*4.5. The E*ff*ect of Vacancy Agglomerates on Strength*

During HPT-processing, vacancies and dislocations are introduced to the sample's lattice. With increasing temperature during heat treatments, the vacancies form agglomerates [65]. Disc-shaped agglomerates form on the close-packed basal planes of the hexagonal Mg lattice [66]. The disc collapses if it is large enough and produces a prismatic dislocation loop. The Burgers vector of such a loop is perpendicular to the plane of the loop, and the loop is therefore immobile. The formed loops are exclusively located on the preferred slip planes of Mg and are therefore strong obstacles for the movement of other dislocations.

Examples of hardening due to the agglomeration of deformation-induced vacancies have already been given some years ago for single and polycrystals of hcp materials for moderate deformation strains [67], and recently even for fcc materials [68,69]. With the latter, the increases in microhardness and yield strength were between 5% and 10% and thereby considerably lower than those observed in this and the recent study of Horky et al. [33]. This may be attributed to the hexagonal lattice of Mg, which makes loop hardening particularly effective because of the coincidence of the loop planes with the preferred dislocation slip plane.

HPT-processed samples provide a significantly higher number of vacancies than non-processed samples in the IS; therefore, the hardness increase during heat treatments is much higher for these samples than for the non-processed ones. The slight hardness increase of the samples in IS is mainly caused by the comparably low number of precipitates (Figure 8) [34].

One interesting fact is that Mg0.3Ca also shows a hardness increase during heat treatments. The peak temperature is around 75 ◦C and it represents a thermally-induced strength increase of 74% for the HPT-processed sample, which is significantly higher than those measured for Mg5Zn0.3Ca and Mg5Zn. For the other samples, heat treatments contributed ~30% to the increase of hardness. Precipitates probably form in the Mg0.3Ca alloy during heat treatments, but further TEM analyses need to be done.

Previous assumptions were that Zn atoms act as trapping sites for vacancies, but the experimental data on Mg0.3Ca show that Zn alone cannot be responsible for the hardness increase due to trapping-induced vacancy agglomeration. Samples with Ca content of 0.15% (Figure 10) showed almost no increase in hardness during heat treatments when furnace-cooled. In this case the vacancies, induced by HPT deformation, may have stayed single and did not agglomerate, even though there was a Zn content of 5%; this is indicated by the vacancy concentrations *cv* for the DSC peaks I and II shown in Tables 6 and 7, respectively. Obviously, the presence of Ca favors the formation of vacancy agglomerates.

As already reported in the previous Section 3, DSC and XPA measurements of furnace-cooled/ HPT-processed/heat-treated samples show very large concentrations of vacancies up to ~10−3; (Tables 6 and 7). These start to migrate/agglomerate/anneal at the very temperature (70–100 ◦C) where the largest values of microhardness appear. The present Kissinger analyses confirm this conclusion as they exhibit an enthalpy between *Q*(*I*) = 0.7–1.3 eV, which agrees well with literature values of vacancy migration enthalpies for Mg and Mg alloys, being 0.8–1 eV [37].

Moreover, it is well known from literature [30,68] that temperatures of peaks representing the annealing of single or double vacancies do not shift with the deformation degree applied. In contrast to that, the peak of dislocation annealing has been often found to shift to lower annealing temperatures the higher the deformation is. This effect can be understood as stress-assisted annihilation of dislocations because of their increasing stress field intensity due to increasing dislocation density at growing plastic deformation (see, e.g., [70]). In the present cases, variations of *Q*(*II*) were indeed found (Figure 23a,c) although occasionally at higher annealing temperatures as well. From this point of view, it is not surprising that the span of measured *Q*(*II*) = 1.3–3.8 ± 0.3 eV for dislocation annealing was much larger than that for vacancy annealing, *Q*(*I*). To substantiate these findings, a comparison with the dependence of annealing peak temperature on the applied deformation strain is helpful, as those values are known to be more accurate than those of activation enthalpies. Here, no such variations could be observed (Figure 23b,d), indicating that there may have been defects other than dislocations, too. When considering the evaluation of defect densities in this peak, vacancy concentrations of the order 10−<sup>4</sup> could still be found, representing a substantial proportion of defects found in this peak II [30].

**Figure 23.** (**a**) Activation enthalpies Q as evaluated from DSC scans and peak temperatures *Tmax* measured by DSC for different HPT-processed alloys: (**a**,**b**) Mg5Zn0.15Ca (full symbols) and Mg5Zn0.15Ca0.15Zr (open symbols), and (**c**,**d**) Mg5Zn0.3Ca (full symbols) and Mg5Zn (open symbols), all alloys as a function of torsional strain γ*T*. The red symbols represent the migration enthalpies of dislocation-type and agglomerate defects (peak II) and the black ones those of single/double vacancies (peak I). Errors are about 0.1 eV for Q, and 5 ◦C for *Tmax*.

*Metals* **2020**, *10*, 1064

A theoretical description of vacancy loop hardening (increase in yield strength Δσ*loops*) of hexagonal metals is given by Kirchner [71]:

$$
\Delta \sigma\_{\text{loops}} = \frac{Gb}{k} N^{\mathfrak{a}} d^{3a-1} \tag{7}
$$

with *N* being the loop density (number of loops/m3); *d* the average loop diameter; and *a* and *k* constants that depend on the ratio of loop distance (λ = *N*−1/3) to diameter *d*. *G* and *b* represent the shear modulus and the Burgers vector, respectively. For a ratio λ/*d* > 10, the constants amount to *a* = 1/2 and *k* = 0.122; otherwise, *a* = 4/3 and *k* = 0.001. According to Equation (7), the strengthening potential of loops is higher if the loop density is larger.

Although we do not know the number and the size of the loops in our HPT-processed and heat-treated samples, we can estimate the amount of loop hardening via Kirchner's equation (Equation (7)) by inserting the previously determined values for the vacancy concentration of HPT-processed materials and by an assumption for *d.* The number of vacancies per loop (*vacloop*) and the loop density *N* when assuming circular loops are given by

$$
tau\_{l\alpha\upsilon\rho} = \frac{d^2 \pi}{4b^2} \tag{8}$$

$$N = \frac{\mathfrak{c}\_V}{\mathfrak{vac}\_{\text{løp}}} \tag{9}$$

where *cv* is the concentration of vacancies assuming that all of them form loops. For simulations of the yield strength using Equation (7), the determined vacancy concentrations (Tables 6 and 7) were used for each alloy, and average loop diameters of 10–100 nm were assumed. Figure 24 shows the calculated dependence of theoretical yield strength on measured vacancy concentration for all alloys, samples in the IS (furnace-cooled) and HPT-processed at room temperature.

**Figure 24.** Illustration of the increase of yield strength as a function of vacancy concentration for the case of Mg5Zn alloy, samples of which were in the (IS (furnace-cooled) + HPT-processed) state.

The theoretical model predicts tremendously high increases in yield strength of up to 98 GPa (at *cv* = 2 <sup>×</sup> 10−<sup>3</sup> being the highest measured vacancy concentration), for the furnace-cooled materials Mg5Zn0.3Ca and Mg5Zn0.15Ca. The lowest increase was found for the Mg0.3Ca alloy with around 16 GPa. Mg5Zn0.15Ca0.15Zr and Mg5Zn showed increases of around 39 GPa.

From a first perspective, those strength increases predicted by the model are far higher than the experimentally measured ones: On the Mg5Zn0.3Ca alloy processed by HPT at low temperatures, one can conclude that from the experimentally determined thermally-induced increase of yield strength of 115 MPa, around 23 MPa were caused by precipitates. This means that about 92 MPa represents hardening from vacancy agglomerates.

Further simulations of strength increases due to loop hardening along Kirchner's model were done assuming fixed loop diameters at various fixed vacancy concentrations (Figure 25). The shaded area represents the field which spans all strength increases measured by tensile tests after HPT and heat treatment, for all materials investigated. The lines and full points represent fixed values of vacancy concentrations (lines) and fixed values of loop diameters (points). In addition to the estimations reported above, it is here even more evident that vacancy concentrations of the order of 10−<sup>5</sup> at maximum lead to the increased values of tensile strength measured. It means that the vacancy concentrations 10−<sup>3</sup> measured by DSC and XPA are far too high to account for these strength increases. Obviously, a major part of the vacancies does not contribute to hardening as they were not part of agglomeration and still stay single.

**Figure 25.** Increase of strength as a function of loop diameter at various vacancy concentrations. The red shaded area represents the values all measured vacancy-induced strength increases found in all materials which were investigated in this work.
