, M. J., pers. comm., December 2015).