Introduction

Monazite and xenotime are common accessory minerals whose compositions and mutual solubility are important geological indicators (Heinrich 1997; Gratz 1997, 1998; Andrehs 1998; Pyle 2001; Seydoux-Guillaume 2002). These materials have also received considerable attention due to their prospective application for nuclear waste management and containment (Ewing 1995). However, it was the co-existence of alumina and monazite in natural minerals that served as a clue to the identification, development, and use of monazite (LaPO4) and other materials with the ABO4 structure as an interlayer in high temperature oxide–oxide ceramic matrix composites (CMCs) (Davis et al. 1993, 1998; Morgan et al. 1993, 1995a, b; Marshall et al. 1998; Kerans et al. 2002). Oxidation resistant fiber–matrix interlayers, capable of providing crack deflection at the fiber–matrix interface, are needed for improved toughness in ceramic composites. Today, monazite is a demonstrated oxidation resistant crack-deflecting fiber coating for oxide CMCs (Keller et al. 2003). Both monazite and xenotime are chemically stable with common structural oxides and bond weakly to them. However, xenotime has a different crystal structure and a lower coefficient of thermal expansion (CTE) than monazite, 6 × 10−6 (Hikichi et al. 1998; Subbarao et al. 1990) versus 9.6 × 10−6 (Morgan et al. 1995b), respectively. Two-phase monazite–xenotime coatings, therefore, might allow the CTE of the coating to be tailored to match the CTE of a wide range of ceramic fibers. The two-phase microstructure would also help control the undesirable grain growth in the coating. However, solid solubility between monazite and xenotime may affect both the thermal expansion and the kinetics of the grain growth in the two-phase coating. The miscibility gaps in a number of monazite–xenotime systems are evaluated based on the regular solid solution model and available experimental data.

Regular solid solution model

Let us consider a monazite–xenotime system with limited mutual solid solubility, Fig. 1.

Fig. 1
figure 1

Schematics of a solid section of a monazite/xenotime phase diagram with limited solubility in the solid state

Full size image

At equilibrium between the monazite and xenotime phases, the chemical potential of both components in both phases can be written as follows:

$$ \left\{{\begin{array}{*{20}c} {{\mu_{m}^{x} = \mu_{x}^{x}}} \\ {{\mu^{x}_{m} = \mu^{m}_{m}}} \\ \end{array}} \right. $$
(1)

or

$$ \left\{{\begin{array}{*{20}c} {{g^{x}_{m} + \hbox{RT}\ln a^{x}_{m} = g^{x}_{x} + \hbox{RT}\ln a^{x}_{x}}} \\ {{g^{m}_{x} + \hbox{RT}\ln a^{m}_{x} = g^{m}_{m} + \hbox{RT}\ln a^{m}_{m}}} \\ \end{array}} \right. $$
(2)

where μ, g, and a denote the chemical potential, the molar Gibbs free energy of pure components, and the activity, respectively. The lower and upper indexes represent the solvent and the solute, respectively. Thus, μ x m and μ x x are the chemical potentials of the xenotime component in the monazite and xenotime phases, g x m and g x x are the molar Gibbs free energies of xenotime in a hypothetical monazite structure and in the native xenotime structure, and a x m and a x x are the activities of xenotime in the monazite and xenotime phases, respectively.

For a regular solution, the excess enthalpy is h ex = Ωx (1− x) and the activity coefficient γ is given by:

$$ \hbox{RT ln} \gamma = \Omega (1 - x)^{2}$$
(3)

where Ω is the interaction energy of the solution and x is the concentration (Gaskell 1995a). Denoting the equilibrium molar concentration of xenotime in monazite as x 1 and that of monazite in xenotime as x 2 (Fig. 1), we obtain:

$$ \left\{{\begin{array}{*{20}c} {{\Omega_{m} (1 - x_{1})^{2} + \hbox{RT}\ln \frac{{x_{1}}} {{1 - x_{2}}} = \Delta h_{x} + \Omega_{x} x^{2}_{2}}} \\ {{\Omega_{x} (1 - x_{2})^{2} + \hbox{RT}\ln \frac{{x_{2}}} {{1 - x_{1}}} = \Delta h_{m} + \Omega_{m} x^{2}_{1}}} \\ \end{array}} \right. $$
(4)

where Ω x and Ω m are the interaction energies of xenotime- and monazite-based solid solutions, and Δh m =  h m m− h x m≈ g m m− g x m and Δh x =  h x x− h m x≈ g x x− g m x are the differences in the enthalpy of formation between the monazite and xenotime (or vice versa) forms of the pure compounds, respectively.

The parameters Δh m and Δh x can be estimated as follows. It has been observed that the formation enthalpy of stable rare earth orthophosphates varies linearly with the rare earth ion radius (Ushakov et al. 2001). The enthalpy of formation of the dimorphic TbPO4 is 286.1 ± 1.9 kJ/mol in the xenotime (stable) structure and is 283.5 ± 1.8 kJ/mol in the monazite (metastable) structure (Ushakov et al. 2001), or \({\Delta h_{{\rm TbPO_{4}}}= 2.6}\)  kJ/mol. Assuming that the formation enthalpy of metastable monazites and xenotimes also varies linearly with the ion radius, and extrapolating to other orthophosphates, we can derive the following equations:

$$ \left\{{\begin{array}{*{20}c} {{\Delta h_{x} = \frac{{R_{{\rm tr}} - R_{x}}} {{R_{{\rm tr}} - R_{{\rm Tb}}}}\Delta h_{{\hbox{TbPO}_{4}}}}} \\ {{\Delta h_{m} = \frac{{R_{m} - R_{{\rm tr}}}} {{R_{{\rm tr}} - R_{{\rm Tb}}}}\Delta h_{{\hbox{TbPO}_{4}}}}} \\ \end{array}} \right. $$
(5)

where R tr is the transition ion radius between the monazite and xenotime structures, at which Δh m and Δh x equal zero, and R x and R m are the ion radii of the xenotime- and monazite-forming cations, respectively. The transition from the stable xenotime to the stable monazite structure occurs between Tb and Gd (Ushakov et al. 2001). Given the ion radii of Tb and Gd, R Tb = 1.04Å, R Gd = 1.053 Å (Shannon 1976 Footnote 1.), this correlates well with the data of Gratz and Heinrich (1998), who reported the maximum solubility of Gd in Y-xenotime to be 0.96, which corresponds to the mean ion radius of 1.0516 Å.

The interaction energies Ω m and Ω x , in turn, consist of two parts: the energy associated with transforming the solute from its stable structure to that of the solvent (−Δh x and −Δh m , respectively), and the elastic energy U el associated with the insertion of a substitution ion into the host lattice:

$$ \Omega_{{x,m}} = - \Delta h_{{m,x}} + U_{{\rm el}} $$
(6)

The elastic energy can be expressed as follows:

$$ U_{{\rm el}} = 4\pi N_{a} E{\left[ {\frac{{R_{0}}} {2}(R_{1} - R_{0})^{2} + \frac{1} {3}(R_{1} - R_{0})^{3}} \right]} = 4\pi N_{a} \hbox{ER}^{3}_{0} \left(\frac{{y^{2}}} {2} + \frac{{y^{3}}} {3}\right) $$
(7)

where E is the Young’s modulus of the host lattice, R 1 is the radius of the substitution ion, R 0 is the radius of the native ions of the host lattice (Brice 1975), and y = (R 1 −R 0)/R 0 is the fractional difference in the ion radius. If the Young’s modulii of the compounds involved are known, the temperature dependence of x 1 and x 2 can be obtained by solving the equations given in (4) numerically for any monazite–xenotime system.

Application to LaPO4–YPO4 and CePO4–YPO4 systems

The calculations are further simplified for the LaPO4–YPO4 and the CePO4–YPO4 systems for which the miscibility gap is strongly asymmetric and the solid solubility of monazite in xenotime is very low (Heinrich et al. 1997; Gratz and Heinrich 1997, 1998; Andrehs and Heinrich 1998; Pyle et al. 2001; Mogilevsky et al. 2006b). Neglecting the small solubility of monazite in YPO4 (x 2 = 0), we obtain the following from the equations given in (4):

$$ \hbox{RT}\ln x_{1} + \Omega_{m} (1 - x_{1})^{2} = \Delta h_{{\hbox{YPO}_{4}}} $$
(8)

According to Eqs. 6 and 8, the difference in solid solubility of YPO4 in LaPO4 and in CePO4 is determined by the difference in the elastic energy. Thus, if the solid solubility of YPO4 in one of the two monazite compounds, e.g. CePO4, at a given temperature is known, then the temperature at which the same level of solid solubility is reached in the other compound can be calculated:

$$ R\Delta T = - \frac{{\Delta U_{{\rm el}} (1 - x_{1})^{2}}} {{\ln x_{1}}} $$
(9)

where ΔU el is the difference in the elastic energy associated with the insertion of a substitution ion (Y) into the two host lattices. The room temperature Young’s modulus of LaPO4 monazite is E = 133 GPa (Morgan et al. 1995b). Although there is no available data on the Young’s modulus of CePO4, it must be quite close to that of LaPO4 Footnote 2. Substituting the ionic radii R 1 = R Y = 1.019 and R 0 = R La = 1.16 Å or R 0 = R Ce = 1.143 Å (Shannon 1976) into Eq. 7 produces U el = 10.7 kJ/mol and U el = 8.2 kJ/mol, respectively, or ΔU el = 2.5 kJ/mol. The temperature dependence of the YPO4 solubility in LaPO4 can now be obtained from the YPO4 solubility in CePO4 by introducing the corresponding temperature correction, as defined by Eq. 9.

The result is shown in Fig. 2, where the data on YPO4 solubility in CePO4 from 300 to 1,000°C, extrapolated to the ambient pressure of 1 atm (Gratz and Heinrich 1997) and adjusted using Eq. 9, are plotted together with the data on YPO4 solubility in LaPO4 from 1,000 to 1,600°C (Mogilevsky et al. 2006b). It is seen that both sets of data form a common trend. Equation 8 was then fitted to the combined set of data, which resulted in the best-fit parameters of \({\Delta h_{{\rm YPO}_{4}}= -8.2}\)  kJ/mol and Ω m =17.4 kJ/mol. These parameters need to be compared to the values predicted by Eqs. 5–7. Substituting R tr = 1.0516 Å (see above Discussion) and R Y  = 1.019 Å (Shannon 1976) into Eq. 5, we obtain \({\Delta h_{{\rm YPO}_{4}} = -7.3}\)  kJ/mol, which is in reasonable agreement with the best fit value of −8.2 kJ/mol. The general agreement between the best fit value of \({\Delta h_{{\rm YPO}_{4}}}\) and that calculated using Eq. 5 is remarkable given the relatively large uncertainties associated with R tr and, especially, \({\Delta h_{{\hbox{TbPO}_{4}}}}.\) Assuming that the value \({\Delta h_{\hbox{TbPO}_{4}}= 2.6}\)  kJ/mol is the correct one, the revised value of R tr that reconciles Eq. 5 with the best fit value of \({\Delta h_{{\hbox{YPO}_{4}}}= -8.2}\)  kJ/mol is R tr = 1.0498 Å, which still falls between the ion radii of Tb and Gd (1.04 and 1.053 Å, respectively). Retaining the previously chosen value R tr = 1.0516Å, the revised value of \({\Delta h_{{\hbox{TbPO}_{4}}}}\) would be −2.92 kJ/mol, which is well within the uncertainty limits associated with its experimental value (Ushakov et al. 2001). Given the value of the elastic energy for YPO4 solution in LaPO4, U el = 10.7 kJ/mol, calculated above, the interaction energy Ω m  = (Δh YPO 4 + U el) = 18.0 kJ/mol, in good agreement with the best fit value of Ω m  = 17.4 kJ/mol.

Fig. 2
figure 2

Fitting of Eq. 8 to the experimental data on solid solubility of YPO4 in LaPO4 (open symbols, Mogilevsky et al. 2006b) combined with the data on solid solubility of YPO4 in CePO4 (Gratz and Heinrich 1997) adjusted using Eq. 9 (filled symbols)

Full size image

Asymmetry of the miscibility gap and distribution coefficient

We can use the approach outlined above to rationalize the asymmetry of the miscibility gap in the LaPO4–YPO4 and the CePO4–YPO4 systems, where there is a much lower solubility of monazite in xenotime than vice versa (Heinrich et al. 1997; Gratz and Heinrich 1997, 1998; Andrehs and Heinrich 1998; Pyle et al. 2001; Mogilevsky et al. 2006b). Substituting Eq. 8 into the second equation given in (4) we obtain:

$$ \Omega_{x} (1 - x_{2})^{2} + \frac{{\Delta h_{x} - \Omega_{m} (1 - x_{1})^{2} }} {{\ln x_{1}}}\ln \frac{{x_{2}}} {{1 - x_{1}}} = \Delta h_{m} + \Omega_{m} x^{2}_{1} $$
(10)

Substituting R La = 1.16 Å (Shannon 1976) into Eq. 5 and again assuming R tr = 1.0516 Å, we obtain Δh m  = −24.3 kJ/mol. Substituting the Young’s modulus of xenotime, E = 152 GPa (Kuo and Kriven 1995; Mogilevsky et al. 2006), and the ionic radii R 1 = R La and R 0 = R Y into Eq. 7, we obtain U el = 12.7 kJ/mol and Ω x  = 37 kJ/mol for the LaPO4 solution in YPO4 xenotime. Solving Eq. 10 numerically, we obtain the dependence of x 2 on x 1, Fig. 3a.

Fig. 3
figure 3

a Calculated solubility of LaPO4 in YPO4 (x2) as a function of the solubility of YPO4 in LaPO4 (x1); b calculated Y distribution coefficient DY versus reciprocal temperature for LaPO4–YPO4 and CePO4–YPO4 (solid lines); symbols are experimental values (Heinrich et al. 1997; Gratz and Heinrich 1997)

Full size image

It is seen that when x 1 = 42% (the maximum solubility of YPO4 in LaPO4 experimentally observed at T = 1,600°C (Mogilevsky et al. 2006b)), the predicted solubility of LaPO4 in YPO4 is much smaller, x 2 = 1.7%, in agreement with the experimental results (Heinrich et al. 1997; Gratz and Heinrich 1997, 1998; Andrehs and Heinrich 1998; Pyle et al. 2001; Mogilevsky et al. 2006b). This is mainly due to the large \({\Delta h_{{\hbox{LaPO}}_{4}}}\) compared to \({\Delta h_{{\hbox{YPO}_{4}}},}\) since the difference between the corresponding elastic energies is relatively small. An Arrhenius plot of the calculated Y distribution coefficient, D Y =  x Y m/x Y x = x 1/(1 − x 2), for the LaPO4–YPO4 and the CePO4–YPO4 systems, along with the experimental data (Heinrich et al. 1997; Gratz and Heinrich 1997), is shown in Fig. 3b. The redistribution of Y between the xenotime-based and the monazite-based solid solutions involves two concurrent reactions with activation energies of Ω x  = 37 kJ/mol and Ω m  = 18.0 kJ/mol for the LaPO4–YPO4 system and Ω x  = 30.2 kJ/mol and Ω m  = 15.4 kJ/mol for the CePO4–YPO4 system. The apparent activation energies for both systems, E a ≈ 32 and E a ≈ 29 kJ/mol, respectively, determined from the slopes of the Arrhenius plots, are indeed very close to the apparent activation energies expected for the process involving two concurrent reactions with activation energies of Ω x and Ω m . These results are in line with the established correlation between the crystal/melt partition coefficients and the elastic strain energy (Wood and Blundy 1997; Blundy and Wood 2002, 2003).

Miscibility gaps in other monazite–xenotime systems

The above analysis shows that for the LaPO4–YPO4 and CePO4–YPO4 systems, Eqs. 5–7 give reasonably accurate estimates for the parameters of the equations given in (4). Therefore, miscibility gaps in other monazite–xenotime systems could be calculated in a similar manner, if the values of the Young’s modulus for these compounds were available. Unfortunately, only limited data on the elastic properties of monazites and xenotimes are available. Therefore, certain assumptions regarding the Young’s modulus of other monazites and xenotimes would have to be made to calculate the miscibility gaps in these systems.

Among monazite compounds, the polycrystalline Young’s modulus is only available for LaPO4 (Morgan et al. 1995b), but a reasonable assumption can be made based on the well-established linear correlation between the melting temperature and the Young’s modulus (Fine et al. 1984; Skinner and Zedalis 1988; Ashby 1998). The melting temperatures of La, Ce, Pr, Nd, and Sm monazites have been reported (Hikichi and Nomura 1987). A near linear correlation between the melting temperatures (Hikichi and Nomura 1987) and the rare earth ion radius has been observed (Mogilevsky et al. 2006b):

$$ T_m(\deg \hbox{C}) = -188.96 + 1951.4 R_m $$
(11)

where R m is the rare-earth ion radius (Å). Thus, given the experimental Young’s modulus of LaPO4, E = 133 GPa (Morgan et al. 1995b), and the La ion radius, R La = 1.16 Å (Shannon 1976), we obtain:

$$ E_m \approx 4.8+110.6 R_m $$
(12)

where E m is in GPa and R m is in Å.

Among xenotime compounds, the polycrystalline Young’s modulus has only been measured and reported for YPO4 (Kuo and Kriven 1995). Data on melting temperatures are equally scarce and the above method can not be used. In general, even though the packing density of the xenotime structure is lower than that of monazite, the average RE–O distances in it are smaller (Ni et al. 1995) and the structure is correspondingly stiffer. Furthermore, it has been generally established that the elastic constants of orthophosphate, orthovanadate, and orthoarsenate xenotimes decrease with the increase in the rare earth ion radius, R x (Armbruster 1976; Mogilevsky et al. 2006a). The experimental value of the polycrystalline Young’s modulus of YPO4 (Kuo and Kriven 1995) is in good agreement with the estimate based on the elastic constants of single crystal YPO4 (Mogilevsky et al. 2006a). Among orthophosphates with xenotime structure, full sets of elastic constants have only been reported for LuPO4 (Armbruster et al. 1974) and YPO4 (Mogilevsky et al. 2006a), leading to the self-consistent estimates of random polycrystalline Young’s modulus (Berryman 2005) of \({E_{{\rm LuPO}_{4}}}\) = 202 and \({E_{{\rm YPO}_{4}}}\)= 152 GPa (Mogilevsky et al. 2006a, in good agreement with the experimental value of Kuo and Kriven (1995)). Assuming, for lack of a better alternative, a linear correlation between the Young’s modulus of xenotimes and the rare earth ion radius, and given the Y and Lu ion radii, R Y  = 1.019 and R Y  = 0.977 Å (Shannon 1976), we obtain:

$$ E_x \approx 1365.1 - 1190.5 R_x $$
(13)

where E x is in GPa and R x is in Å. The dependence of the Young’s modulus of pure xenotime and monazite compounds on the rare earth cation radius, as described by Eqs. 12 and 13, is shown in Fig.  4.

Fig. 4
figure 4

Young’s modulus—rare earth ion radius correlations of pure xenotime and monazite rare earth orthophosphates used in this work. Filled symbols are experimental values; open symbols are extrapolations according to Eqs. 12 and 13

Full size image

Admittedly, the linear correlations between the Young’s modulus and the rare earth ion radius proposed here (Eqs. 12 and 13) are not backed up by extensive experimental evidence. It appears noteworthy, therefore, that the linear trends for monazite and xenotimes in Fig. 4 intersect at the ion radius R = 1.0455 Å, once again between the ion radii of Tb and Gd (1.04 and 1.053 Å, respectively) and close to the previously discussed estimates of R tr = 1.0498 and R tr = 1.0516 Å. Although far from validating Eqs. 12 and 13, it shows that they are at least consistent with the other data used here.

Solving the equations given in (4) numerically, we can obtain the temperature dependence of x 1 and x 2 for any monazite–xenotime system. Calculations were conducted for a number of monazite–YPO4 and monazite–LuPO4 systems. The choice of YPO4 and LuPO4 xenotimes was due to the fact that the Young’s modulus is known for these two compounds. On the other hand, Lu is the smallest rare earth element, while Y is one of the largest (after Dy and Tb) tri-valent elements forming xenotime orthophosphates. In addition, calculations were conducted for the SmPO4–DyPO4 and the EuPO4–TbPO4 systems. The results of these calculations are shown in Fig. 5.

Fig. 5
figure 5

Calculated miscibility gaps for a number of monazite–xenotime systems: a monazite–YPO4; b monazite–LuPO4; c SmPO4–DyPO4 and EuPO4–TbPO4 systems

Full size image

In addition to the experimental data on the CePO4–YPO4 (Gratz and Heinrich 1997) and the LaPO4–YPO4 (Mogilevsky et al. 2006b) systems that were discussed earlier, limited solubility data are also available for LaPO4–YPO4, NdPO4–YPO4, SmPO4–YPO4, NdPO4–YbPO4 (Van Emden et al. 1996), GdPO4–YPO4 (Gratz and Heinrich 1998), and CePO4–LuPO4 (Lempicki et al. 1993). The available experimental data are compared to the values predicted by the present model in Table 1. There is a generally good agreement between the calculated values and the experimental ones. For the NdPO4–YPO4 system, the calculated solubility in both phases is somewhat lower than the experimental values. However, it must be noted that the experimental data on this system were obtained on samples precipitated from solution and heat treated at 1,000°C for 48 h (Van Emden et al. 1996). It has been found that at 1,000°C, longer heat treatments (100 h) were needed to equilibrate similar samples of the LaPO4–YPO4 system (Mogilevsky et al. 2006b). Accordingly, the NdPO4–YPO4 samples (Van Emden et al. 1996) might not have reached equilibrium and thus the corresponding solid solutions might have concentrations in excess of the equilibrium solubility level. Another discrepancy is for the SmPO4–YPO4 system where the calculations predict a considerably higher solubility of SmPO4 in YPO4 than the reported experimental value (0.36 vs. 0.15), even though there is a good agreement for the solubility of YPO4 in SmPO4. In the CePO4–LuPO4 system, 0.898% Ce for Lu substitution has been reported in Ce-doped LuPO4 crystals grown from the melt at 1,360°C (Lempicki et al. 1993). The present calculations suggest that the level of Ce for Lu substitution in LuPO4 at 1,360°C should be 0.93%, in good agreement with the reported experimental data.

Table 1 Comparison of calculated and experimental solubility of xenotime in monazite (x 1) and monazite in xenotime (x 2)
Full size table

Lastly, the very large (∼96%) and virtually temperature-independent solubility of GdPO4 in most xenotimes (Fig. 5a, b) should be pointed out. This is in agreement with the experimental data for GdPO4–YPO4 (Gratz and Heinrich 1998), Table 1. In addition, at least 75% solubility of GdPO4 in TbPO4 (Mullica et al. 1992) and in YbPO4 (Mullica et al. 1990) has been reported, although the maximum solubility was not explored in those studies. In part, this has been built into the model by the choice of R tr in Eq. 5, which was based on the experimental solubility of Gd in YPO4 (Gratz and Heinrich 1998). However, since R tr is only one of the model’s parameters and its value is tightly bound by other experimental observations as well (see Sects. 3, 10), this agreement with the experimental results appears to be a significant additional validation of the model.

Implications for two-phase monazite–xenotime fiber coatings for CMCs

As mentioned in the Introduction, solid solubility between monazite and xenotime may affect both the thermal expansion and the kinetics of the grain growth in the two-phase coating. The temperature-dependent solid solubility will lead to temperature-induced variations in phase volume fractions, as well as (though less significant) changes in molar volumes, Young’s modulus, and intrinsic CTE of the constituent phases (solid solutions), all of which affect the CTE of the material (Mogilevsky et al. 2006b). These effects will be particularly severe for systems with considerable asymmetry of the miscibility gap, such as LaPO4–YPO4 and CePO4–YPO4, Fig. 5a. In these systems, the CTE of the two-phase monazite–xenotime coating will increase with temperature because of the decrease of the volume fraction of the low CTE phase (xenotime), and this change is estimated to be comparable to the temperature dependence of the CTE of pure monazite (Mogilevsky et al. 2006b). In systems with low solubility in monazite but high solubility in xenotime, such as EuPO4–YPO4, EuPO4–LuPO4, and SmPO4–LuPO4 (Fig. 5a, b), the volume fraction of the low CTE phase (xenotime) will increase with temperature, and the CTE will tend to decrease, possibly offsetting the intrinsic temperature dependence of the CTE of the constituent phases. The very limited mutual solid solubility between LaPO4 and LuPO4 (the largest monazite-forming light rare earth element (LREE) and the smallest xenotime-forming heavy rare earth elements (HREE)), Fig. 5b, will lead to a weak temperature dependence for the phase composition and volume fractions, and the associated change in CTE will be minimal for this system. Systems with a large mutual solid solubility but a nearly symmetrical miscibility gap, such as SmPO4–YPO4 (Fig. 5a), and SmPO4–DyPO4 and EuPO4–TbPO4 (Fig. 5c) should also have relatively small CTE variations, because the phase volume fractions will be nearly independent of temperature. Among these, the EuPO4–TbPO4 system also has the extra benefit of having nearly the lowest Young’s modulus (Fig. 4) and possibly the lowest hardness among the monazite/xenotime-based compositions, which may make it a better fiber coating for CMCs (Kerans et al. 2002). However, the disadvantage of compositions with large mutual solubility is that very little, if any, grain growth inhibition may be expected in these two-phase materials as compared to the single-phase materials.

The very large (∼96%), virtually temperature-independent solubility of GdPO4 in most xenotimes was pointed out in the previous section. These results mean that only 4 at.% of HREE elements or Y are sufficient to completely transform GdPO4 into the xenotime structure, and even smaller amounts (<1 at. of Y or <0.1 at.% of Lu at 1,000°C) would shift the equilibrium into the two-phase (monazite + xenotime) region. This, as well as the slow transformation kinetics, may be partially responsible for the observation of both monazite and xenotime modifications in nominally pure GdPO4 (Gratz and Heinrich 1998; Ushakov et al. 2001; Celebi and Kolis 2002). Such large solubility in xenotimes makes GdPO4 an unlikely candidate for the design of two-phase monazite/xenotime materials. However, it could be used to soften (achieve the lowest modulus, Fig. 4, and possibly hardness) stable xenotime compounds, such as YPO4.

The choice of a particular monazite/xenotime coating system for CMCs will depend on the priorities set for the particular application, such as CTE and its temperature dependence, grain growth inhibition, or mechanical properties (elastic modulus and/or hardness). The present results can assist in choosing the optimal coating system depending on the application and design priorities.

Solid solubility in monazite/monazite and xenotime/xenotime systems

Solid solubility in monazite/monazite and xenotime/xenotime systems can be obtained from the equations given in (4). In this case, both phases have the same structure and the interaction energy of the solid solution is dominated by the elastic energy. Given the rather weak dependence of the Young’s modulus of monazite on the rare earth ion radius (Fig. 4) and the nearly symmetric nature of Eq. 7, we can assume that the interaction energy (Ω) of monazite/monazite solid solutions is independent of concentration over the entire composition range, as is the case for regular solid solution. In this case, x 1 = x 2 and the equations given in (4) converge into the usual equation describing phase separation in regular solid solutions:

$$ T_{{\rm ps}} = \frac{{2T_{c} (1 - 2x)}}{{\ln (1/x - 1)}} $$
(14)

where T ps is the temperature of phase separation and T c  = Ω/2R is the critical temperature above which the solid solution is stable over the entire range of compositions (Gaskell 1995b). For xenotime/xenotime systems, the Young’s modulus depends strongly on the rare earth ion radius (Fig. 4), and consequently, the elastic energy must depend on composition. Strictly speaking, in this case, the symmetrical regular solution model and the equations given in (4) do not apply. However, the limiting case of the solution behavior can be readily obtained if the interaction energy is assumed to be independent of the composition, but is set to its highest possible value for the given system.

Figure 6 shows the elastic energy of LaPO4-based monazite and LuPO4-based xenotime solid solutions calculated using Eq. 7 as a function of the fractional difference in the rare earth ion radius. All data fall on a common parabolic curve. Even for solid solutions consisting of the end members of both series, LaPO4–GdPO4 and LuPO4–TbPO4, the fractional difference in the rare earth ion radius is below 0.1 (Fig. 6), indicating the likelihood of complete solid solubility. The system with the highest elastic energy is LaPO4–GdPO4 with U el ∼6.3 kJ/mol, which corresponds to a critical temperature of T c ∼ 104°C. This means that in practice, all xenotime/xenotime and monazite/monazite systems should indeed exhibit the complete range of solid solution. This is in agreement with the data on CePO4–GdPO4 systems (Gratz and Heinrich 1998; Vance et al. 2000). Among xenotimes, at least 90% solubility in the LuPO4–TbPO4 system has been reported (Mullica et al. 1992).

Fig. 6
figure 6

Elastic energy of LaPO4-based monazite and LuPO4-based xenotime solid solutions

Full size image

Pressure dependence of the miscibility gap

Pressure dependence of the miscibility gap, which is particularly important for geological applications (Heinrich et al. 1997; Gratz and Heinrich 1997, 1998; Andrehs and Heinrich 1998; Pyle et al. 2001; Seydoux-Guillaume et al. 2002), can be evaluated in the following manner. The monazite structure has a higher packing density (lower molar volume) than the xenotime structure (Aldred 1984; Ni et al. 1995). The cell volume of the dimorphic TbPO4 is 275.9 Å3 in the monazite (metastable) structure and 292.7  Å3 in the xenotime (stable) structure (Ushakov et al. 2001). For GdPO4 the corresponding values are 280.32 Å3 in the monazite structure and 296.98 Å3 in the xenotime structure (Gratz and Heinrich 1998). Therefore, external pressure will increase the relative stability of the monazite structure and decrease the stability of the xenotime structure. As a zero approximation of the pressure dependence of the thermodynamic stability of each structure, the elastic energy of hydrostatic compression should be added to Δh x and subtracted from Δh m in the equations given in Eq. 4:

$$ \left\{{\begin{array}{*{20}c} {{\Delta {h}'_{x} = \Delta h_{x} + \frac{{P^{2}}} {{2K_{x}}}V_{x}}} \\ {{\Delta {h}'_{m} = \Delta h_{m} - \frac{{P^{2}}} {{2K_{m}}}V_{m}}} \\ \end{array}} \right. $$
(15)

where K x and K m are the bulk modulus and V x and V m are the molar volumes of the xenotime and monazite phases, respectively.

The excess elastic energy associated with the insertion of a substitution ion into the host lattice is also pressure dependent. External pressure should increase the excess elastic energy associated with the insertion of a larger substitution ion into the host lattice (solid solution of monazite in xenotime) and decrease it for a smaller substitution element (solid solution of xenotime in monazite). Considering this, along with the effect of hydrostatic compression on the relative stability of monazite and xenotime structures (Eq. 15), hydrostatic compression should increase the solubility of xenotime in monazite and decrease the solubility of monazite in xenotime, shifting the miscibility gap towards the xenotime side of the phase diagram.

Again as a zero approximation, we can assess the effect of external pressure on the excess elastic energy associated with ion substitution through its effect on the effective ion radius of the host lattice, R 0 in Eq. 7. Both monazite and xenotime structures consist of rigid PO4 tetrahedra and eight- or nine-coordinated oxygen-bonded rare earth ions (Ni et al. 1995). The energy of the P–O bonds is considerably higher than that of the rare earth–O bonds (Mogilevsky et al. 2006), and the mean P–O distances remain nearly unchanged for all rare-earth orthophosphates from La to Lu (Ni et al. 1995). Furthermore, although there appears to be no high pressure structural data for either xenotime or monazite orthophosphates, PO4 tetrahedra in aluminum, boron, and lead orthophosphates are rigid and undergo little structural change with pressure (Haines et al. 2003; Angel et al. 2001). Therefore, for this analysis, it can be assumed that, at least at room temperature, the PO4 tetrahedra in both monazite and xenotime structures are incompressible, and under hydrostatic pressure, the compression of the lattice results from the contraction of the rare earth–oxygen bonds, which can be interpreted as a decrease of the effective radius of the rare earth ion.

The cube root of the cell volume of both monazite and xenotime orthophosphates depend linearly on the rare earth ion radius (Aldred 1984):

$$ V^{1/3}= a + bR $$
(16)

where V is the cell volume, R is the rare earth ion radius, and a and b are constants whose values for rare earth orthophosphates with monazite and xenotime structures are given in Table 2.

Table 2 Parameters a and b of Eq. (16) for rare earth orthophosphates with monazite and xenotime structures
Full size table

For relatively low pressures (P << K), V = V 0(1−P/K), where K is the bulk modulus, and P is the pressure. Substituting this into Eq. 16 and taking (1− P/K)1/3 ≈ (1 − P/3K), we obtain:

$$ {R}'_{0} \approx R_{0} - (R_{0} + a/b)\frac{P}{{3K}} $$
(17)

where R 0 is the standard effective ion radius of the host lattice at ambient pressure.

Substituting Eq. 17 into Eq. 7, we can obtain the dependence of U el on the external hydrostatic pressure for any monazite–xenotime system. The results for the LaPO4–YPO4 and the CePO4–YPO4 systems are shown in Fig. 7. The bulk modulii of LaPO4 and CePO4 monazites (K m=Em/3(1-2ν) ≈ 98 GPa (Morgan et al. 1995b)) and YPO4 xenotime (K x=132 GPa (Mogilevsky et al. 2006a)) were assumed for these calculations.

Fig. 7
figure 7

Pressure dependence of the elastic energy Uel of a monazite in xenotime and b xenotime in monazite solid solutions for LaPO4–YPO4 and CePO4–YPO4 systems

Full size image

Substituting the pressure-dependent parameters Δh x , Δh m , and U el into the equations given in (4), we can calculate miscibility gaps in monazite–xenotime systems as a function of temperature and pressure. Figure 8 compares the results of such calculations to the available experimental data for the CePO4–YPO4 system at 800°C (Gratz and Heinrich 1997). Although in the experimentally studied pressure range of 2–15 kbar the model predicts a slightly accelerating non-linear pressure dependence of solubility as compared to the linear empirical correlation (Gratz and Heinrich 1997, the dashed line in Fig. 8), the overall agreement between the calculated and experimental values is good. However, at higher pressures, the model predicts a much faster increase of the solubility with pressure as compared to the extrapolation of the empirical linear correlation (see inset in Fig. 8).

Fig. 8
figure 8

Calculated (solid line) and experimental (symbols, Gratz and Heinrich 1997) pressure dependence of solubility of YPO4 in CePO4. The dashed line is the linear correlation (Gratz and Heinrich 1997)

Full size image

The increase of xenotime solubility with pressure indicates that the excess volume of solution, V ex =  ∂Ω m /∂P, is negative. The excess volume can be calculated based on the Eqs. 6, 7, 15, and 17 and at ambient pressure is −1.64 and −1.9 cm3 for the solution of YPO4 in CePO4 and in LaPO4, respectively. Given the molar volumes of YPO4, CePO4, and LaPO4 (42.96, 45.26, and 46.17 cm3, respectively (Aldred 1984)), the volumetric deviation from the ideal behavior is ∼1% in both cases. This roughly corresponds to the maximum deviation from Vegard’s law for the lattice parameters of ∼0.3% (at 0.5 concentration). Since most measurements are for more dilute solutions, the deviation from Vegard’s law becomes even less apparent. Accordingly, linear dependence of lattice parameters on composition is commonly observed for monazite and xenotime solid solutions (Gratz and Heinrich 1997, 1998; Andrehs and Heinrich 1998; Mogilevsky et al. 2006b) and these solid solutions are sometimes described as ideal (see Kolitsch and Holtstam 2004 for a review of studies on solid solutions in REEXO4 systems). It should be noted, however, that at least in one study (Van Emden et al. 1996), a considerable positive V ex was reported for the NdPO4–YPO4 system. However, as was pointed out in Sect. 5, the data for this system might have been collected from solution-precipitated specimens that had not reached equilibrium and thus contained Y in excess of the equilibrium solubility level.

Application to multicomponent systems

The agreement with the available experimental data demonstrates that the approach presented above is applicable to binary monazite–xenotime systems. This may be sufficient for the prospective applications in CMCs. However, the systems of interest in geological research are multicomponent. Rigorous treatment of a multicomponent system would require solving large systems of equations with multiple interaction parameters. A simplified treatment of multicomponent systems can be achieved with the present approach, provided some a-priori information is available on the distribution coefficients of the components involved.

Let us consider, as an example, the synthetic multicomponent system described by Andrehs and Heinrich (1998), whose composition was chosen to match a natural sedimentary rock (metapelite) sample (Heinrich et al. 1997). Elemental analysis of the monazite and xenotime phases indicated that monazite-forming light rare earth elements (LREE) from La to Sm were predominantly incorporated into the monazite phase, while xenotime-forming heavy rare earth elements (HREE), as well as Y and Gd, were predominantly incorporated into the xenotime phase. The mean ion radii of LREE and HREE (plus Y and Gd) in the monazite phase were nearly independent of temperature [4] and equal to \({\bar{R}_{m}=1.1352\pm 0.0002}\)  Å and \({\bar{R}_{x}=1.0193 \pm 0.0004}\)  Å (the uncertainties are the standard deviations based on 10 original composition measurements, (Andrehs and Heinrich 1998)). Figure 9 shows the comparison between the solid solubility calculated using the equations given in (4) for a hypothetical pseudo-binary LREE monazite/HREE xenotime system having the effective ionic radii \({\bar{R}_{m}}\) and \({\bar{R}_{x}}\) and the experimental data (Heinrich et al. 1997; Andrehs and Heinrich 1998) expressed as the mole fraction of HREE in monazite and the mole fraction of LREE in xenotime. It is seen that the model captures the experimental trend for HREE solubility in monazite fairly well. The solubility of LREE in xenotime is underestimated by a factor of 2–3.

Fig. 9
figure 9

Solid solubility (mole fraction of HREE in monazite and mole fraction of LREE in xenotime) in a multicomponent system. Symbols are experimental data (Heinrich et al. 1997; Andrehs and Heinrich 1998), solid lines are equations given in (4)

Full size image

Partitioning of trace elements between monazite and xenotime

Partitioning of trace elements between monazite and xenotime is an important geological indicator (Gratz and Heinrich 1998; Andrehs and Heinrich 1998; Seydoux-Guillaume et al. 2002; Wood and Blundy 1997; Blundy and Wood 2002, 2003). The partition (distribution) coefficient of a trace element i between monazite and xenotime (D i = x m i /x x i ) can be expressed in the form:

$$ D_{i} = \exp {\left[ {\frac{{\Omega_{x} - \Omega_{m}}}{{RT}}} \right]} $$
(18)

where Ω m and Ω x are the interaction energies of the solid solution of the trace element in the monazite and xenotime phases, respectively. These can be calculated as a function of temperature and pressure, as described above, for any trace element and any monazite–xenotime pair. Figure 10 shows a comparison of the calculated distribution coefficients and the experimental values from Andrehs and Heinrich (1998). As in the previous section, the calculations were conducted using the mean ion radius of monazite as \({\bar{R}_{m}=1.135}\)  Å and that of xenotime as \({\bar{R}_{x}=1.019}\)  Å. It should be noted that the calculated distribution coefficients depend strongly on the parameters R tr and \({\Delta h_{{\rm TbPO_{4}}}}\) in Eq. 5. For that reason, the calculated values of the partition coefficients were best-fit to the experimental data using R tr as a fitting parameter (\({\Delta h_{{\rm TbPO_{4}}}}\) was fixed at −2.6 kJ/mol). This resulted in the best-fit value of R tr =  1.04836 Å, once again between the ion radii of Tb and Gd and very close to the values of R tr = 1.0498 and R tr = 1.0455 Å that were evaluated in Sects. 3 and 5, respectively.

Fig. 10
figure 10

Calculated (Eq. 18) and experimental (Andrehs and Heinrich 1998) distribution coefficients of rare earth elements plus Y between monazite and xenotime versus REE ion radius

Full size image

The temperature dependence of the calculated distribution coefficients of Gd, Nd, and Y is shown in Fig. 11a along with the experimental data (Heinrich et al. 1997; Gratz and Heinrich 1997; Andrehs and Heinrich 1998). Note that while the agreement with the experimental values is very good for Gd and Nd, there is a systematic difference for Y. This is due to the fact that the experimental data for Y were obtained on systems where Y was a major component and not a trace element (Heinrich et al. 1997; Gratz and Heinrich 1997; Andrehs and Heinrich 1998). In this case, Eq. 18 does not apply and the data agree much better with the equations given in (4) (dashed line in Fig. 11a). The distribution coefficient decreases with temperature for monazite-forming elements (La–Eu), increases for xenotime-forming elements (Tb–Lu plus Y), and is virtually temperature independent for Gd. Because of its relative temperature insensitivity, the partition coefficient of Gd may not be a suitable indicator for geothermometry (Wood and Fraser 1978). D Nd, on the other hand, varies by a factor of ∼3 from 450 to 800°C, and D Y varies by a factor of ∼7 from 300 to 850°C. It should be noted, however, that both the present results and the experimental data (Andrehs and Heinrich 1998) on the temperature dependence of D Gd are in disagreement with other data (Heinrich et al. 1997; Gratz and Heinrich 1998), where a substantial temperature dependence of D Gd has been reported between 450 and 1,000°C. There is no clear explanation for this difference (Andrehs and Heinrich 1998). It seems unlikely, however, that it is related to the incorporation of U, Th, and Si in the xenotime phase in natural samples (Andrehs and Heinrich 1998), because similar results were obtained for synthetic CePO4–YPO4–GdPO4 samples that presumably contained no U, Th, or Si (Gratz and Heinrich 1998).

Fig. 11
figure 11

a Temperature dependence of DNd, DGd, and DY. Solid lines correspond to Eq. 18, symbols are experimental values (Heinrich et al. 1997; Gratz and Heinrich 1997; Andrehs and Heinrich 1998). Dashed line is DY calculated according to the equations given in (4); b Calculated pressure dependence of the distribution coefficients of La, Nd, Gd, Y, and Lu according to the present model

Full size image

The pressure dependence of the distribution coefficient, relative to its value at normal pressure, for a number of rare earth elements is shown in Fig. 11b. The distribution coefficients of smaller elements (HREE) are somewhat more pressure-sensitive, but compared to the difference in the absolute values, the difference in the pressure dependence of the distribution coefficient is relatively small. In all cases, the distribution coefficient increases by about 15–40% between 0 and 15 kbar. As a pressure indicator, therefore, Gd has the advantage of providing temperature independent measurements (Wood and Fraser 1978). Also, because its partition coefficient is close to 1, its value is better assessed than for La and Ce, and Yb and Lu, which have very low concentrations in either xenotime or monazite, respectively (Andrehs and Heinrich 1998).

Conclusions

A regular solution model has been applied to solid solubility in monazite–xenotime systems and its pressure dependence. The parameters of the model were chosen based on the available thermodynamic data and elastic properties of pure compounds. Where no experimental data were available, linear dependence of the thermodynamic and elastic properties of monazites and xenotimes on the rare earth ion radius was assumed. The model has been verified by comparison with the available experimental data on solid solubility in a number of monazite–xenotime systems.

Good agreement between the model and the available experimental results suggests that the model could be used to predict the solubility in other monazite–xenotime systems, as well as its pressure dependence. The degree of mutual solid solubility and the asymmetry of the miscibility gaps depend on the difference in the ion radii in the usual way, but also on how close the particular ions are to the “transition point” between the monazite and xenotime structures. Pressure generally increases the solubility of xenotime in monazite and decreases the solubility of monazite in xenotime, shifting the miscibility gap towards the xenotime side of the phase diagram.

The model indicated complete solid solubility in all xenotime/xenotime and monazite/monazite systems, in agreement with the available experimental data for the CePO4–GdPO4 system.