Reverse Monte Carlo simulations of an amorphous Cr25Nb75 alloy produced by mechanical alloying

Journal of Non-Crystalline Solids 352 (2006) 109–115 www.elsevier.com/locate/jnoncrysol

Reverse Monte Carlo simulations of an amorphous Cr25Nb75 alloy produced by mechanical alloying J.C. de Lima *, T.O. Almeida, A.R. Jeroˆnimo, S.M. Souza, C.E.M. Campos, T.A. Grandi Departamento de Fı´sica, Universidade Federal de Santa Catarina, Trindade, C.P. 476, Cep 88040-900, Floriano´polis, Santa Catarina, Brazil Received 25 May 2005; received in revised form 11 November 2005 Available online 27 December 2005

Abstract The local atomic structure of an amorphous Cr25Nb75 alloy produced by mechanical alloying was determined using only one X-ray total structure factor S(K) as input data for reverse Monte Carlo simulations. The results showed that the amorphous alloy has a local atomic structure similar to that predicted by the additive hard sphere model for a Cr and Nb mixture with same composition of the alloy, and quite different of those found in the cubic and hexagonal Cr2Nb crystals. Ó 2005 Elsevier B.V. All rights reserved. PACS: 61.10.Eq; 61.10.Ht; 61.43.Bn; 05.10.Ln Keywords: Amorphous metals; Metallic glasses; Mechanical alloying; Monte Carlo simulations; Structure

1. Introduction Alloys based upon Cr2Nb have been investigated for high temperature structural applications [1–3] because of the high melting temperature (
1730 °C), reasonable oxidation resistance [4,5], appreciable creep resistance [4] and high temperature strength [6] of the ordered intermetallic compound. The Nb–Cr phase diagram [7] shows only the Cr2Nb compound between 33 and 37 at.% Nb, which can be formed in the cubic and hexagonal phases. The cubic phase has the Cu2Mg compound as a prototype, space group Fd˚ , while the hexagonal 3m and lattice parameter a = 6.989 A phase has the Zn2Mg compound as a prototype, space ˚, group P63/mmc and lattice parameter a = 4.976 A ˚. c = 8.059 A Recently, Thoma et al. [8] reported the formation of metastable bcc phase in the Nb–Cr system by mechanical

*

Corresponding author. E-mail address: [email protected] (J.C. de Lima).

0022-3093/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2005.11.020

alloying (MA). They started from a mixture of Cr and Nb elemental powder with Cr67Nb33 nominal composition. To understand the formation of amorphous and crystalline alloys, knowledge of two key considerations is necessary: (i) the two elements must have a large negative relative heat of mixing, and (ii) either one of them must be an anomalously fast diffuser (to produce an amorphous phase). If the two elements have similar diffusion coefficients, a simultaneous diffusion of them occurs, and usually a crystalline phase is formed. Of course, the knowledge of other physical mechanisms is necessary. Recently, de Lima et al. [9,10] developed a thermodynamic approach that has been used to describe the formation of intermetallic binary alloys by MA. This approach assumes that during the milling a composite powder is formed and that its interfacial component is composed by the mixture of the interfacial components of the elemental powders, which are formed during milling process. This approach also assumes that the nucleation and growing of new phases occur in the interfacial component of composite powder at average temperatures not greater than 373 K. From a thermodynamic point of view, the mixture

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of the interfacial components has been treated as an ideal solution. Thus, the Gibbs free energy and equilibrium volume equations have been used together with the results obtained for the excess Gibbs free energy for the metals, in nanometric form, to estimate the theoretical activation energy value associated with diffuse process responsible for grain growth, grain boundary migration, atomic migration, and nucleation of new phases. It has been observed that for activation energy values around 0.03 eV the final product of milling shows an amorphous structure; while for values around 0.06 eV the final product shows a nanocrystalline structure. This thermodynamic approach was applied to estimate theoretical values of the activation energy and excess volume theoretical values for different ideal solutions of Cr and Nb at 300 and 323 K average temperatures. The obtained values are listed in Table 1. From this table one can see that for CrxNb1x ideal solutions, with 10 at.% 6 x 6 80 at.%, the activation energy values suggest the possibility of obtaining amorphous phases. Then, in order to confirm these predictions, a Cr25Nb75 mixture was prepared and submitted to MA. After 53 h of milling an amorphous phase was observed. This paper describes the local atomic structure of this amorphous phase determined from only one X-ray total structure factor S(K), which is used as input data for the reverse Monte Carlo (RMC) simulations. The local atomic structure of the alloy was compared with that predicted by the additive hard sphere (AHS) model, for a Cr and Nb mixture with same composition of the alloy, and those present in the cubic and hexagonal Cr2Nb crystals.

three independent S(K) factors are necessary. However, there are several difficulties to obtain independent S(K) factors, and two of them are the following: (1) for neutron scattering measurements, samples can be prepared by using the isotope substitution method. The problem is that there is no disposable isotope for all chemical elements or, otherwise, they are very expensive. (2) For conventional X-ray measurements, the commercial tubes have fixed energy and obtain only one independent S(K) factor. These difficulties started to be overcome with the development of synchrotron sources, which allow tuning the energy of incoming beam near the K-absorption edge of the alloy components. This fact has permitted to carry out the anomalous X-ray scattering (AWAXS) measurements. By using AWAXS, De Lima et al. [10,11] determined the three Sij(K) factors for the amorphous Ni2Zr and NiZr2 alloys. Recently, with the development of the reverse Monte Carlo (RMC) method [12–15] a number of local atomic structures of the amorphous alloys were modelled. Several papers are reported in the literature describing those [16–22]. According to Faber and Ziman [23], S(K) is obtained for a binary alloy from the X-ray scattered intensity per atom Ia(K) through 2

SðKÞ ¼ SðKÞ ¼

I a ðKÞ  bhf 2 ðKÞi  hf ðKÞi c hf ðKÞi2 2 X

W ij ðKÞS ij ðKÞ;

ci cj fi ðKÞfj ðKÞ

hf ðKÞi2 X hf 2 ðKÞi ¼ ci fi2

2.1. Faber and Ziman partial structure factors

ð2Þ

i;j¼1

W ij ðKÞ ¼

2. Theoretical background

ð1Þ

;

ð3Þ

;

i

The structure of the amorphous binary alloy is described by three pair correlation functions Gij(r), which are the Fourier transformation of the three partial structure factors Sij(K). The total structure factor S(K) is a weighted sum of Sij(K), and it can be derived from the X-ray and/ or neutron scattering measurements. Usually, to determine the Sij(K) factors for an amorphous binary alloy at least

and " 2

hf ðKÞi ¼

X

#2 ci fi ðKÞ ;

i h where K ¼ 4p sin , ci is the concentration of atoms of type i k and f(K) is the atomic scattering factor.

Table 1 Theoretical estimated values of the activation energy (Ea) and excess volume (DVIS = DV/V0) for different ideal solutions of Cr and Nb at 300 and 323 K Cr (at.%)

DVIS

300 K

323 K 20

Ea (10 10 20 30 40 50 60 70 80 90

0.443 0.443 0.443 0.442 0.392 0.392 0.392 0.392 0.392

0.1947 0.1143 0.0493 0.0077 0.0379 0.1527 0.3273 0.5541 0.8279

J/at)

Ea (eV)

Ea (1020 J/at)

Ea (eV)

0.012 0.007 0.003 0.001 0.002 0.009 0.020 0.035 0.052

0.2932 0.1880 0.0991 0.0025 0.0802 0.2309 0.4426 0.7072 1.0197

0.018 0.012 0.006 0.001 0.005 0.014 0.028 0.044 0.064

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The total and partial reduced distribution functions c(r) and cij(r) are related to S(K) and Sij(K) by Fourier transformation and they are written as Z 1 cðrÞ ¼ ð2=pÞ K½SðKÞ  1 sinðKrÞdK; ð4Þ 0 Z 1 cij ðrÞ ¼ ð2=pÞ K½S ij ðKÞ  1 sinðKrÞdK. ð5Þ 0

The total and partial pair probability functions G(r) and Gij(r) are related to the c(r) and cij(r) through the expressions, cðrÞ ¼ 4pq0 r½GðrÞ  1

ð6Þ

and cij ðrÞ ¼ 4pq0 r½Gij ðrÞ  1.

ð7Þ

The total and partial radial distribution functions RDF(r) and RDFij(r) are written as RDFðrÞ ¼ 4pq0 r2 GðrÞ

ð8Þ

and RDFij ðrÞ ¼ 4pq0 cj r2 Gij ðrÞ;

ð9Þ

˚ . where q0 is the atomic density of the alloy in atoms/A 3

2.2. Reverse Monte Carlo method The RMC method is based in statistical mechanics, and its basic idea and the algorithm are described elsewhere [12–15]. Its application for modeling the structure of noncrystalline materials uses as starting point one or more experimental S(K) factors, and has as objective to generate static atomic configurations by a procedure explicitly designed to give best agreement with experimental data. It bypasses the need for any representations of the interatomic forces or atomic potentials. The best match between the simulated and experimental S(K) factors indicates that the static atomic configuration generated can be representative of reality, and can also allow an understanding of the atomic structure as well as the search for insights behind the raw data. The RMC method allows taking into account properly the effects of experimental resolution [24]. Thus, it is important to determine the experimental S(K) factors in a ˚ 1) in order to K-range as large as possible (typically 50 A achieve the best possible resolution in G(r) function. The resolution Dr is given as 2p/Kmax, where Kmax is the maximum value of K achieved in the measurement of S(K). The forward Fourier transforms of K[S(K)  1] (see Eq. (4)) to obtain c(r) is straightforward, but is subject to a problem that arises from the fact that the data for S(K) extend only to some maximum value of K. The effect of the finite range of K on the Fourier transform is to introduce spurious ripple peaks into the computed c(r) function. This problem can be avoided by multiplying K[S(K)  1] by a modification function M(K) that decreases smoothly to zero at Kmax. However, the Fourier transform of the

111

modified form of K[S(K)  1] will then be convoluted with the Fourier transform of M(K), which means that the peaks in c(r) will be artificially broadened. In order to overcome this problem the RMC method can be used to obtain the G(r), whose Fourier transform best matches the experimental K[S(K)  1] data. It is important to note that, although the spurious ripple peaks into the computed G(r) function are overcome, the effects of experimental resolution remain. The starting point of the RMC method to study noncrystalline materials is to generate an initial configuration of atoms (a random distribution of atoms) without unreasonably short interatomic distances. In this study, the initial configuration was generated by considering a cubic ˚ , 5000 particles (1250 and 3750 reprebox of edge L = 46 A senting the Cr and Nb atoms, respectively) and a density of ˚ 3. q0 = 0.0516 atoms/A A RMC simulation will evolve to maximize the amount of disorder (entropy) in the configurations generated. Thus, it will give the most disordered atomic configurations that are consistent with the experimental data. There may be a range of configurations that match the data, with different degrees of disorder. Only by maximizing the range of experimental data can this problem be minimized. In order to obtain the most consistent atomic configurations, more than one S(K) factors should be used. In this study we have used only one S(K) factor, which was derived from a X-ray diffraction measurement using CuKa radiation (k = ˚ ). Thus, the K maximum value reached was 1.5406 A ˚ 1. Trying to obtain the most disordered final atomic 7A configuration that is consistent with our experimental data, two preliminary procedures were adopted: (i) in the initial atomic configuration, at the beginning, minimum approach (cutoff) distances between the atomic centers of rCr–Cr = ˚ and rCr–Nb = rNb–Nb = 2.0 A ˚ were fixed to act as 1.97 A constraints on the short-range structure, and (ii) entropy maximization of atomic configuration (tempere process). This step will ensure that the model does not get trapped in a local minimum, and instead the model will converge on the global minimum. The individual pair distribution Gij(r) functions are defined as Gij ðrÞ ¼

N ij ðrÞ ; 4pr2 qi dr

where Nij(r) is the number of atoms of type j lying within the range of distances between r and r + dr from any atom of type i, and qi = ciq0. During the RMC simulation process the following function is minimized: w2 ¼

m
RMC 2 1X S ðK i Þ  SðK i Þ . d i¼1

The sum is over m experimental points and d is related to the experimental error in S(K). In order to minimize the w function, atoms are selected at random, and moved small

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random distances. If the move reduces, it is accepted. If the move increases, it is accepted with the probability

3

P ¼ expðDw2 =2Þ. 2

S(K)

As the process is iterated w2 decreases until it reaches a global equilibrium value. Thus, in principle, the final atomic configuration corresponding to the equilibrium should be the most disordered and consistent with our experimental data. Using GRMC ðrÞ and S RMC ðKÞ functions ij ij corresponding to the final atomic configuration, the coordination numbers, the interatomic distances and the bond-angle distributions can be calculated.

1

0

3. Experimental procedures -1 0

1

2

3

4

5

6

7

-1

K (Å ) Fig. 1. Total structure factor for the as-milled Cr25Nb75 powder after 53 h of milling: experimental (noise line) and RMC simulated (thick line).

are 3%, 27% and 70%. Thus, to get information about the Cr–Cr correlation pairs is very difficulty. The S(K) factor was used as input data together with computational RMC programs to obtain the most representative atomic configuration. The simulated SRMC(K) factor is also shown in Fig. 1 (thick line), and one can see an excellent agreement between them. RMC Fig. 2 shows the partial S RMC Cr–Cr ðKÞ, S Cr–Nb ðKÞ and RMC S Nb–Nb ðKÞ factors obtained from the RMC simulation. From this figure one can see that the S RMC Cr–Cr ðKÞ shows noise levels greater than the others due to its small contribution to S(K) factor. All the S RMC ðKÞ factors show an intense difij ˚ 1 and a second wide and less fuse halo at about K = 2.70 A ˚ 1. intense one at about K = 4.60 A

SNbNb

S ij (K)

A binary mixture of high-purity elemental powders of chromium (particle size < 10 lm) and niobium (particle size < 10 lm) with nominal composition of Cr25Nb75 was sealed together with several steel balls into a cylindrical steel vial under an argon atmosphere. The ball-to-powder weight ratio was 6:1. A Spex Mixer/Mill model 8000 was used to perform MA at room temperature. A ventilation system was used to keep the vial temperature close to room temperature. After 10, 25, 41 and 53 h of milling the process was stopped and the powder analyzed via X-ray diffraction (XRD) technique. The XRD patterns were recorded on a powder Rigaku Miniflex diffractometer, and that corresponding for 53 h of milling on a powder Philips X-Pert diffractometer equipped with a graphite monochromator. Both diffractometers used the CuKa line. The analysis of the as-milled powder after 53 h of milling by energy dispersive X-rays analysis in a Philips scanning electron microscope showed a content of Nb and Cr of 75 at.% and 25 at.%, respectively. Small iron contamination (less than 3 at.%) was also observed and not considered. Therefore, the final milling product will be considered as being an amorphous Cr25Nb75 alloy. The S(K) was computed from the XRD pattern after corrections for polarization, absorption, and inelastic scattering following the procedure described by Wagner [25]. The f 0 and f00 values were taken from a table compiled by Sasaki [26]. The density of the amorphous alloy was calculated from the slope of the straight line (4pq0r) fitting the initial part of the computed c(r) function (until the first mini˚3 mum) [25]. The value obtained is q0 = 0.0516 atoms/A 3 (7.1 g/cm ).

SCrNb

0

4. Results Fig. 1 shows the total structure factor S(K), derived from the XRD pattern measured for the sample milled for 53 h. From this figure one can see the absence of sharp peaks indicating the amorphous nature of the as-milled powder. The S(K) function is a weighted sum of the SCr–Cr(K), SCr–Nb(K) and SNb–Nb(K) structure factors (see Eq. (2)), ˚ 1, and their respective values of Wij(K), at K = 0.50 A

SCrCr

1

2

3

4

5

6

7

-1

K (Å ) Fig. 2. Partial structure factors: RMC simulations and AHS model (dashed line).

Fig. 3 shows the partial reduced distribution functions cij(r) obtained directly from the RMC simulations. Due to the poor contribution of S RMC Cr–Cr ðKÞ to S(K) factor, the cCr–Cr(r) has also high noise levels. The interatomic distances for first neighbors were obtained from the first peak of cij(r) functions and they are listed in Table 2. Fig. 4 presents the partial radial distribution functions RDF RMC ðrÞ obtained from the cij(r) functions. The coordiij nation numbers for the first neighbors were obtained by integrating the first shell displayed in the RDF RMC ðrÞ funcij tions and the obtained values are listed also in Table 2. Fig. 5 presents the Nb–Nb–Nb, Nb–Cr–Nb and Cr– Nb–Cr bond-angle distributions (the angle is centered at the middle atom) obtained from the final atomic configuration.

RDFij (r)

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113

NbNb

CrNb

CrCr

0

2

4

5. Discussion

6

8

10

r (Å)

From Table 2 one can see that Cr and Nb atoms have on average 12 nearest neighbors. This result indicates that Cr and Nb play equivalent roles, on a topological network formed by spheres of very similar sizes. This fact is supported by the similarity of the Nb–Nb–Nb and Nb–Cr– Nb bond-angle distributions showed in Fig. 5. Other evi-

Fig. 4. Partial radial distribution functions RDFij(r): RMC simulations and AHS model (dashed line).

γCrCr

γij(r)

γCrNb -0.8

γNbNb

0

2

4

6

8

10

r (Å) Fig. 3. Partial reduced distribution functions cij(r) obtained by the RMC simulations.

Table 2 Structural parameters obtained for amorphous Cr25Nb75 alloy Bond type Cr–Nb

Nb–Cr

Nb–Nb

3.1 2.88

9.1 2.91

3.0 2.91

9.3 2.94

AHS model N 2.3 ˚) r (A 2.70

7.8 2.82

2.6 2.82

8.8 3.01

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

cos (bond-angle) Fig. 5. Bond-angle distributions (the angle is centered at the middle atom) obtained from the RMC simulations: Nb–Nb–Nb (solid line), Nb–Cr–Nb (square + line) and Cr–Nb–Cr (crosses + line).

dence is that the Cr–Cr neighbor peak at cos a
0.4 in the Cr–Nb–Cr bond-angles is less than that for Nb–Cr– Nb, which suggests that Cr has a special behavior in the network. The chemical short-range order (CSRO) in the amorphous alloy is obtained using the Warren parameter aw given by [27] aw ¼ 1:0 

Cr–Cr RMC N ˚) r (A

-0.6

N 12 ; c2 ½c1 ðN 22 þ N 21 Þ þ c2 ðN 11 þ N 12 Þ

where Nij are the coordination numbers listed in Table 2. The aw parameter is null for a random distribution. If there is a preference for forming unlike pairs in the alloy, it becomes negative. Otherwise, it is positive if homopolar pairs are preferred. In this study, the obtained value is aw = 0.008. Based on the calculated aw value, it is interesting

J.C. de Lima et al. / Journal of Non-Crystalline Solids 352 (2006) 109–115

to compare the functions obtained in this work with those predicted by the additive hard sphere (AHS) model for a Cr and Nb mixture with same composition of the amorphous alloy. The Percus–Yevick (PY) equation [27] is applied to systems in which short-range forces are dominant. This equation has an exact solution for a hard sphere potential, making it possible to obtain analytical expression for the pair distribution and structure factor functions. Initially, the PY equation solution was used to determine the structure of single metals in liquid state. The solution was based on a hard sphere potential. Later, this model was extended to binary alloys in the liquid state. Weeks [28] used the PY model to study the atomic structure of metallic glasses, assuming an isotropic and homogeneous phase that does not occur in the solid state. The extension of the PY model to study the glass state is based on the structure of an amorphous state, which can be considered as an extrapolation of the atomic structure of the liquid state. A review of the PY model is given in Ref. [29]. We used a subroutine given in Ref. [29] to calculate the Sij(K) predicted by the AHS model as part of a computer program to calculate the S(K) for a binary amorphous alloy. It is well known that the intensity of the main halo of the SAHS(K), generated by the AHS model, is larger than the experimental one. Thus, S(K) has to be multiplied by an exp(r2K2) function in order to introduce a ‘thermal’ effect. The best results were reached by considering ˚ 2 and a packing fraction of 0.70. Fig. 6 shows r2 = 0.045 A the experimental S(K) and SAHS(K) (thin line) functions, where a good agreement between them can be observed. Fig. 7 shows the c(r) and cAHS(r) (thin line) functions. The density of the AHS mixture was determined by using the same procedure already described for the amorphous

10

8

S(K)

6

4

4 3 2 1

γ (r)

114

0 -1 -2 -3 0

2

4

6

8

10

r (Å) Fig. 7. Experimental and AHS (dashed line) reduced distribution functions.

alloy and the obtained value is qAHS = 7.0 g/cm3, which agrees with that obtained for the amorphous alloy. In order to compare the SAHS(K) and cAHS ðrÞ (thin line) ij functions with those obtained from the RMC simulations, they are shown in Figs. 2 and 4, respectively. From these figures one can see reasonable agreement among them. The calculated coordination numbers and interatomic distances for first neighbors from RMC and AHS simulations, are listed in Table 2. According to AHS model characteristics, smaller interatomic distances than those obtained from RMC results are expected. Using the coordination numbers obtained from the AHS model, the CSRO parameter obtained was aAHS ¼ 0:002, which is of the same order w of that obtained from experimental results (RMC). The cubic and hexagonal Cr2Nb phases have the six Cr– ˚ and 2.49 A ˚ , respectively; six Cr first neighbors at 2.47 A ˚ , 12 Nb–Cr neighbors at 2.89 A ˚ and four Cr–Nb at 2.89 A ˚ . The CSRO parameter calcuNb–Nb neighbors at 3.03 A lated for both crystals was aw = 0.24, indicating them preference to form unlike pairs. Furthermore, both crystals structures have: (i) Nb–Nb first neighbors’ peaks in the Nb–Nb–Nb bond-angle distributions at cos a
0.333; (ii) Cr–Cr neighbors peak in the Cr–Nb–Cr bond-angle distributions at cos a
0.636, and (iii) Cr–Nb neighbors peak in the Nb–Cr–Nb bond-angle distributions at cos a
0.45. All these results show that the local atomic structure of the amorphous Cr25Nb75 alloy and Cr2Nb crystals are different.

2

6. Conclusions Several conclusions are taken from the study. The main ones are:

0 0

1

2

3

4

5

6

7

-1

K (Å )

Fig. 6. Experimental and AHS (dashed line) total structure factors.

1. The formation of an amorphous Cr25Nb75 phase by mechanical alloy was predicted by the thermodynamic

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approach mentioned in Section 1 and confirmed experimentally. 2. Using only one X-ray total structure factor S(K) for the amorphous Cr25Nb75 alloy as input data for RMC simulations, the partial S RMC ðKÞ, GRMC ðrÞ and RDF RMC ðrÞ ij ij ij functions were obtained. From the first peak maximum of GRMC ðrÞ functions and by integration of first peak of ij RDF RMC ðrÞ functions, the interatomic distances and ij coordination numbers for the first neighbors were obtained. Using these coordination numbers the CSRO parameter was calculated, showing that the atomic structure of the amorphous Cr25Nb75 alloy is similar to that predicted by the AHS model for a mixture with same chemical composition of the alloy. 3. The local atomic structure of the amorphous Cr25Nb75 alloy is different of those present in the cubic and hexagonal Cr2Nb crystals.

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