Heterogeneous Zr solute segregation and Al3Zr dispersoid distributions in Al–Cu–Li alloys

Available online at www.sciencedirect.com

ScienceDirect Acta Materialia 93 (2015) 73–86 www.elsevier.com/locate/actamat

Heterogeneous Zr solute segregation and Al3Zr dispersoid distributions in Al–Cu–Li alloys ⇑

D. Tsivoulas and J.D. Robson School of Materials, The University of Manchester, Manchester M13 9PL, UK Received 27 December 2014; revised 14 March 2015; accepted 28 March 2015

Abstract—During the homogenisation treatment of Al alloys the Al3Zr phase is known to form heterogeneously in interdendritic areas where the Zr supersaturation is low. Several types of clusters were observed in the present Al–Cu–Li alloy. Although some clusters resemble the shape of the h0 Al2Cu lath-shaped particles, it is explained here that there is no direct nucleation on these particles and neither is Zr contained in them at amounts detectable via TEM-EDX. These planar arrays of Zr dispersoids were established to form via repeated precipitation on dislocations. Nucleation on dislocations was the dominant mechanism for individual Al3Zr dispersoids also in the dendrite centre. This was explained on the grounds of the large atomic size misfit between Zr and the Al matrix which leads to segregation of the former atoms to dislocations and was verified experimentally by EDX. It is noteworthy that although Zr did not interact with the h0 phase, it did so with the equilibrium h phase and produced two different types of particles, one containing only Zr, and another having both Zr and Mn. It was also seen to be contained within Al20Cu2Mn3 dispersoids in agreement with previous findings. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Homogenisation; Microsegregation; Precipitation; Al3Zr dispersoids; Al–Cu–Li alloys

1. Introduction Zirconium additions in Al alloys aim at minimising the degree of recrystallisation and improving mechanical properties via the formation of fine and coherent Al3Zr dispersoids. The distribution of this phase is crucial to the material’s mechanical properties, thus the homogenisation process needs to be carefully tailored to produce a large volume fraction of these fine-sized particles that would occupy the majority of the grain volume. However, such a task is difficult due to the strong microsegregation of Zr solute within a dendrite during casting [1]; the dendrite core is enriched in Zr, while a much lower concentration is present towards the dendrite edge. The direct effect of Zr microsegregation within a dendrite is the large variation of the dispersoid density across a grain even after homogenisation from the as-cast state. An additional characteristic is that the actual microsegregation pattern of Zr solute across a homogenised grain is not monotonic and some fluctuations in concentration are observed when measured experimentally, due to the growth of secondary or tertiary dendrite arms that contribute towards the retention of this variation at a finer scale [2]. Regarding the formation of the Al3Zr dispersoids, they tend to produce a homogeneous distribution in the dendrite centre where the Zr supersaturation is high and a

⇑ Corresponding author.

heterogeneous distribution in interdendritic areas where the supersaturation is significantly lower. Heterogeneous nucleation and various cluster morphologies have been observed by several researchers in the past. Robson and Prangnell [2] mentioned as a side observation the presence of Al3Zr clusters with similar shape to the g0 phase in interdendritic regions of a 7050 alloy, which were attributed to preferential dispersoid nucleation on the g0 precipitates. Knipling et al. [3] employed the theory on morphological changes of a surface of revolution proposed by Nichols and Mullins [4] to support that the presence of straight, linear clusters of spherical Al3Zr dispersoids resulted from the spheroidisation of rod-shaped particles of the same composition. On the other hand, curved, linear clusters of spherical Al3Zr dispersoids were alleged to form on dislocations, whereas plate-shaped Al3Zr particles were mentioned but not discussed [3]. Jia et al. [5] observed straight, linear arrays of Al3Zr along h0 0 1iAl, which could be what Nes [6] noticed for rod-shaped Al3Zr dispersoids that partially break up into spherical particles, since the alloy composition was similar in these two studies. Jia et al. [5] also reported the presence of two more types of clusters oriented along h0 0 1iAl; helical and lath. The former was said to be associated with dislocations but the latter was claimed to be related to the h0 -Al2Cu phase, although no specific mechanism was given. Another type of heterogeneous Zr segregation is its inclusion in various types of particles and this can also have a detrimental effect to the final dispersoid distribution. The

http://dx.doi.org/10.1016/j.actamat.2015.03.057 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

74

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

most striking example was shown previously by the present authors for the case of Al20Cu2Mn3 dispersoids containing small amounts of Zr and decreasing the recrystallisation resistance of an AA2198 sheet [7]. Other phases reported to contain Zr are the Al7Cu3Zr2 and Al3CuZr2 [8], as well as the Al3Cr and Al6Mn [9]. Such anomalies in the nucleation process of the Al3Zr phase during homogenisation will inevitably affect its distribution in a dendrite. The result is the presence of areas with a low dispersoid density within the grains in the final form of the material, which render it more prone to recrystallisation due to the localised reduction in Zener pinning. Hence the scope of the present work is to provide with a deeper understanding of the heterogeneous segregation patterns of zirconium solute and of the subsequent heterogeneous distribution of Al3Zr dispersoids. Several types of dispersoid clusters are analysed, together with the interaction of Zr with the equilibrium h-Al2Cu phase and the Al20Cu2Mn3 dispersoids. 2. Experimental For the requirements of the present work two experimental, third generation Al–Cu–Li-based alloys with a Zr content of 0.11 wt.% were studied; an AA2050 as-cast billet slice and an AA2198 6 mm thick sheet in the F temper (i.e. as-hot rolled). Both alloys were supplied by Constellium, Centre de Recherches de Voreppe, France. The AA2050 alloy has a very similar composition to the AA2198, as shown in Table 1, and was used here to investigate the Al3Zr formation since the latter alloy was not available in the as-cast condition. The as-cast AA2050 material was homogenised with a ramp heating pattern of 10 h up to 505 °C, followed by an isothermal plateau of 12 h and finally water quenched. On the other hand, the AA2198 material in the F temper was isothermally annealed for 4 min at 535 °C and water quenched. Most of the analysis was performed in a FEG-TEM FEI Tecnai F30 operating at 300 kV, fitted with an Oxford Instruments X-Max 80 energy-dispersive X-ray (EDX) Si-drift detector (SDD), a high angle annular dark field detector for scanning transmission electron microscopy (HAADF-STEM), and an electron energy loss spectrometer (EELS). Additionally, EDX was also carried out with a more advanced XFEGTEM FEI G2 80-200 ChemiSTEM operating at 200 kV and fitted with four Bruker windowless SDD X-ray detectors.

3. Results 3.1. Types of heterogeneous Al3Zr distributions A careful inspection of the grain interior in the as-cast and fully homogenised AA2050 material can give an indication of the differences in the Al3Zr distribution between the dendrite centre and dendrite edge (Fig. 1). This is not

Fig. 1. HAADF-STEM images of Al3Zr distributions taken near the h1 0 0iAl zone axis in a fully homogenised AA2050 sample; (a) dendrite centre, and (b) further away from the dendrite centre.

unexpected since a strong microsegregation pattern within the grains is retained from the as-cast microstructure even after homogenisation. In the former region, a homogeneous Al3Zr dispersoid distribution is apparent, while in the latter there are clusters of various shapes. The area further from the dendrite centre, where the Zr supersaturation is low, is the main point of focus here. From a first glance at Fig. 1b the main observations on the Al3Zr distribution away from the dendrite centre are; (i) mainly rectangular clusters of fine spherical dispersoids appear and they are mostly situated adjacent to elongated Al20Cu2Mn3 dispersoids, both being oriented in the same direction, i.e. h0 0 1iAl, (ii) the Al3Zr diameter is much smaller and the local number density is much higher compared to the dendrite centre, (iii) in certain cases the clusters are filled

Table 1. Nominal compositions of the alloys used in this work (wt.%).

AA2050 AA2198

Cu

Li

Mg

Ag

Zn

Fe

Si

Zr

Mn

3.20–3.90 2.90–3.50

0.70–1.30 0.80–1.10

0.20–0.60 0.25–0.80

0.20–0.70 0.10–0.50

<0.25 <0.08

<0.10 <0.10

<0.08 <0.08

0.110 0.110

0.30 0.30

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

throughout with dispersoids, whereas some others are primarily void, and (iv) occasionally few, discrete, coarse Al3Zr particles are seen at a distance from other dispersoids of the same type. Several different types of Al3Zr clusters were seen in areas further from the dendrite centre, where the Zr supersaturation starts to decrease considerably. One of the most interesting cluster types was of rectangular shape as shown in Fig. 2a. Their shape is identical to that of the coarse h0 Al2Cu plate-like particles (Fig. 2b) which are known to precipitate on the {1 0 0}Al habit plane [10] and are present in the microstructure since the as-cast stage but dissolve upon homogenisation. Thus one might assume that they are related to them, but the actual nucleation mechanism is not as simple as will be explained later on. Such clusters were frequently attached to Mn dispersoids, but they were also observed far from them. Rod-shaped Al3Zr particles were also met occasionally in the fully homogenised microstructure. One such particle is marked with an arrow in Fig. 2a and it seems to have just started spheroidising,

75

judging by the small spherical particles right below its bottom side that are aligned along the same axis. In order to obtain more information on the habit plane of the rectangular clusters, the TEM specimen had to be tilted appropriately to view these arrays edge-on. Fig. 3a presents several rectangular and linear clusters of Al3Zr dispersoids when the sample was tilted to the h2 1 1iAl zone axis. After tilting to the h1 1 0iAl zone axis most clusters became straight and linear, which suggests that they are now viewed edge-on, although the morphology of some did not change and they still appeared linear (Fig. 3b). Several different cases were identified according to their orientation and features. Clusters such as those labelled as A are planar and form on the {0 2 0}Al planes. Clusters B are planar and form on the {1–11}Al planes. Cluster C is linear and with orientation along the h1 1 1iAl. Cluster D consists of two closely-spaced parallel arrays on {0 2 0}Al planes, however they are believed to be separated as one is longer than the other. Finally, cluster E is linear and oriented along a higher index crystallographic axis. All the above are related to dislocations as will be discussed later on. Other types of clusters included ones with very few or no dispersoids in their interior again along the h0 0 1iAl direction (Fig. 4a). In certain areas several blocky Al3Zr clusters were observed, the majority of which consisted of typically less than 10 particles, apart from few cases where more particles could be seen (Fig. 4b). When the TEM specimen was tilted to a two-beam condition near the h1 0 0iAl zone axis, it was observed that all these clusters were attached to dislocation loops and helices (Fig. 4c). These clusters might resemble the cauliflower-shaped Al3Zr dispersoids that Knipling et al. [3] observed at temperatures between 375 and 425 °C, but those were all interconnected and formed one particle, in contrast to the present case where all particles may be closely-spaced but are clearly separated. Apart from these blocky clusters, few individual and very coarse Al3Zr particles could also be seen in the same area to be linked with dislocation loops and helices. In order to quantify the Zr dispersoid distributions locally in areas where heterogeneous nucleation prevails, STEM imaging was combined with EELS for dispersoid size and specimen thickness measurements respectively. The area in Fig. 5 was selected for this analysis and the large cluster was divided into three subsets which were then processed individually. The results are presented in Table 2 and show a remarkable increase locally in dispersoid number density and volume fraction. Compared to the relative values for the dendrite centre, these clusters can have approximately five times higher number density and three times higher volume fraction. On the other hand, the average particle diameter was reduced from 25 to 20 nm within the clusters. This is an indication of a very high local Zr supersaturation, much higher than what is present in the dendrite centre. 3.2. Characterisation of the plate-like particles

Fig. 2. Similarity in the shape of some rectangular Al3Zr clusters with that of the of h0 -Al2Cu particles. (a) Rectangular Al3Zr cluster in the fully homogenised condition, and (b) h0 -Al2Cu particles in the as-cast microstructure.

The next crucial stage was to characterise the plate-like phase in the as-cast microstructure. TEM-EDX maps from an area in the as-cast material are shown in Fig. 6. It is clear that the plates were rich in Cu, but interestingly also contained small amounts of Mn and Fe. Semi-quantitative composition results from selected locations in the matrix and in the Cu-rich particle are presented in Table 3.

76

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

Fig. 3. HAADF-STEM images showing various Al3Zr clusters while tilting the TEM specimen (fully homogenised condition). (a) Rectangular clusters reveal their two-dimensional structure when viewed off their plane axis (h2 1 1iAl z.a.), and (b) rectangular clusters viewed edge-on appear to be linear (h1 1 0iAl z.a.).

Detection of trace Zr in the particles is due to the higher X-ray counts arising from their larger atomic number compared to the matrix. Further evidence was obtained by performing selected area electron diffraction on a particle and from the relevant pattern (Fig. 7) it was confirmed that the plate-like phase was indeed the h0 -Al2Cu [11]. It was mentioned earlier that the h0 -Al2Cu particles have been assumed to be related by some researchers to the formation of rectangular Al3Zr clusters in interdendritic areas. Therefore, two immediate assumptions could be made while bearing this correlation in mind; either Zr solute is contained within the h0 phase and upon its dissolution it leaves behind Zr solute in the matrix which in turn leads to the subsequent precipitation of the Al3Zr dispersoids, or the Zr dispersoids nucleate heterogeneously at the interface of the plate-like h0 particles. The first possibility was

ruled out earlier via the EDX analysis presented in Fig. 6 and in Table 3, after examining a large number of h0 plates. In order to verify the validity of the second assumption, it is of vital importance to monitor the dissolution of the h0 phase during the homogenisation process. To this end, several stages throughout the homogenisation treatment were examined starting from the as-cast microstructure (Fig. 8). In more detail, it can be seen that the as-cast microstructure in (a) contains a large density of coarse h0 particles near the dendrite edge which start dissolving upon increasing the temperature during the ramp heating stage. At 443 °C, in (b), a large fraction of the h0 -Al2Cu particles has dissolved and the remaining plates continue to dissolve as is evident from their irregularly rounded edges. At the end of the ramp heating at 505 °C in (c), there is no more h0 phase remaining and the elongated particles in

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

77

Fig. 5. Calculation of local volume fractions for the Al3Zr clusters in the fully homogenised AA2050 further from the dendrite centre. (a) Location of the analysed area, and (b) three clusters for which Al3Zr volume fractions were calculated.

it proves that it is impossible for the Al3Zr dispersoids to nucleate heterogeneously on the h0 particles, as the latter have dissolved several hours earlier during the homogenisation process. Hence a different nucleation mechanism needs to be sought. Fig. 4. HAADF-STEM images showing events of heterogeneous Al3Zr precipitation slightly further from the dendrite centre after full homogenisation (near h1 0 0iAl z.a.). (a) Rectangular cluster without any particles in its interior, (b) blocky clusters of Zr dispersoids, and (c) dislocation helices and loops are linked to each cluster and to coarse individual particles from (b).

the microstructure are Al20Cu2Mn3 dispersoids. Careful examination of the dendrite centre in (d) proved that no Al3Zr dispersoids had formed by the end of the ramp heating stage. The Al3Zr phase starts precipitating only after 8 h during the isothermal plateau at 505 °C, as shown in (e, f), which is much later than the dissolution of the Al2Cu phase. This observation is of huge significance since

3.3. EDX analysis of Zr solute segregation Following the EDX findings that showed Zr solute not being contained within any h0 particles out of a large number that was analysed, an investigation was carried out on the likelihood for the presence of locally high Zr concentrations in the matrix prior to the precipitation of the Al3Zr phase. Such evidence would help define the dominant nucleation mechanism for Al3Zr. Fig. 9 presents EDX maps obtained with an FEI Titan ChemiSTEM XFEGTEM equipped with four windowless SDDs for enhanced chemical analysis sensitivity. The sample analysed was at the end of the ramp homogenisation stage at 505 °C, where the h0 particles had completely dissolved and no Al3Zr

78

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

Table 2. Quantification of Al3Zr dispersoids in the dendrite centre and in the three clusters in an interdendritic area from Fig. 5. The required local Zr content was predicted using a precipitation model developed by Robson [2,17,18]. The bottom two rows contain the measured data normalised to represent the actual values in a cluster of one particle layer of 20 nm thickness.

Diameter (nm) Num. density (lm3) fv Predicted average local Zr content required (at.%) Normalised num. density (lm3) Normalised fv

Dendrite centre

Cluster 1

Cluster 2

Cluster 3

25.1 ± 0.3 311 0.0026 0.0912 — —

20.0 ± 0.5 807 0.0038 0.1211 22,593 0.1051

19.4 ± 0.7 1,615 0.0077 0.2185 45,184 0.2148

19.9 ± 0.5 1,177 0.0053 0.1586 32,915 0.1479

Fig. 6. EDX maps of the plate-shaped precipitates in the as-cast microstructure.

Table 3. EDX results from two areas selected from the matrix and a rectangular particle (all contents in wt.%).

Particle Matrix

Al

Cu

Mn

Fe

Zr

79.3 97.4

20.0 2.4

0.4 0.1

0.2 0.0

0.1 0.0

dispersoids had formed yet. By comparing the HAADFSTEM image to the Zr map, it is readily visible that Zr solute was found to be contained in the two elongated Al20Cu2Mn3 dispersoids in Fig. 9, as well as in dislocations. Regarding the two Mn dispersoids, Zr was contained fairly heterogeneously within their volume as some areas had higher concentration than others. It should also be noted that not all Mn dispersoids contained Zr; this is evident from the elliptical dispersoid right below the two elongated ones. Since the two elongated dispersoids are connected with dislocations, one could argue that the Zr solute is contained inside these interfacial dislocations. However, this does not seem to be the case judging from the particle on the left; despite the lack of Zr across its whole length, there is high Zr content across its entire width. In terms of the second incident of Zr local enrichment in this map, there is high solute content in the dislocation protruding from the interface of the globular Mn dispersoid.

Fig. 7. Electron diffraction pattern of a h0 -Al2Cu plate-shaped precipitate in the as-cast microstructure (pattern corresponds to particle 1).

The second Zr cluster seen near the top of the map appears not to be connected to any dislocations at first sight. However, with a careful look at the HAADF-STEM image

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

79

Fig. 8. HAADF-STEM images showing the stages of dissolution of the h0 -Al2Cu phase prior to Al3Zr precipitation during the homogenisation treatment (near h1 0 0iAl z.a.). (a) h0 plates formed during solidification in the dendrite edge (as-cast), (b) gradual dissolution of the h0 plates (ramp, 443 °C), (c) complete dissolution of h0 (end of ramp, 505 °C), (d) no Al3Zr present in the dendrite centre (end of ramp, 505 °C), (e) initiation of Al3Zr precipitation in the dendrite centre (8 h isothermal plateau, 505 °C), and (f) higher magnification of linear arrays of dispersoids from (e).

a faint dislocation can be discerned. The reason for its low contrast with the matrix is probably the large thickness of the TEM specimen in that particular location and the dislocation is most likely situated at a depth from the outer surface which affects the clarity of the STEM image. Several similar maps were recorded in areas further from the dendrite centre in the same condition, but also in the as-cast state. Quantification of the EDX maps helped to construct the concentration plot in Fig. 10. The graph is showing clearly that a large amount of Zr can be dissolved within the Mn dispersoids, with the approximate stoichiometry of the phase becoming Al20Cu1.57Mn2.89Zr0.53. It is thus evident that Zr substitutes mainly for Cu in the Al20Cu2Mn3 phase. Finally, another interesting observation on the behaviour of zirconium was its detection in the interior of Cucontaining particles of globular and random morphologies in both the fully homogenised AA2050 and the as-hot rolled AA2198 (Fig. 11). For this purpose composition plots were constructed by analysing a large number of such particles. The majority of the data points for Cu fall on straight lines that approach the chemical composition of the Al2Cu phase. The line corresponding to the atomic ratio of the Al2Cu phase has been superimposed on the data for means of comparison. Straight lines were fitted carefully to the experimental data points for both Cu and Zr in order to derive the equations that best described them. Occasionally, a small number of outlier data points

at the higher end of the Al atomic content had to be masked for optimum fitting to be achieved. The concentration plots proved that the phase in question is the equilibrium h-Al2Cu. From the lines fitted to all the detected elements, the stoichiometry of each phase was determined. Two variants of the equilibrium h-Al2Cu phase were found in the AA2198 material after annealing for 4 min at 535 °C from the as-hot-rolled F temper; Al2Cu0.94Zr0.06 and Al2Cu0.87Mn0.13 (Fig. 11b and c). In the latter phase trace Zr was detected only in very limited occasions, but its content was particularly low and has been omitted from the stoichiometry. It should be noted that in every case where such particles were detected in the rolled sheet, they were associated with recrystallised grains by being located either in their interior or on GBs. It is also important to note that aggregates of these coarse particles were usually aligned along the rolling direction. This fact suggests that they were previously located on GBs in the as-cast microstructure and later became aligned during rolling. For this reason, the study was extended to the fully homogenised AA2050 where globular particles on GBs were analysed. The graphs in Fig. 11d and e show that the phases observed in this case were similar to the 2198 sheet but with somewhat different compositions, namely Al2Cu0.93Zr0.07 and Al2Cu0.95Mn0.03Zr0.02. It is evident that the Zr content is slightly higher than in the rolled sheet in this case for both phases and in particular for the latter where the Mn content was found to be notably lower.

80

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

Fig. 9. EDX maps obtained with a Titan ChemiSTEM XFEG-TEM showing Zr solute contained in Al20Cu2Mn3 dispersoids and in dislocations in interdendritic regions (end of ramp, 505 °C). The bottom right image is an overlay of the Mn and Zr maps.

Fig. 10. EDX concentration plot in the fully homogenised AA2050 showing Zr content in the Al20Cu2Mn3 dispersoids.

4. Discussion 4.1. Heterogeneous Zr solute segregation The most significant finding so far has been the presence of Zr solute clusters in dislocations in the matrix prior to the precipitation of the Al3Zr phase (Fig. 9). Solute segregation to dislocations is promoted by the tendency to reduce the misfit between the solute and solvent atoms. Table 4 presents atomic sizes for all the alloying elements in the two alloys used here. The difference between the atomic size of Al and Zr is 60% which verifies that Zr atoms

are more likely to segregate to dislocations. Atom probe work has shown dislocations in an AA2024 to be enriched with Mg, Cu, Si, Zn [12] and in an AA2198 to contain Cu, Li, Mg, Ag [13]. In AA3003 dislocations become enriched with Mn after deformation [14]. From all these elements only Ag has higher atomic misfit than Zr (70% and 60% respectively – Table 4). Unfortunately no reference on Zr segregation to dislocations in Al alloys was found in the literature. In contrast, Zr segregation to dislocations has already been observed in Ni–Al [15,16] and Ti–Al [16]. This is rather confusing since there is a smaller misfit between either Ni or Ti and the Zr atoms as compared to Al and Zr; one would certainly expect to see the same phenomenon in Al alloys as well. Fine scale atomic clustering of Zr in the matrix could not be detected with the techniques employed in this work, since atom probe tomography is required. Nevertheless, the detection of Zr within dislocations suggests that an extremely heterogeneous solute distribution exists locally in the microstructure. This heterogeneity is supported by the observation that the Al3Zr dispersoids form with a larger size in the dendrite centre as compared to within clusters in interdendritic regions. In addition, local volume fractions are much higher away from the dendrite centre even up to five times, when measured within the clusters (Table 2). This is a rather unexpected phenomenon at first sight, since it is anticipated that all Al3Zr particles would be coarser in the interdendritic regions where the Zr supersaturation is much lower than in the dendrite centre. This controversial observation can be explained on the grounds of the higher local supersaturation within the dislocations, where

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

81

Fig. 11. Zr inclusion in the equilibrium h-Al2Cu phase. (a) Examples of AlCuZr particles in the F temper AA2198 sheet (after annealing for 4 min at 535 °C). EDX concentration plots in the F temper AA2198 sheet after annealing for 4 min at 535 °C for (b) globular AlCuZr particles, and (c) globular AlCuMn particles. EDX concentration plots in the fully homogenised AA2050 for (d) globular AlCuZr particles on GBs, and (e) globular AlCuMnZr particles on GBs.

Table 4. Values of atomic radius and atomic size misfit relative to the Al atom for the main elements present in the Al–Cu–Li alloys investigated here (quoted from Slater [40]).

˚) Atomic radius (A Atomic misfit (%)

Al

Si

Cu

Zn

Mn

Fe

Li

Mg

Zr

Ag

1.25 —

1.1 30

1.35 20

1.35 20

1.40 30

1.40 30

1.45 40

1.50 50

1.55 60

1.60 70

concentrations of up to 0.5 at.% Zr were detected. Using a precipitation model that was developed by Robson [2,17,18], the required local Zr content would be between 0.12 and 0.22 at.% in order to produce the dispersoid distributions in the clusters that were analysed in Table 2. It has to be noted that the measured volume fractions of particle clusters and at.% content of Zr solute lumps are lower due to the effect of sample thickness. If the layer thickness is assumed to be 20 nm, equal to the average Al3Zr diameter in

the cluster, the normalised volume fractions would lie between 0.10 and 0.21 for the three measured areas (Table 2). The predicted concentration of 0.12 and 0.22 at.% is markedly higher than the average alloy Zr content and can justify the measured levels of 0.5 at.% within solute clusters in the matrix. Thus it is easily understood that Zr solute is largely segregated prior to precipitation, especially in interdendritic regions. High Zr concentrations in the close vicinity of dislocations have been previously

82

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

observed in Ni–Al via atom probe measurements [15] and they are consistent with Cottrell solute equilibrium atmospheres [19]. According to Headley and Hren [20] solute clustering may be occurring on quenching and also solute can be trapped at jogs and kinks. Solute enrichment of kinks was recently verified via atom probe experiments by Araullo-Peters et al. [13], who noted the presence of Cu, Mg, Li, and Ag in them in an AA2198 alloy. Additional effects on the Al3Zr precipitation behaviour could arise from interactions with other solute elements trapped in dislocations. Cu, Li, Mg, Ag, and Mn are all known to segregate to dislocations [12–14] and are also reported to decrease the metastable Zr solubility [18,21], thus potentially incurring a much higher dispersoid volume fraction locally. Further evidence of Zr segregation to dislocations is given by Fig. 8e and f where Al3Zr dispersoids start nucleating on them even in areas of high supersaturation such as the dendrite centre and not just in interdendritic regions. This is in agreement to the findings of Nes [6] who supports that Al3Zr precipitate nucleation is considered to take place mainly on dislocations and subgrain boundaries following diffusion-controlled kinetics. Dispersoid precipitation is also reported to take place heterogeneously on dislocations in Al–Sc alloys [22]. Furthermore, Robson et al. [23] using the same precipitation model which was mentioned earlier concluded that a high dislocation density can increase the frequency of heterogeneous precipitation of L12-type dispersoids. 4.2. Zr inclusion in particles Zirconium was detected via EDX in this work to be contained in certain particles, such as the equilibrium h-Al2Cu on grain boundaries and the Al20Cu2Mn3 dispersoids in the grain interior (Figs. 9–11). However, it was not traced in the metastable h0 -Al2Cu phase (Fig. 6), at least not in quantities that can be detected even by state-of-the-art EDX detectors. The equilibrium h phase appeared with two different variants, depending on the elements that it contained; one variant consisted of Al, Cu, and Zr, while the other of Al, Cu, Mn, and Zr (Fig. 11). These phases were met in both the AA2198 sheet and the AA2050 billet. The calculated stoichiometry for the AlCuZr particles was almost identical in both alloys, namely Al2Cu0.94Zr0.06 and Al2Cu0.93Zr0.07. On the other hand, the AlCuMnZr phases differed since their stoichiometry was equal to Al2Cu0.87Mn0.13 and Al2Cu0.95Mn0.03Zr0.02. The former phase in the rolled AA2198 contained more Mn and trace Zr which was too low and omitted from the formula, whereas the latter phase in the AA2050 billet contained less Mn but higher Zr. Since the two alloys had the same Zr and Mn contents, these compositional differences do not necessarily indicate effects from interactions with the other alloying elements, but they are likely to reflect the microstructural variability of the as-cast material and the wealth of GB phases that are present. Especially in the case of the AlCuMnZr particles in the AA2050, the number of points in the composition plot in Fig. 11e is not exhaustive to give the precise stoichiometry. Nonetheless, the main conclusion is that Zr can be contained within the equilibrium h-Al2Cu particles of globular morphologies located on GBs either with or without the presence of Mn. Additionally it appears that the Mn content in the h phases increases at the expense of Zr and vice versa. Other phases

reported in the literature to contain Zr together with Al and Cu, are the Al7Cu3Zr2 and Al3CuZr2 [8], but are fairly different from the phase observed in the present work. On the other hand, Zr inclusion in the Al20Cu2Mn3 dispersoids in the grain interior led to a stoichiometry of Al20Cu1.57Mn2.89Zr0.53. However, this segregation phenomenon has been analysed in detail elsewhere [7] and two main conclusions need to be carried forward; (i) Zr can also be contained in Al20Cu2Mn3 dispersoids, and (ii) Zr solute is not evenly distributed within the Mn dispersoids but concentrated in parts of their volume, which indicates that some of these dispersoids are quite likely nucleated on Zr solute clusters in dislocations. The latter conclusion is supported by the large scatter in the measured Zr content in the Al20Cu2Mn3 phase (Fig. 10), since the Zr solute clusters are not expected to have a fixed composition. In support of the present results, Zr atoms have been previously shown to segregate to particles other than Al3Zr, such as Al3Cr and Al6Mn [9]. These findings verify an early estimation made by Ohashi et al. [24], that Zr atoms were expected to be present in the Al6Mn phase. Zr segregation in such particles may remain even in the final sheet form after homogenisation, hot-rolling and solution treatment, thus imposing a detrimental effect on recrystallisation resistance [7,25]. Equilibrium phases are known to precipitate preferentially on GBs [26], hence the decoration of these sites with h particles in both the homogenised AA2050 and rolled AA2198. In the latter, an obvious reduction in the Al3Zr number density can be seen locally around the coarse h particles, due to the inclusion of Zr in them (Fig. 11a). The effect of this phenomenon on recrystallisation resistance is very important, since the Zener pinning is reduced in that location adjacent to the large particles. Particle stimulated nucleation (PSN) will thus be easier to occur in such regions of low pinning pressure, provided that the PSN nucleus size is sufficiently large and a minimum strain is exceeded. The AlCuZr particles were always included in recrystallised grains in the microstructure of the annealed AA2198 sheet that just started recrystallising after 4 min at 535 °C. Considering that the large particles in Fig. 11a lay on, or right next to, the GB of a recrystallised grain, it is fairly possible that this aggregate of h particles is associated with the introduction of a highly misoriented front that migrated and consumed the surrounding microstructure. 4.3. Mechanisms of Al3Zr cluster formation Several types of Al3Zr clusters were mentioned earlier in this work; (i) blocky clusters most frequently consisting of up to 10 particles, (ii) rectangular clusters filled with dispersoids throughout, (iii) rectangular clusters having dispersoids only on their perimeter and not in their interior, and (iv) straight, linear clusters, sometimes appearing as closely-spaced parallel rows. All these clusters were related to dislocations. The most complicated case was the elongated and filled clusters. The blocky clusters were always associated with dislocation loops and helices, while the elongated and non-filled clusters appeared due to dispersoids precipitating on dislocations with such shapes. Similarly, the straight, linear arrays were always connected to dislocations. In order to obtain some first answers on the precipitation mechanism of the rectangular and filled Al3Zr clusters,

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

one needs to carefully monitor the microstructure throughout the entire homogenisation sequence. To summarise the transition steps from the casting stage to the fully homogenised condition, it all starts with the h0 phase forming upon quenching after casting, prior to the precipitation of the two dispersoid families. After reaching the solvus temperature of the h0 -Al2Cu phase at the homogenisation stage, these particles will start dissolving in an irregular manner without maintaining a straight interface. At the end of the ramp heating stage after 10 h, the temperature has reached 505 °C and there are no more h0 particles present in the microstructure and only the Al20Cu2Mn3 phase can be seen on the {1 0 0}Al habit plane [27]. The Al3Zr dispersoids only start precipitating after 8 h at the isothermal plateau at 505 °C. This significant time lapse leaves no hesitation in rejecting the possibility of direct heterogeneous nucleation of the Al3Zr dispersoids on the h0 -Al2Cu particles. An alternative possibility for the mechanism of cluster formation, that Zr could be contained within the h0 -Al2Cu phase and lead to Al3Zr formation upon the dissolution of the Cu-containing particles, was dismissed on the grounds of EDX measurements that failed to detect any significant amount in them (Fig. 6, Table 3) even when using state-of-the-art X-ray detectors and analysing a very large number of particles. Even if trace amounts of Zr could be potentially present within the h0 particles and were below the detection limit of the technique, such a minor quantity would still be insufficient for the formation of the rectangular clusters. By employing the precipitation model developed by Robson [2,17,18], it was found that a very large Zr concentration would be required locally, in the order of 0.2 at.%, to produce the measured volume fractions from Fig. 5 (Table 2). Although large Zr contents of the order of 0.5 at.% were measured in dislocations in interdendritic regions at the end of the ramp heating stage at 505 °C, they were fairly concentrated in small volumes and did not appear to have resulted upon the h0 dissolution. This statement can be further supported by considering that the measured Zr content corresponds to the analysed column through the thickness of the TEM specimen. Since the Zr clusters in Fig. 9 are close to 40 nm long, which is at most one eighth of the sample thickness in that area, it is understandable that the actual Zr concentration is much higher inside the volume occupied by the cluster, as indicated by the normalised particle volume fractions in Table 2. Thus it is not possible that the mechanism of Zr inclusion within the h0 plates could be valid. A contradictory observation now remains to be explained; some Al3Zr clusters resemble the shape of h0 -Al2Cu but they do not seem to originate from them. If they were directly related to them there would be a much higher number of such clusters due to the very high density of h0 in the as-cast material (Fig. 8a). It was shown in Fig. 7 that dislocations are present at the interface of large h0 particles in the as-cast microstructure. Although Zr may be contained in these dislocations, it is not granted that the h0 dissolution will leave behind dislocations of the same shape as the particle. Diffusion of solute atoms and vacancies away from the dissolving interface is fast at high temperature and the h0 phase dissolves non-uniformly as can be seen from Fig. 8b. In fact, other researchers have observed dislocations being left behind in the matrix after the dissolution of h0 -Al2Cu in aluminium [28] and of cementite in steel [29], but in neither case did those have the same

83

shape as the pre-existing particles. These remnant dislocations are generated in order to account for the volume difference that arises upon particle dissolution and which is equal to 4% for h0 [20,30]. In addition to the above discussion on the shape of dislocations, generally it is not uncommon that some of them appear with ordered shapes and specific orientations in the matrix as can be seen from Fig. 12, but this is not due to the pre-existing h0 particles. Some of these dislocations have straight segments or may form rectangles either when in contact with Al20Cu2Mn3 dispersoids as in (a) or even when there are no such particles to pin them from all sides as in (b). All of them have segments parallel to h0 0 1iAl and h0 1 0iAl. The overall distribution of oriented dislocation segments and rectangular loops in these directions can be seen in (c), at the end of the ramp heating stage several hours prior to Al3Zr precipitation. Thus there is a high probability that they may be responsible for various forms of clusters, such as those shown in Fig. 3. Another possibility that needs to be addressed is the effect of local solute supersaturation on the solubility of Zr in the matrix. It does not seem likely that the remnant Cu, Mn, or Fe solute in the location of dissolved h0 -Al2Cu reduces the Zr solubility locally so as to induce precipitation comparable to the Al3Zr clusters observed here. EDX analyses of the areas around dissolving h0 particles did not reveal any enrichment locally. Diffusion of vacancies and solute at 505 °C is fast and these atoms would not remain in the same location for a period of more than 8 h which is required for the initiation of Al3Zr nucleation. Most likely the majority of the Mn and Fe atoms would eventually segregate to dislocations in order to moderate the misfit with the matrix, while the excess Cu would mainly diffuse into solid solution. This motion would be accelerated by the high vacancy flux during h0 dissolution. One mechanism for particle cluster formation reported in the literature is the “autocatalytic nucleation” of h0 plate-shaped precipitates [31]. However, in the Al–Cu alloy studied by Perovic et al. [31] only few clusters were seen to be associated with dislocations, in contrast to what is observed here for the Al3Zr dispersoids. Additionally, these clusters were not subject to growth or coarsening. Here the clusters consist of a large number density of very fine Al3Zr spherical particles in the fully homogenised condition (Fig. 13a). However, extended isothermal holding at the 505 °C plateau for 144 h allowed the clusters to maintain their orientation relationships with the matrix along h0 0 1iAl while the dispersoids in them coarsened significantly (Fig. 13b). This suggests that the cluster size was unaffected by time irrespective of the duration of annealing. But the most important difference is that autocatalytic nucleation refers to lath-shaped particle interactions due to the high strain field generated by the high misfit along the c-axis of the h0 phase. In contrast, the Al3Zr phase has a generally low misfit with the Al matrix of the order of 0.8% [6] and thus no elastic strain interactions are expected to take place. Hence the autocatalytic nucleation mechanism cannot be valid in the present case. The prevailing nucleation mechanism is based on an interaction between Zr solute and dislocation climb and has been previously described as “repeated precipitation on dislocations” by several researchers working on various alloy systems [20,30,32–34]. Zr atoms were shown earlier to diffuse to dislocations in order to moderate their large atomic misfit with the Al matrix. Furthermore, a

84

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

Fig. 12. BF-TEM images of dislocations with rectangular shapes at the interfaces of Mn dispersoids (AA2050, end of ramp). (a) Rectangular dislocation arrangement due to contact with Mn dispersoids on all sides, (b) rectangular dislocation arrangement in contact with Mn dispersoids only on one side, and (c) numerous rectangular dislocation arrangements and straight dislocations oriented along h0 0 1iAl.

percentage of Zr atoms are likely to be retained in solid solution by linking with vacancies since they are strongly attracted to them [35]. Once Al3Zr precipitation is initiated, the dislocations which are acting as substrates will try and free themselves from the dispersoids by climb while precipitation continues, provided that there is sufficient Zr supersaturation remaining in them. The driving force for

Fig. 13. HAADF-STEM images of Al3Zr clusters further from the dendrite centre in an AA2050 sample homogenised at 505 °C for (a) 12 h, (b) 144 h, and (c) 144 h (overall distribution).

dislocation climb is known to be the reduction of quenched-in vacancies [20]. The solute content in the dislocation is increased during climb since it drags Zr atoms from the solid solution and Al3Zr forms by fast pipe diffusion [30], consequently leading to a rectangular and filled cluster. One of the main features of repeated precipitation on dislocations is that a minimum solute supersaturation is required for this precipitation mode to take place [32]. For this reason it is sensible to support that the mechanism

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

of precipitation on dislocations depends strongly on the concentration of Zr solute in them; relatively small amounts will produce linear or non-filled Al3Zr clusters, whereas high Zr contents will create larger and filled clusters such as those in Figs. 2a and 3. Such high local concentrations can also arise from Zr clusters in the matrix, or at dislocations, which form upon quenching [20]. Their presence was experimentally verified in this work both for the as-cast condition and at the end of ramp during homogenisation at 505 °C. Thus a higher Zr content implies that a larger number of such solute clusters are present in the matrix. This can explain the fact that an Al–0.18 wt.% Zr alloy studied by Jia [5] had more dispersoid clusters than the present AA2050 which has a lower content of 0.11 wt% Zr. Dislocation motion plays a vital role during repeated precipitation on dislocations. In Al alloys dislocations are expected to move on the close packed {1 1 1}Al planes, as is the case for the planar cluster B and the linear cluster C in Fig. 3. However, a large number of planar clusters were seen to form on {0 1 0}Al in the same images. Dislocation climb on non-densely packed crystallographic planes such as the {0 1 0}Al, or the higher index plane corresponding to cluster E in Fig. 3, is possible at high temperatures due to the activation of a kink-pair mechanism [36], according to which a kink enables the dislocation to move on a non-packed plane and then a second kink shifts it back to the compact plane. Indeed a high density of dislocations on {0 1 0}Al were seen prior to Al3Zr precipitation in Fig. 12. Similar precipitation phenomena on non-packed planes were previously observed by other researchers as well [37–39]. Moreover, curved interfaces of clusters are due to the variation in the climb path of the dislocations [20], which explains the presence of the helical dispersoid arrays observed by Jia et al. [5]. As a final point, another characteristic feature of the repeated precipitation on dislocations is that it yields an initially heterogeneous distribution of clusters whose particles only undergo growth during ageing, but the overall distribution eventually becomes more homogeneous with time due to nucleation of particles outside these clusters [32]. In agreement to these observations, the present study shows that the Al3Zr number density increased notably after extended annealing for 144 h at 505 °C so that their distribution became more homogeneous in interdendritic areas. Fig. 13c verifies that interdendritic regions contained at this stage a much higher density of Al3Zr particles compared to the standard fully homogenised condition (12 h at 505 °C). The reason is that in the case of extended annealing diffusion has proceeded to a significant degree and precipitation on dislocations was more extensive, alongside coarsening of the early nucleated dispersoids. Once precipitation has advanced to such a degree, it is more difficult to discern oriented clusters in interdendritic regions. With regard to viewing the findings of the present work from a wider angle, it is evident that the heterogeneous segregation of Zr solute, and subsequently of Al3Zr dispersoids, within a dendrite is detrimental to the recrystallisation resistance of Al alloy sheet. Zr solute clusters should be minimised from the casting stage, which would potentially lead to a slightly more uniform Zr solute distribution in interdendritic regions. This fact would imply a potentially larger coverage of the grain interior with Al3Zr dispersoids and, thus, slightly higher recrystallisation resistance, since there would be fewer Al3Zr localised

85

clusters and less Zr contained in other particles such as Mn dispersoids or equilibrium h-Al2Cu. 5. Conclusions A detailed study was carried out in this work to interpret the formation of various shapes of Al3Zr clusters in a homogenised Al–Cu–Li alloy (AA2050). The types observed were; blocky clusters most frequently consisting of up to 10 particles, rectangular clusters filled with dispersoids throughout, rectangular clusters having dispersoids only around their perimeter and not in their interior, and straight linear clusters sometimes appearing in pairs as parallel rows. Previous theories claiming a direct relationship of planar Al3Zr clusters with the h0 -Al2Cu particles had to be discarded on the grounds of systematic evidence which proved that the h0 particles dissolve upon homogenisation at a much earlier stage than the initiation of the Al3Zr dispersoid precipitation. Additionally, no Zr was detected in the h0 phase which could have potentially led to the formation of rectangular clusters. The formation of Al3Zr clusters was determined to be due to repeated precipitation on climbing dislocations mainly on {1 0 0}Al and {1 1 1}Al planes in the case of the planar arrays, whereas linear arrays formed simply owing to nucleation on quenched-in dislocations. Blocky clusters of generally up to 10 particles nucleated on dislocation loops and helices. Al3Zr precipitation on dislocations was intense even in the dendrite centre where the Zr supersaturation is high. This observation verified the EDX findings that Zr solute is contained in dislocations and occasionally segregated as coarse solute clusters upon quenching. The preferential presence of Zr in dislocations was explained on the grounds of minimising its atomic size misfit with the Al matrix. In contrast to what was evidenced for the h0 phase, zirconium was found to interact with the equilibrium h-Al2Cu phase in the as-cast microstructure by producing two different types of grain boundary phases, one containing Al, Cu, Zr and another with Mn as well. Such particles were also located in recrystallised grains in the microstructure of the hot rolled and annealed AA2198 sheet, but with slightly different compositions. Acknowledgments The authors would like to thank LATEST, the University of Manchester EPSRC Light Alloys Portfolio Partnership (EP/ D029201/1), for the financial support to this project. Additional financial contribution by Alcan CRV is also gratefully acknowledged. Also thanks to Dr Christophe Sigli and Dr Bernard Be`s of Alcan CRV for providing the materials.

References [1] C. Sigli, L. Maenner, C. Sztur, R. Shahani, in: ICAA-6: 6th International Conference on Aluminium Alloys, Toyohashi, Japan, 1998, p. 87. [2] J.D. Robson, P.B. Prangnell, Acta Mater. 49 (2001) 599. [3] K.E. Knipling, D.C. Dunand, D.N. Seidman, Acta Mater. 56 (2008) 114. [4] W.W. Mullins, F.A. Nichols, J. Appl. Phys. 36 (1965) 1826. [5] Z.-H. Jia, J.-P. Couzinie´, N. Cherdoudi, I. Guillot, L. ˚ sholt, S. Brusethaug, B. Barlas, D. Massinon, Arnberg, P. A Trans. Nonferrous Met. Soc. China 22 (2012) 1860.

86

D. Tsivoulas, J.D. Robson / Acta Materialia 93 (2015) 73–86

[6] E. Nes, Acta Metall. 20 (1972) 499. [7] D. Tsivoulas, J.D. Robson, C. Sigli, P.B. Prangnell, Acta Mater. 60 (2012) 5245. [8] P. Vermaut, P. Ruterana, Acta Mater. 44 (1996) 2445. [9] R. Kaibyshev, F. Musin, D.R. Lesuer, T.G. Nieh, Mater. Sci. Eng. A 342 (2003) 169. [10] R. Yoshimura, T.J. Konno, E. Abe, K. Hiraga, Acta Mater. 51 (2003) 4251. [11] A. Wiengmoon, J.T.H. Pearce, T. Chairuangsri, S. Isoda, H. Saito, H. Kurata, Micron 45 (2013) 32. [12] G. Sha, R.K.W. Marceau, X. Gao, B.C. Muddle, S.P. Ringer, Acta Mater. 59 (2011) 1659. [13] V. Araullo-Peters, B. Gault, F.D. Geuser, A. Deschamps, J.M. Cairney, Acta Mater. 66 (2014) 199. [14] S.P. Chen, N.C.W. Kuijpers, S. van der Zwaag, Mater. Sci. Eng. A 341 (2003) 296. [15] R. Jayaram, M.K. Miller, Scr. Metall. Mater. 33 (1995) 19. [16] D.J. Larson, M.K. Miller, Mater. Charact. 44 (2000) 159. [17] J.D. Robson, Mater. Sci. Eng. A 338 (2002) 219. [18] J.D. Robson, P.B. Prangnell, Mater. Sci. Eng. A 352 (2003) 240. [19] E.A. Marquis, J.M. Hyde, Mater. Sci. Eng.: R: Rep. 69 (2010) 37. [20] T.J. Headley, J.J. Hren, Philos. Mag. 34 (1976) 101. [21] C. Sigli, in: ICAA-9: 9th International Conference on Aluminium Alloys, Brisbane, Australia, 2004, p. 1353.

[22] E.A. Marquis, D.N. Seidman, Acta Mater. 49 (2001) 1909. [23] J.D. Robson, M.J. Jones, P.B. Prangnell, Acta Mater. 51 (2003) 1453. [24] T. Ohashi, L. Dai, N. Fukatsu, Metall. Mater. Trans. A 17 (1986) 799. [25] D. Tsivoulas, P.B. Prangnell, Acta Mater. 77 (2014) 1. [26] K.C. Russell, H.I. Aaronson, J. Mater. Sci. 10 (1975) 1991. [27] C. Li, S.C. Wang, Y. Jin, M. Hua, M. Yan, in: Aluminium Alloys ‘90. Second International Conference on Aluminium Alloys-Their Physical and Mechanical Properties, Beijing, China, 1990, p. 504. [28] P. Hewitt, E.P. Butler, Acta Metall. 34 (1986) 1163. [29] M. Nemoto, Acta Metall. 22 (1974) 847. [30] P. Guyot, M. Wintenberger, J. Mater. Sci. 9 (1974) 614. [31] V. Perovic, G.R. Purdy, L.M. Brown, Acta Metall. 29 (1981) 889. [32] T.J. Headley, J.J. Hren, J. Mater. Sci. 11 (1976) 1867. [33] U. Dahmen, K.H. Westmacott, Phys. Status Solidi A 80 (1983) 249. [34] E. Nes, Acta Metall. 22 (1974) 81. ¨ zbilen, H.M. Flower, Acta Metall. 37 (1989) 2993. [35] S. O [36] M. Carrard, J.L. Martin, Philos. Mag. A 58 (1988) 491. [37] H.M. Miekk-oja, R. Ra¨ty, Philos. Mag. 24 (1971) 1197. [38] U. Dahmen, K.H. Westmacott, Scr. Metall. 17 (1983) 1241. [39] J.K. Solberg, E. Nes, J. Mater. Sci. 13 (1978) 2233. [40] J.C. Slater, J. Chem. Phys. 41 (1964) 3199.