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The Fermi energy and the enthalpy of mixing
Journal of
J~d~Y5 AND COM?OL~D5 ELSEVIER
Journal of Alloys and Compounds 220 (1995) 193-196
The Fermi energy and the enthalpy of mixing Erhard Hayer a, Jean-Pierre Bros b Institute o f Inorganic Chemistry, Wdhringerstr. 42, 1090 I/'tenna, Austria b Universit~ de Provence, IUSTI-UA 1168, Centre de Saint Jdr6me, 13397 Marseille C~dex 20, France
Abstract
In this paper we describe a new function to determine the number of electrons transferred by alloying. HF, the new function proposed, is given by the difference of the partial enthalpies of the two constituents, i.e. H F = h ( i ) - h ( ] ) - h ( i ) ° and is called the Fermi enthalpy. It is retrieved from data calorimetrically measured. In some way, it describes the change in the enthalpic part of the Fermi energy on alloying. This function is determined for P d + G a , P d + I n , N i + G a , N i + I n , P t + I n , A u + A ! and Ag+AI. The corresponding number of electrons transferred is calculated on stoichiometric considerations. Keywords: Fermi energy; Electron transfer; Enthalpy of mixing; Partial enthalpies; Liquid alloys
1. Introduction
In the: last few years the precision of high temperature calorimetry has advanced decisively. Integral and partial enthalpies of mixing of binary alloys have been determined for quite a lot of liquid alloy systems with high reliability. In some of the systems a very peculiar shape of the graphs has been revealed, e.g. triangleshaped curves of the integral enthalpy vs. composition or nearly constant partial enthalpies over a broad concentration range of about 30 at.% or higher. Usually, the thermodynamic properties have been discussed by association models [1-5], i.e. by the formation of small heteroclusters or compounds. However, in most of the experimentally investigated systems an asymmetry exists either in the partial data or in the position of the minimum. Sometimes, the values of the partial enthalpies of mixing correspond to the limiting partial enthalpies over an extended concentration range, where the partial enthalpy is nearly non-dependent on the concentration. In the association model these effects were discussed in terms of a second associate present. Nevertheless, the real presence of small associates could not be verified, until now. In the following another approach is assumed.
2. The "Fermi enthalpy"
It is widely accepted [6] that a metal consists of a lattice of ions filled with an electron liquid. The same
0925-8384;/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSD1 0925-8388(94)06020-7
model is valid for a liquid metal. The stability of the metal is given by the interaction of electrons and metal ions. The chemical bond in a metal or an alloy is due to the non-localized upper electrons of the constituent metals, which are energetically at different but very close levels forming a so-called energy band. The uppermost occupied energy level of the electrons at 0 K is called the Fermi energy EF. All energy levels below EF are occupied and all higher energy levels are empty at 0 K. At higher temperatures some electrons can move to higher energies; nevertheless, most of the electrons are not excited. Metals and alloys exhibit a good electrical conductivity owing to the overlap of filled valence bands with unoccupied conduction bands or owing to half-filled valence bands. Since the valence and conduction bands are influenced by the formation of an alloy the density of states (DOS) will change significantly. In Fig. 1 the model of the DOS in Ni and Ni alloys with electropositive metals is shown [7]. In the alloy, the d band is filled and the DOS at Ev is reduced. In the absence of an external field the Fermi energy or the electrochemical potential/ze of the electrons is equivalent to the chemical potential or the partial Gibbs energy g(i). The Gibbs-Helmholtz equation gives g(i) =h(i) - Ts(i), in which way we can separate entropy and enthalpy. The partial enthalpy h(i) or h(j) of mixing of i or j reveals the energy change when we take one atom of bulk i or j respectively and put it into the bulk alloy under isobaric conditions. In this way, the energy for
E. Hayer, J.-P. Bros I Journal of Alloys and Compounds 220 (1995) 193-196
194
DENSITIESOF STATESIN Ni ALLOYS .°-,..
[" --"]
N!
~~Total 03 ILl
3. Discussion
( ..... o.oo--"
03
Ni ALLOY
°°°°°%
LL 0 >-
l-tO
Z ILl a
..........jl <
...ii i . ".......... BE
the partial excess Gibbs energies of the elements 1 and 2 respectively, and g(2) ° is the partial excess Gibbs energy of element 2 at infinite dilution. However, they had to take into account an elastic part of the Gibbs energies, which was estimated. These authors assumed that the electronic and the elastic parts of the Gibbs energy were separable, with the electronic part determined by the change in the Fermi energy.
In Fig. 2 the HF function vs. the number of electrons transferred for the liquid alloys Ni + In [9], Pd + In [10], and P t + I n [11] is shown. (However, the noble metal rich regions in these systems are extrapolated.) Looking at P d + I n first, it is seen that the change in the In concentration, e.g. by increasing the In content of the alloy, leads to a change in the Fermi energy by filling up the electron bands of the Pd + In alloy. A small increase in the energy corresponds to a large DOS, whereas a steep slope indicates a low DOS. However, we do not know which bands are filled up. In Fig. 2 a continuous slope of the Fermi enthalpy is easily
400
E F(eV)
Fig. 1. Model of the DOS in Ni and Ni alloys after Ref. [7]. In the alloy the d states are occupied and the DOS at EF is very small. BE is the bonding energy.
the exchange of one atom j for one atom i of the bulk, corresponding to the alloying procedure, is given by h(i)-h(j). In a first approximation, we assume that this energy is only due to electronic effects on alloying, as elastic effects are negligible in liquid alloys. The internal energy AU is given by AU= A H - p AV, but AU is equal to AH, approximately, because of the small volume change on alloying. Therefore, we suggest to represent the change in the Fermi energy on alloying by monitoring h(i)-h(j), the difference in the energies measuring the input and the output of atoms i and j respectively. However, for a better comparison of different alloys, one should start at one pure metal side with AEF= 0 independent of the particular system investigated. Therefore, we define a function Hr:
I
350 300 250
I'/
o
g 200
~.. 150
/
! i [
IO0
/1
so
H~=h(i)-h(j)-h(i)
°
where h(i) ° is the limiting enthalpy of i in pure j. This function HF, the "Fermi enthalpy", describes the change in the energetic part of the Fermi energy. Schaller and Brodowsky [8] have tried to determine the change in the Fermi energy for solid alloys by the function gF =g(2) --g(1) --g(2) °, where g(2) and g(1) are
i
'i
IN'+'rl
o
0
0.5
1
1.5
2
2.5
3
N u m b e r s of e l e c t r o n s of In Fig. 2. Fermi enthalpy vs. the numbers of electrons transferred from In. The number corresponds to three times the mole fraction of In (stoichiometric contribution).
E. Hayer, J.-P. Bros / Journal of Alloys and Compounds 220 (1995) 193-196
recognized going up to a particular concentration followed by a constant range. Regarding the number of electrons added, it is evident that the Fermi enthalpies for Ni, Pd and Pt alloys with In increase to values of 2.5, 2, and 2.5 electrons respectively. Then no further energy change occurs. The energetic interaction stays constant although a higher number of electrons is available for alloying. The present authors cannot imagine that additional electrons are involved in the bonding without a release of an energy corresponding to a change in the Fermi energy. Therefore, we conclude that the limiting number of electrons involved in the bonding of the abovementioned systems is given by the stoichiometric number of electrons indicated by the break point of the Fermi enthalpy curve. Similar effects were found for the Ga alloys of Ni [9], Pd [12], and Au [13], shown in Fig. 3. Of course, the model proposed is a very rough approximation, since constant partial enthalpies are not really constant, as in the case of two-phase field. Fortunately, the systems investigated show very strong interactions re-
195
vealed by the very high negative enthalpies, e.g. for Ga + Pd liquid alloys A m i x H ° = - 60 kJ mol- a. The high energies found bring to mind the Engel-Brewer phases such as PtZr and PtTi; their high negative enthalpies of formation have been predicted by the Engel-Brewer theory [14]. Schaller and Brodowsky, who observed a similar flattening of the Fermi energy curves above some definite valence electron concentrations in solid Pd-In [8] and Pd-Sn [15] alloys, attributed this change of slope to the higher DOS of the 5p band with respect to the 5s band. Liquid A g + A l [16] alloys show a very particular shape of the Fermi enthalpy curve (Fig. 4). A distinct peak appears on the HF curve at the concentration corresponding to 2 electrons transferred. According to the definition of HF, a transfer of electrons from Ag to Al is indicated for high A1 concentrations. Although that signifies a transfer of electrons towards the more electropositive metal, the interpretation could be correct. The positive integral enthalpies of mixing measured in the region close to AI could be caused by the transfer in the "wrong" direction, which is energy consuming. A similar effect was found for Ag + Ga [16].
450
50
__........------400
40 350
0 E
J
300
30
1I
m
0
250
"-~20
e-.
t
200 I-
*t o ¢..
,/"
'.A..
!
i
<~10 e-
p. 150
t
l
100
50
I
!
i
0.5
1
-10 0
0.5
1
1.5
2
2.5
3
Number of electrons of Ga Fig. 3. Fermi enthalpy vs. the numbers of electrons transferred from Ga. The n u m b e r corresponds to three times the mole fraction of Ga (stoichiometric contribution).
0
1.5
2
2.5
3
N u m b e r of electrons of AI
Fig. 4. Fermi enthalpy vs. the numbe rs of electrons transferred from AI. The number corresponds to three times the mole fraction of AI (stoichiometric contribution).
196
E. Hayer, J.-P. Bros / Journal o f Alloys and Compounds 220 (1995) 193-196
4. Conclusion (i) The investigation of the Fermi enthalpy function of liquid alloys gives some insight into the change in the Fermi energy by alloying, which helps us to understand what happens with the electrons at the formation of a liquid alloy. (ii) The horizontal part of the partial enthalpy vs. mole fraction curve reveals a reduced number of electrons transferred. This conclusion is based on stoichiometric considerations. (iii) The function H F = h ( i ) - h ( j ) - h ( i ) ° is a good description for the change in the Fermi energy on alloying, at least for liquid alloys. (iv) the breakpoint in the Hv vs. concentration curve can be an indication of very interesting materials. References [1] A.S. Jordan, Metall. Trans., 1 (1970) 239.
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
B. Prede| and G. Oehme, Z. MetaUkd., 67 (1976) 826. F. Sommer, Z. Metallkd., 73 (1982) 72. M. Hoch and I. Arpshofen, Z. Metallkd., 75 (1984) 23. E. Hayer, Z. Phys. Chem., N.F., 156 (1988) 611. C. Kittel, Einfiihrung in die FestkOrperphysik, Oldenbourg, Munich, 4 t h edn., 1976, p. 303. F.U. Hillebrecht, J.C. Fuggle, P.A. Bennet, Z. Zolnierek and Ch. Freiburg, Phys. Rev. B, 27 (1983) 2179. H.J. Schaller and H. Brodowski, Ber. Bunsenges. Phys. Chem., 82 (1978) 773. D. El Allam, Thdse de l'Universitd de Provence, Marseille, 1989. D. El Allam, M. Gaune-Escard, J.P. Bros and E. Hayer, submitted to Metall. Trans. P. Anres, M. Gaune-Escard, J.P. Bros and E. Hayer, submitted to J. Alloys Compd. D. El Allam, M. Gaune-Escard, J.P. Bros and E. Hayer, Metall. Trans. B, 23 (1992) 39. E. Hayer, K.L. Komarek, M. Gaune-Escard and J.P. Bros, Z. Metallkd., 81 (1990) 233. L. Brewer, J. Chem. Educ., 61 (1984) 101. H.J. Schaller and H. Brodowski, Z. Metallkd., 69 (1978) 87. E. Hayer, Calorimetric results on Ag-alloys with AI, Ga and In, Discussion Meet. on Thermodynamics of Alloys, Genova, April 1994.
J~d~Y5 AND COM?OL~D5 ELSEVIER
Journal of Alloys and Compounds 220 (1995) 193-196
The Fermi energy and the enthalpy of mixing Erhard Hayer a, Jean-Pierre Bros b Institute o f Inorganic Chemistry, Wdhringerstr. 42, 1090 I/'tenna, Austria b Universit~ de Provence, IUSTI-UA 1168, Centre de Saint Jdr6me, 13397 Marseille C~dex 20, France
Abstract
In this paper we describe a new function to determine the number of electrons transferred by alloying. HF, the new function proposed, is given by the difference of the partial enthalpies of the two constituents, i.e. H F = h ( i ) - h ( ] ) - h ( i ) ° and is called the Fermi enthalpy. It is retrieved from data calorimetrically measured. In some way, it describes the change in the enthalpic part of the Fermi energy on alloying. This function is determined for P d + G a , P d + I n , N i + G a , N i + I n , P t + I n , A u + A ! and Ag+AI. The corresponding number of electrons transferred is calculated on stoichiometric considerations. Keywords: Fermi energy; Electron transfer; Enthalpy of mixing; Partial enthalpies; Liquid alloys
1. Introduction
In the: last few years the precision of high temperature calorimetry has advanced decisively. Integral and partial enthalpies of mixing of binary alloys have been determined for quite a lot of liquid alloy systems with high reliability. In some of the systems a very peculiar shape of the graphs has been revealed, e.g. triangleshaped curves of the integral enthalpy vs. composition or nearly constant partial enthalpies over a broad concentration range of about 30 at.% or higher. Usually, the thermodynamic properties have been discussed by association models [1-5], i.e. by the formation of small heteroclusters or compounds. However, in most of the experimentally investigated systems an asymmetry exists either in the partial data or in the position of the minimum. Sometimes, the values of the partial enthalpies of mixing correspond to the limiting partial enthalpies over an extended concentration range, where the partial enthalpy is nearly non-dependent on the concentration. In the association model these effects were discussed in terms of a second associate present. Nevertheless, the real presence of small associates could not be verified, until now. In the following another approach is assumed.
2. The "Fermi enthalpy"
It is widely accepted [6] that a metal consists of a lattice of ions filled with an electron liquid. The same
0925-8384;/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSD1 0925-8388(94)06020-7
model is valid for a liquid metal. The stability of the metal is given by the interaction of electrons and metal ions. The chemical bond in a metal or an alloy is due to the non-localized upper electrons of the constituent metals, which are energetically at different but very close levels forming a so-called energy band. The uppermost occupied energy level of the electrons at 0 K is called the Fermi energy EF. All energy levels below EF are occupied and all higher energy levels are empty at 0 K. At higher temperatures some electrons can move to higher energies; nevertheless, most of the electrons are not excited. Metals and alloys exhibit a good electrical conductivity owing to the overlap of filled valence bands with unoccupied conduction bands or owing to half-filled valence bands. Since the valence and conduction bands are influenced by the formation of an alloy the density of states (DOS) will change significantly. In Fig. 1 the model of the DOS in Ni and Ni alloys with electropositive metals is shown [7]. In the alloy, the d band is filled and the DOS at Ev is reduced. In the absence of an external field the Fermi energy or the electrochemical potential/ze of the electrons is equivalent to the chemical potential or the partial Gibbs energy g(i). The Gibbs-Helmholtz equation gives g(i) =h(i) - Ts(i), in which way we can separate entropy and enthalpy. The partial enthalpy h(i) or h(j) of mixing of i or j reveals the energy change when we take one atom of bulk i or j respectively and put it into the bulk alloy under isobaric conditions. In this way, the energy for
E. Hayer, J.-P. Bros I Journal of Alloys and Compounds 220 (1995) 193-196
194
DENSITIESOF STATESIN Ni ALLOYS .°-,..
[" --"]
N!
~~Total 03 ILl
3. Discussion
( ..... o.oo--"
03
Ni ALLOY
°°°°°%
LL 0 >-
l-tO
Z ILl a
..........jl <
...ii i . ".......... BE
the partial excess Gibbs energies of the elements 1 and 2 respectively, and g(2) ° is the partial excess Gibbs energy of element 2 at infinite dilution. However, they had to take into account an elastic part of the Gibbs energies, which was estimated. These authors assumed that the electronic and the elastic parts of the Gibbs energy were separable, with the electronic part determined by the change in the Fermi energy.
In Fig. 2 the HF function vs. the number of electrons transferred for the liquid alloys Ni + In [9], Pd + In [10], and P t + I n [11] is shown. (However, the noble metal rich regions in these systems are extrapolated.) Looking at P d + I n first, it is seen that the change in the In concentration, e.g. by increasing the In content of the alloy, leads to a change in the Fermi energy by filling up the electron bands of the Pd + In alloy. A small increase in the energy corresponds to a large DOS, whereas a steep slope indicates a low DOS. However, we do not know which bands are filled up. In Fig. 2 a continuous slope of the Fermi enthalpy is easily
400
E F(eV)
Fig. 1. Model of the DOS in Ni and Ni alloys after Ref. [7]. In the alloy the d states are occupied and the DOS at EF is very small. BE is the bonding energy.
the exchange of one atom j for one atom i of the bulk, corresponding to the alloying procedure, is given by h(i)-h(j). In a first approximation, we assume that this energy is only due to electronic effects on alloying, as elastic effects are negligible in liquid alloys. The internal energy AU is given by AU= A H - p AV, but AU is equal to AH, approximately, because of the small volume change on alloying. Therefore, we suggest to represent the change in the Fermi energy on alloying by monitoring h(i)-h(j), the difference in the energies measuring the input and the output of atoms i and j respectively. However, for a better comparison of different alloys, one should start at one pure metal side with AEF= 0 independent of the particular system investigated. Therefore, we define a function Hr:
I
350 300 250
I'/
o
g 200
~.. 150
/
! i [
IO0
/1
so
H~=h(i)-h(j)-h(i)
°
where h(i) ° is the limiting enthalpy of i in pure j. This function HF, the "Fermi enthalpy", describes the change in the energetic part of the Fermi energy. Schaller and Brodowsky [8] have tried to determine the change in the Fermi energy for solid alloys by the function gF =g(2) --g(1) --g(2) °, where g(2) and g(1) are
i
'i
IN'+'rl
o
0
0.5
1
1.5
2
2.5
3
N u m b e r s of e l e c t r o n s of In Fig. 2. Fermi enthalpy vs. the numbers of electrons transferred from In. The number corresponds to three times the mole fraction of In (stoichiometric contribution).
E. Hayer, J.-P. Bros / Journal of Alloys and Compounds 220 (1995) 193-196
recognized going up to a particular concentration followed by a constant range. Regarding the number of electrons added, it is evident that the Fermi enthalpies for Ni, Pd and Pt alloys with In increase to values of 2.5, 2, and 2.5 electrons respectively. Then no further energy change occurs. The energetic interaction stays constant although a higher number of electrons is available for alloying. The present authors cannot imagine that additional electrons are involved in the bonding without a release of an energy corresponding to a change in the Fermi energy. Therefore, we conclude that the limiting number of electrons involved in the bonding of the abovementioned systems is given by the stoichiometric number of electrons indicated by the break point of the Fermi enthalpy curve. Similar effects were found for the Ga alloys of Ni [9], Pd [12], and Au [13], shown in Fig. 3. Of course, the model proposed is a very rough approximation, since constant partial enthalpies are not really constant, as in the case of two-phase field. Fortunately, the systems investigated show very strong interactions re-
195
vealed by the very high negative enthalpies, e.g. for Ga + Pd liquid alloys A m i x H ° = - 60 kJ mol- a. The high energies found bring to mind the Engel-Brewer phases such as PtZr and PtTi; their high negative enthalpies of formation have been predicted by the Engel-Brewer theory [14]. Schaller and Brodowsky, who observed a similar flattening of the Fermi energy curves above some definite valence electron concentrations in solid Pd-In [8] and Pd-Sn [15] alloys, attributed this change of slope to the higher DOS of the 5p band with respect to the 5s band. Liquid A g + A l [16] alloys show a very particular shape of the Fermi enthalpy curve (Fig. 4). A distinct peak appears on the HF curve at the concentration corresponding to 2 electrons transferred. According to the definition of HF, a transfer of electrons from Ag to Al is indicated for high A1 concentrations. Although that signifies a transfer of electrons towards the more electropositive metal, the interpretation could be correct. The positive integral enthalpies of mixing measured in the region close to AI could be caused by the transfer in the "wrong" direction, which is energy consuming. A similar effect was found for Ag + Ga [16].
450
50
__........------400
40 350
0 E
J
300
30
1I
m
0
250
"-~20
e-.
t
200 I-
*t o ¢..
,/"
'.A..
!
i
<~10 e-
p. 150
t
l
100
50
I
!
i
0.5
1
-10 0
0.5
1
1.5
2
2.5
3
Number of electrons of Ga Fig. 3. Fermi enthalpy vs. the numbers of electrons transferred from Ga. The n u m b e r corresponds to three times the mole fraction of Ga (stoichiometric contribution).
0
1.5
2
2.5
3
N u m b e r of electrons of AI
Fig. 4. Fermi enthalpy vs. the numbe rs of electrons transferred from AI. The number corresponds to three times the mole fraction of AI (stoichiometric contribution).
196
E. Hayer, J.-P. Bros / Journal o f Alloys and Compounds 220 (1995) 193-196
4. Conclusion (i) The investigation of the Fermi enthalpy function of liquid alloys gives some insight into the change in the Fermi energy by alloying, which helps us to understand what happens with the electrons at the formation of a liquid alloy. (ii) The horizontal part of the partial enthalpy vs. mole fraction curve reveals a reduced number of electrons transferred. This conclusion is based on stoichiometric considerations. (iii) The function H F = h ( i ) - h ( j ) - h ( i ) ° is a good description for the change in the Fermi energy on alloying, at least for liquid alloys. (iv) the breakpoint in the Hv vs. concentration curve can be an indication of very interesting materials. References [1] A.S. Jordan, Metall. Trans., 1 (1970) 239.
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
B. Prede| and G. Oehme, Z. MetaUkd., 67 (1976) 826. F. Sommer, Z. Metallkd., 73 (1982) 72. M. Hoch and I. Arpshofen, Z. Metallkd., 75 (1984) 23. E. Hayer, Z. Phys. Chem., N.F., 156 (1988) 611. C. Kittel, Einfiihrung in die FestkOrperphysik, Oldenbourg, Munich, 4 t h edn., 1976, p. 303. F.U. Hillebrecht, J.C. Fuggle, P.A. Bennet, Z. Zolnierek and Ch. Freiburg, Phys. Rev. B, 27 (1983) 2179. H.J. Schaller and H. Brodowski, Ber. Bunsenges. Phys. Chem., 82 (1978) 773. D. El Allam, Thdse de l'Universitd de Provence, Marseille, 1989. D. El Allam, M. Gaune-Escard, J.P. Bros and E. Hayer, submitted to Metall. Trans. P. Anres, M. Gaune-Escard, J.P. Bros and E. Hayer, submitted to J. Alloys Compd. D. El Allam, M. Gaune-Escard, J.P. Bros and E. Hayer, Metall. Trans. B, 23 (1992) 39. E. Hayer, K.L. Komarek, M. Gaune-Escard and J.P. Bros, Z. Metallkd., 81 (1990) 233. L. Brewer, J. Chem. Educ., 61 (1984) 101. H.J. Schaller and H. Brodowski, Z. Metallkd., 69 (1978) 87. E. Hayer, Calorimetric results on Ag-alloys with AI, Ga and In, Discussion Meet. on Thermodynamics of Alloys, Genova, April 1994.