research papers
Misfitinduced changes of lattice parameters in twophase systems: coherent/incoherent precipitates in a matrix
^{a}Max Planck Institute for Intelligent Systems (formely MPI for Metals Reseach), Heisenbergstrasse 3, Stuttgart, 70569, Germany, ^{b}Robert Bosch GmbH Heat Treatment Processes and Heat Treatment Technology (CR/APM4), Postbox 300240, Stuttgart, 70442, Germany, and ^{c}Institute for Materials Science, University of Stuttgart, Heisenbergstrasse 3, Stuttgart, 70569, Germany
^{*}Correspondence email: m.akhlaghi@is.mpg.de
Elastic accommodation of precipitationinduced or thermally induced misfit leads to latticeparameter changes in crystalline multiphase systems. Formulae for calculation of such misfitinduced latticeparameter changes are presented for the aggregate (matrix + secondphase particles) and for the individual matrix and second phase, recognizing the occurrence of either coherent or incoherent diffraction by the matrix and secondphase particles. An overview and an (re)interpretation on the above basis is presented of published latticeparameter data, obtained by Xray diffraction analyses of aggregates of matrix plus secondphase particles. Examples for three types of systems consisting of a matrix with misfitting secondphase particles are dealt with, which differ in the origin of the misfit (precipitation or thermally induced) and in the type of diffraction (coherent or incoherent diffraction of matrix plus secondphase particles). The experimental data are shown to be in good to very good agreement with predictions according to the current treatment.
Keywords: latticeparameter change; precipitationinduced misfit; thermally induced misfit; elastic/plastic misfit accommodation; coherent/incoherent diffraction.
1. Introduction: micro and macrostrains; coherent and incoherent diffraction
The complex interplay of the chemical Gibbs energy change driving solidstate phase transformations and the deformation energy associated with the elastic/plastic accommodation of the misfit of parent and product phases hinders the prediction of the course and the kinetics of phase transformations (Starink et al., 1993; Mittemeijer, 2010; Song et al., 2014). Accordingly, unusual nonequilibrium phenomena can occur: occurrence of metastable phases, nonmonotonic variation of transformation rate and unusual morphologies (Thompson et al., 1994; Biglari et al., 1995; Sennour & Esnouf, 2003; Liu et al., 2003; Meka et al., 2013; Loewy et al., 2014; Villa et al., 2014). Therefore, it is essential to acquire a fundamental understanding of the effects of developing misfitstrain energies on the thermodynamics and kinetics of phase transformations. Only by doing so can the resulting microstructure and properties of engineering components be optimized (Abbaschian et al., 2010; Porter & Easterling, 1982; Guo & Sha, 2005). Thus the evolution of shape changes during precipitate growth in an elastically anisotropic, infinite matrix has been investigated for different twophase systems [e.g. nickelbased alloys (Wang & Khachaturyan, 1995; Ratel et al., 2006, 2010; Ardell, 2014) and piezoelectric piezomagnetic composites (Ni & Khachaturyan, 2007)].
The elastic strain field in the matrix surrounding very small (of the order of 1 nm) misfitting particles/point defects can be investigated by highresolution ). Larger sized misfitting particles can be associated with more extended misfitstrain fields, which evoke contrast in diffractioncontrast images recorded by (Bor et al., 2002) and broadening of diffraction lines [van Mourik et al., 1988; van Berkum et al., 1996; see in particular ch. 4 of Mittemeijer & Welzel (2012)].
and by Xray diffraction and diffuse scattering (static Debye–Waller factor; Krivoglaz, 1969The application of continuum elasticity theory to describe the elastic accommodation of the misfit of the (parent) matrix and the (product) secondphase particles (Eshelby, 1954, 1956; Khachaturyan, 1983) allows the calculation of microstrains, say the variance of the local strain 〈e^{2}〉 (van Mourik et al., 1988; van Berkum et al., 1996; Mura, 1982), and of the average (hydrostatic) macrostrain 〈e〉 (van Mourik et al., 1983). The microstrain reveals itself through (extensive) diffractionline broadening. The hydrostatic macroscopic strain causes a (distinct) change of the (average) lattice constant of a phase. Countless observations of diffractionline broadening due to microstrain have been reported; in fact, analysis of diffractionline broadening has become a standard method of microstructural characterization (Mittemeijer & Welzel, 2012). However, little experimental work has been devoted to the change of the (average) macroscopic lattice parameter by misfit strain (Mittemeijer et al., 1981; Mittemeijer & van Gent, 1984; Zhou et al., 1991; Müller et al., 1997; Timmermans & Froyen, 1999; Chechenin et al., 2002; Bruno et al., 2003; Lu et al., 2007). Against the above background, this manuscript presents a review of misfitinduced (average, macroscopic) latticeparameter changes, as predicted and as observed. In this context in particular the distinction of cases of coherent and incoherent diffraction by secondphase/matrix assemblies is emphasized.
The coherent or incoherent nature of the diffraction by a second phase (such as precipitate particles) with respect to the diffraction by the matrix needs not coincide with the occurrence of a coherent or incoherent nature of the interfaces between the secondphase particles and the matrix. The coherency of diffraction depends not only on the specimen microstructure but also on the diffraction conditions, such as the length of the diffraction vector (van Berkum et al., 1996; Rafaja et al., 2004). Coherency/incoherency of diffraction depends on whether or not constructive interference of waves scattered by separate parts of the diffracting material (matrix and second phase) occurs. For a detailed discussion on the origin of coherent and incoherent diffraction of a secondphase particle and the matrix the reader is referred to van Berkum et al. (1996) [see also Mittemeijer (2010)]. The effect of latticeparameter variation/the distribution of lattice spacings in a diffracting material on the occurrence of coherent or incoherent diffraction (and intermediates thereof) is considered most generally by Leineweber & Mittemeijer (2010).
Provided the matrix and secondphase particles diffract independently, i.e. incoherent diffraction of the matrix and of the secondphase particles occurs, the changes of the lattice parameters of the matrix and of the secondphase particles can be determined from measurements of the positions of the (separate) Xray diffraction peaks of the matrix and of the second phase. If a coherent nature of the secondphase particles/matrix interfaces prevails, then, depending on the length of the diffraction vector and the extent of misfitstrain variation, coherent diffraction by secondphase particles and the matrix can occur.
2. Latticeparameter changes due to elastically accommodated misfit
The continuum theory for the fully elastic accommodation of the misfit of a point imperfection in a matrix was originally developed by Eshelby (1954, 1956). This theory is more likely to be applicable to the case of precipitation of misfitting entities of larger dimensions, such as secondphase particles (i.e. a block of atoms), which are more likely to exhibit elastic characteristics as for bulk materials. The formulae presented below describe the changes of the relative volume/lattice parameters of the matrix, of the misfitting particles and of the assembly, i.e. matrix plus misfitting particles [for more information see Akhlaghi et al. (2015)].
2.1. Change of the lattice parameter of the matrix
For the case of an isotropic cubic finite matrix A with a (strainfree) lattice parameter a_{A}, the change of lattice parameter, , of the finite matrix due to the introduction of misfitting isotropic inclusions B can be written as (Christian, 2002; Mittemeijer et al., 1981)
with
where represents the linear misfit parameter (see below), K_{} and represent the bulk modulus and the respectively, and y_{B} is the of inclusions. It is essential to recognize that the matrix is of finite dimensions (Akhlaghi et al., 2015).
2.2. Change of the lattice parameter of the misfitting phase
For the case of isotropic secondphase particles of cubic a_{B}, the change of the lattice parameter, , of the misfitting second phase in a finite matrix is (van Mourik et al., 1985)
with a (strainfree) lattice parameter2.3. Change of the lattice parameter of the assembly (matrix plus misfitting phase)
For a finite aggregate of cubic a, the resulting change of the lattice parameter, , of the aggregate for a y_{B} of inclusions B is given by (Christian, 2002)
and a (strainfree) lattice parameter of the matrix,with
2.4. The misfit parameter
The linear misfit parameter appearing in the above equations can be given an explicit analytical expression in the following cases:
(i) Thermal misfit. This misfit originates from a change in the temperature of the two (or multi)phase composite (matrix + secondphase particles) and is due to the difference in linear
coefficients of the matrix () and the secondphase precipitate (). This type of misfit can be described by the linear thermal misfit parameter, :where T_{h} and T_{r} stand for the elevated heattreatment temperature and the considered low (e.g. room) temperature, where measurements are made, respectively.
(ii) Precipitationinduced misfit. This misfit originates from the different specific volumes of (solute and solvent) atoms in the second phase (precipitate) and the (solute and solvent) atoms in the matrix. This type of misfit can be described by the linear volume misfit parameter, :
where v_{B}^{0} and v_{A}^{0} are the volume of the strainfree precipitate particle and the volume occupied by the atoms of the precipitate particle as previously incorporated in the matrix, respectively.
3. Observations and discussion
At elevated (aging/annealing) temperature (T_{h}), development of a misfitting second phase can occur from a supersaturated matrix. The volume misfit () of the two phases can initially be accommodated elastically under preservation of a coherent interface. Upon prolonged annealing, leading to an increasing size of the secondphase particles, this misfit can no longer be accommodated fully elastically, and the interface can become semi or fully incoherent (Porter & Easterling, 1982; Quek et al., 2011; Meka et al., 2013; Geslin et al., 2014).
Upon cooling from elevated temperature (T_{h}) to room temperature (T_{r}), irrespective of whether the precipitationinduced misfit at elevated temperature is accommodated elastically or plastically, thermal misfit arises which has to be accommodated either elastically (Mittemeijer et al., 1981) or plastically (van Berkum et al., 1991). In many cases, upon fast cooling of the specimens, the thermal misfit accommodation occurs mainly elastically (even if an incoherent matrix/secondphase particle occurs), and it can remain in that state for a long time owing to very slow rates of plastic relaxation processes at low (room) temperature.
As long as the (transformation or thermal) misfit is accommodated elastically, its effect on the expansion/contraction of the system consisting of the matrix and misfitting secondphase particles can be quantified by considering the model description presented in §2. Thus, in the following, three types of system consisting of a matrix with misfitting secondphase particles are dealt with (see Table 1), which differ in the origin of misfit (transformation induced or thermally induced) and in the type of diffraction (coherent or incoherent diffraction of matrix plus secondphase particles). For the different systems described in the following, the values of the constants and misfit parameters used for the calculation of latticeparameter changes of the matrix [equation (1)], the precipitate [equation (3)] and the assembly [equation (4)] are presented in Tables 2 and 3, respectively.
‡b.c.t. – bodycentred tetragonal. 

3.1. Coherent diffraction by the assembly matrix plus secondphase particles: transformation misfit
3.1.1. Misfitting coherent alloying element nitrides in ferrite
Upon nitriding of ferritic Fe–Me (Me = Cr, V, Ti) alloys, tiny nitride precipitates (MeN) develop (Schacherl et al., 2002; Miyamoto et al., 2006; Vives Díaz et al., 2008; Jack, 1976; Rickerby et al., 1986). At least initially these nitrides possess coherent interfaces with the matrix. Matrix/nitrideparticle misfit in this system, as considered at room temperature, originates from (i) the specific volume misfit induced by nitride precipitation from a supersaturated matrix as a consequence of elastic accommodation of the misfit while the precipitate/matrix interface remains coherent and (ii) the thermal misfit induced by cooling after nitriding. The specific volume misfit between the MeN precipitates and the ferrite matrix can be calculated from equation (7), where v_{B}^{0} and v_{A}^{0} can now be taken as the molar volumes of the MeN precipitates and the matrix, respectively. After complete precipitation, the molar volume of the MeN precipitates can be calculated from the volume of the precipitate divided by the number of metal atoms occupying one of the precipitate, and the molar volume of the matrix (in this case, containing only Fe atoms) can be calculated from the volume of the matrix divided by the number of iron atoms occupying one of the matrix. In this case the thermal misfit (corresponding to the difference of the coefficients of the Fe matrix and the developed nitrides) is negligible as compared to the precipitationinduced misfit (i.e. and for cooling from 673 K to room temperature for nitrided Fe–Cr alloy and and for cooling from 673 to 298 K for nitrided Fe–V alloys; Basinski et al., 1955; Samsonov, 1964).
This system is an example of coherent diffraction by the entity matrix plus precipitates in the presence of coherent interfaces between the Fe matrix [of bodycentred cubic (b.c.c.) crystal structure] and the MeN precipitates (of NaCltype crystal structure) . Hence, the overall expansion of the assembly (composed of MeN nitride precipitates and ferrite matrix), expressed in terms of the change in the lattice parameter deduced from the position of the `ferritic' peak maxima, can be calculated using equation (4) as a function of the of alloying element nitrides. Experimental values for such changes of lattice parameter upon nitriding of Fe–Cr and Fe–V alloys were obtained by employing Xray diffraction on homogenously nitrided (thus macroscopically strainfree) Fe–Me specimens (Akhlaghi et al., 2015).^{1} The of (CrN and VN) nitride precipitates was determined (by weightchange measurements) from the N content of denitrided thin foils [i.e. after removing from the specimen, by denitriding, the dissolved (excess) N and the excess N adsorbed at the nitrideplatelet faces during nitriding; Somers et al., 1989, Podgurski & Davis, 1981]. The of precipitates was varied by varying the nitriding time (for times larger than the minimal time to achieve a homogenously nitrided specimen) and/or varying the amount of Me (Cr, V) in the alloy (realizing full precipitation throughout the specimen). The results are shown in Fig. 1 (the single dots). The predictions on the basis of equation (4) (solid lines in Fig. 1) are in good agreement with the experimental data. Evidently, in this case equation (1) (implying incoherent diffraction of matrix and precipitates; dashed lines in Fig. 1) does not at all correctly predict the ferritic (Xray) latticeparameter changes observed for the nitrided Fe–Cr and Fe–V alloys.
3.1.2. Misfitting coherent cobalt precipitates in copper
A thin film of a metastable Cu–Co et al., 1992; Michaelsen, 1995).
can be prepared by codeposition of Cu and Co by magnetron sputtering up to a maximum of 12 at.% Co. For Co contents larger than 12 at.%, clusters/tiny precipitates of Co occur in the alloy (BerkowitzThe Xray diffraction patterns recorded from such Cu–Co alloy films of different Co concentrations show the occurrence of a single facecentred cubic (f.c.c.) Cu(Co) phase up to 65 at.% Co. Only beyond this Co concentration can additional hexagonal closepacked (Co) reflections be observed [see Fig. 5 of Michaelsen (1995)]. Evidently, the Co inclusion/particles present in the Cu–Co alloy films with more than 12 at.% Co and up to 65 at.% Co diffract coherently with the matrix.
The experimentally obtained lattice parameters of the f.c.c. Cu(Co) films as a function of Co concentration are shown as single dots in Fig. 2 for Co concentrations between 12 and 65 at.%. The predictions of the latticeparameter values of the aggregate Cu matrix (Cu 12 at.% Co solid solution) and secondphase Co particles as a function of the volume percentage of Co precipitates in the Cu–12 at.% Co matrix according to equation (4) (coherent diffraction of matrix and inclusions) and equation (1) (incoherent diffraction of matrix and inclusions) are also shown in Fig. 2. Evidently, the experimentally determined latticeparameter values and the predictions on the basis of equation (4) (solid black line in Fig. 2) are in very good agreement, implying occurrence of coherent diffraction of matrix and precipitates. In this case predictions according to equation (1) (dashed line in Fig. 2) are not at all in agreement with the experimental data. The prediction of the lattice parameter of f.c.c. Cu(Co) can also be made on the basis of Vegard's law [linear interpolation between Cu and f.c.c. Co lattice parameters as shown by Michaelsen (1995)].
3.2. Incoherent diffraction of the matrix and secondphase particles: transformation misfit
3.2.1. Misfitting incoherent nitrides in ferrite
Upon nitriding of pure αFe (ferrite) in an atmosphere of a certain nitriding potential, N dissolves in the ferrite matrix. The formation of this leads to an increase of the lattice parameter of the ferrite matrix. By quenching after nitriding the can be retained. The system is then metastable. During aging of the αFe matrix supersaturated with dissolved N at room temperature, formation of α′′Fe_{16}N_{2} precipitates occurs in association with depletion of N from the ferrite matrix. Experimentally, a decrease of the ferrite lattice parameter is observed at room temperature during development of the α′′ precipitates (Mittemeijer et al., 1980; Mittemeijer, 1981; Ferguson & Jack, 1983; Mittemeijer & van Gent, 1984). Upon prolonged aging, after complete precipitation of α′′, a constant latticeparameter value is established that is larger than the one expected for pure αFe (the equilibrium solubility of N in αFe at room temperature is practically nil) [see Fig. 4 of Mittemeijer (1981), Fig. 1 of Mittemeijer et al. (1980) and Fig. 7 of Ferguson & Jack (1983)].
The values of the constant lattice parameter of the ferrite matrix at this stage of prolonged aging at room temperature, i.e. after all N has precipitated, as recorded from specimens containing different amounts of nitrogen, are shown as a function of the of developed α′′ precipitates in Fig. 3. Predictions for the change of the ferritic lattice parameter are also shown in Fig. 3: (i) for the case of incoherent diffraction of matrix and precipitates [equation (1); dotted line in Fig. 3] and (ii) for the case of coherent diffraction of the assembly matrix plus precipitates [equation (4); full line in Fig. 3]. The experimental results very well agree with the predictions according to equation (1): evidently, in this case of prolonged ageing, the αFe matrix and the α′′nitride precipitates diffract incoherently. The separate α′′ reflections are very weak,^{2} because of the small amounts of N (and thus α′′) in the specimens, and cannot be observed by conventional laboratory Xray diffraction [as is the case here; synchrotron radiation would be required as shown by van Genderen et al. (1993)].
As mentioned above, before complete precipitation of α′′, i.e. in the intermediate stages of aging, the lattice parameter of ferrite decreases continuously as the ferritelattice contraction by the depletion of N, dissolved on the interstitial (octahedral) sites of the ferrite matrix, is larger than the ferritelattice expansion due to the formation of misfitting α′′ precipitates diffracting either coherently or incoherently with the matrix. Thus, during precipitation of α′′, a (possibly occurring) change of the diffraction conditions from coherent to incoherent will only lead to a change in magnitude of the resulting latticeparameter decrease. Corresponding calculations for the change of the ferritic lattice parameter as a function of the fraction transformed (i.e. the precipitated fraction of N), for a fixed N content of the specimen, are shown in Fig. 4: (i) for the case considering both the effect of the depletion of dissolved N from the [equation (1) of Mittemeijer et al. (1980)] and incoherent diffraction of matrix and precipitate [equation (1); dashed line in Fig. 4], (ii) for the case considering both the effect of the depletion of dissolved N from the [equation (1) of Mittemeijer et al. (1980)] and coherent diffraction of assembly matrix plus precipitates [equation (4); full line in Fig. 4], and (iii) for the case considering only the effect of the depletion of dissolved N from the [equation (1) of Mittemeijer et al. (1980); dotted line in Fig. 4]. Evidently, ignorance of the effect of misfit strain on latticeparameter changes results in a strongly erroneous (too small) precipitated fraction of N as derived from the decrease of lattice parameter.
3.3. Incoherent diffraction of the matrix and secondphase particles: thermal misfit
3.3.1. Misfitting incoherent silicon in aluminium
During hightemperature annealing/aging of an Al matrix supersaturated with dissolved Si, precipitation of incoherent Si occurs. Initially, lattice expansion of the Al matrix happens owing to partially elastic accommodation of the precipitate/matrix misfit [cf. equation (1)] and depletion of Si from the [see Fig. 9 of van Mourik et al. (1985)]. This same process also results in lattice contraction for the Si precipitates [cf. equation (3); see Fig. 7 of van Mourik et al. (1985)]. Upon prolonged aging at high temperature, this precipitationinduced misfit relaxes fully, i.e. is accommodated entirely plastically. Then a case of incoherent diffraction of matrix and precipitates in the presence of incoherent interfaces between the Al matrix (with f.c.c. crystal structure) and the Si precipitates (with diamond cubic crystal structure) is established.
Now, upon rapid cooling of such a relaxed twophase (pure Al matrix + incoherent Si particles) specimen to room temperature, thermal misfit develops, owing to the difference of the (a) (dots). The prediction (now) on the basis of equation (1) [dashed line in Fig. 5(a)] is in good agreement with the experimental data. Evidently, in this case equation (4) [implying coherent diffraction of matrix and precipitate; solid line in Fig. 5(a)] does not at all correctly predict the latticeparameter change of the aluminium matrix. An only reasonable agreement occurs for the theoretical [equation (3)] and experimental values for the latticeparameter change of the silicon precipitates (Fig. 5b); significant experimental errors are inherent in the determination of the minorityphase lattice parameter of the Si precipitates, in particular for the alloy of the lowest Si content of 2.8 vol.% (cf. Fig. 5b).
coefficients of the Al matrix and Si precipitates; this misfit ( for cooling from 425 K to room temperature) is accommodated elastically. The changes of the lattice parameters of the matrix and the precipitates were measured as a function of the of developed precipitates (Si), utilizing the separate diffraction peaks from the matrix and precipitates as recorded from fastcooled (to suppress any relaxation of the thermal misfit during cooling) twophase specimens of different Si contents. The thus experimentally obtained latticeparameter change for the Al matrix is shown as a function of the of Si precipitates in Fig. 53.3.2. Misfitting incoherent alloying element nitrides in ferrite
As described in §3.1, upon nitriding Fe–Cr alloys, misfitting CrN particles develop which initially have a coherent interface with the matrix, leading to an overall lattice expansion as observed from the change of the `ferrite' lattice parameter (cf. Fig. 1). Upon aging (at the nitriding temperature), coarsening of the precipitates occurs [see Fig. 5 of Steiner et al. (2015)] in association with the development of incoherent matrix/precipitate interfaces: plastic accommodation of the precipitationinduced misfit. Consequently, the system in such conditions exhibits incoherent diffraction of the matrix and precipitates [i.e. the matrix and coarsened precipitates now diffract separately: indeed both matrix and precipitate reflections are detected; see Fig. 11 of Steiner et al. (2015)].
Upon cooling of this relaxed twophase composite (Fe matrix + incoherent CrN nitrides) to room temperature, thermal misfit develops, owing to the difference of the . A very good agreement occurs (Table 4). The change of the lattice parameter of the precipitates has also been measured by Xray diffraction and has been predicted on the basis of equation (3). As predicted, the experimental change of the lattice parameter of the precipitates has a sign opposite to that of the matrix (cf. Table 4). The quantitative agreement here is less good, which can be ascribed to uncertainty concerning the strainfree lattice parameter of CrN (ICDD, 2002) and limited accuracy in the experimental determination of the lattice parameter of the minority phase.
coefficients of the Fe matrix and CrN nitrides; this misfit ( for cooling from 773 K to room temperature) gets accommodated elastically. The change of the lattice parameter of the ferrite matrix by elastic accommodation of this thermal misfit has been measured by Xray diffraction and has been predicted on the basis of equation (1)4. Conclusions
Formulae have been presented for the calculation of latticeparameter changes induced in misfitting matrix/secondphaseparticle systems.
Diffraction analysis of such latticeparameter changes requires distinction of (i) coherent diffraction by the matrix/secondphaseparticle aggregate and (ii) incoherent diffraction of the matrix and secondphase particles.
A number of examples presented for cases of precipitationinduced specific volume misfit and coolinginduced thermal misfit show good to very good agreement of theoretical predictions and experimental observations.
The discussed misfitinduced latticeparameter/volume changes are often ignored in the diffraction (and also in the dilatometric) analysis of phasetransformation kinetics. The present treatment provides a direct route for correction of these effects.
Footnotes
‡Current address: Metallurgical and Materials Engineering Department, Indian Institute of Technology Roorkee, Roorkee 247667, India.
^{1}No separate nitride reflections could be discerned; the strongly broadened ferritic peaks represent coherent diffraction by the distorted (by elastic misfitstrain accommodation) ferrite matrix and the nitride precipitates [for details see Akhlaghi et al. (2015)].
^{2}The α′′ can be conceived as a developing in nitrogen ferrite by ordering of N on the of octahedral interstices. The intensity of the α′′structurespecific reflection is governed by the weak scattering power of the N atoms.
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