III. CONSIDERATION OF POSSIBLE THERMODYNAMIC STABILIZATION SYSTEMS

An alloy system that has been studied extensively is nanocrystalline Pd with Zr additions.Reference Krill, Ehrhardt and Birringer^18, Reference Krill, Klein, Janes and Birringer²⁰ The results have been summarized by Krill et al.Reference Krill, Ehrhardt and Birringer¹⁸ The latter authors derive an expression for the grain boundary energy as affected by solute segregation as follows:

$${\rm{\gamma }} = {\rm{\gamma }}_{\rm{0}} - \Delta H^{{\rm{seg}}} \left( {{{n_{\rm{B}}^{{\rm{seg}}} } \over A}} \right)\quad,$$

where γ₀ is the specific grain boundary energy of the pure element, ΔH ^seg is the enthalpy of segregation of B solute atoms to the grain boundary, $n_{\rm{B}}^{{\rm{seg}}} $ is number of moles of B solute atoms segregated to the grain boundary phase, and A is total grain boundary area. Insertion of typical numbers for these quantities gives a value for γ ∼ 0 when ΔH ^seg =25 kJ/mol or more. The segregation enthalpy of Zr in Pd is estimated to be 31 kJ/mol.Reference De Boer, Boom, Mattens, Miedema and Niessen²¹ However, the solid solubility of Zr in Pd at 600 °C is 12 at.% and increases up to 17 at.% at the peritectic temperature.Reference Okamoto²² Furthermore, the equation derived for γ is valid only for the dilute segregation limit. In their initial studies of the Pd–Zr alloys, Krill et al.Reference Krill, Klein, Janes and Birringer²⁰ found that ball milling induced metastable solid solutions up to 20 at.% Zr. At concentrations of 21 at.% Zr and higher, the intermetallic Pd₃Zr phase formed. The lattice parameters of the metastable Pd–Zr alloys (>12 at.% Zr) showed an approximately linear increase with Zr concentration. The lattice parameters decreased on annealing at 500 °C, which the authors attributed to the segregation of Zr atoms to the grain boundaries. Annealing for 2 h at 500 °C increased the grain sizes, but the increase was less as Zr concentration increased. The approximately linear decrease in grain size with at.% Zr for the annealed samples has a higher slope than that of the as-milled alloys. The authors subsequently foundReference Krill, Ehrhardt and Birringer¹⁸ that raising the Zr concentration increased the onset temperature for significant grain growth and it suppressed the rate at which coarsening progressed. They attributed these effects to the segregation of Zr atoms to the grain boundaries. This segregation may lead to either kinetic stabilization by solute drag on the boundaries, or to thermodynamic stabilization by lowering the grain boundary energy, or a combination of the two. From their x-ray data for grain size and lattice parameter changes with Zr concentration and annealing temperatures up to 1125 °C, the authorsReference Krill, Ehrhardt and Birringer¹⁸ estimated the areal density of Zr atoms segregated to the grain boundaries, which increased with annealing temperature, and the specific grain boundary energy, which decreased with annealing temperature. The authors suggest that while the enhanced stabilization at low and moderate temperatures may be partly due to solute drag, only the thermodynamic stabilization can provide a credible explanation for the enhanced stability at high temperatures. Subsequently, annealing of the 15, 19, and 20 at.% Zr nanocrystalline alloys was conducted at 1500 °C, a temperature ∼ 95% of the melting temperatures. The grain size of the 15 at.% Zr sample grew slightly, while those of the 19 and 20 at.% Zr samples remained approximately at the values obtained at 1125 °C. However, the grain sizes in these studies were mainly determined by XRD line broadening analysis. Subsequent work by VanLeeuwen et al.Reference VanLeeuwen, Darling, Koch, Scattergood and Butler²³ using microscopy [TEM and focused ion beam channeling contrast imaging (FIB-CCI)] indicated that significant grain growth occurred for a Pd–19 at.% Zr nanocrystalline alloy at temperatures >600 °C. As previously suggested,Reference Michels, Krill, Ehrhardt, Birringer and Wu⁶ it appears that the thermal stabilization for temperatures up to ∼600 °C is due to solute drag, and there is no effective stabilization above this temperature. Models for thermodynamic stabilization,Reference VanLeeuwen, Darling, Liu, Koch and Scattergood^16, Reference Saber, Kotan, Koch and Scattergood¹⁷ which include chemical effects as well as elastic enthalpy, predict that the Pd–Zr system should not exhibit thermodynamic stabilization. Therefore, it appears that both experimental evidence and model predictions show that the Pd–Zr alloys do not exhibit thermodynamic stabilization.

A very detailed study of grain boundary segregation has been carried out by Detor and SchuhReference Detor and Schuh^24–Reference Detor, Miller and Schuh²⁶ in nanocrystalline Ni–W alloys made by electrodeposition. The elastic enthalpy for W solute in Ni is estimated to be 60 kJ/mol,Reference De Boer, Boom, Mattens, Miedema and Niessen²¹ although there is significant solid solubility of W in Ni of >10 at.% W.Reference Okamoto²⁷ From precise lattice parameter measurements as a function of W compositionReference Detor and Schuh²⁴ and from atomistic computer simulationsReference Detor and Schuh²⁵ and atom probe tomography,Reference Detor, Miller and Schuh²⁶ it was determined that the segregation tendency for W to grain boundaries in Ni is weak. Segregation energy decreased with W addition from a dilute limit value owing to solute–solute interactions in the grain boundaries. The driving force for grain growth decreased with W additions but did not reach zero for any composition or grain size considered. Detor and SchuhReference Detor and Schuh²⁸ have subsequently measured the grain growth in their electrodeposited nanocrystalline Ni–W alloys. The as-deposited grain size was a function of composition, with the smallest grain sizes (3 nm) corresponding to the highest W content (21 at.%) and the largest grain size (70 nm) for the lowest W concentration (6 at.%). The grain size for three Ni–W samples is plotted versus reduced annealing temperature (T/T _M) in Fig. 1. These data were obtained by analysis of XRD line broadening. TEM studies on these samples showed good agreement with the x-ray data and gave no evidence for abnormal grain growth. While good stabilization compared to pure Ni occurs for these alloys up to about T/T _M = 0.4, significant grain growth begins by T/T _M ∼ 0.5, and the grains grow to above 100 nm (out of the nanoscale regime) by 0.6–0.7. The initial grain sizes of the electrodeposited alloys, and the stabilization, are strong functions of the W concentration. The Ni–21 at.% W alloy with as-deposited grain size of 3 nm is the most stable. It does precipitate the intermetallic compound Ni₄W at the highest annealing temperature of 900 °C. The lower W content alloys maintain the fcc solid solution structure up to this temperature. The grain growth curves for the three alloys shown in Fig. 1 appear to be almost parallel. This suggests that the stabilization effectiveness for the various alloys may be similar if they were normalized to the same as-deposited grain size. Whether the stabilization is partially due to solute drag or thermodynamic stabilization is not clear.Reference Detor, Miller and Schuh²⁶

FIG. 1.

Grain size versus reduced annealing temperature (T/T _m) for Ni–6 at.% W (▼), Ni–13 at.% W (O), Ni–21 at.% W (●); data from Ref. Reference Detor and Schuh28.

The dependence of as-deposited grain size on composition in nanocrystalline alloys prepared by electrodeposition had been clearly noted in Ni–P alloys.Reference Farber, Cadel, Menand, Schmitz and Kirchheim²⁹ The authors studied the microstructures of Ni–P alloys including the grain boundary segregation with tomographic atom probe analysis. The composition dependence of the grain size was explained by assuming a metastable grain boundary phase, of 15 at.% P, and thickness of 0.8 nm and using a mass balance calculation. It was possible to predict the grain size–P concentration relationship in as-deposited and annealed alloys. For low P content (less than about 5 at.%), the grain size increased upon annealing, while at the higher concentrations (>10 at.%), there was little difference between the as-deposited and annealed grain size. These were for annealing temperatures around 300 °C, which are below the critical temperature where Ni₃P formation starts. At 15 at.% P, the samples become amorphous—in a sense, become “all grain boundary.” This same approach might also apply to the Ni–W nanocrystalline alloys made by electrodeposition, as discussed above. Whether similar ideas can be applied to nanocrystalline alloys made by other methods such as ball milling is problematic.

Nanocrystalline Co–1.1 at.% P was prepared by pulsed electrodeposition.Reference Choi, da Silva, Klement, Al-Kassab and Kirchheim³⁰ Its thermal stability was studied by a variety of characterization methods including tomographic atom probe analysis. The thermal stability of the alloy at temperatures below about 733 K was attributed to the segregation of P to the grain boundaries. At temperatures between 673 and 733 K, the allotropic phase transformation, hcp Co to fcc Co, occurs and abnormal grain growth is observed. It was suggested that there is a synergistic relationship between the abnormal grain growth and the phase transformation.

The thermal stabilization of Fe and Fe–Cr alloys with the addition of Zr was studiedReference Darling, Chan, Wong, Semones, Scattergood and Koch^31–Reference Saber, Kotan, Koch and Scattergood³³ since the recent models for thermodynamic stabilizationReference VanLeeuwen, Darling, Liu, Koch and Scattergood^16, Reference Saber, Kotan, Koch and Scattergood¹⁷ predicted that Zr should be an effective solute addition. The initial research,Reference Darling, Chan, Wong, Semones, Scattergood and Koch³¹ which used XRD line broadening analysis, indicated that a nanocrystalline grain size (∼50 nm) was retained on annealing up to a homologous temperature (T/T _m) of 0.92. However, measurements of hardness as a function of annealing temperature, Fig. 2, suggested that grain growth must have occurred at lower annealing temperatures. This was subsequently confirmedReference Darling, VanLeeuwen, Koch and Scattergood³² with optical, TEM, and FIB-CCI. It was found that the thermal stability increased with increasing Zr content, and the highest temperature where a nanoscale grain size was observed, 913 °C, was for a Fe–4 at.% Zr composition. At higher temperatures, grain sizes grew into the micrometer size range and precipitates of Fe₂Zr were observed by TEM. Furthermore, detailed TEM also revealed precipitates at the lower annealing temperatures, 900 °C for the Fe–4 at.% Zr samples.Reference Darling³⁴ Thus, the results of additions of Zr to nanocrystalline Fe provide apparent evidence for thermodynamic stabilization at elevated temperatures, but eventually, precipitation of intermetallic particles occurs so at the higher temperatures (∼900 °C), both Zener pinning and thermodynamic stabilization may be contributing to the thermal stability. The possibility of both mechanisms being operable was also suggested by the results of Zr additions to Fe–10 at.% Cr and Fe–18 at.% Cr nanocrystalline alloys.Reference Saber, Kotan, Koch and Scattergood³³ Analysis based on the Hall–Petch grain size strengthening and the Orowan particle strengthening, Fig. 3, for the Fe–10 Cr–2 at.% Zr alloy annealed at 900 °C was used to extract the volume fraction of intermetallic particles having a mean particle size of 20 nm. The stabilized volume average grain size (from TEM) was 82 nm, whereas the Zener pinning model predicts a grain size of 168 nm. The additional stabilization may be attributed to the thermodynamic mechanism. For these Fe-based alloys, it appears that thermodynamic stabilization may be operative for annealing temperatures below homologous temperature of about 0.65. At about this temperature, both Zener pinning by precipitates and thermodynamic stabilization may be providing stabilization. At higher temperatures, as the precipitates coarsen, and Zr is taken out of the grain boundaries, thermal stabilization is lost.

FIG. 2.

Vickers hardness (25 g load for 15 s) versus 1-h isochronal annealing temperature for pure Fe and Fe–Zr alloys; data from Ref. Reference Darling, VanLeeuwen, Koch and Scattergood32.

FIG. 3.

Hall–Petch plot of Fe–10 Cr and Fe–10 Cr + 2 at.% Zr; data from Ref. Reference Saber, Kotan, Koch and Scattergood33.

Other examples of the possible thermodynamic stabilization of nanocrystalline grain sizes are given in Table I along with the homologous temperature for apparent maximum stability (grain size remaining <100 nm) and the reference.

TABLE I.

Thermal stabilization of nanocrystalline materials by the thermodynamic mechanism of solute segregation to grain boundaries.

IV. THERMAL STABILIZATION BY ZENER PINNING OF GRAIN BOUNDARIES

There have been a number of examples of grain boundary pinning in nanocrystalline materials by the presence of small second phase particles—the Zener drag mechanism.Reference Humphreys and Hatherly³⁸ The expression for the pinning pressure exerted on the grain boundary by small particles is:

$$P_Z = {{3F{\rm{\gamma }}} \over {2r}}\quad,$$

where P_Z is the pinning pressure exerted by the particles on unit area of the boundary, F is the volume fraction of randomly distributed spherical particles of radius r, and γ is the specific grain boundary energy. It is clear that a high volume of small particles is desirable to increase the pinning pressure and thus to impede grain boundary mobility. It is also desirable for the nanoscale particles to resist coarsening at elevated temperatures. Several well-known properties such as coherent interface with the matrix (minimum interfacial energy), low solubility in the matrix, and low diffusivity in the matrix are needed but often not found in the same particles.

The stabilization in 5-nm grain size Ni–P alloys, which were produced by electrodeposition, did not take effect until after an increase of the grain size by a factor of 2–3.Reference El-Sherik, Boylan, Erb, Palumbo and Aust³⁹ The solid solution of the Ni–1.2% P alloy did not measurably grow and was stable as a solid solution up to an annealing temperature of 473 K. For annealing temperatures of 523 and 623 K, the grain size initially increased by a factor of 2–3, before the grain size stabilized, which coincided with the precipitation of a second phase, Ni₃P. Although other retarding effects in reducing the grain boundary mobility are considered (solute drag), the pinning of the grain boundaries by Ni₃P as the second phase was believed to be the major factor of the grain size stabilization at 573 and 623 K. Another example of nanocrystalline grain size stabilization by second phase particles is the work of Perez et al.Reference Perez, Huang, Sharif, Lavernia and Bourell⁴⁰ The authors studied the thermal stability of cryomilled Fe–10 wt% Al and found that a grain size of about 20 nm was maintained after annealing to about 600 °C, and even after annealing at 1100 °C, the grain size only grew to about 50 nm. This excellent grain size stability was attributed to the presence of nanoscale Al₂O₃ and AlN dispersoids. Another example of small particle-induced grain size stabilization is for an Al-based alloy by Shaw et al.Reference Shaw, Luo, Villegas and Miracle⁴¹ An Al₉₃Fe₃Cr₂Ti₂ alloy was prepared by mechanical alloying. The as-milled alloy was a solid solution of Al. After annealing at various temperatures, the internal strain was released, intermetallic compound particles precipitated, and there was some grain growth. The precipitation of a variety of intermetallics with Cr, Fe, or Ti was followed with several analytical techniques. There was essentially no grain growth at temperatures up to about 300 °C with the grain size distribution from TEM being about 6–45 nm. Even after heating to 450 °C (0.77 T _m), the grain size distribution was still nanoscale, i.e., from 20 to 100 nm. The authors believe that the inhibition of grain growth at the lower annealing temperature was due to solute drag on the grain boundaries, but at the higher temperatures, it was due to pinning of the boundaries by the nanoscale intermetallic precipitates. Morris and MorrisReference Morris and Morris⁴² reported stabilization of nanocrystalline grain sizes (<100 nm) at annealing temperatures up to 900 °C in nanocrystalline Cu with ZrC particles pinning the grain boundaries. The carbide particles were 5–10 nm in diameter. TEM was used to measure the grain size and particle sizes. This provides a very high value of the homologous temperature, 0.86, where a nanocrystalline grain size is maintained. Other examples of probable thermal stabilization of nanocrystalline grain sizes by Zener pinning are listed with the above in Table II. As in Table I, the maximum homologous temperatures are given along with the references.

TABLE II.

Thermal stabilization of nanocrystalline materials by the kinetic mechanism of Zener Pinning by nanoscale particles.

V. COMPARISON OF THERMODYNAMIC VERSUS ZENER PINNING STABILIZATION

The data from Table I and Table II are plotted in Fig. 4 to illustrate the temperature ranges where thermodynamic stabilization or kinetic stabilization by Zener pinning are likely. This strongly suggests that the Zener pinning mechanism may provide the best thermal stabilization at the higher homologous temperatures.

FIG. 4.

Maximum homologous temperature (T/T _m) for grain sizes <100 nm. Possible thermodynamic stabilization (●) and possible Zener pinning (O).