**5. Heterogeneous Nucleation**

Another group of nanocrystalline alloys under the most extensive study is light Al-based alloys. These alloys are light high-strength materials, which opens up extensive possibilities of their practical application. These alloys also crystallize by the nucleation and growth mechanism, and nanocrystals are formed by the primary crystallization reaction. Nanocrystal growth under the primary crystallization was investigated in a number of works, and it was shown, for example, for alloys of Al-Ni-Ce system [55] that it is diffusion-controlled. In literature, there are controversial data on the mechanisms of nanocrystal nucleation. There are literature data [56] stating that the process of Al nanocrystal nucleation is homogeneous. According to [57], this process is heterogeneous.

It is known that the homogeneous crystallization related to the fluctuation nucleation of nuclei with a critical size can take place only above the glass transition temperature Tg. At T < Tg, the viscosity of an amorphous phase is too high for such fluctuations, and nucleation can occur by heterogeneous mechanism only [58]. In light metallic glasses, it is difficult to define in which temperature range relative to Tg the process of nucleation occurs since the value of Tg in these alloys is unknown. Consequently, it is impossible to conclude correctly on the nucleation mechanism, based on the temperature range of the transformation.

To conclude reliably on the mechanism of nucleation under nanocrystallization, the data on nanocrystal distribution over size under different durations of isothermal exposure and the corresponding analysis of the distribution are required. This issue was studied in most details for alloys of Al86Ni11Yb3 system [59]. As a result of primary crystallization, a structure is formed in an alloy, which consists of an amorphous matrix containing Al fcc crystals randomly distributed over it. Al fcc crystals have a size of several nanometers. An example of this structure is shown in Figure 5. Al nanocrystals are generally isolated by the amorphous matrix from each other.

**Figure 5.** Microstructure of Al86Ni11Yb3 sample after annealing at 473 K for 30 min: (**a**) bright-field and (**b**) dark-field images.

A change in the average nanocrystal size measured by dark-field electron microscope images depending on the exposure duration is presented in Figure 6. The average size changes from 8 nm (under exposure for 5 min) to 12 nm (under exposure for 60 min). As one can see, the sharpest changes in the average nanocrystal size occur at initial transformation stages. The experimentally obtained nanocrystal distributions over size for exposures for 5 and 15 min are illustrated in Figure 7. Note that the fraction of the smallest crystals may be too low due to difficulty of the observation. This is particularly important for the obtained distribution over size under exposure for 5 min, when the distribution is shifted towards the region of small sizes.

**Figure 6.** Dependence of an average nanocrystal size (triangles) and fraction of the nanocrystals (asterisks) on the duration of annealing at 473 K for Al86Ni11Yb3 sample [59] [reproduced from Physics of The Solid State 2001, 43, 2003 with permission from Pleiadis Publishing, 2020].

**Figure 7.** Nanocrystal distribution over size after annealling for (**a**) 5 and (**b**) 15 min. The theoretical data are marked with solid curves the experimental data are marked with columns [59] [reproduced from Physics of The Solid State 2001, 43, 2003 with permission from Pleiadis Publishing, 2020].

The obtained experimental distributions of nanocrystals over size have some specific features. To analyze the features, let us consider theoretically possible crystal distributions over size and compare them with those observed experimentally. Figure 8 shows all the possible size distributions of crystals, which crystallize by the nucleation and growth mechanisms, for homogeneous and heterogeneous nucleation [60].

If one compares these distributions with those observed experimentally, it is clear that heterogeneous nucleation with a latent period takes place in the alloys under study. This is indicated by the following:


Thus, at the annealing beginning, there is a time period (τ), during which the stationary distribution of subcritical nuclei over size is reached that corresponds to classical theory. According to [61], in this case, the time-dependent rate of nucleus formation I(t) is determined by the equation

$$\mathbf{I(t)} = \mathbf{I\_{st}} \{ 1 + 2 \sum (-1)^{n} \exp \left[ -n^{2} (\mathbf{t}/\tau) \right] \}\tag{1}$$

where the summation is performed over n in the range between 1 and <sup>∞</sup>, τ is the latent period which increases sharply with decreasing temperature, and Ist is the nucleation rate under stationary conditions, which in turn is described by the equation

$$\mathbf{I}\_{\rm st} = \mathbf{I}\_0 \cdot \exp(-\mathbf{L} \Delta \mathbf{G}\_\mathbf{c} / \mathbf{R} \mathbf{T}) \cdot \exp(-\mathbf{Q}\_\mathbf{N} / \mathbf{R} \mathbf{T}) \tag{2}$$

where L is the Loschmidt number, QN is the activation energy of transfer of an atom through the crystallization front surface, ΔGc is the free energy necessary for the formation of a critical nucleus. Equation (2) is an approximate solution of the Fokker–Planck equation which was obtained for the first time in [62].

At strong supercooling, the value of ΔGc is very low, then

$$\mathbf{I}\_{\rm st} = \mathbf{I}\_0 \exp(-\mathbf{Q}\_{\rm N}/\mathbf{RT}).\tag{3}$$

Let us examine the growth of the formed nanocrystals. Since, in the case under consideration, the concentration gradient of other alloy components arises in the amorphous matrix in front of growing Al nanocrystals, the matrix gets enriched with Ni and Yb, the atoms of which diffuse over larger distances. Then, the growth rate of the formed crystals decreases with annealing duration. At the same time, it is known [58] that under primary crystallization, the radius of the growing crystals depends parabolically on the duration of isothermal exposure. In our case, the growth of the crystals is determined by the bulk diffusion of Ni and Yb components in the amorphous matrix

$$\mathbf{R} = \alpha \text{ (Dt)}^{0.5} \tag{4}$$

where D is the bulk diffusion coefficient, t is the time of isothermal exposure, R is the radius of a growing crystal, and α is the dimensionless parameter of an order of unity. Meanwhile, we consider the parameter α to be independent of the fraction of the crystalline phase.

This time dependence of the crystal size results in that nanocrystal distribution over size in the case of heterogeneous nucleation becomes narrower with time. Thus, for the distribution shown in Figure 7a the dispersion was 15.76 nm2, and for that presented in Figure 7b it was 4.96 nm2. In principle, the Ostwald coalescence described by the Lifshitz–Slyozov theory [63] can also lead to narrowing of the histograms of nanocrystal distribution over size and shift it.

In order to specify the heterogeneous mechanism of nanocrystal nucleation and growth during isothermal exposure, it is necessary to perform computer calculations and plot the histograms of nanocrystal distribution over size for heterogeneous nucleation and diffusion-controlled growth with the purpose of comparing them with the experimental data.

To carry out these calculations using Equations (1)–(4), it is necessary to know the following parameters:

N0, which is the number of nuclei, limited under heterogeneous crystallization;

τ, which is the latent period (duration of the nonstationary stage);

Ist, which is the rate of nucleation under stationary conditions;

I0, which is the constant determining the stationary rate of nucleus formation, and

QN, which is the activation energy of transfer of an atom through the crystallization front surface.

The value of I0 can be considered to be 3 × 10<sup>30</sup> m s<sup>−</sup>1; such a value is typical of the nucleation of Al nanocrystals in alloys of Al-Ni-REM systems [56]. The value of N0 can be calculated from the experimental data as follows. Under exposures for more than 30 min, the fraction of the crystallize phase does not significantly change (it is about 0.23). An average size of the nanocrystals is about 12 nm. Then, the number of nanocrystals, N0 (by assuming that all the regions of heterogeneous nucleation are implemented and by neglecting the possible nanocrystal coalescence), will be about 2 × 10<sup>23</sup> m<sup>−</sup>3. Such an estimate agrees well with the known literature data. Thus, in [56], it is reported that under nanocrystallization N0 can reach 10<sup>25</sup> m<sup>−</sup>3. According to [64], in order to estimate

nanocrystal distribution over size it is necessary to divide the time of isothermal exposure into short time intervals Δt and to calculate the number of nanocrystals crystallized during each interval Δt. Then, for heterogeneous crystallization at the limited number N0 of active nuclei

$$\mathbf{N}\_{\rm i} = \mathbf{I} \left( \mathbf{t} \right) (1 - \mathbf{x}\_{\rm i-1}) (1 - \mathbf{N}\_0^{-1} \sum \mathbf{N}\_{\rm j}) \Delta \mathbf{t}\_{\rm \prime} \tag{5}$$

where the summation is performed over j in the range between 1 and i at Nj ≤ N0; Ni = 0 for all the other values of i. xi is the volume fraction of a material crystallized during the time interval Δt by the mechanism of primary crystallization (R = (Dt)<sup>1</sup>/2):

$$\mathbf{x}\_{\mathbf{i}} \approx \mathbf{D}^{3/2} \sum \mathbf{N}\_{\mathbf{j}} \left\{ \Delta \mathbf{t} (\mathbf{i} + \mathbf{1} - \mathbf{j}) \right\}^{3/2},\tag{6}$$

where the summation is performed over j in the range between 1 and i.

The shape and location of the theoretical curve on the axes of size distribution depend on the values of the parameters substituted into the formulas. The correction of the parameters Q N and τ and the di ffusion coe fficient (D) enables the best approximation of the theoretical curve to the distribution obtained experimentally.

**Figure 8.** Histograms of the grain distribution over size for (**<sup>a</sup>**,**b**) homogeneous and (**<sup>c</sup>**,**d**,**<sup>e</sup>**) heterogeneous nucleation; (**b**,**<sup>e</sup>**) cases of nonstationary nucleation [59]. [reproduced from Physics of The Solid State 2001, 43, 2003 with permission from Pleiadis Publishing, 2020].

The two experimental curves of nanocrystal distribution over size for Al86Ni11Yb3 alloy, presented above, should be compared with those calculated theoretically. When estimating both histograms, one should use the same crystallization parameters. A change in their shape and location should be related to the di fferent process durations (5 and 15 min) only. This requirement is rather rigid. It was shown that both experimental curves can correspond quite well to the theoretical ones at the same values of Q N and τ, but di fferent di ffusion coe fficients D (Figure 9). For good correspondence of the curves, the di ffusion coe fficient D should diminish with a rise in the exposure time (and, correspondingly, in the fraction of the formed crystalline phase). A decrease in D under primary crystallization of amorphous alloys is rather usual at in increase in the crystallized material fraction [58]. The same phenomenon is apparently observed in the case under consideration. From the comparison of calculated and experimental data it follows that at 473 K the effective coefficient of diffusion of Ni and Yb in amorphous Al86Ni11Yb3 alloy is 1.4 × 10−<sup>19</sup> m2s−1, and the latent period is 150 s. Since usually the diffusion coefficient of Ni is significantly higher than that of Yb, one may suppose that it is the Yb removal rate from growing crystal that limits the nanocrystal growth. Then, the obtained diffusion coefficient value is related to Yb diffusion. The value of Yb diffusion coefficient which is 1.4 × 10−<sup>19</sup> m2s−<sup>1</sup> at 473 K seems to be rather realistic. If one compares it with the available data on the diffusion coefficients of rare-earth metals in Al-based alloys, their similarity should be noted. Thus, the diffusion coefficient of Y in Al88Fe7Y5 alloy is 9 × 10−<sup>20</sup> m2s−<sup>1</sup> at 518 K [65]. This value was obtained using the Frank approach, where it is assumed that the parameter α in the equation R = α (Dt)0.5 is 1.5.

**Figure 9.** Dependence of an average size of the nanocrystals in Al86Ni11Yb3 alloy on the exposure duration at 473 K. The calculated results are marked with a solid line, the experimental values are shown with rhombi [59].

Note that Hono et al. [66] detected a rise in the Ce concentration in front of growing Al nanocrystals under the crystallization of alloys of Al-Ni-Ce systems, while Ni was distributed uniformly over the amorphous phase. Considering the similarity of the properties of Ce and Yb atoms, the above assumption on the ratio of the diffusion coefficients of Ni and Yb elements seems to be rather reasonable.

Another difference between the experimental and calculated distributions over size is, as one can clearly see in Figure 9, a "tail" of large particles presented in the experimental histograms. The existance of larger particles is probably related to the presence of a small amount of the so-called "frozen-in crystallization centers" in as-prepared alloy. The formation of crystals is facilitated in this case. The particles grow earlier (up to the completion of the latent period of attaining the stationary distribution of subcritical nuclei over size) and reach larger sizes. The particles grow earlier (by the end of the latent period of reaching the stationary distribution of subcritical nuclei) and grow to larger sizes.

When calculating above, it was assumed that the parameter α in the equation R = α (Dt)0.5 determines the time dependence of the radius of a growing crystal under primary crystallization, independently of the composition of the residual amorphous matrix. Therefore, when it turned out that the rate of crystal growth decreases with time more sharply than it follows from the dependence R = α (Dt)0.5, an agreemen<sup>t</sup> between the calculations and the experiment was reached by decreasing the diffusion coefficient with the process duration. However, it seems to be expedient to consider another approach which relates a more significant than it follows from the equation R = α (Dt)0.5 decrease in the growth rate with time to a decrease in the driving force of the process. If one takes into account a change in the matrix composition, according to the Ham's approach [67] the time dependence of the size will be determined as

$$\mathbf{R(t)} = \left[2(\mathbf{C\_m} - \mathbf{C(t)})/\mathbf{C\_m} - \mathbf{C\_P}\right]^{1/2} \left(\mathbf{D} \,\mathrm{t}\right)^{1/2} \tag{7}$$

where C m and Cp are the concentration of a redistributed component in an amorphous matrix and precipitate at the interface; C(t) = C0/1 − x(t) is the concentration of a component in an amorphous matrix the time t after the process onset.

Ham's model considers a sequence of the identical particles; the initial sizes of a growing particle are negligibly small, the concentration of an admixture (alloying element) in a matrix near the particle is constant along the entire interface, at the interface the condition of di ffusion equilibrium is satisfied, and an average size of the particle is about a half of the distance between the particles at the moment of reaction completion (i.e., at their maximum quantity). In [68], an expression was obtained which relates C m, C(t), Cp, t, and R0 to each other, where R0 is the distance between the particle centers.

$$\begin{aligned} \text{(D t/(R\_0)^2) } \left[ (\text{C}\_0 - \text{C}\_{\text{m}}) \langle \text{C}\_{\text{m}} - \text{C}\_{\text{P}} \rangle \right]^{1/3} &= \text{(1/6)} \ln \left[ (\text{u}^2 + \text{u} + 1) \langle \text{u}^2 - 2\text{u} + 1 \rangle \right] - \\ &\quad \text{(1/3)} ^{1/2} \tan^{-1} \left[ (2\text{u} + 1) / 3^{1/2} \right] . \end{aligned} \tag{8}$$

where u<sup>3</sup> = 1 − C(t)/C0.

When obtaining this expression, it was assumed that C(t = 0) = C0, the initial radius of a particle is zero.

The essence of this consideration is that at an early crystallization stage the enrichment of a matrix is low due to the concentration redistribution, (C(t) ≈ C0) and R ~ (Dt)<sup>1</sup>/2. At times corresponding to the final stages of the reaction, an average matrix composition approaches C m and dR/dt → 0. The rate of precipitate growth converges to zero since the driving force of the precipitate process converges to zero. Taking into account the concentration dependence of the parameter α, the dependences of an average size of Al nanocrystals in Al86Ni11Yb3 alloy were calculated. N0 was 2 × 1023, D = 1 × 10−<sup>19</sup> m<sup>2</sup> s<sup>−</sup>1, x(t), the fraction of a crystallized material, was determined. The obtained results are demonstrated in Figure 9, where the results of the calculations based on Ham's approach are marked with a solid line and the experimental points are shown as rhombi. It is seen that a good agreemen<sup>t</sup> between the calculated and experimental data is observed. Note also that, in this case, the correspondence is observed when using one value of the di ffusion coe fficient for all durations of isothermal exposure in calculations. This value almost coincides with that obtained earlier. A slight di fference is caused only by the coe fficients used in the estimates. In the first case we assumed R = (Dt)<sup>1</sup>/<sup>2</sup> and in the second case we calculated R = α (Dt)<sup>1</sup>/2, where

$$\alpha = \left[ 2(\mathbf{C}\_{\rm m} - \mathbf{C}(\mathbf{t})) \mathbf{C}\_{\rm m} - \mathbf{C}\_{\rm P} \right]^{1/2}. \tag{9}$$

In principle, Ham s analysis is related to binary systems. However, based on the data presented in [69] one can consider it to be rather applicable to the case of light three-component alloys.

One should also analyze a possible e ffect of the Ostwald coalescence on the observed histograms of nanocrystal distribution over size. In the Lifshitz–Slyozov theory, the evolution of crystal sizes is described by the following equation:

$$\text{R}^3 - \overline{\text{R}}\_0 \text{ $^3 = 8 \text{ D } \sigma \text{ V}$ \text{ $^{\circ}$ C}(\text{\infty}) \text{ t/ 9 N k T} \tag{10}$$

where R is the average particle size, R0 is the initial average size, V m is the molar volume of precipitates, σ is the energy of the particle-matrix interface, C( ∞) is the equilibrium solubility of a component at a grea<sup>t</sup> distance from a particle, k is the Boltzmann constant, and N is the Avogadro number.

The maximum growth rate caused by the coalescence is

$$\text{(dR/dt)}\_{\text{max}} = 8 \text{ D } \sigma \text{ V}\_{\text{m}} \text{ C(\text{\textdegree\textdegree C})/27 \text{ N } \text{k T R}^2 \text{\textdegree R}^2 \tag{11}$$

The maximum growth rate estimated by Equation (11), which was caused by the coalescence, was about 0.2 nm/h (for 473 K, R ≈ 4 nm after the exposure for 5 min, and D =1.4 × 10−<sup>19</sup> m<sup>2</sup>/s). For R ≈ 5.5 nm (after the exposure for 15 min), the maximum growth rate is <0.03 nm/h. One can see that these rates are insignificant in the considered time interval of the nucleation and evolution of the nanocrystals (exposure for up to 60 min). Ardell [70] made the corrections in Equation (8), allowing for the volume fraction of crystals. Ardell obtained an equation which di ffers from the Lifshitz–Slyozov equation in the parameter K, which is the function of the crystal volume fraction only:

$$
\overline{R}^3 - \overline{R}\_0^3 = 8 \text{ K D } \sigma \text{ V\\_C(\infty) } \text{t/} \beta \text{ N k T} \tag{12}
$$

K = 1 at zero volume fraction of precipitates, K ≈ 6 at 15% fraction (exposure of the alloy under investigation for 5 min), and K = 10 at 25% fraction, which approximately corresponds the exposure for 15 min in our case. Then, the maximum growth rates caused by the coalescence are ~1 nm/h and 0.3 nm/h for nanocrystals in the alloy after the exposure for 5 and 15 min, respectively. Note that these estimates are too high since the di ffusion coe fficient decreases with time (in this calculation, the dependence of the parameter α in the equation dR/dt = (α/2)(D/t)<sup>1</sup>/<sup>2</sup> on the crystalline phase fraction was considered, and it was taken to be constant and equal to 1). The obtained values can be regarded as the estimates "above" the values of an instantaneous growth rate (under the exposure for 5 and 15 min) decreasing with time. Therefore, in this case, even consideration of the parameter K related to the volume fraction of precipitates does not make these rates significant for the evolution of nanocrystal distribution over size.

Furthermore, in the Lifshitz–Slyozov theory it is considered that the system is in equilibrium, ib this case the formation and growth of particles of the second phase do not occur due to the matrix (the fraction of particles of the second phase is constant). Thus, a change in the nanocrystal size, caused by coalescence processes, which is described by the Lifshitz–Slyozov theory, may be significant after the completion of nanocrystal growth from the amorphous phase during the existence of metastable equilibrium of the nanocrystals – amorphous matrix. For coalescence processes to occur, this equilibrium should be persist for a long time, and no further crystallization of the amorphous phase should take place. It is important to note that the particle distributions over size were obtained for the stages at which a metastable equilibrium between the amorphous and nanocrystalline phases was not reached ye<sup>t</sup> and the fraction of the nanocrystalline phase continued to increase.

Thus, nanocrystal nucleation under the crystallization of amorphous Al86Ni11Yb3 alloy occurs by the heterogeneous mechanism from "frozen-in" crystallization centers. A good agreemen<sup>t</sup> between the experimental data on a change in the nanocrystal size with the time of isothermal annealing and the calculations by the method described above was observed also for alloys of Al-Ni-Y system [71].

### **6. Some Features of Nanocrystal Formation (Free Volume)**

The interrelation of an amorphous state and the crystalline phases emerging under its decay would be more evident if under crystallization there were no e ffects having a significant influence on the formed structure. The bulk crystallization e ffect belongs to these e ffects. The point is that the density of amorphous metallic alloys is 1–5% lower than their density in a crystalline state. Larger di fferences in the density of the amorphous phase and the arisen crystalline phases are observed in the case when the amorphous phase has semiconductor properties, for example, in the Al–Ge system. Therefore, it seems to be important to study how the compensation of bulk phase mismatch occurs under crystallization and how this a ffects the formed structure.

Several methods of the compensation of bulk mismatch are possible:


Obviously, since the densities of the amorphous and emerged crystalline phases differ, elastic stresses and, consequently, deformation of both the crystalline and amorphous phases occur in the front. The values of the elastic stresses and deformation can determine the relative position of the phases, their morphology and structure, as well as the formation sequence. Obviously, the methods of compensation of the bulk effect act in an integrated way.

Let us consider some examples which demonstrate different methods of the bulk effect compensation.

### *6.1. Dependence of the Sequence of Phase Formation and Crystal Morphology on the Bulk E*ff*ect Demonstration in Alloys of Fe-B, Fe-Co-Si-B Systems*

As stated in [72], the method of compensation of bulk mismatch under crystallization can determine the morphology of precipitates. In order to investigate the bulk effect demonstration, a comparative study of crystallization in the samples of "thick" (with an initial thickness) and "thin" cross-sections was carried out. Polished samples for electron microscopy are meant by thin cross-sections. Under the in-situ studies of crystallization in an electron microscope column, phase transformations are usually investigated in samples with a thickness of ≈100 nm. The thin cross-sections were studied to ge<sup>t</sup> rid of the influence of effects related to thick cross-sections on crystallization. The remoteness of the sinks of elementary free volume carriers from the reaction front, difficulty of the compensation of bulk mismatch, and others belong to these effects.

The investigations of samples of Fe100−<sup>x</sup>Bx (16 < x < 20) alloys showed [73] that at the first stage of crystallization of samples of all the studied compositions, α-Fe crystals (or rather crystals of a solid solution of B in α-Fe) are precipitated. In the thin cross-sections, precipitates of α-Fe have the form of commas, needles, plates, differing in the thickness, of an irregular shape with a length of several tens of nanometers and thickness of 5–10 nm (Figure 10). Precipitate chains are formed, and a correlation in the arrangemen<sup>t</sup> of first and subsequent precipitates is observed. Then, grains of Fe3B arise, and a structure consisting of Fe and Fe3B grains is formed.

**Figure 10.** Microstructure of Fe84B16 alloy sample annealed in an electron microscope (in-situ) at 623 K for 90 min: (**a**) bright-field and (**b**) dark-field images [reproduced from Mat. Sci. [73]. Eng., 1978, 36, 193 with permission from Elsevier, 2020].

Under the crystallization of ribbons with an initial thickness, the sequence of phase precipitation and morphology of the precipitates differ. At the first stage, in alloys of a hypoeutectic composition (at.% of B < 17.5), as well as in the thin cross-sections, α-Fe crystals are the first to nucleate and grow. They have faceting and form crystallites with a distinctly dendritic shape (Figure 11). The crystallites

are randomly distributed over the amorphous matrix, and no correlation in their arrangemen<sup>t</sup> is observed. At the second crystallization stage, colonies are formed, which consist of α-Fe and Fe3B.

**Figure 11.** Microstructure of Fe84B16 alloy sample with an initial thickness, annealed for 60 min at 623 K.

Thus, in hypoeutectic alloys, primary α-Fe crystals were not formed in the thick cross-sections. In the thin cross-sections, α-Fe crystals were always first to be precipitated. In the thick cross-sections, colonies were formed, which consisted of α-Fe and Fe3B, while in the thin cross-sections no colonies (i.e., simultaneous formation of α-Fe and Fe3B) were observed, and Fe3B grains emerged at the second crystallization stage. Fully crystallized samples of thick cross-sections contained dendritic α-Fe crystals and colonies consisting of α-Fe and Fe3B, and in the thin cross-sections there was a structure consisting of Fe and Fe3B grains.

It is reasonable to relate the observed difference in the morphology and arrangemen<sup>t</sup> of crystals in the thin and thick cross-sections to the proximity in thin surface cross-sections. In the thick cross-sections, the compensation of the bulk effect occurs, most probably, with the viscous flow of a matrix and deformation of precipitates and a matrix, and in some cases, probably, with the formation of micropores in the reaction front.

The formed α-Fe crystals in the thick cross-sections grow anisotropically in the field of tensile stresses, which explains their growth in the {111} planes in the {110} direction, but not in the close-packed {110}, as was observed in [74]. In the case of thin cross-sections, the compensation of the bulk effect has time to proceed by the diffusion of excess volume carriers to the surface due to proximity of the surface. Deformation fields around α-Fe have lower values, and the crystal shape is more equilibrium than that in the case of thick cross-sections. According to the literature data, the density of Fe crystals is 7.48 × 10<sup>3</sup> kg m<sup>−</sup><sup>3</sup> [74], and that of Fe3B is 7.48 × 103kg m<sup>−</sup><sup>3</sup> [75]. The density of amorphous Fe83B17 alloy is 7.3110<sup>3</sup> kg m<sup>−</sup><sup>3</sup> [75]. Consequently, under the formation of the same fraction of these phases, the arisen level of stresses, related to crystallization, will be higher in the case of Fe formation than that in the case of Fe3B formation. However, the compensation of bulk mismatch and, consequently, a decrease in the internal stresses are carried out more easily in the thin cross-sections. Therefore, the formation of primary Fe crystals and Fe3B occurs in all the alloys under study after the completion of Fe precipitation. The formation of colonies in the thick cross-sections, which have a complex structure and structural components nano-sized in two directions, may be associated, in this case, with a decrease in the stress level in the crystallization front and avoidance of the disturbance of material continuity.

In principle, there can be another explanation of the e ffects observed, i.e., chemical composition change in the thin cross-sections as compared with the thick ones. However, the study of distribution of element concentrations over sample depth, performed by Auger electron spectroscopy, did not reveal any significant di fferences.

### *6.2. Formation of a Nanocrystalline Structure as Demonstration of the Bulk E*ff*ect*

A vivid demonstration of the influence of the bulk crystallization e ffect on the morphology and structure of the formed phases is the formation of a nanocrystalline structure in Al32Ge68 alloy. A feature of this material is that Al and Ge phases are formed under the decomposition of Al32Ge68 alloy, with Al being less dense and Ge being denser than the amorphous matrix. In this case, one could expect the formation of a specific structure in the region of the crystallization front. One can assume that the compensation of the bulk e ffects (with opposite signs) of phase formation should lead to the formation of a highly dispersed mixture of two formed phases in the reaction front in order to avoid the disturbance of sample continuity. In this case, bulk e ffects with opposite signs will be compensated, and the sample can remain continuous. The studies carried out [76] demonstrated that at an initial crystallization stage a nanocrystalline structure is formed, the nanocrystal size is about 10 nm. Crystallization begins in the depth and propagates to the sample edge. The formed nanocrystalline structure is very unstable. At that, equilibrium phases of Al and Ge are formed. Recrystallization under isothermal exposure occurs very quickly; the nanocrystalline region is followed immediately by the region of larger crystals. According to the data of di fferential scanning calorimetry, crystallization proceeds in one stage. Thus, at initial crystallization stages, a nanocrystalline structure is formed, which was detected near the crystallization front only. A significant grain growth occurs at a distance of 100 nm from the front.

The formation of a nanocrystalline structure, as a consequence of the compensation of bulk mismatch under the amorphous phase crystallization, is caused by the necessity of elastic stress compensation at a nanolevel in the crystallization front in order to avoid the disturbance of material continuity (the emergence of additional free surfaces under the decomposition will increase the free energy of a system and can make this process unprofitable). In this case, such demonstration of the bulk mismatch compensation is related to large di fferences in the density of an initial amorphous phase and formed crystalline phases. The specific molar volume of an amorphous matrix is, according to di fferent data, within the limits of (11.2–12.6) 10−<sup>6</sup> m3mol−1, and that of crystalline Al and Ge is 10.0 × 10–<sup>6</sup> m3mol–1 and 13.6 × 10−<sup>6</sup> m3mol−1, respectively. That is, the density of an amorphous matrix is in an intermediate position between Al and Ge densities, and the di fference is approximately 20% for both elements. Thus, the formation of one phase under crystallization results in a very high level of stresses in the reaction front, which is compensated immediately under the formation of the second phase. Thereby, one succeeds in avoiding sample decomposition. The formation of phases in a nanocrystalline state results in the stress compensation not only at macrolevel, but at the microlevel (nanolevel). An important fact is that the amorphous phase exists in a narrow concentration rage only, and precipitation of one of the phases will result in a change in its composition and, consequently, to the decomposition. Therefore, the simultaneous formation of two phases is observed, and their size is determined by the necessity of compensation of the bulk e ffect of phase formation in order to preserve the sample continuity.

The formed nanocrystalline structure turns to be very unstable. In this structure, the area of interphase interfaces and the corresponding energy are high, that is why grain coarsening is necessary to diminish the free energy of the system, which happens in reality. In this case, nothing impedes the coalescence processes: the nanocrystals are separated by high-angle boundaries, but not by the special low-energy ones, and there is no layer of an amorphous matrix isolating the crystals from each other. Under the formation of a nanocrystals by the primary crystallization mechanism, the amorphous matrix changes its composition as the nanocrystals nucleate and grow. In the case of amorphous Al-Ge alloy, this process is impossible. The reason is that, as stated, the amorphous alloy of Al32Ge68 nominal

content has a very narrow concentration region of existence [77], and the layers of an amorphous phase with a changed composition cannot exist between the grains. Consequently, there are no obstacles to coalescence, and the nanocrystalline structure will decompose easily. Al should be the leading phase here, for which the corresponding homologous temperature is lower. As shown experimentally, the prevailing growth of Al crystals is observed under the nanocrystalline phase separation.

### *6.3. Compensation of Structural Mismatch by Pore Fformation and Nanocrystal Formation in the Shear Bands*

A number of works, for example, [78–80] were devoted to the study of shear band formation and structure in amorphous alloys. Plastic deformation in alloys is strongly localized and is realized by the formation and propagation of di fferent shear bands. The rate of shear band propagation does not depend on the deformation rate in a range of 2 × 10−4–10−<sup>2</sup> s<sup>−</sup>1. High shear stresses are localized in shear bands, as a result, a large amount of free volume is concentrated in them; the shear bands have more random structure compared to the surrounding amorphous phase [78,81]. The structure of the main part of an amorphous matrix also can change under deformation; it becomes anisotropic under certain conditions [29–31]. The number of shear bands depends on the alloy chemical composition and deformation conditions. Their size can be from several tens of nanometers to several micrometers in width and from tens to hundreds of micrometers in length.

As a result of plastic deformation, the amorphous phase structure becomes non-uniform: an amorphous structure in the region of a shear band di ffers from the structure in a surrounding amorphous matrix. The relaxation of a structure in the shear band can lead to di fferent e ffects.

In literature there are several models which describe the processes responsible for the shear band formation. All of them include, to varying degrees, the concept of free volume. Thus, for example, in the work by Spaepen [82] it is assumed that the motion of a material in a shear band consists of the formation and disappearance of free volume regions. An interest in shear bands is caused by several factors. Firstly, a lot of mechanical properties of amorphous alloys depend on the presence and characteristics of shear bands. Secondly, as was demonstrated in [83–85], the nucleation of nanocrystals begins in shear bands or their vicinity. In [86], it was demonstrated that pores also grow in shear bands (Figure 12). The arrow indicates the reflex in which the dark-field image is obtained.

**Figure 12.** Microstructure of deformed Al88Ni2Y10 sample (40%): (**a**) bright-field and (**b**) dark-field images.

(**a**)

The analysis of pore growth kinetics (Figure 13) allowed estimating the e ffective di ffusion coe fficient. When calculating, it was assumed that pores grow due to di ffusion of free volume from a shear band to the bulk of the surrounding matrix. It was determined that the di ffusion coe fficient decreases with time at room temperature RT (from 3 × 10−<sup>24</sup> m<sup>2</sup>/s for ageing for 1.78 × 10<sup>7</sup> s, to 7 × 10−<sup>25</sup> m<sup>2</sup>/s for ageing for 2.95 × 10<sup>7</sup> s). A decrease in the di ffusion coe fficient with time is caused by free volume depletion in shear zones adjacent to pores. As mentioned above, shear bands can be

the regions of facilitated nanocrystal formation. The formation of nanocrystals in a shear band can be observed in Figure 12.

**Figure 13.** Shear bands in amorphous Al88Ni2Y10 after deformation at room temperature (**a**), nanopores (marked with arrows) in shear bands after ageing at room temperature for 1.78 × 10<sup>7</sup> s (**b**), nanopores in shear bands after ageing at RT for 2.96 × 10<sup>7</sup> s (**c**) [reproduced from Mechanics of Materials, 2017, 113, 19 with permission from Elsevier, 2020] [86].

(**c**)

A number of experimental results also are an evidence of high diffusion coefficient values in these regions. In [30,87] the formation of nanocrystals was observed in deformation bands under the subsequent exposure of samples at room temperature. The preferred crystallization in shear bands was observed also in [88]. The authors found out the preferred nanocrystal formation in shear bands in Al88Y7Fe5 alloy samples deformed by tension. The authors related the nanocrystal formation in shear bands to an increase in the local mass transfer rate in these regions. Similar results were obtained in [89], where the authors demonstrated the formation of Al nanocrystals in rolled amorphous Al85.1Ni6Co2Gd6Si0.9 alloy at room temperature. Figure 14a shows the microstructure of an alloy deformed by multiple rolling. Figure 14b illustrates the microstructure of the same alloy after ageing at room temperature for ~6000 h. In Figure 14a, one can clearly see brighter regions with an extended shape, which are oriented along some direction. The size of the regions is up to 100 nm in length and 20–40 nm in width. These regions have rather sharp boundaries with the amorphous matrix and represent regions (shear bands) with a high level of deformation, i.e., shear bands. They contain a small number of nanocrystals formed during deformation. The surrounding matrix remains amorphous. Thus, the formation of Al nanocrystals under rolling occurred in the places of plastic deformation localization. Under the ageing of a sample at room temperature, its structure changed significantly. In shear bands, the number of nanocrystals increased strongly, and shear bands became completely filled with nanocrystals. Meanwhile, a small number of nanocrystals emerged in other parts of the amorphous matrix, too. The average size of nanocrystals in shear bands increased slightly.

**Figure 14.** Microstructure of Al85.1Ni6Co2Gd6Si0.9 alloy sample (**a**) after rolling and (**b**) after ageing at room temperature ) [reproduced from Acta Materialia, 2008, 56, 2834 with permission from Elsevier, 2020] [83].

For comparison, Figure 15 shows the microstructure of a sample after rolling and heating to 245 ◦C. One can see that a considerable number of nanocrystals were formed in the sample during heat treatment; however, it is seen that their major part is concentrated in the region of the shear bands. The rates of diffusion providing a slight growth of Al nanocrystals at about 60 ◦C were estimated to be 10−<sup>24</sup> m<sup>2</sup>/s. These results agree with the data of [90], where it was demonstrated that the formation of excess volume during plastic flow can lead to an increase in the diffusion coefficient in shear bands by 4–6 orders of magnitude. The formation of nanocrystals in local shear bands in deformed amorphous alloys was observed in both Fe–B alloys (Figure 16) and deformation bands of Al88Ni2Y10 alloy (Figure 12). An important conclusion of the authors of the above works is that high atomic mobility in the zones of plastic deformation localization makes a crucial contribution in the nanocrystallization process stimulated by deformation.

**Figure 15.** Microstructure of a sample after rolling and heating to 245 ◦C [reproduced from Physics of The Solid State 2011, 53, 229 with permission from Pleiadis Publishing, 2020] [89].

**Figure 16.** Nanocrystals in deformed amorphous Fe-B alloy.

### **7. Nanocrystal Formation in Amorphous Phase**

The parameters of the crystallized structure depend on the processing conditions. As a result, many properties of materials change dramatically with a change in structure. So, for example, the properties of nanocrystalline alloys differ both from the properties of polycrystals and from the properties of amorphous alloys [3,4]. As a rule, the nanocrystalline structure forms by the primary crystallization reaction. The nanocrystalline structure in most cases is two-phase and consists of nanocrystals formed by the primary crystallization reaction, and interlayers of the remaining amorphous phase of a changed composition. The nanocrystalline structure was first obtained in an alloy of the Fe-Cu-Nb-Si-B system, called Finemet [91].

The principle of obtaining a nanocrystalline structure in this alloy was based on the fact that small amounts of copper and niobium were added to the base composition of the amorphous Fe–Si –B alloy. An amorphous Fe–Si–B alloy crystallizes upon heating to form a usual structure with a grain size noticeably exceeding nanosizes. Copper addition leads to the formation of microsegregations, which serve as sites of facilitated nucleation of crystals, and the addition of slowly diffusing niobium helps to slow down the growth of crystals. As a result, during crystallization of the amorphous Fe–Cu–Nb–Si–B alloy, a microstructure with a crystal size of about 10 nm was obtained. Nanocrystalline alloys of the Fe-CuNb–Si–B system have excellent magnetic properties and a huge amount of work has been devoted to their study [92–97]. Later, a number of alloys were obtained in the nanocrystalline state by the controlled crystallization of the amorphous phase. To date, the nanostructure has been obtained in a wide group of metal systems; there is a number of data on the parameters of the nanocrystalline structure obtained by different methods [98–101]. As was mentioned above, depending on chemical composition, nanocrystalline materials have good plasticity and high viscosity, high strength and hardness, low moduli of elasticity, higher diffusion coefficients, larger values of thermal expansion coefficient, and better magnetic properties as compared with traditional crystalline materials. Research on nanocrystalline materials is actively being carried out at the present time.

### **8. Nanocrystal Formation under Heating and Deformation**

As already mentioned, the amorphous phase is crystallized at an increase in the temperature or duration of heat treatment. At that, metastable phases are generally formed at an initial crystallization stage, with some of these phases being formed only under the crystallization of the metallic glasses

and not being formed under other conditions [102–104]. A transition from the amorphous to the equilibrium state is often carried out by successive structural transformations [35,105]. An important feature of the amorphous phase crystallization is the formation of the crystalline phases at an initial stage, the short-range order of which corresponds to that of these ordered regions. Therefore, a structural state of amorphous phase before beginning of the crystallization can have a decisive e ffect on the morphology, phase composition, crystallographic characteristics of a structure formed under crystallization. In [106–108], the e ffect of an initial amorphous state on the parameters of a crystalline structure formed under the crystallization of the metallic glasses was investigated. It was determined that deformation or annealing within an amorphous state leads to the formation of a non-uniform amorphous structure (nanoglass). This a ffects the parameters of a crystalline structure formed under the subsequent treatment. It was shown that the state of the amorphous phase before the crystallization onset can significantly a ffect the characteristics of the formed crystalline structure. The formation of an inhomogeneous amorphous structure in Al-based alloys accelerates the crystallization processes, a ffects the nanocrystal size and the fraction of a nanocrystalline component in amorphous-nanocrystalline alloys. The history of samples turned to be important, too, that is, under which conditions nanocrystals were nucleated: under deformation or heat treatment. The investigations of a large group of Al-based alloys shows that the highest fraction of the nanocrystals and the smallest nanocrystal size were observed in the case when the heterogeneous amorphous phase was formed during deformation. Table 1 lists some data on the parameters of the structure formed under the crystallization of the uniform and non-uniform amorphous phases [106–108].


**Table 1.** The nanocrystal size (D) and the nanocrystalline phase fraction (f) formed in the uniform and non-uniform amorphous phases.

As one can see from the table, the largest Al nanocrystals formed under the separation of the amorphous phase and their lowest volume fraction are observed under the crystallization of the homogeneous amorphous phase (without any treatment of the amorphous phase before the crystallization onset). Under nanocrystallization of the heterogeneous amorphous phase with preliminary heat treatment, the listed parameters were intermediate.

Thus, the formation of the nanocrystals depends essentially on the conditions of an action on an amorphous structure, and the use of combined treatments permits obtaining structure with di fferent structural parameters.

Note that the use of heat treatment allows obtaining a nanocrystalline structure not in all systems. For a nanostructure to be formed from the amorphous phase, a high rate of crystal nucleation and

a low rate of crystal growth are necessary. This depends, particularly, on the di ffusion rate and, of course, cannot be provided in all materials. Severe plastic deformation turned to be the other method, which is e ffective in view of the nanocrystallization initiation [84,89,102,109–113]. The use of this method enabled obtaining an amorphous-nanocrystalline structure in alloys where it is not formed under the crystallization by heat treatment [112,113]. The formation of a nanostructure under plastic deformation generally occurs in the zones of plastic deformation localization (shear bands) or in the regions surrounding them. The formation of nanocrystals in these regions is caused by high values of the parameters of di ffusion mass transfer. The reasons for an increase in the di ffusion coe fficient by several orders of magnitude are not fully understood yet. In general, an increase in the di ffusion coe fficient in deformation bands is related to one of the two processes (or their combination): a local strong, but short (~30 ps), increase in the temperature in this region [114–118] and a change in the structure of the amorphous phase in a shear band (an increase in the free volume fraction) [119–122]. Today, it is unclear which of the reasons constitutes a deciding factor [123]. Both of the factors obviously promote the di ffusion acceleration, and one of them can prevail in di fferent cases. The formation of shear bands occurs under the action of shear stresses. Therefore, shear stresses play an important role in nanocrystalline structure formation.

### **9. Mechanical Properties of Metallic Glasses near Shear Bands**

The results above and the available literature data show that a material in shear bands is softened due to high concentration of free volume [78] and a disordered structure. This results in a significant diminution in the di ffusion coe fficient in an amorphous phase (by several orders of magnitude), and shear bands become the regions of facilitated nanocrystal formation under subsequent heating or even aging at room temperature. Since these factors turn to be important to find out the conditions of the nanocrystal formation [124,125], the issue of definition of the mechanical properties of the material near shear bands has become prominent. As is well known, a thickness of shear bands is from several tens to hundreds of nanometers; steps with the sizes depending on the deformation level and elastic parameters of alloys formed on the sample surface in the places where shear bands come to the surface [78,126]. The mechanical properties of the alloys with shear bands were studied in a number of works by nanoindentation [125–127]. At that, it was determined [127] that near a shear band, there is a region of low hardness. The size of this region is more than 100 μm, which is significantly greater than the size of the shear band. In [128] it was found out that both hardness and the Young's modulus diminish in the shear bands and in the zones around them. However, the size of the region with changed Young's modulus was smaller than that in [127].

The local distribution of zones with di fferent Young's moduli in deformed amorphous Al87Ni8La5 alloy was investigated in [128]. PeakForce QNM (DimensionFastScanTM Atomic Force Microscope (AFM), Bruker) was used for the investigation of local mechanical properties. The measurements were performed with standard probes, and the load was varied from 0.3 to 0.85 μN. After the deformation of an amorphous alloy by multiple rolling, it was discovered that non-uniformity of the distribution of local mechanical properties over the alloy surface (non-uniform distribution of the e ffective Young's modulus) appears with a rise of the load. At that, the non-uniformity in a band system was also observed. The band thickness is 50–250 nm. Figure 17 shows the maps of mechanical property distribution, observed under di fferent loads (in Figure 17 the bands are marked with arrows). The heterogeneities were most noticeable under the load of 0.85 μN. The values of thickness of the bands, as well as their shape and distribution over the surface, were fully consistent with the shear bands observed on the surface of deformed samples using a scanning electron microscope (Figure 18).

The results obtained in [128] are an evidence that the subsurface region of a deformed amorphous alloy contains a lot of deformation bands, where the material is characterized by a lower Young's modulus. The size of the zones with lower Young's modulus is tenths of a micrometer. These zones are distributed uniformly over the sample. The cross-sectional dimensions of such zones (bright bands in Figure 17) can change from tens nanometers up to 0.5 μm. A sharp contrast between the amorphous matrix and deformation bands indicates a significant change in the properties. It should be noted that the properties change insignificantly in the region with a width of several tenths of a micrometer. The difference between these results and the data from [125–127] is apparently caused by different deformation level: single shear bands in the mentioned works and numerous shear bands in [128]. Note, however that that the size of the zones with lower Young's modulus in vicinity of shear bands in amorphous Al87Ni8La5 alloy is noticeably less than that observed in Zr-based alloys. This difference can be caused by different reasons, for example, different deformation conditions (compression of Zr-based alloys and rolling of Al-based alloy) or a different composition of the alloys.

**Figure 17.** Maps of distribution of mechanical properties for Peak Force = 0.85 μN.

**Figure 18.** SEM image of the surface of deformed Al87Ni8La5 alloy [reproduced from Mater. Let. 2019, 252, 114 with permission from Elsevier, 2020] [128].
