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Prediction of amorphous formation in binary transition metal alloys
Materials Science and Engineering, A 179/A 1,~'0(1994) 238-242
238
Prediction of amorphous formation in binary transition metal alloys Yukio Makino Welding Research Institute, Osaka University, 1-1 Mihogaoka, lbaraki-City, 567 Osaka (Japan)
Abstract An explanation of the formation of an amorphous phase in binary transition metal alloys using two band parameters (H and S) is proposed on the basis of Zunger's orbital electronegativities and the bond orbital model. It is indicated that the formation of the amorphous phase is controlled by the ratio of H to S which expresses the ionic and covalent characters, and the formability increases with increasing ratio HIS. When the ratio HIS of an AB binary alloy exceeds 1.45, the amorphous forming range can be predicted by an energy function constructed from two band parameters and the compositional factor including the valence electron number. With decreasing ratio of H to S, the predicted range of the amorphous phase deviates widely from the experimental result. The disagreement occurs because formation of the amorphous phase is not controlled predominantly by the ratio of ionic character H to covalent character S, and a different picture is required in binary alloys with ratio HIS less than 1.2.
1. Introduction
Since the early work of Duwez et al. [1], the formation of metallic glasses by rapid quenching has been reported in a large number of alloys [2]. Several models have been proposed to explain the formability of metallic glasses. These models can be briefly divided into kinetic and descriptive models. In the kinetic models (see for example ref. 3), although the importance of the glass transition temperature Tg is indicated, the value of Tg has to be determined experimentally. However, it has been reported that the ratio of atomic radii, thermodynamic properties such as the heat of mixing, and the valence electron concentration are useful parameters for explaining the formability of metallic glasses [4-6]. In models based on the ratio of atomic radii, experimental values in twelve coordination are used. However, the atomic radius depends on the coordination number, especially in non-metallic elements, and both AHf and the atomic radius (related to the interatomic distance) are related to the ionic and covalent characters [7], so it is more desirable to consider the formability of the amorphous state from the standpoint of the character of the chemical bonding. The formability of metallic glasses also depends on the quenching velocity, i.e. on the method of producing metallic glasses. Discrepancies in the region of glass formation between mechanical alloying and rapid quenching methods have been discussed [8], although amorphization using the former method is strongly affected by impure elements and other experimental 0921-5093/94/$7.00 SSDI 0921-5093(93)05515-Q
factors. Further, it is suggested that the extreme rapid quenching such as of the order of 1014 K- ~s is realized in the processes related to ion beam techniques [9]. Thus, amorphous alloys have been obtained using various methods, so it is worthwhile reconsidering the physical meaning of the formability of amorphous alloys. In the present study, we tried to construct a twoparameter representation of the formability of amorphous alloys in binary transition metals, based on the bond orbital model and orbital electronegativities. 2. Construction of two band parameters
According to the bond orbital model [ 10], the energy gap at the center of the Brilliouin zone (k = 0)in zinc blende type compounds can be expressed by the following formula: E g = - [( ep ~ - e ~ ) + ( e p ~' - e ~")]/2 + [(e ~ - e ~")2/4
+ (4Es~)2]'/2 + [(epc - ep")+ (4Exx)2]'/2
(1)
where gpC, fpa, gsc and e~~ are the electronic energies of s and p electrons of cationic and anionic atoms respectively, and E,~ = V,,,~,, Exx = Vppo/3 + 2 Vpp~/3. Further, V,~o, Vpp, and Vpp~ are the off-diagonal matrix elements for ssa, ppa and pp~ bondings respectively. In the previous papers [1 1, 12], it was indicated that a linear relation holds between Eg (obs) and (AX/nav) 1/2, where AX is the difference in Pauling's electronegativity of different atoms and n,v is the average principal © 1994 - Elsevier Sequoia. All rights reserved
Y. Makino / Predictionof amorphous phase formation quantum number of these atoms. Eg(obs) is the observed band gap of the binary compound. In the present study, application of two parameters related to the A-B bond character to the formability of a metallic amorphous phase in transition metal compounds is attempted. The band parameters are constructed using Zunger's orbital electronegativities [13] based on the bond orbital model. Zunger's orbital electronegativities are defined a s (Z/ri) 1/2 (i is s, p or d), where r/is a sort of pseudopotential radius (exactly classical crossing point) of s, p or d valence electrons and Z is the valence. The first term in eqn. (1) corresponds to the band gap reduction. The second and third terms are the contributions of s and p electrons to energy separation between bonding and antibonding levels. Because the last two terms are related to hybridization, we call the sum of these terms a hybrid function. Taking into account the functional form relating Eg(obs) and Pauling's electronegativity, an empirical relation between orbital electronegativities and the band gap has been determined on the basis of the bond orbital model [11], in which the relation for sp-bonded compounds is expressed as follows:
239
For simplicity, it is assumed that S(dd) can be substituted by S(sd). Further, ¢td is not simply given by the absolute value of the difference between d orbital electronegativities but by the following equations modified for the effect of s electrons (sd interaction) when the compound contains 4d and/or 5d transition metals: case (a), both atoms are 3d transition metals, ad = I Xo(A) - xa(B)l ; case (b), the B atom is 3d transition metal, I[xs(A )+ xd(A )]- Xd(B)l ;
ad =
case (c), neither atoms is a 3d transition metals, ad = I [xs(A )+ Xd(A )]- [xs(B)+ Xd(B)] I. Subsequently, H and S values for each compound are calculated by assuming that the bonding between transition metals can be divided into the following three cases: case (1), bonding between early transition metals, H = H(dd), S = S(sd);
(2)
case (2), bonding between 3d and 3d late transition metals and between 3d late and 4d (or 5d) early transition metals, H = 1 [H(sd) + H(sp)], S = ½[S(sd)+ S(sp)];
H(sp) = (t~s/r/av) 1/2 + (ap/nav) 1/2
(3)
case (3), bonding in other combinations, H = H(sd), S = S(sd),
S- S(sp)=[{Ssp(A )+ Ssp(B)}/nav]1/2
(4)
Eg = 7.26[H(sp)- S ] - 2.87 where
or
-= S(spd) = [{Ssp(A )+ Ssd(B)}/nav] '/2 and
ai=lxi(A )-xi(B)l Ssp=lxs(M)-xp(M) I Ssd=[Xs(M)--Xd(M)[
(iis s or p)
(MisAorB)
(5)
(6)
x s, Xp and x d are the orbital electronegativities of s, p and d electrons. Because H and S correspond to the hybridization effect and gap reduction in eqn. (1), we call H and S the hybrid function and gap reduction parameter. S(spd) is used as the gap reduction parameter when the effect of d electrons has to be considered. Details are given elsewhere [11]. In compounds formed of transition metals, we firstly assume three sorts of bonding model, i.e. sd, dd and sp bonding models. Subsequently, similar to the above expression for sp bonding (eqns. (3)-(5)), hybrid functions and gap reduction parameters for sd and dd bonding are defined as follows: H(sd) =
( as/nav)1/2+ (rid~nay) 1/2
H(dd) = (tXd/nav)1/2 S(sO) = S(dd)= [{Ssd(A )+
(7)
Early and late transition metals refer to transition metals with valence electrons from 3 to 6 and from 7 to 10 respectively. Lastly, a strong d character is assumed in V, Cr and Pd. The strong d character of Pd is based on the results of Nieminen and Hodges [14]. When the compound containing these elements corresponds to case (2) or (3), the average value of H(dd) and the value of H in case (2) or (3) is taken as the hybrid function for those compounds. Further, in Y and La compounds with 3d transition metals, donation of valence electrons to d levels of 3d transition metals is assumed owing to the strongly donating character of these elements irrespective of the difference between orbital electronegativities. Thus, two band parameters are determined using Zunger's orbital electronegativities and the average principal quantum number according to the bond orbital model. In practice, fnvH and fnv S are used as the two band parameters to examine the formability of amorphous alloys between transition metals, where finv is the compositional factor which is explained in Section 3. 3. Selection of the energy function for the heat of formation
(8)
Ssa(B) }/nav]1/2
(9)
In this section, an energy function based on the two band parameters is proposed to estimate the heat of
240
Y. Makino /
Predictionof amorphous phaseformation
formation because it has been indicated that the heat of formation AHf is important for understanding the formability of metallic glasses [4, 6]. Firstly, the relation between ( H - S) and AHf must be determined. According to a serial paper [12], a linear relation is indicated between lily(x, N) (H-S)2r, where fnv(X, N) and r are the compositional factor and the sum of pseudopotential radii of the two constituent atoms. Further, it has been indicated that the dependence of the difference between valence electron numbers of the two constituent atoms (AN) on AHf is linear in binary transition metal systems [15]. Taking these results into account, the formula ANfinv(X,N) (H-S)2r is expected to be approximate for examining the relation with AHf. As described above, AN is generally given by the difference between valence electron numbers of the constituent atoms. When unidirectional donation of valence electrons occurs, however, the valence electron number of the donor atom is taken as AN. As expected, unidirectional donation is assumed when both orbital electronegativities of s and d electrons in one atom (atom A) are larger than those of the other atom (atom
B). The next problem is the method of selecting the sum of the pseudopotential radii. Firstly, we select the pseudopontential radius of an s electron for the effective radii of 4d and 5d late transition metals because the assumption of sd interaction suggests that the effect of valence electrons reaches to the distance r~. Secondly, the average r~d of r~ and r d is used as the effective radius for 3d (late), 4d (early) and 5d (early) transition metals. Lastly, the effective radius of 3d early transition metals is assumed to be r d in 3d/3d alloys and I",d in other cases. The use of r d in the former cases is based on the assumption that direct donation of valence electron occurs predominantly between 3d levels. Finally, we consider the compositional factor. According to the regular solution model (see for example ref. 16), compositional dependence is expressed by 4XAXB, where XA and x B are the molar fractions of A and B atoms in the binary system respectively. However, the expression does not include the bonding character of the valence electrons. Evidence is found in the phase diagram and from thermodynamic results. For example, an AB type compound is not always the most stable and the maximum heat of formation is not always obtained in AB type compounds. Although details of the compositional factor including the valence electron number are not discussed here [12], inverse proportionality against the valence electron number is introduced into the regular solution model as follows:
fnv =
4XAXBNANB/(XANB 4- XBNA )2
(10)
where finv, NA and N~ are the compositional factor and numbers of valence electrons of A and B atoms respectively.
4. Results and discussion
When the data on amorphous formation in various binary transition metal alloys are plotted taking fin,.H and fnvS as the coordinates, we obtain the graph shown in Fig. 1. The data for amorphous formation are quoted from refs. 2 and 6. The result indicates that formability of the amorphous phase increases with increasing ratio of H to S. A high ratio of H to S corresponds to a high stability of the A-B bond in the binary transition metal system, so that it is expected that formation of the amorphous transition metal alloy is closely related to compound formation. When the formation of an AB type compound between transition metals is arranged using fnv H and fi~vS, we obtain Fig. 2. All the data for compound formation are quoted from ref. 17. Although there is an intermixed domain, compound formation can be divided into two domains by the lines showing constant ratios of H and S: in domain I the compound has a melting point; in domain II the compound has no melting point and decomposes below the
HIS
= i ,45
/
/ q3o o
o
/
./" f / /
//
/
/
//
1,0
o eo "%.. % / /
,4L~
/
0,5 /
/ /
/
/
/
/
/
/
~,2
/
/
/
/,/
..° " ~ . ~ s -
/
/
/
/
I
I
0.4
0,6
I 0,8
.0
f~ ,,,, s
Fig. 1. Plot of amorphous phases observed for vapour-deposited and melt-spun alloys using two band parameters H and S and compositional factor f,v. Triangles are the data for vapour-deposited alloys. The broken lines are same as shown in Fig. 2, dividing various AB compounds into two domains and an intermixed domain.
Y. Makino Intermixed region o
Oo
do~ln I
/
/
/
a
%
HIS = 1.45
/
~:~iI / / / / /
0,5 /
'~ '~
/
,~
o
MnPd
o ~°e°~ o,~nao / o o /~ ~ / ~ o/O~z~o ° l I~ I I A I iz~ ~x A ~x
~r
io0
d0r~ I n II
/ /
~zx
i
oo
/ /Ooo
A
/
H/S = 1.2
// // I
I
0,5
1,0
o°
o
50
// // //
0
/
o
o~ A
//
//
241
aH e- RTm(OV) = 34finvAN(H-S)Zr-5
/
/
/ d?o
/ /
01
8
o
1,0
//
Predictionof amorphous phaseformation
°-5
finv S
Fig. 2. Plot of AB compounds against the two band parameters multiplied by the compositional factor. Open circles and triangles are compounds with and without melting points respectively: domain I, compound with melting point; domain II
compound without melting point.
solidus temperature. Data for the compounds without melting points are excluded from Fig. 2 when another compound with a melting point can be formed in the same system. Subsequently, we consider amorphous formability from the standpoint of the heat of formation. As described before, we choose the function ANfi.v(HS):r as the energy function. When the dependence of [AHf(obs)-RTr~(av) on the energy function is examined in an AB type compound, Fig. 3 is obtained, where R and Tm(av) are the gas constant and the average melting temperature of the constituent metals. As shown in Fig. 3, we obtain good linearity with a correlation coefficient of 0.91. All data in this figure are quoted from ref. 18. The average values are used when more than two data are available. When AHf is selected as the ordinate instead of [AHf-RTm(aV)], a linear relation is also obtained but the straight line passes far from the origin, in contrast to the relation in Fig. 3. As reported in the previous paper [15], it can be interpreted that the RTm(av ) term in the following relation derived from Fig. 3 is the covalent contribution to the heat of formation A H f = K[AUfnv(H- S)2r] + RTm(aV )
(11)
where K is a constant and the empirical value in this study is 34. The first term in eqn. (11) corresponds to the ionic contribution to AHf. If this interpretation is
0
2 finv AN(tt-S)2r
Fig. 3. Relation between (AHf-RTm(aV)) and the energy function fnvAN(H-S)2r in AB type compounds; symbols are explained in the text.
correct then it can be seen from Figs. 1 and 2 that amorphous alloys can be formed easily in binary transition metal systems when the ionic contribution to AHf is larger than the covalent contribution. In amorphous alloys, some decrease in the average coordination number between different atoms has been reported in several binary systems and the decrease is of the order of about 10% at most [19]. Accordingly, if the ionic energy term reduced by about 10% (first term of eqn. (11)) is larger than the covalent term, then the amorphous state can be formed because the ionic energy being larger than the covalent energy can control the nearest neighbour environment (i.e. the coordination number) predominantly. Thus, the amorphous-forming range can be predicted by comparing the first term reduced by 10% with the second term in eqn. (11). The predicted ranges in several systems are given in Table 1. These ranges are in good agreement with previously predicted ranges for binary transition metal systems which have at least one compound with a melting point. The proposed method based on ionic and covalent energies, however, cannot predict the amorphousforming range in systems with less stable intermetallic compounds without melting points (for example, Mo-Re, W-Os, etc.). In these systems, the predominant factor for the formability of the amorphous phase may not be controlled only by the relation between ionic
242
Y. Makino
/
Prediction of amorphous phase formation
TABLE 1. Amorphous-forming ranges of several binary transition metal systems System
Experimental range
Ni-Ti Ni-Zr Ni-Hf Co-Zr Co-Hf Co-Ta Fe-Zr
25-65 22-90 2(/-89 20-90 22-91 30-70 18-91
(%)
Ni Ni Ni Co Co Co Fe
Predicted range
(%)
20-96 20-96 21-96 20-95 21-96 41-89 34-90
Ni Ni Ni Co Co Co Fe
and covalent energies. In all these systems, an amorphous state has been produced by a vacuum evaporation method. Accordingly, it is thought that the amorphous state in these systems is attained by a more rapid quenching velocity. Therefore, it is suggested that another interpretation is required for formation of the a m o r p h o u s phase due to rapid quenching at m o r e than 108 K/s J, although the details are not discussed here owing to limited space. Finally, we indicate in addition that the two band parameters are semi-empirically constructed f r o m Zunger's orbital electronegativities and the b o n d orbital model so that various crystal structures of binary c o m pounds can be separated, and the ratio of H to S is substantially related to the ionicity of the A - B b o n d [12, 20] which is a very important factor for separating crystal structures in c o m p o u n d s [7]. Therefore, the a m o r p h o u s formability is closely related to the ionicity of the A - B bond as well as to the stabilization of the crystal structure.
5. Summary A new a p p r o a c h to determine the a m o r p h o u s formability was tried using two band p a r a m e t e r s H and S, which are constructed using the b o n d orbital model and Z u n g e r ' s orbital electronegativities. H and S express the ionic and covalent characters and the ratio of H to S controls the a m o r p h o u s formability of transition metal alloys as well as c o m p o u n d formation. Although the p r o p o s e d method can predict the a m o r p h o u s ° forming range in binary alloys with ratio H I S of m o r e
than 1.45, disagreement between the predicted range of a m o r p h o u s formation and the experimental range increases with decreasing ratio because the relation between H and S is then not the controlling factor for formation of the a m o r p h o u s phase. Different methods must be considered for applying the two band parameters to all binary transition metal alloys.
References 1 E Duwez, R. H. Williams and W. Klement, Jr., J. Appl. Phys., 31 (1960) 1136. 2 U. Mizutani, Y. Hoshino and H. Yamada, Handbook of Amorphous Alloy Production by the Rapid Quenching Method (Binary Systems), AGNE, Tokyo, 1988 (in Japanese). 3 T. B. Massalski, in T. Masumoto and K. Suzuki (eds.), Proc. 4th Int. Conf. on Rapidly Quenched Metals Vol. 1, Japan Institute of Metals, Sendai, 1982, p. 209. 4 B.C. Giessen, in T. Masumoto and K. Suzuki (eds.), Proc. 4th Int. Conf. on Rapidly Quenched Metals, Vol. 1, Japan Institute of Metals, Sendai, 1982, p. 213. 5 T. Egami and Y. Waseda, J. Non-Cryst. Solids, 64 (1984) 113. 6 G. J. Van der Kolk, A. R. Miedema and A. K. Niessen, J. Less- Common Met., 145 ( 1988) 1. 7 J.C. Phillips, Rev. Mod. Phys., 42 (1970) 317. 8 E. Hellstern, L. Schultz and J. Eckert, J. Less-Common Met., 140(1988)93. 9 L. Guzman, Proc. 6th Symp. on Surface Layer Modification by Ion Implantation (SM12), Japanese Society of SMI2, Tokyo, 1990, p. 7. 10 W. A. Harrison, Electronic Structure and the Properties of Solids, Freeman, San Francisco, CA, 1980, Chap. 3. 11 Y. Makino and N. Iwamoto, Proc. 6th Japan Institute of Metals ,~vmp., Japan Institute of Metals, Sendai, 1991, p. 123. 12 Y. Makino, lntermetallics, 2 (1994) in press. 13 A. Zunger, Phys. Rev., B22 (1980) 5839. 14 R. M. Nieminen and C. H. Hodges, J. Phys. Met. Phys., 6 (1976)573. 15 Y. Makino and N. lwamoto, Proc. 6th Japan Institute of Metals Syrup., Japan Institute of Metals, Sendai, 1991, p. 233. 16 E. A. Guggenheim, Mixtures, Oxford University Press, Oxford, 1952, Chapter 4. 17 T. B. Massalski (ed.), Binary Alloy Phase Diagrams, ASM International, Metals Park, OH, 1990. 18 F. R. de Boer and D. G. Pettifor (eds.), Cohesion in Metals, Vol. I, North-Holland, Amsterdam, 1988. 19 H. S. Chen and Y. Waseda, Phys. Status Solidi A51 (1979) 593. 20 Y. Makino, lntermetallics, 2(1994) in press.
238
Prediction of amorphous formation in binary transition metal alloys Yukio Makino Welding Research Institute, Osaka University, 1-1 Mihogaoka, lbaraki-City, 567 Osaka (Japan)
Abstract An explanation of the formation of an amorphous phase in binary transition metal alloys using two band parameters (H and S) is proposed on the basis of Zunger's orbital electronegativities and the bond orbital model. It is indicated that the formation of the amorphous phase is controlled by the ratio of H to S which expresses the ionic and covalent characters, and the formability increases with increasing ratio HIS. When the ratio HIS of an AB binary alloy exceeds 1.45, the amorphous forming range can be predicted by an energy function constructed from two band parameters and the compositional factor including the valence electron number. With decreasing ratio of H to S, the predicted range of the amorphous phase deviates widely from the experimental result. The disagreement occurs because formation of the amorphous phase is not controlled predominantly by the ratio of ionic character H to covalent character S, and a different picture is required in binary alloys with ratio HIS less than 1.2.
1. Introduction
Since the early work of Duwez et al. [1], the formation of metallic glasses by rapid quenching has been reported in a large number of alloys [2]. Several models have been proposed to explain the formability of metallic glasses. These models can be briefly divided into kinetic and descriptive models. In the kinetic models (see for example ref. 3), although the importance of the glass transition temperature Tg is indicated, the value of Tg has to be determined experimentally. However, it has been reported that the ratio of atomic radii, thermodynamic properties such as the heat of mixing, and the valence electron concentration are useful parameters for explaining the formability of metallic glasses [4-6]. In models based on the ratio of atomic radii, experimental values in twelve coordination are used. However, the atomic radius depends on the coordination number, especially in non-metallic elements, and both AHf and the atomic radius (related to the interatomic distance) are related to the ionic and covalent characters [7], so it is more desirable to consider the formability of the amorphous state from the standpoint of the character of the chemical bonding. The formability of metallic glasses also depends on the quenching velocity, i.e. on the method of producing metallic glasses. Discrepancies in the region of glass formation between mechanical alloying and rapid quenching methods have been discussed [8], although amorphization using the former method is strongly affected by impure elements and other experimental 0921-5093/94/$7.00 SSDI 0921-5093(93)05515-Q
factors. Further, it is suggested that the extreme rapid quenching such as of the order of 1014 K- ~s is realized in the processes related to ion beam techniques [9]. Thus, amorphous alloys have been obtained using various methods, so it is worthwhile reconsidering the physical meaning of the formability of amorphous alloys. In the present study, we tried to construct a twoparameter representation of the formability of amorphous alloys in binary transition metals, based on the bond orbital model and orbital electronegativities. 2. Construction of two band parameters
According to the bond orbital model [ 10], the energy gap at the center of the Brilliouin zone (k = 0)in zinc blende type compounds can be expressed by the following formula: E g = - [( ep ~ - e ~ ) + ( e p ~' - e ~")]/2 + [(e ~ - e ~")2/4
+ (4Es~)2]'/2 + [(epc - ep")+ (4Exx)2]'/2
(1)
where gpC, fpa, gsc and e~~ are the electronic energies of s and p electrons of cationic and anionic atoms respectively, and E,~ = V,,,~,, Exx = Vppo/3 + 2 Vpp~/3. Further, V,~o, Vpp, and Vpp~ are the off-diagonal matrix elements for ssa, ppa and pp~ bondings respectively. In the previous papers [1 1, 12], it was indicated that a linear relation holds between Eg (obs) and (AX/nav) 1/2, where AX is the difference in Pauling's electronegativity of different atoms and n,v is the average principal © 1994 - Elsevier Sequoia. All rights reserved
Y. Makino / Predictionof amorphous phase formation quantum number of these atoms. Eg(obs) is the observed band gap of the binary compound. In the present study, application of two parameters related to the A-B bond character to the formability of a metallic amorphous phase in transition metal compounds is attempted. The band parameters are constructed using Zunger's orbital electronegativities [13] based on the bond orbital model. Zunger's orbital electronegativities are defined a s (Z/ri) 1/2 (i is s, p or d), where r/is a sort of pseudopotential radius (exactly classical crossing point) of s, p or d valence electrons and Z is the valence. The first term in eqn. (1) corresponds to the band gap reduction. The second and third terms are the contributions of s and p electrons to energy separation between bonding and antibonding levels. Because the last two terms are related to hybridization, we call the sum of these terms a hybrid function. Taking into account the functional form relating Eg(obs) and Pauling's electronegativity, an empirical relation between orbital electronegativities and the band gap has been determined on the basis of the bond orbital model [11], in which the relation for sp-bonded compounds is expressed as follows:
239
For simplicity, it is assumed that S(dd) can be substituted by S(sd). Further, ¢td is not simply given by the absolute value of the difference between d orbital electronegativities but by the following equations modified for the effect of s electrons (sd interaction) when the compound contains 4d and/or 5d transition metals: case (a), both atoms are 3d transition metals, ad = I Xo(A) - xa(B)l ; case (b), the B atom is 3d transition metal, I[xs(A )+ xd(A )]- Xd(B)l ;
ad =
case (c), neither atoms is a 3d transition metals, ad = I [xs(A )+ Xd(A )]- [xs(B)+ Xd(B)] I. Subsequently, H and S values for each compound are calculated by assuming that the bonding between transition metals can be divided into the following three cases: case (1), bonding between early transition metals, H = H(dd), S = S(sd);
(2)
case (2), bonding between 3d and 3d late transition metals and between 3d late and 4d (or 5d) early transition metals, H = 1 [H(sd) + H(sp)], S = ½[S(sd)+ S(sp)];
H(sp) = (t~s/r/av) 1/2 + (ap/nav) 1/2
(3)
case (3), bonding in other combinations, H = H(sd), S = S(sd),
S- S(sp)=[{Ssp(A )+ Ssp(B)}/nav]1/2
(4)
Eg = 7.26[H(sp)- S ] - 2.87 where
or
-= S(spd) = [{Ssp(A )+ Ssd(B)}/nav] '/2 and
ai=lxi(A )-xi(B)l Ssp=lxs(M)-xp(M) I Ssd=[Xs(M)--Xd(M)[
(iis s or p)
(MisAorB)
(5)
(6)
x s, Xp and x d are the orbital electronegativities of s, p and d electrons. Because H and S correspond to the hybridization effect and gap reduction in eqn. (1), we call H and S the hybrid function and gap reduction parameter. S(spd) is used as the gap reduction parameter when the effect of d electrons has to be considered. Details are given elsewhere [11]. In compounds formed of transition metals, we firstly assume three sorts of bonding model, i.e. sd, dd and sp bonding models. Subsequently, similar to the above expression for sp bonding (eqns. (3)-(5)), hybrid functions and gap reduction parameters for sd and dd bonding are defined as follows: H(sd) =
( as/nav)1/2+ (rid~nay) 1/2
H(dd) = (tXd/nav)1/2 S(sO) = S(dd)= [{Ssd(A )+
(7)
Early and late transition metals refer to transition metals with valence electrons from 3 to 6 and from 7 to 10 respectively. Lastly, a strong d character is assumed in V, Cr and Pd. The strong d character of Pd is based on the results of Nieminen and Hodges [14]. When the compound containing these elements corresponds to case (2) or (3), the average value of H(dd) and the value of H in case (2) or (3) is taken as the hybrid function for those compounds. Further, in Y and La compounds with 3d transition metals, donation of valence electrons to d levels of 3d transition metals is assumed owing to the strongly donating character of these elements irrespective of the difference between orbital electronegativities. Thus, two band parameters are determined using Zunger's orbital electronegativities and the average principal quantum number according to the bond orbital model. In practice, fnvH and fnv S are used as the two band parameters to examine the formability of amorphous alloys between transition metals, where finv is the compositional factor which is explained in Section 3. 3. Selection of the energy function for the heat of formation
(8)
Ssa(B) }/nav]1/2
(9)
In this section, an energy function based on the two band parameters is proposed to estimate the heat of
240
Y. Makino /
Predictionof amorphous phaseformation
formation because it has been indicated that the heat of formation AHf is important for understanding the formability of metallic glasses [4, 6]. Firstly, the relation between ( H - S) and AHf must be determined. According to a serial paper [12], a linear relation is indicated between lily(x, N) (H-S)2r, where fnv(X, N) and r are the compositional factor and the sum of pseudopotential radii of the two constituent atoms. Further, it has been indicated that the dependence of the difference between valence electron numbers of the two constituent atoms (AN) on AHf is linear in binary transition metal systems [15]. Taking these results into account, the formula ANfinv(X,N) (H-S)2r is expected to be approximate for examining the relation with AHf. As described above, AN is generally given by the difference between valence electron numbers of the constituent atoms. When unidirectional donation of valence electrons occurs, however, the valence electron number of the donor atom is taken as AN. As expected, unidirectional donation is assumed when both orbital electronegativities of s and d electrons in one atom (atom A) are larger than those of the other atom (atom
B). The next problem is the method of selecting the sum of the pseudopotential radii. Firstly, we select the pseudopontential radius of an s electron for the effective radii of 4d and 5d late transition metals because the assumption of sd interaction suggests that the effect of valence electrons reaches to the distance r~. Secondly, the average r~d of r~ and r d is used as the effective radius for 3d (late), 4d (early) and 5d (early) transition metals. Lastly, the effective radius of 3d early transition metals is assumed to be r d in 3d/3d alloys and I",d in other cases. The use of r d in the former cases is based on the assumption that direct donation of valence electron occurs predominantly between 3d levels. Finally, we consider the compositional factor. According to the regular solution model (see for example ref. 16), compositional dependence is expressed by 4XAXB, where XA and x B are the molar fractions of A and B atoms in the binary system respectively. However, the expression does not include the bonding character of the valence electrons. Evidence is found in the phase diagram and from thermodynamic results. For example, an AB type compound is not always the most stable and the maximum heat of formation is not always obtained in AB type compounds. Although details of the compositional factor including the valence electron number are not discussed here [12], inverse proportionality against the valence electron number is introduced into the regular solution model as follows:
fnv =
4XAXBNANB/(XANB 4- XBNA )2
(10)
where finv, NA and N~ are the compositional factor and numbers of valence electrons of A and B atoms respectively.
4. Results and discussion
When the data on amorphous formation in various binary transition metal alloys are plotted taking fin,.H and fnvS as the coordinates, we obtain the graph shown in Fig. 1. The data for amorphous formation are quoted from refs. 2 and 6. The result indicates that formability of the amorphous phase increases with increasing ratio of H to S. A high ratio of H to S corresponds to a high stability of the A-B bond in the binary transition metal system, so that it is expected that formation of the amorphous transition metal alloy is closely related to compound formation. When the formation of an AB type compound between transition metals is arranged using fnv H and fi~vS, we obtain Fig. 2. All the data for compound formation are quoted from ref. 17. Although there is an intermixed domain, compound formation can be divided into two domains by the lines showing constant ratios of H and S: in domain I the compound has a melting point; in domain II the compound has no melting point and decomposes below the
HIS
= i ,45
/
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o
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./" f / /
//
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/
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,4L~
/
0,5 /
/ /
/
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/
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/
/,/
..° " ~ . ~ s -
/
/
/
/
I
I
0.4
0,6
I 0,8
.0
f~ ,,,, s
Fig. 1. Plot of amorphous phases observed for vapour-deposited and melt-spun alloys using two band parameters H and S and compositional factor f,v. Triangles are the data for vapour-deposited alloys. The broken lines are same as shown in Fig. 2, dividing various AB compounds into two domains and an intermixed domain.
Y. Makino Intermixed region o
Oo
do~ln I
/
/
/
a
%
HIS = 1.45
/
~:~iI / / / / /
0,5 /
'~ '~
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,~
o
MnPd
o ~°e°~ o,~nao / o o /~ ~ / ~ o/O~z~o ° l I~ I I A I iz~ ~x A ~x
~r
io0
d0r~ I n II
/ /
~zx
i
oo
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// // I
I
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o°
o
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241
aH e- RTm(OV) = 34finvAN(H-S)Zr-5
/
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01
8
o
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Predictionof amorphous phaseformation
°-5
finv S
Fig. 2. Plot of AB compounds against the two band parameters multiplied by the compositional factor. Open circles and triangles are compounds with and without melting points respectively: domain I, compound with melting point; domain II
compound without melting point.
solidus temperature. Data for the compounds without melting points are excluded from Fig. 2 when another compound with a melting point can be formed in the same system. Subsequently, we consider amorphous formability from the standpoint of the heat of formation. As described before, we choose the function ANfi.v(HS):r as the energy function. When the dependence of [AHf(obs)-RTr~(av) on the energy function is examined in an AB type compound, Fig. 3 is obtained, where R and Tm(av) are the gas constant and the average melting temperature of the constituent metals. As shown in Fig. 3, we obtain good linearity with a correlation coefficient of 0.91. All data in this figure are quoted from ref. 18. The average values are used when more than two data are available. When AHf is selected as the ordinate instead of [AHf-RTm(aV)], a linear relation is also obtained but the straight line passes far from the origin, in contrast to the relation in Fig. 3. As reported in the previous paper [15], it can be interpreted that the RTm(av ) term in the following relation derived from Fig. 3 is the covalent contribution to the heat of formation A H f = K[AUfnv(H- S)2r] + RTm(aV )
(11)
where K is a constant and the empirical value in this study is 34. The first term in eqn. (11) corresponds to the ionic contribution to AHf. If this interpretation is
0
2 finv AN(tt-S)2r
Fig. 3. Relation between (AHf-RTm(aV)) and the energy function fnvAN(H-S)2r in AB type compounds; symbols are explained in the text.
correct then it can be seen from Figs. 1 and 2 that amorphous alloys can be formed easily in binary transition metal systems when the ionic contribution to AHf is larger than the covalent contribution. In amorphous alloys, some decrease in the average coordination number between different atoms has been reported in several binary systems and the decrease is of the order of about 10% at most [19]. Accordingly, if the ionic energy term reduced by about 10% (first term of eqn. (11)) is larger than the covalent term, then the amorphous state can be formed because the ionic energy being larger than the covalent energy can control the nearest neighbour environment (i.e. the coordination number) predominantly. Thus, the amorphous-forming range can be predicted by comparing the first term reduced by 10% with the second term in eqn. (11). The predicted ranges in several systems are given in Table 1. These ranges are in good agreement with previously predicted ranges for binary transition metal systems which have at least one compound with a melting point. The proposed method based on ionic and covalent energies, however, cannot predict the amorphousforming range in systems with less stable intermetallic compounds without melting points (for example, Mo-Re, W-Os, etc.). In these systems, the predominant factor for the formability of the amorphous phase may not be controlled only by the relation between ionic
242
Y. Makino
/
Prediction of amorphous phase formation
TABLE 1. Amorphous-forming ranges of several binary transition metal systems System
Experimental range
Ni-Ti Ni-Zr Ni-Hf Co-Zr Co-Hf Co-Ta Fe-Zr
25-65 22-90 2(/-89 20-90 22-91 30-70 18-91
(%)
Ni Ni Ni Co Co Co Fe
Predicted range
(%)
20-96 20-96 21-96 20-95 21-96 41-89 34-90
Ni Ni Ni Co Co Co Fe
and covalent energies. In all these systems, an amorphous state has been produced by a vacuum evaporation method. Accordingly, it is thought that the amorphous state in these systems is attained by a more rapid quenching velocity. Therefore, it is suggested that another interpretation is required for formation of the a m o r p h o u s phase due to rapid quenching at m o r e than 108 K/s J, although the details are not discussed here owing to limited space. Finally, we indicate in addition that the two band parameters are semi-empirically constructed f r o m Zunger's orbital electronegativities and the b o n d orbital model so that various crystal structures of binary c o m pounds can be separated, and the ratio of H to S is substantially related to the ionicity of the A - B b o n d [12, 20] which is a very important factor for separating crystal structures in c o m p o u n d s [7]. Therefore, the a m o r p h o u s formability is closely related to the ionicity of the A - B bond as well as to the stabilization of the crystal structure.
5. Summary A new a p p r o a c h to determine the a m o r p h o u s formability was tried using two band p a r a m e t e r s H and S, which are constructed using the b o n d orbital model and Z u n g e r ' s orbital electronegativities. H and S express the ionic and covalent characters and the ratio of H to S controls the a m o r p h o u s formability of transition metal alloys as well as c o m p o u n d formation. Although the p r o p o s e d method can predict the a m o r p h o u s ° forming range in binary alloys with ratio H I S of m o r e
than 1.45, disagreement between the predicted range of a m o r p h o u s formation and the experimental range increases with decreasing ratio because the relation between H and S is then not the controlling factor for formation of the a m o r p h o u s phase. Different methods must be considered for applying the two band parameters to all binary transition metal alloys.
References 1 E Duwez, R. H. Williams and W. Klement, Jr., J. Appl. Phys., 31 (1960) 1136. 2 U. Mizutani, Y. Hoshino and H. Yamada, Handbook of Amorphous Alloy Production by the Rapid Quenching Method (Binary Systems), AGNE, Tokyo, 1988 (in Japanese). 3 T. B. Massalski, in T. Masumoto and K. Suzuki (eds.), Proc. 4th Int. Conf. on Rapidly Quenched Metals Vol. 1, Japan Institute of Metals, Sendai, 1982, p. 209. 4 B.C. Giessen, in T. Masumoto and K. Suzuki (eds.), Proc. 4th Int. Conf. on Rapidly Quenched Metals, Vol. 1, Japan Institute of Metals, Sendai, 1982, p. 213. 5 T. Egami and Y. Waseda, J. Non-Cryst. Solids, 64 (1984) 113. 6 G. J. Van der Kolk, A. R. Miedema and A. K. Niessen, J. Less- Common Met., 145 ( 1988) 1. 7 J.C. Phillips, Rev. Mod. Phys., 42 (1970) 317. 8 E. Hellstern, L. Schultz and J. Eckert, J. Less-Common Met., 140(1988)93. 9 L. Guzman, Proc. 6th Symp. on Surface Layer Modification by Ion Implantation (SM12), Japanese Society of SMI2, Tokyo, 1990, p. 7. 10 W. A. Harrison, Electronic Structure and the Properties of Solids, Freeman, San Francisco, CA, 1980, Chap. 3. 11 Y. Makino and N. Iwamoto, Proc. 6th Japan Institute of Metals ,~vmp., Japan Institute of Metals, Sendai, 1991, p. 123. 12 Y. Makino, lntermetallics, 2 (1994) in press. 13 A. Zunger, Phys. Rev., B22 (1980) 5839. 14 R. M. Nieminen and C. H. Hodges, J. Phys. Met. Phys., 6 (1976)573. 15 Y. Makino and N. lwamoto, Proc. 6th Japan Institute of Metals Syrup., Japan Institute of Metals, Sendai, 1991, p. 233. 16 E. A. Guggenheim, Mixtures, Oxford University Press, Oxford, 1952, Chapter 4. 17 T. B. Massalski (ed.), Binary Alloy Phase Diagrams, ASM International, Metals Park, OH, 1990. 18 F. R. de Boer and D. G. Pettifor (eds.), Cohesion in Metals, Vol. I, North-Holland, Amsterdam, 1988. 19 H. S. Chen and Y. Waseda, Phys. Status Solidi A51 (1979) 593. 20 Y. Makino, lntermetallics, 2(1994) in press.