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V. I. Tkatch*, K. A. Svyrydova, S. V. Vasiliev, and O. V. Kovalenko. Galkin Donetsk Physicotechnical Institute, ul. Rosy Lyuksemburg 72, Donetsk, 83114 Ukraine.
ISSN 0031-918X, Physics of Metals and Metallography, 2017, Vol. 118, No. 8, pp. 764–772. © Pleiades Publishing, Ltd., 2017. Original Russian Text © V.I. Tkatch, K.A. Svyrydova, S.V. Vasiliev, O.V. Kovalenko, 2017, published in Fizika Metallov i Metallovedenie, 2017, Vol. 118, No. 8, pp. 806–814.

STRUCTURE, PHASE TRANSFORMATIONS, AND DIFFUSION

Relation between the Structural Parameters of Metallic Glasses at the Onset Crystallization Temperatures and Threshold Values of the Effective Diffusion Coefficients V. I. Tkatch*, K. A. Svyrydova, S. V. Vasiliev, and O. V. Kovalenko Galkin Donetsk Physicotechnical Institute, ul. Rosy Lyuksemburg 72, Donetsk, 83114 Ukraine *e-mail: [email protected] Received August 16, 2016; in final form, December 20, 2016

Abstract—Using the results of differential scanning calorimetry and X-ray diffractometry, an analysis has been carried out of the initial stages of the eutectic and primary mechanisms of crystallization of a series of metallic glasses based on Fe and Al with the established temperature dependences of the effective diffusion coefficients. Analytical relationships, which relate the volume density of crystallites formed in the glasses at the temperatures of the onset of crystallization with the values of the effective diffusion coefficients at these temperatures have been proposed. It has been established that, in the glasses, the crystallization of which begins at the lower boundary of the threshold values of the effective diffusion coefficients (~10–20 m2/s), structures are formed with the volume density of crystallites on the order of 1023–1024 m–3 and, at the upper boundary (10–18 m2/s), of the order of 1018 and 1020 m–3 in the glasses that are crystallized via the eutectic and primary mechanisms, respectively. Good agreement between the calculated and experimental estimates indicates that the threshold values of the effective diffusion coefficients are the main factors that determine the structure of glasses at the initial stages of crystallization. Keywords: metallic glasses, onset crystallization temperature, threshold values of the diffusion coefficients, volume density of crystallites DOI: 10.1134/S0031918X17080142

INTRODUCTION The term thermal stability (TS) of metallic glasses refers to the boundary temperatures or lifetimes of the existence of an amorphous state. As a rule, this term can be adapted to the conditions of heating at a constant rate, and the values of the temperatures of the onset Tons or of the maximum of the rate of crystallization Tx are used as the quantitative characteristics of the TS. From a physical viewpoint, the temperature Tons seems to be a more correct characteristic of the TS, which also is used as the parameter in the thermodynamic criteria that describe the tendency of the melts toward amorphization [1]. The values of Tons are experimentally determined by thermographic methods (differential thermal analysis (DTA), differential scanning calorimetry (DSC)) or are based on the changes in physical properties. However, as was established in [2], the values of Tons determined based on the DSC data and from the changes in the electrical resistance and magnetic susceptibility, differ essentially. The observed scatter is mainly caused by two factors. Firstly, the procedure of estimating Tons as a temperature at which the measured characteristic differs from the initial dependence (background), which

is used in this and other works, seems to be incorrect, since its results significantly depend on the sensitivity of the equipment. Secondly, the values of Tons determined based on the changes in the fraction of the transformed volume X (or some property proportional to X), differ from the estimates made from the changes in the rate of the transformation, dX/dT. At present, in the DSC method, the Tons is determined with a special program as the point of intersection of the tangent at the point of inflection of the lowtemperature branch of the maximum rate of heat release (proportional to dX/dT) with the background [3]. This procedure significantly levels off the dependence of the results of the estimates of Tons on the equipment employed and is more correct. It should be noted that, despite the importance of the problem and large number of related publications, the factors that determine the TS of metallic glasses up to now remain the object of discussions. Since the temperature threshold of the stability of an amorphous structure is determined by the process of crystallization, it is most logical when analyzing the TS to use approaches based on estimating the parameters that control this process. This is particularly common for the rates of nucleation and growth, e.g., the diffusion coefficient at the inter-

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face between the crystalline and amorphous phases [4]. However, in contrast to the coefficients of self-diffusion and heterodiffusion, the experimental estimates of the diffusion rate at the interphase boundaries are impossible and no a priori theoretical models have been developed, even for pure metals [5]. Nevertheless, the values of the diffusion coefficients, which control the transfer of the atoms through an interphase boundary, can be determined via the comparison of experimental data on the nucleation and growth of crystals or on the kinetics of crystallization with the calculations within the framework of the appropriate theoretical models. One of the first estimations of this type [6] showed that the diffusion coefficients at the interphase boundary of pure Pb and Sn are approximately 15 and 8 times larger than the coefficients of self-diffusion of these elements. Similar estimations of the diffusion coefficients at the interphase boundary were carried out, also, for the process of crystallization of metallic glasses, but, before examining these results, we should note the following. According to the classification proposed by Köster and Herold [7], the growth of crystals in metallic glasses can occur via three mechanisms: polymorphous, eutectic, and primary. In the first two cases, the composition of the growing crystals (or eutectic colonies) coincides with the composition of the amorphous phase, and the process itself is controlled by diffusion at the interface [4]. The composition of the crystals that grow via the primary mechanism differs from the composition of the matrix, and the process is controlled by the bulk diffusion in the matrix [4]. This means that the physical sense of the diffusion coefficients that controls the different mechanisms of growth is different and the diffusion coefficients, which are the parameters of the corresponding models, are effective quantities. The simplest method of evaluating the effective diffusion coefficients Deff is the approximation of the experimentally measured changes in the average sizes of crystals that grow via the primary [8–14] or eutectic [15, 16] mechanisms. The procedures for estimating Deff based on the data on the kinetics of crystallization are somewhat more complex [13, 14, 17–20], and their results, as was established in [13, 14], coincide in the margins of the error with estimates based on the growth of crystals. In most cases, the values of Deff are of the same order of magnitude as the coefficients of the heterodiffusion of separate components in the appropriate metallic glasses, and their temperature dependences are satisfactorily described by Arrheniustype equations. Using Deff(T) dependences for a series of Al-based metallic glasses and for the amorphous alloy Fe73.5Si13.5B9Cu1Nb3 (Finemet), it has been established in [13] that, despite the wide diapason of Tons (453–778 K), the values of Deff (Tons) lie in a narrow range (1.7–4.7) × 10–20 m2/s. Note that all metallic glasses investigated in [13] are crystallized by the PHYSICS OF METALS AND METALLOGRAPHY

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primary mechanism with the formation of nanophase composites, the structure of which represents a population of nanocrystals of the basic element (Al or Fe) with sizes of 10–35 nm and a volume density of ~1023 m–3 that are dispersed in the residual amorphous matrix. The existence of the narrow range of threshold values of Deff(Tons) gave grounds to assume the decisive inf luence of the effective diffusion coefficients on the TS of amorphous structures. However, a later analysis [14] showed that the primary crystallization of the amorphous alloy Fe80B14Si6 begins at a substantially higher threshold value of Deff (2.9 × 10–18 m2/s). To explain the observed difference, it was assumed that the TS of the amorphous alloy Fe80B14Si6 is determined by the process of nucleation, while the TS of the metallic glasses with low values of Deff(Tons) is determined by the process of the growth of crystallization centers that exist in the system. It is obvious that, in order to check the correctness of this assumption, it is necessary to compare the structural parameters of glasses at Tons and those at the final stages of the process for a wide variety of glasses. The establishment of the relation between these results and the values of Deff(Tons) is the main purpose of this work. As the objects of studies and analysis in this work, we selected two groups of metallic glasses with the Deff(T) dependences established earlier. The first group consists of three glasses, Fe40Ni40P14B6, Fe40Co40P14B6, and Fe80B20, which are crystallized by the eutectic mechanism. The crystallization of these glasses have been studied in sufficient detail (see, e.g., [15–19, 21]) and, as was established, the first two glasses are crystallized by the mechanism of nucleation and growth, while the dominating process of the transition of the amorphous alloy Fe80B20 into the crystalline state is the growth of quenched-in nuclei. However, the data on the structural state of these glasses at the initial stages of transformation and even the values of Tons are absent in the literature. The second group of glasses consists of nine alloys (five based on Al, and four based on Fe), whose compositions are given in the Table 1. The crystallization of these glasses occurs via the primary mechanism [10, 11, 13, 14] and, as was noted above, it begins at essentially different threshold values of Deff(Tons). EXPERIMENTAL Since metallic glasses, the TS of which is analyzed in this work (see Table 1) were studied earlier, the conditions of obtaining the samples and the procedure of studying their crystallization and structure were described here only briefly with references to the appropriate works. The Fe-based alloys were prepared by the induction melting of mixtures of the chemically pure (not lower than 99.9%) elements of Fe, Ni, Co,

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Table 1. Onset crystallization temperatures, the parameters of the temperature dependences of the effective diffusion coefficients, their threshold values, and the structural parameters at Tons of glasses that are crystallized by (1–3) the eutectic and (4–12) primary mechanisms (Tons ± 1.5), K

D0, m2/s

QD, K

1 Fe40Ni40P14B6

666

1.1 × 1011

43800

2 Fe40Co40P14B6

730

57800

Alloy

3 Fe80B20

698

4 Fe85B15

665

5 Fe80B14Si6

781

6 Fe45Ni19.4Co8.5Cr5.7-Мо1.9B14Si5.5

741

7 Fe73.5Si13.5B9Cu1Nb3

778

8 Al87Ni8Gd5 9 Al86Ni6Co2Gd6

16

6.4 × 10 71.8

31900

23.5

28950

Deff(Tons),

Xons

Nons, m–3

Source

3.0 × 10–18

0.05

3.0 × 1017

[18]

–18

0.05

17

[19]

17

[18]

m2/s

2.6 × 10

–18

0.04

2.9 × 10

–18

0.008

2.9 × 10

–18

0.017

–20

1.0 × 10

4.2 × 10 9.9 × 10

20

[11]

20

5.8 × 10

[14]

0.01

1.3 × 1023

[14]

5.1 × 10

5

41940

–2

1.4 × 10

29600

6.3 × 10

16 400

1.9 × 10–20

0.009

1.9 × 1023

[13]

453

2.7 × 10–11 5.96

20 970

4.7 × 10–20

0.006

3.3 × 1023

[13]

473

2.9 × 10–4

17590

2.1 × 10–20

0.006

8.9 × 1023

[13]

10 Al87Ni8Y5

484

4.7 × 10–3

19 023

4.0 × 10–20

0.006

3.0 × 1023

[13]

11 Al86Ni6Co2Gd3Y2Tb1

500

2.2 × 10–2

20850

1.7 × 10–20

0.008

1.7 × 1024

[13]

505

–9

13020

–20

0.006

23

[13]

12 Al87Ni4Fe4Gd5

6.0 × 10

3.3 × 10

Si, B, Nb, Cu, and of the preliminarily prepared master alloys Fe73P27 and Co2P [14, 17, 19]. The Al-based alloys containing Ni, Co, Fe, were prepared using master alloys enriched in rare-earth elements by the arc melting under atmosphere of purified argon [13]. The rapidly cooled ribbons of the Fe-based alloys with thicknesses of 23–30 μm and Al-based alloys with thicknesses of 40–50 μm were obtained by the method of melt spinning by the ejection of a jet of melt through a quartz nozzle onto the rotating copper roller in air (Fe-based alloys [14, 19]) and in a helium atmosphere (Al-based alloys [13]). The structure of the as-quenched and heat-treated ribbons was from the X-ray diffraction patterns obtained using an automated DRON-3M diffractometer in filtered Co Kα radiation. The average size of nanocrystals in partially crystallized samples was determined from the broadening of the diffraction lines B as L ≈ λ/(B cos θ) [22], where λ is the wavelength of the X-ray radiation, and θ is the angular position of the maximum. The fraction of the crystalline phase X in the samples with the amorphous-crystalline structure was evaluated as X = Acr/(Acr+ Aam) [23, 24], where Aam and Acr are the integrated intensities of reflections from the amorphous and crystalline phases, respectively; the volume density of nanocrystals was calculated as N = 6X/(πL3). The TS of amorphous phases in the investigated alloys was studied using the DSC thermograms registered on Perkin-Elmer DSC7 and NETZSCH DSC404 calorimeters at the rate of heating equal to 0.167 K/s. The values of Tons were determined as the temperatures that correspond to the intersection of the

2.1 × 10

3.9 × 10

tangents drawn at the point of inflection of the maxima of the heat-release rate with the background [3]. As the additional characteristic of the initial stages of crystallization, we determined the fractions of the crystalline phases at the Tons by the comparison of the experimentally measured curves of the rate of transformation, dX/dT, and the fraction of the transformed volume, X(T), obtained by the integration of the DSC signal. For the glasses that are crystallized by the eutectic mechanism, the integrated X(T) curves were normalized to 1; for the glasses that are crystallized by the primary mechanism, these curves were normalized to the fractions Xf determined from the X-ray diffraction data of the samples heated to the temperatures of the completion of the first stage of the transformation [11, 13, 14]. EXPERIMENTAL RESULTS The amorphous nature of the structure of the rapidly cooled ribbons of all the alloys investigated in this work was established based on the shape of X-ray diffraction patterns (see, e.g., Fig. 1) typical of the metallic glasses. In the DSC thermograms of the ribbons of the Fe40Ni40P14B6, Fe40Co40P14B6, and Fe80B20 alloys, a single maximum of the heat-release rate is present (Fig. 2a). The structure of the ribbons heated to the temperatures of the completion of crystallization consists of two phases, i.e., the intermetallic compounds M3X (M stands for metals and X stands for metalloids), and solid solutions based on γ-Fe in the Fe40Ni40P14B6 alloy (XRD pattern 3 shown in Fig. 1a) or based on α-Fe in the Fe40Co40P14B6 and Fe80B20 alloys (not shown). The two-phase structure formed in the sin-

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767

(a)

2 1 30

40

50 60 2θ, deg

70

500 mW/mg

3

2500 mW/mg

Heat flow, mW/mg

exo

Intensity, arb. units

(a)

3 1

2

(b) 600 700 Temperature, К (b)

800

Heat flow, mW/mg

3 2 1 30

40

50 2θ, deg

60

Fig. 1. X-ray diffraction patterns of rapidly cooled ribbons of (a) Fe40Ni40P14B6 and (c) Al87Ni4Fe4Gd5 alloys: (1) initial (as-quenched) state; (3) completely or partially (after the first stage) crystallized states; and combinations of these diffraction patterns calculated for the amorphous structures containing (a, curve 2) 5% and (b, curve 2) 0.6% of crystalline phases.

500

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600 Temperature, К

700

Fig. 2. DSC thermograms obtained at a rate of heating of 0.167 K/s from (a) rapidly cooled ribbons of (1) Fe40Ni40P14B6, (2) Fe40Co40P14B6, and (3) Fe80B20 (dashed line) alloys; and (b) Al87Ni4Fe4Gd5 alloy.

gle-stage process is in agreement with the experimentally established eutectic mechanism of crystallization of glasses [15, 16, 19]. Metallic glasses, the TS of which is limited by mechanism of primary crystallization, are transformed into a completely crystalline state via two or more sequential stages, as is shown in Fig. 2b based on the example of the Al87Ni4Fe4Gd5 amorphous alloy. According to the data of X-ray diffraction analysis (X-ray diffraction pattern 3 in Fig. 1b), at the first stage of the transformations, nanoscale crystals of pure Al are formed, the average size of which is 24 nm, while the volume fraction is 40%. The similar (twoand three-stage) behavior was characteristic of the

0.4 mW/mg

exo

Intensity, arb. units

500

DSC thermograms of all amorphous alloys given in the Table 1 under numbers 4–9, and the samples after the first stage of crystallization had a nanocomposite structure, which consisted of nanocrystals of α-Fe (with sizes of 10–80 nm) or Al (13–35 nm) and of a residual amorphous matrix [11, 13, 14]. At the second stage of crystallization of these glasses, in the residual amorphous matrix, crystals of intermetallic compounds were formed. At the same time as the estimation of Tons from the DSC data, we determined the fractions of the crystal-

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(a) 1.0

Heat flow, mW/mg

0.8

X

0.6 0.4 0.2 0

Tons 660

670 Temperature, К (b)

680

0.4

0.2

X

Heat flow, mW/mg

0.3

0.2

0

Tons 500

520 540 560 Temperature, К

580

Fig. 3. Examples of determining onset crystallization temperatures and the fraction of the transformed volume at Tons(Xons) from the DSC thermograms measured at a rate of heating of 0.167 K/s for (a) Fe40Ni40P14B6 alloy, which is crystallized by the eutectic mechanism (Tons= 666 K, Xons= 0.05) and (b) Al87Ni4Fe4Gd5 alloy, which is crystallized by the primary mechanism (Tons= 505 K, Xons= 0.006).

lized volume at these temperatures (Fig. 3). As can be seen from the data presented in Fig. 3a, the Tons temperature of the Fe40Ni40P14B6 glass with a rate of heating of 0.167 K/s is 666 K and, at this temperature, the fraction of the crystalline phase in the amorphous phase is 5%. A similar analysis of the DSC thermograms of the metallic glasses Fe40Co40P14B6 and Fe80B20 showed that their values of Tons are 730 and 698 K, while those of Xons are 0.05 and 0.04, respectively.

It follows from the performed analysis that, at Tons, the metallic glasses that are crystallized by the eutectic mechanism contain a 4–5% crystalline phase. Unfortunately, the conditions of calorimetric experiments do not make it possible to retain the structural states formed at Tons at room temperature. For this reason, the diffraction patterns of partially crystalline structures were calculated theoretically as the linear combinations of the experimentally measured diffraction curves of the amorphous (Ia(θ)) and crystalline (Icr(θ)) phases, i.e., I(θ) = (1 – X) Ia(θ) + X Icr(θ) [25]. The calculations showed (Fig. 1a) that the presence of 5% crystalline phases does not lead to visually noticeable changes in the diffractograms compared with the patterns of the initial samples. A similar analysis (comparison of changes in dX/dT and X(T)) of the DSC thermograms of the glasses that are crystallized by the primary mechanism (Fig. 3b) has shown that, in this group of alloys, the fraction of the crystalline phase that is formed upon the heating to Tons is essentialy lower and lies within the range of 1.7–0.6 at % (see Table 1). As can be seen from a comparison of the results given in Fig. 3, the observed difference in the values of Xons has nonrandom behavior, but rather caused by the different shapes of dX/dT curves of glasses that are crystallized via different mechanisms. As in the preceding case, the presence of the estimated quantities of primarily crystallizing phases does not lead to visually noticeable changes in the diffraction patterns compared with patterns from as-quenched samples (Fig. 1b). An additionally carried out processing of a series (of more than 20) DSC thermograms showed that the accuracy of estimates of Tons based on the DSC thermograms is relatively low (is no less than 1.5 K, see Table 1), which is caused by the instrumental errors and by the procedure of determining the background. ANALYSIS AND DISCUSSION OF RESULTS The presence in metallic glasses of a certain fraction of the crystallinity Xons formed in the process of heating to Tons, which is established from the analysis of DSC thermograms, makes it possible to estimate the average sizes of crystallites, Lons, and their volume density, Nons, at this temperature and to relate the values of these parameters with the threshold values of the effective diffusion coefficients. The procedure of this estimation consists of the following. In the first approximation, the fraction Xons of the crystallites of spherical form comprises Xons =

( π 6) N ons L3ons, regardless of the mechanism of crystallization, and the density of crystallites selected for the subsequent analysis can be calculated from the relationship

(

)

N ons = 6 X ons πL3ons .

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To calculate the average size of crystallites that grow under the conditions of heating at a constant rate, we in 2 this work used the parameter t eff (Tons ) = Tons (q +Q ) , which has the physical meaning of the effective time of a thermally activated process (Q is the activation energy of diffusion in kelvins) upon the heating at a rate of q+. This parameter, which has the dimension of time, was introduced in [17] and, as was shown in [11, 18, 26], the use of teff instead of the real time in the equations for the isothermal processes, makes it possible to correctly describe changes in the rates of nucleation and growth of crystals and the kinetics of crystallization upon the continuous heating. In particular, to calculate the average size of a eutectic colony in glass heated to Tons, an equation of the following form was used:

Lons = 2U (Tons ) t eff (Tons ) ,

(2)

where U is the rate of linear growth. To describe the rate of growth controlled by diffusion on the phase boundary, the classical equation of the following form is used [4, 15–18]:

U (T ) = [( D0 a0 ) exp ( −Q T )] × {1 − exp [−Δ G m (T ) ( RT )]} = ( Deff a0 ) F (T ) ,

(3)

which correctly describes the growth of crystals (colonies) in metallic glasses [15, 17, 18]. Here, Deff is the effective diffusion coefficient, a0 is the length of a diffusion jump (average atomic diameter), ∆Gm is the molar difference in the thermodynamic Gibbs potentials, and R is the universal gas constant. The substitution of relationships (2) and (3) into (1) gives the following expression:

a03 X ons eut = 3 3 , N ons 3 4π Deff (Tons ) F 3 (Tons ) t eff (Tons )

(4)

which relates the volume density of crystallites at Tons in the glasses that are crystallized by the eutectic or polymorphous mechanisms. As was noted above, the crystallization of the Fe40Ni40P14B6, Fe40Co40P14B6, and Fe80B20 glasses has been studied in sufficient detail, including the quantitative assessments of the growth rates [15–19]. The results of these works are fairly close, and the U(T) dependences for these alloys given in [17–19] were used in the Eq. (3) to determine the parameters D0 and Q of the temperature dependences of Deff(T). To calculate the values of the thermodynamic factor designated as F(T), the following relationship [27] was used:

2Δ H mT (Tm − T ) Δ G m (T ) = , Tm (Tm + T )

where ∆Hm and Tm are the latent heat and the temperature of melting of the alloys, respectively. The PHYSICS OF METALS AND METALLOGRAPHY

validity of the application of the relationship (5) to describe the crystallization of metallic glasses was confirmed, in particular in [17–19], from which we borrowed the values of the temperatures and heats of melting (1180 K and 10 270 J/mol for Fe40Ni40P14B6; 1262 K and 13 700 J/mol for Fe40Co40P14B6; and 1543 K and 12 820 J/mol for Fe80B20, respectively). The parameters D0 and Q of thus-obtained equations of the Arrhenius type that describe the Deff(T) dependences are given in the Table 1, and the threshold values of the Deff(Tons) of Fe40Ni40P14B6, Fe40Co40P14B6, and Fe80B20 glasses estimated using these parameters were equal to 3.0 × 10–18, 2.6 × 10–18, and 1.0 × 10–18 m2/s, respectively. In turn, the substitution of such estimated values of Deff(Tons) into relationship (4) showed that the volume densities of crystallites in the Fe40Ni40P14B6, Fe40Co40P14B6, and Fe80B20 alloys are 3.0 × 1017, 4.2 × 1017, and 9.9 × 1017 m–3, respectively. The calculated values of Nons in the Fe40Ni40P14B6 and Fe40Co40P14B6 glasses are essentialy higher than the estimated densities of quenched-in nuclei (2 × 1014 and 4 × 1016 m–3) and somewhat lower than the density of crystallites in the completely crystallized samples (7.5 × 1017 and 1.7 × 1019 m–3, respectively) given in [19]. This means that, at the initial stages of crystallization of the Fe40Ni40P14B6 and Fe40Co40P14B6 glasses, the process of the nucleation, the rate of which determines their TS, plays an important role. At the same time, the value of Nons in the Fe80B20 glass is close to estimates of the density of quenched-in nuclei ((0.9–3.6) × 1018) [16] and the density of crystallites in the crystallized samples (1.9 × 1018 m–3) [18], which is in agreement with the dominant role of quenched-in nuclei established for this glass. This circumstance probably explains the approximately 2.5–3 times lower value of Deff(Tons) of this glass compared with the Fe40Ni40P14B6 and Fe40Co40P14B6 glasses (see Table 1).

Similar estimates of the values of Nons were carried out for the group of the metallic glasses that are crystallized by the primary mechanism, the threshold values of which Deff(Tons) lie in a wide range of 10–18 to 10–20 m2/s [13, 14]. In the case of the diffusion-controlled growth of particles, the average size of crystallites is determined by the known mechanism of parabolic growth L ∝ 2 Dt [28], which for the case of heating at a constant rate takes on the following form [26]: Lons = 2λ Z [Deff (Tons ) t eff (Tons )] , 12

(6)

where λ Z (= [2 (C I − C M ) (C I − C P )] ) is a dimensionless parameter depending on the concentration of the alloying elements in the growing crystallite (CP) and in the matrix at the interface (CI) and far from the interface (CM), the values of which in the investigated 12

(5)

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1024

11 12

1023 Nons, m–3

10

9 10 7

8 6

22

1021 10

4

5

20

1019 3

1018

2 1

10–20

10–19 10–18 Deff(Tons), m2/s

Fig. 4. Relation between threshold diffusion coefficients and volume density of crystallites at Tons in metallic glasses that are crystallized by (d) eutectic and (j, h) primary mechanisms; (j) Fe-based alloys; (h) Al-based alloys. Dashed line represents linear approximation of the data for primarily crystallizing alloys with a slope of ‒1.43 ± 0.1. Numbers in the plot correspond to the numbers of the alloys in the Table 1.

alloys are close to unity [8, 13, 14]. Accordingly, Eq. (1) for the growth of the primary crystals is written as follows: pr N ons = 3 3 3 2 X 3 2 . 4π λ Z Deff (Tons ) t eff (Tons )

(7)

The group of glasses that are crystallized by the primary mechanism, for which estimates of Nons were made based on relationship (7), includes four alloys based on Fe and five alloys based on Al (Table 1, alloys 4–12). The calculations, the results of which are presented in the Table 1 and graphically in Fig. 4, showed that as Deff(Tons) decreases from approximately 10–18 to 10–20 m2/s, the volume density of crystallites in the glasses at Tons increases from ~1020 to ~1023 m–3. As can be seen from Fig. 4, the dependence of Nons on Deff(Tons) in the double logarithmic coordinates is satisfactorily approximated by a straight line, the slope of which is determined by the least-squares method, is 1.43 ± 0.1. Note that the calculated values of teff for alloys 4–12 lie in a relatively wide range (59–221 s) and do not correlate with Tons. This means that the proximity of the slope of the linear dependence in Fig. 4 to the exponent of the Deff(Tons) dependence in (7) indicates the dominant effect of this value on the parameters of the microstructure of glasses at the initial stages of crystallization. It should be noted that the density of crystallites in the alloys with Deff(Tons) ≈ 10–18 m2/s, which are crys-

tallized by the eutectic mechanism, is approximately two orders of magnitude lower than in glasses with a similar level of the threshold diffusion coefficients in which the TS is determined by the primary crystallization (Fig. 4). This difference is related with the fact that the sizes of eutectic colonies Lons are much larger (420–680 nm) than those of the primary crystals (2.1– 41.0 nm). It was interesting to compare values of Nons given in the Table 1 with estimates of the number density Nf of nanocrystals in the samples of these alloys at temperatures of the completion of nanocrystallization. In particular, the value of Nf in the Fe85B15 alloy was equal to 9.3 × 1020 m–3 [11]; in Fe80B14Si6 and Fe45Ni19.4Co8.5Cr5.7Mo1.9B14Si5.5 alloys, it was equal to 9.9 × 1021 and 6.8 × 1022 m–3, respectively [14]; and, in the group of Al-based glasses (nos. 8–12), this value was in the range of (1.1–4.4) × 1023 m–3 [13]. In addition, the analysis of the XRD pattern of the ribbon of the Fe73.5Si13.5B9Cu1Nb3 alloy heated to the temperature of the completion of nanocrystallization, which was carried out in the present work, gave a value Nf = 2.1 × 1023 m–3. It follows from a comparison of these estimations with the results given in the Table 1that, in the process of the crystallization of glasses with high (~10–18 m2/s) values of Deff (Tons), a noticeable contribution comes from the process of nucleation. At the same time, in the glasses in which the threshold values of the diffusion coefficients are of the order of 10–20 m2/s, the volume density of crystallites in the process of crystallization remains almost constant. Good agreement between the calculated values of Nons and the experimentally measured values of Nf for the majority of alloys indicates the correctness of the analysis. The close values of Nons and Nf in nanocomposite structures agree with the data of [29, 30], in which it was experimentally established that the maximum density of nanocrystals in the Al-based glasses is achieved at the initial stages of nanocrystallization. This means that by the prevailing process of the nanocrystallization of glasses with low values of Deff(Tons) is the diffusion-controlled growth, which therefore determines the TS of these glasses. The eutectic crystallization of glasses investigated in this work (nos. 1–3 in the Table 1) begins with the threshold values of the effective diffusion coefficient of the order of 10–18 m2/s and, as was shown by calculations, the density of crystals in them is less by approximately two orders of magnitude than in the primarily crystallizing glasses with similar values of Deff(Tons), Fig. 4. The main reason for the noted difference in Nons is caused by the higher values of the rates of eutectic growth at the identical values of the effective diffusion coefficient. On the other hand, as follows from relationship (4), the Nons in the glasses that

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RELATION BETWEEN THE STRUCTURAL PARAMETERS OF METALLIC GLASSES

are crystallized by the eutectic or polymorphous mechanisms, should exhibit a stronger dependence on Deff(Tons). However, as a result of the relatively high values of the rates of growth controlled by diffusion at the interphase boundary, nanoscale structures in these glasses are formed extremely rarely. An example of this is Fe33Zr67 glass, in which crystals of the intermetallic compound FeZr2 are formed (by the mechanism of polymorphous growth) with an average size of 11– 24 nm [31] and an average volume density of ~8 × 1023 m–3. The substitution of this value of N into relationship (4) in combination with the parameters for the Fe40Ni40P14B6, Fe40Co40P14B6, and Fe80B20 glasses gave the values of Deff(Tons) in the range of (1.1–2.0) × 10–20 m2/s, which nearly coincide with the values of the threshold diffusion coefficients for primarily crystallizing alloys with similar values of Nons. Undoubtedly, the analysis of the relation between Deff(Tons) and Nons for the mechanism of growth controlled by diffusion at the interphase boundary, is of a fairly approximate nature. Nevertheless, the obtained result, which is in agreement with the above-established dominating effect of Deff(Tons) on the volume density of nanocrystals that grow by the primary mechanism (Eq. (7)), seems to be physically reasonable. The threshold values of the effective diffusion coefficients estimated for the group of metallic glasses investigated in this work lie within the limits of two orders of magnitude, i.e., from 3 × 10–18 to 1.7 × 10–20 m2/s (see Table 1), and the corresponding values of Nons, lie in the range of 1017–1018 to 1023–1024 m–3. In turn, the range of the computed values of Nons, which corresponds to the sizes of crystallites in partially or completely crystallized glasses of 1 μm to 10 nm, includes almost the entire spectrum of structural parameters known from the literature. Since the formation of structures with a density of crystallites higher than ~1023 m–3 occurs predominantly via the growth of nuclei that are formed at Tons, which means that the boundary values of Deff(Tons) established in the present work are characteristic of a wide spectrum of metallic glasses. Thus, the analysis of the initial stages of crystallization of metallic glasses carried out in this work shows that an effective diffusion coefficient that controls the process of crystallization of metallic glasses is the most important parameter in determining both the thermal stability of amorphous phases and the structure of the partially or completely crystallized samples.

ning of crystallization, which lie in the range of 10–18 to 10–20 m2/s. The processing of DSC thermograms showed that the fraction of the crystallized volume at the onset crystallization temperatures depends on the mechanism of process and in the alloys that are crystallized by the eutectic and primary mechanisms, these fractions are 4–5% and 0.6–1.7%, respectively. Relationships that connect the threshold values of the effective diffusion coefficients and the volume density of crystallites formed in the glasses at the onset crystallization temperatures have been derived. The calculations showed that, as the threshold diffusion coefficients decrease, the contribution from the process of nucleation decreases, the volume density of crystallites increases, and the thermal stability (TS) of glasses with Deff(Tons) ~ 10–20 m2/s and Nons ~1023–1024 m–3 is limited by the process of growth of the crystallites. The results of an analysis of the calculated and experimental data indicate the determining influence of the effective diffusion coefficients on the TS of glasses and their structural parameters at the initial stages of crystallization. The relations between the threshold values of the effective diffusion coefficients and the volume density of crystallites at Tons of the glasses that are crystallized by different mechanisms, which were established for the first time in this work, present new opportunities for a quantitative analysis of the processes of crystallization of metallic glasses and, therefore, for controlling the structure and physical properties of partially or completely crystallized samples.

CONCLUSIONS An analysis was carried out of the initial stages of crystallization of a group of metallic glasses with the previously established temperature dependences of the effective diffusion coefficients and their threshold values were determined at the temperatures of the beginPHYSICS OF METALS AND METALLOGRAPHY

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