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Atomic bonding and mechanical properties of Al–Li–Zr alloy
Materials Science and Engineering A 499 (2009) 299–303
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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Atomic bonding and mechanical properties of Al–Li–Zr alloy Y.J. Gao a,b,∗ , Q.F. Mo a,b , H.N. Chen a,b , C.G. Huang a,b , L.N. Zhang a,b a b
School of Physics Science and Engineering, Guangxi University, Nanning 530004, China Central of International materials and Physics, Chinese Academy of Science, Shenyang 150016, China
a r t i c l e
i n f o
Article history: Received 10 April 2007 Received in revised form 11 July 2007 Accepted 14 November 2007 Keywords: Al–Li–Zr alloys Atomic bonding Mechanical property Simulation
a b s t r a c t The atomic bonding of Al–Li alloy with minor Zr is calculated according to the “Empirical Electronic Theory in Solids”. The result shows that the stronger interaction between Al and Zr atoms, which leads to form the Al–Zr segregation regions, promotes the precipitation of Al3 Zr particles and produces a remarkable refinement of Al3 Li grains in the alloy. Because there are the strongest covalent Al–Zr bonds in Al3 Zr and Al3 (Zr, Li) particles, these covalent bonds can cause a great resistance for dislocation movement, and is favorable to strengthen the alloy. On the other hand, with precipitating the Al3 (Zr, Li) particles, it causes the coherent interphase boundary energy of Al/Al3 Li to decrease, and atomic bonding is well matched in between the interface of two phases. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Al–Li-based alloys are very attractive, e.g., in aerospace industry, because of their low density, high specific strength, high elastic modulus, and good corrosion resistance [1,2]. In recent years, it was discovered by experimental research that short-range ordered structures [3] and GP zones were formed in quenched Al–Li alloy with more than 5.5 at.% Li. In the ageing process these segregation structures of solute atoms in alloy gradually evolve into the ordered structure Al3 Li, in which solute atoms interacting with vacancy will strongly influence the formation of ␦ phase [4,5]. Minor Zr addition to Al–Li alloy can be used to refine the Al3 Li particles [4]. Experimental research [6,7] also reveals that the high specific strength, high elastic modulus are closely related with the Li atom and its atomic bonding in surrounding. Now more attention has been paid to revealing the micro-mechanisms underlying the excellent physical properties of the alloy on atomic bonding level. The empirical electron theory (EET) [8] in solid, which is established on the basis of Pauling’s valence bond theory [9] and the energy band theory, offers a simple, direct and practical empirical method, the bond length difference (BLD) method, to deal with valence electron structures of complicated systems. It has been applied to many fields, such as solute atom segregation [10], phase transformation [11], magnetic properties [12], design of diamond [13] and
∗ Corresponding author. Tel.: +86 771 3232666; fax: +86 771 3237734. E-mail address: [email protected] (Y.J. Gao). 0921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.11.113
alloy design [14] successfully. In this paper the atomic bonding of the Al–Li–Zr alloy is calculated by using the EET of solid and molecule, then its influence on the properties of alloy is discussed. 2. Cell structure In the case of small additions of Li and Zr in Al–Li–Zr alloy, there is a great difference of chemical valence between Al and Li atoms, which will cause a strong chemical affinity between the two atoms. This will lead to non-uniform distribution of Li and Zr atoms in solid solution of alloy on atom scale, and form local heterogeneous microstructures in alloy. The heterogeneous microstructure can be approximately described by the “mixture cell model” according to EET theory. As pointed out in Ref. [15], the solid solution of Al–Li–Zr alloy in micro-scale should be made of some kinds of segregated cells by mixing: (1) the pure Al cell (FCC structure) shown in Fig. 1(a); (2) the Al–Li cell, in which Li atoms take the place of Al atoms in face center positions in Fig. 1(b); (3) the Al–Zr cell, in which Zr atoms take the place of Al atoms in face center positions in Fig. 2(a); (4) the Al–Li–Zr cell, in which Li and Zr atoms take some places of Al atom positions in Fig. 2(b). These heterogeneous microstructures are distributed disorderly in matrix. In addition, during the solid solution treatment for Al–Li–Zr alloy, Al3 Li and Al3 Zr phase with L12 structure given in Ref. [16] are precipitated in supersaturated solid solution by quenching. Beside Al–Li cell, Al–Zr cell, Al–Li–Zr cell, the solid solution should include the particle of Al3 Li and Al3 Zr, of which crystal structures are shown
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Fig. 1. Cell structure: (a) pure Al cell; (b) Al–Li cell; (c) Al3 Li cell, where () Al atom; (䊉) Li atom.
Fig. 2. Cell structure with Zr: (a) Al–Zr cell; (b) Al–Li–Zr cell; (c) Al3 Zr cell, where () Al atom; (䊉) Li atom; () Zr atom.
in Fig. 1(c) and Fig. 2(c). The lattice constants of Al3 Zr and Al3 Li are 0.4050 nm and 0.4046 nm, respectively.
3. Theoretical method and results 3.1. State of Al atom According to the fundamental hypothesis of EET in Ref. [8], the electron configuration relating to ground state of element Al in the IIIA group is 3s2 3p1 , which is a one-covalence state. Because of the metallic properties of the elements under solid conditions, the s electron is considered to be the lattice electron (like free electron) to form the h (head) state from the ground state, while the p electron is considered to be the electron to form covalent bond in order to ensure the one-covalence state. Let one lattice electron in an orbit be denoted by (Ф), the covalent electron by (䊉), and the empty orbit by (); then, the h state can be represented as
The t state is chosen by considering that there is one part covalence electron is in s state, and the other part in p state, which often appears in solids and molecules. The related electron configuration is 2sx 2p1−x , where the x is a percentage. To ensure the covalence state, all of them should be covalent electrons in order that the Li metal has enough cohesive energy, then the t state are shown as above. 3.3. State of Zr atom
where ( ) represents two lattice electrons in an orbit. The t (tail) state is selected by considering the three-covalence state, which often appears in solids and molecules. The related electron configuration is 3s1 3p2 . To ensure the three-covalence state, all of electrons should be covalent electrons. So the t state can be represented as above.
3.2. State of Li atom Similar to the Al atom state, the electron configuration relating to ground state of element Li in IA group is 2s1 2p0 , which is of ones electron state. Based on the metallic properties of the element under solid conditions, the s electron is acted as a lattice electron in the h state from the ground state. Let one lattice electron be denoted by (Ф), and the empty orbit by (), then, the h state can be represented as follows:
Here the electron configuration relating to ground state of element Zr in IVB group is 3d2 4s2 . Owing to the metallic properties of the elements, the s electron is regarded as the lattice electron in the h state at the ground state. Then the h state can be represented as follows with one lattice electron in s orbital, and two covalent electrons in p orbit and one covalent electron in d orbit which are all to contribute to cohesive energy of metal, respectively, in p and d orbit.
The t state is of the two-covalence electron state in s, p orbit, and two-covalence electrons in d orbit to contribute to cohesive energy which often appears in solids and molecules. The related electron configuration is 3d2 4s1 4p1 . So the t state is given as above. 3.4. Calculation and results According to EET [8], valence electron structure of the metal determines the states of the atoms that form the alloy and the
Y.J. Gao et al. / Materials Science and Engineering A 499 (2009) 299–303 Table 3 Atomic bonding of Al–Zr cell
Table 1 Atomic bonding of Al cell Bond
I˛
Al–Al DnA Al–Al DnB
Dn˛ (nm)
12 6
301
0.28635 0.40496
¯ n˛ (nm) D 0.28633 0.40494
D (nm)
nA 0.2086 0.0045
0.00002 0.00002
a0 = 0.40496 nm; Al: = 4, nc = 2.5296, R(1) = 0.119 nm.
Bond
I˛
Dn˛ (nm)
¯ n˛ (nm) D
nA
D (nm)
Al–Zr DnA Al–Al DnB Al–Al DnD Zr–Zr DnC
24 24 18 6
0.28635 0.28635 0.40496 0.40496
0.28554 0.28554 0.40415 0.40415
0.30411 0.16128 0.00170 0.00679
0.0003 0.0003 0.0003 0.0003
a = 0.40496 nm; Al: Al = 5, nc = 2.8970, R(1) = 0.119 nm; Zr: Zr = B13, nc = 3.8430, R(1) = 0.12637 nm.
electron distribution of the covalent bonds formed by these atoms. Covalent electrons are distributed in the bonds of the nearest neighbor, the second nearest neighbor and sth nearest neighbor. The amount of covalent electrons of each bond (namely bond orders ns ) can be represented by the following bond length formula: u
v
D(ns ) = R (1) + R (1) − ˇ ln ns
(1)
where D(ns ) is the bond length, Ru (1) and Rv (1) are the single bond radii of u and v atoms, respectively, ˇ is constant, where ˇ = 0.710 nm. The amount of covalent electron in a cell can be represented by the following formula: K1 nuc + K2 nvc =
Is ns
(2)
s
where the numbers of u and v atoms in the cell are represented by K1 , K2 , respectively. nuc and nvc are the amount of covalent electrons of u and v atoms, respectively. Is is the equivalence-bond number of ns . The equivalence-bond number can be calculated according to Refs. [8,14]. For a given structure, if the calculated atom states in solid are ¯ n˛ determined by atom state correct, the theoretical bond lengths D parameters nc and R(1) should all accord with the experimental bond lengths Dn˛ . Therefore, by comparing the theoretical val¯ n˛ , which can calculated from certain atom state with the ues D experimental one of all covalent bonds in a structure unit, we can determine if the given atom states accord with the actual states ¯ n˛ , accord in solid. To determine if the theoretical bond lengths D with the experimental one Dn˛ , quantitatively, Yu [8] suggested that the absolute value of their difference should be less than 0.005 nm under first-order approximation
¯ n˛ − Dn˛ < 0.005 nm Dn˛ = D
(3)
where Eq. (3) is the BLD criterion. According to Ref. [8], the atomic bonding of the cells can be calculated one by one with BLD criterion, and the more detailed procedure has been shown in Refs. [10–15]. The calculated results of the covalent bond of atom in pure Al cell and the segregated cells are shown in Tables 1–6. The sign in these tables is the hybrid level. 4. Discussion Comparing the results from Tables 1–6, it can be seen that the covalent bonds in the cell with Zr atoms are the strongest, the Al–Zr covalent bond nA in the Al–Zr and Al3 Zr cells are 0.30411 and 0.30611, respectively, about 50% greater than the Al–Al bond
Table 4 Atomic bonding of Al3 Li cell Bond
I˛
Dn˛ (nm)
¯ n˛ (nm) D
nA
D (nm)
Al–Al DnA Al–Li DnB Al–Al DnC Li–Li DnD
24 24 18 6
0.28355 0.28355 0.40100 0.40100
0.28288 0.28288 0.40033 0.40033
0.2333 0.1204 0.0052 0.0014
0.00066 0.00066 0.00066 0.00066
a = 0.4010 nm; Al: R(1) = 0.0986 nm.
Al = 4,
nc = 2.5296,
R(1) = 0.119 nm;
Li:
Li = 4,
nc = 1,
(nA = 0.2086) of FCC Al cell. It indicates that the coalescent incline of Al and Zr is strongest, and the covalent bond with p–d electron is easily formed. In the phase diagram [16] of Al–Zr alloy, the precipitation temperature of Al3 Zr in the Al–0.17(at.%)Zr solid solution is 1250 K, which is much higher than that of Al–8(at.%)Li alloy. So there are quite a lot the primary Al3 Zr particles to form before Al–Li–Zr alloy solidifies. Because Al3 Zr has a L12 structure as that of metastable Al3 Li phase and both phases have excellent lattice matching relation, and thus a much low coherent interphase energy. The strong Al–Zr covalent bond in Al3 Zr is greater than the Al–Li covalent bond in Al3 Li will lead to Al3 Zr using as an inhomogeneous nucleus for Al3 Li, play a great role in grain refinement and improve the toughness of alloy. The calculated results of atomic bonding of Al–Li–Zr segregation cell in Table 6 indicate that the Al–Li bond and Li–Zr covalent bond are much stronger, therefore it is easily to form an Al3 Zr core and Al3 Li shell, or to say, a composite phase Al3 Li/Al3 Zr. In recent years, the investigation [17–18] has shown that the coherent composite Al3 Li/Al3 Zr phase precipitates in matrix and enhances the thermal stability, and strengthens the alloy. These can be well explained by the calculated atomic bonding in this paper.
Table 5 Atomic bonding of Al3 Zr cell Bond
I˛
Dn˛ (nm)
¯ n˛ (nm) D
nA
D (nm)
Al–Zr DnA Al–Al DnB Al–Al DnC Zr–Zr DnD
24 24 18 6
0.28638 0.28638 0.40500 0.40500
0.28554 0.28554 0.40417 0.40417
0.30611 0.16128 0.00170 0.00679
0.0003 0.0003 0.0003 0.0003
a = 0.4050 nm; Al: Al = 5, nc = 2.8970, R(1) = 0.119 nm; Zr: Zr = 13, nc = 2.9893, R(1) = 0.13705 nm.
Table 6 Atomic bonding of Al–Li–Zr cell Table 2 Atomic bonding of Al–Li cell Bond
I˛
Dn˛ (nm)
¯ n˛ (nm) D
nA
D(nm)
Al–Al DnA Al–Li DnB Al–Al DnD Li–Li DnC
24 24 18 6
0.28591 0.28591 0.40434 0.40434
0.28287 0.28287 0.40130 0.40130
0.2262 0.1714 0.0050 0.0013
0.0003 0.0003 0.0003 0.0003
a = 0.40434 nm; Al: R(1) = 0.11046 nm.
Al = 6,
nc = 3,
R(1) = 0.119 nm;
Li:
Li = 3,
nc = 0.6511,
Bond
I˛
Dn␣ (nm)
¯ n˛ (nm) D
nA
D (nm)
Al–Zr DnA Al–Al DnB Li–Zr DnC Al–Li DnD Al–Al DnE Zr–Zr DnG
16 8 8 16 12 6
0.28591 0.28591 0.28591 0.28591 0.40434 0.40434
0.28571 0.28571 0.28571 0.28571 0.40414 0.40414
0.2683 0.2113 0.2034 0.1602 0.0014 0.0051
0.00020 0.00020 0.00020 0.00020 0.00020 0.00020
a = 0.40434 nm; Al: Al = 5, nc = 2.8970, R(1) = 0.119 nm; Li: Li = 3, nc = 0.6511, R(1) = 0.11046 nm; Zr: Zr = B13, nc = 3.8430, R(1) = 0.12637 nm.
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In aging process of Al–Li–Zr alloy, a lot of tiny secondary Al3 Zr particles which structure are coherence with matrix, also precipitate in solid solution matrix due to the stronger interaction between Al and Zr atoms. Because the Al–Zr covalent bond network is much stronger in Al3 Zr particles, which are not easy cut by dislocations. It can repress the grain boundary moving when grains grow up, and hence increase the thermal stability of alloy. Because the particles Al3 Zr and Al3 (Li, Zr) are coherently existed in matrix, the mismatch of lattice constant at interphase boundary are so small that to form a strong covalence bond network in matrix. The stronger covalence Al–Zr bond in Al3 Zr is not easily to be broken down by dislocation cutting and can be hinder dislocation moving. For this reason, the matrix strengthening of alloy is owing to the stronger Al–Zr covalent bond network. The precipitation of composite particle Al3 (Li, Zr) can reduce the mismatch of coherence interface atoms and decrease interface energy, make the interface of phases with uniform atomic bonding distribution, and lead to the uniform atomic bonding network in between the matrix and precipitations. It is also the reason that strength and toughness of the alloy can be enhanced by better match of covalent bonding on interface among Al, Li and Zr atoms and the higher modulus are obtained for Al–Li alloy. 5. Conclusion (i) Because the combination incline of Al and Zr is the strongest, there is quite a lot of Zr and Al atoms to form the primary Al3 Zr particles before Al–Li–Zr alloy solidifies. The strong Al–Zr covalent bond in Al3 Zr is greater than the Al–Li covalent bond in Al3 Li will lead to Al3 Zr using as an inhomogeneous nucleus for Al3 Li, play a great role in grain refinement and improve the toughness of alloy. (ii) The stronger covalence Al–Zr bond in Al3 Zr is not easily to be broken down by dislocation cutting and can be hinder dislocation moving. For this reason, the matrix strengthening of alloy is owing to the stronger Al–Zr covalent bond network. (iii) The precipitation of composite particle Al3 (Li, Zr) reducing the mismatch of coherence interface atoms and decreasing interface energy, are owing to uniform atomic bonding distribution on interface between matrix and precipitations. It is also the reason that strength and toughness of the alloy can be enhanced by better match of covalent bonding on interface among Al, Li and Zr atoms. Acknowledgments This work was financially supported by the Natural Science Foundation of China under Project numbers 50061001 and 50661001; supported by the Science Foundation of Guangxi Province under Project numbers 0823029 and 0639004. Appendix A. Appendix Fundamental hypothesis in the empirical electron theory (EET) of solid and molecule:
means the valence electron, which is in the space enclosed by two or more atoms in a solid system formed by many atoms. A.2. Fundamental hypothesis 2 Normally, the hybridization of states is not continuous. Let Ct represent the content of the t states in the hybrid state, and Ch represent that of the h state. In most structures, Ct and Ch can be given with the following formulas: Ct = k=
1 , 1 + k2
Ct + Ch = 1
l + n + m l + n + m
(A1)
√ √ l + n + m l ± 5n ± 3m √ √ l + n + m l ± 5n ± 3m
(A2)
which l, n, m and l , n , m represent the sum of covalent electron number and lattice electron number of s, p, d electron of the h and t states, respectively. The terms and are parameters for the h and t states, respectively, and value 1 when the s electron is covalent electron or 0 when the s electron is lattice electron. The terms k = ∞ and k = 0 represent the h and t states, respectively, the number of various k values is called hybrid level number. The character parameters describing the atom state at hybrid level can be given with the following formulas: nT = nTh Ch + nTt Ct = (l + m + n)Ch + (l + m + n )Ct
nl = nlh Ch + nlt Ct = (1 − )lCh + (1 − )l Ct
(A3) (A4)
nc = nch Ch + nct Ct = (l + m + n)Ch + ( l + m + n )Ct
(A5)
R(1) = R(1)h Ch + R(1)t Ct
(A6)
where Ch and Ct represent the contents of the h and t states at hybrid level, respectively; R(1)h and R(1)t represent the bond radii of them, respectively. A.3. Fundamental hypothesis 3 Except in some special conditions, there will always be covalent electrons pairs between two adjacent atoms u and v. The number of this covalent electron pair is represented by n˛ and the distance between these two atoms is called covalent bond length, which is uv , according to Pauling‘s research, the following represented by Dn˛ uv , Ru (1), Rv (1) and n : relation exists among Dn˛ ˛ uv = Ru (1) + Rv (1) − ˇ ln n˛ Dn˛
(A7)
where u and v can be the same kind or different kind of atoms; n˛ can be an integer or a fraction; ˛ = A, B, C, . . ., N represents all bonds that cannot be neglected in a structure. The bond cannot be neglected means the bond with so long a bond length that the n˛ calculated with formula (A7) cannot be neglected compared with the possible of the largest nM ˛ in the structure. The value of ˇ should observe the following rule: ˇ=
0.710,
M when nM ˛ < 0.25 or n˛ > 0.75 0.600, when 0.300 ≤ nM ˛ ≤ 0.700 M 0.710−2.2ε, when nM ˛ = 0.25+ε, or, n˛ = 0.750−ε, where 0 < ε < 0.050.
(A8)
A.1. Fundamental hypothesis 1 In solids and molecules, an atom is normally formed by the hybridization of two atom states. These two states are called the h (head) state and the t (tail) states. At least one of them is the ground state or the excited state that is nearest to the ground state. Both of the states have their own total valence electron numbers nT , covalent electron numbers nc , lattice electron numbers nl , and bond radius R(1). Here, lattice electron is a new concept, which
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Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Atomic bonding and mechanical properties of Al–Li–Zr alloy Y.J. Gao a,b,∗ , Q.F. Mo a,b , H.N. Chen a,b , C.G. Huang a,b , L.N. Zhang a,b a b
School of Physics Science and Engineering, Guangxi University, Nanning 530004, China Central of International materials and Physics, Chinese Academy of Science, Shenyang 150016, China
a r t i c l e
i n f o
Article history: Received 10 April 2007 Received in revised form 11 July 2007 Accepted 14 November 2007 Keywords: Al–Li–Zr alloys Atomic bonding Mechanical property Simulation
a b s t r a c t The atomic bonding of Al–Li alloy with minor Zr is calculated according to the “Empirical Electronic Theory in Solids”. The result shows that the stronger interaction between Al and Zr atoms, which leads to form the Al–Zr segregation regions, promotes the precipitation of Al3 Zr particles and produces a remarkable refinement of Al3 Li grains in the alloy. Because there are the strongest covalent Al–Zr bonds in Al3 Zr and Al3 (Zr, Li) particles, these covalent bonds can cause a great resistance for dislocation movement, and is favorable to strengthen the alloy. On the other hand, with precipitating the Al3 (Zr, Li) particles, it causes the coherent interphase boundary energy of Al/Al3 Li to decrease, and atomic bonding is well matched in between the interface of two phases. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Al–Li-based alloys are very attractive, e.g., in aerospace industry, because of their low density, high specific strength, high elastic modulus, and good corrosion resistance [1,2]. In recent years, it was discovered by experimental research that short-range ordered structures [3] and GP zones were formed in quenched Al–Li alloy with more than 5.5 at.% Li. In the ageing process these segregation structures of solute atoms in alloy gradually evolve into the ordered structure Al3 Li, in which solute atoms interacting with vacancy will strongly influence the formation of ␦ phase [4,5]. Minor Zr addition to Al–Li alloy can be used to refine the Al3 Li particles [4]. Experimental research [6,7] also reveals that the high specific strength, high elastic modulus are closely related with the Li atom and its atomic bonding in surrounding. Now more attention has been paid to revealing the micro-mechanisms underlying the excellent physical properties of the alloy on atomic bonding level. The empirical electron theory (EET) [8] in solid, which is established on the basis of Pauling’s valence bond theory [9] and the energy band theory, offers a simple, direct and practical empirical method, the bond length difference (BLD) method, to deal with valence electron structures of complicated systems. It has been applied to many fields, such as solute atom segregation [10], phase transformation [11], magnetic properties [12], design of diamond [13] and
∗ Corresponding author. Tel.: +86 771 3232666; fax: +86 771 3237734. E-mail address: [email protected] (Y.J. Gao). 0921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.11.113
alloy design [14] successfully. In this paper the atomic bonding of the Al–Li–Zr alloy is calculated by using the EET of solid and molecule, then its influence on the properties of alloy is discussed. 2. Cell structure In the case of small additions of Li and Zr in Al–Li–Zr alloy, there is a great difference of chemical valence between Al and Li atoms, which will cause a strong chemical affinity between the two atoms. This will lead to non-uniform distribution of Li and Zr atoms in solid solution of alloy on atom scale, and form local heterogeneous microstructures in alloy. The heterogeneous microstructure can be approximately described by the “mixture cell model” according to EET theory. As pointed out in Ref. [15], the solid solution of Al–Li–Zr alloy in micro-scale should be made of some kinds of segregated cells by mixing: (1) the pure Al cell (FCC structure) shown in Fig. 1(a); (2) the Al–Li cell, in which Li atoms take the place of Al atoms in face center positions in Fig. 1(b); (3) the Al–Zr cell, in which Zr atoms take the place of Al atoms in face center positions in Fig. 2(a); (4) the Al–Li–Zr cell, in which Li and Zr atoms take some places of Al atom positions in Fig. 2(b). These heterogeneous microstructures are distributed disorderly in matrix. In addition, during the solid solution treatment for Al–Li–Zr alloy, Al3 Li and Al3 Zr phase with L12 structure given in Ref. [16] are precipitated in supersaturated solid solution by quenching. Beside Al–Li cell, Al–Zr cell, Al–Li–Zr cell, the solid solution should include the particle of Al3 Li and Al3 Zr, of which crystal structures are shown
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Fig. 1. Cell structure: (a) pure Al cell; (b) Al–Li cell; (c) Al3 Li cell, where () Al atom; (䊉) Li atom.
Fig. 2. Cell structure with Zr: (a) Al–Zr cell; (b) Al–Li–Zr cell; (c) Al3 Zr cell, where () Al atom; (䊉) Li atom; () Zr atom.
in Fig. 1(c) and Fig. 2(c). The lattice constants of Al3 Zr and Al3 Li are 0.4050 nm and 0.4046 nm, respectively.
3. Theoretical method and results 3.1. State of Al atom According to the fundamental hypothesis of EET in Ref. [8], the electron configuration relating to ground state of element Al in the IIIA group is 3s2 3p1 , which is a one-covalence state. Because of the metallic properties of the elements under solid conditions, the s electron is considered to be the lattice electron (like free electron) to form the h (head) state from the ground state, while the p electron is considered to be the electron to form covalent bond in order to ensure the one-covalence state. Let one lattice electron in an orbit be denoted by (Ф), the covalent electron by (䊉), and the empty orbit by (); then, the h state can be represented as
The t state is chosen by considering that there is one part covalence electron is in s state, and the other part in p state, which often appears in solids and molecules. The related electron configuration is 2sx 2p1−x , where the x is a percentage. To ensure the covalence state, all of them should be covalent electrons in order that the Li metal has enough cohesive energy, then the t state are shown as above. 3.3. State of Zr atom
where ( ) represents two lattice electrons in an orbit. The t (tail) state is selected by considering the three-covalence state, which often appears in solids and molecules. The related electron configuration is 3s1 3p2 . To ensure the three-covalence state, all of electrons should be covalent electrons. So the t state can be represented as above.
3.2. State of Li atom Similar to the Al atom state, the electron configuration relating to ground state of element Li in IA group is 2s1 2p0 , which is of ones electron state. Based on the metallic properties of the element under solid conditions, the s electron is acted as a lattice electron in the h state from the ground state. Let one lattice electron be denoted by (Ф), and the empty orbit by (), then, the h state can be represented as follows:
Here the electron configuration relating to ground state of element Zr in IVB group is 3d2 4s2 . Owing to the metallic properties of the elements, the s electron is regarded as the lattice electron in the h state at the ground state. Then the h state can be represented as follows with one lattice electron in s orbital, and two covalent electrons in p orbit and one covalent electron in d orbit which are all to contribute to cohesive energy of metal, respectively, in p and d orbit.
The t state is of the two-covalence electron state in s, p orbit, and two-covalence electrons in d orbit to contribute to cohesive energy which often appears in solids and molecules. The related electron configuration is 3d2 4s1 4p1 . So the t state is given as above. 3.4. Calculation and results According to EET [8], valence electron structure of the metal determines the states of the atoms that form the alloy and the
Y.J. Gao et al. / Materials Science and Engineering A 499 (2009) 299–303 Table 3 Atomic bonding of Al–Zr cell
Table 1 Atomic bonding of Al cell Bond
I˛
Al–Al DnA Al–Al DnB
Dn˛ (nm)
12 6
301
0.28635 0.40496
¯ n˛ (nm) D 0.28633 0.40494
D (nm)
nA 0.2086 0.0045
0.00002 0.00002
a0 = 0.40496 nm; Al: = 4, nc = 2.5296, R(1) = 0.119 nm.
Bond
I˛
Dn˛ (nm)
¯ n˛ (nm) D
nA
D (nm)
Al–Zr DnA Al–Al DnB Al–Al DnD Zr–Zr DnC
24 24 18 6
0.28635 0.28635 0.40496 0.40496
0.28554 0.28554 0.40415 0.40415
0.30411 0.16128 0.00170 0.00679
0.0003 0.0003 0.0003 0.0003
a = 0.40496 nm; Al: Al = 5, nc = 2.8970, R(1) = 0.119 nm; Zr: Zr = B13, nc = 3.8430, R(1) = 0.12637 nm.
electron distribution of the covalent bonds formed by these atoms. Covalent electrons are distributed in the bonds of the nearest neighbor, the second nearest neighbor and sth nearest neighbor. The amount of covalent electrons of each bond (namely bond orders ns ) can be represented by the following bond length formula: u
v
D(ns ) = R (1) + R (1) − ˇ ln ns
(1)
where D(ns ) is the bond length, Ru (1) and Rv (1) are the single bond radii of u and v atoms, respectively, ˇ is constant, where ˇ = 0.710 nm. The amount of covalent electron in a cell can be represented by the following formula: K1 nuc + K2 nvc =
Is ns
(2)
s
where the numbers of u and v atoms in the cell are represented by K1 , K2 , respectively. nuc and nvc are the amount of covalent electrons of u and v atoms, respectively. Is is the equivalence-bond number of ns . The equivalence-bond number can be calculated according to Refs. [8,14]. For a given structure, if the calculated atom states in solid are ¯ n˛ determined by atom state correct, the theoretical bond lengths D parameters nc and R(1) should all accord with the experimental bond lengths Dn˛ . Therefore, by comparing the theoretical val¯ n˛ , which can calculated from certain atom state with the ues D experimental one of all covalent bonds in a structure unit, we can determine if the given atom states accord with the actual states ¯ n˛ , accord in solid. To determine if the theoretical bond lengths D with the experimental one Dn˛ , quantitatively, Yu [8] suggested that the absolute value of their difference should be less than 0.005 nm under first-order approximation
¯ n˛ − Dn˛ < 0.005 nm Dn˛ = D
(3)
where Eq. (3) is the BLD criterion. According to Ref. [8], the atomic bonding of the cells can be calculated one by one with BLD criterion, and the more detailed procedure has been shown in Refs. [10–15]. The calculated results of the covalent bond of atom in pure Al cell and the segregated cells are shown in Tables 1–6. The sign in these tables is the hybrid level. 4. Discussion Comparing the results from Tables 1–6, it can be seen that the covalent bonds in the cell with Zr atoms are the strongest, the Al–Zr covalent bond nA in the Al–Zr and Al3 Zr cells are 0.30411 and 0.30611, respectively, about 50% greater than the Al–Al bond
Table 4 Atomic bonding of Al3 Li cell Bond
I˛
Dn˛ (nm)
¯ n˛ (nm) D
nA
D (nm)
Al–Al DnA Al–Li DnB Al–Al DnC Li–Li DnD
24 24 18 6
0.28355 0.28355 0.40100 0.40100
0.28288 0.28288 0.40033 0.40033
0.2333 0.1204 0.0052 0.0014
0.00066 0.00066 0.00066 0.00066
a = 0.4010 nm; Al: R(1) = 0.0986 nm.
Al = 4,
nc = 2.5296,
R(1) = 0.119 nm;
Li:
Li = 4,
nc = 1,
(nA = 0.2086) of FCC Al cell. It indicates that the coalescent incline of Al and Zr is strongest, and the covalent bond with p–d electron is easily formed. In the phase diagram [16] of Al–Zr alloy, the precipitation temperature of Al3 Zr in the Al–0.17(at.%)Zr solid solution is 1250 K, which is much higher than that of Al–8(at.%)Li alloy. So there are quite a lot the primary Al3 Zr particles to form before Al–Li–Zr alloy solidifies. Because Al3 Zr has a L12 structure as that of metastable Al3 Li phase and both phases have excellent lattice matching relation, and thus a much low coherent interphase energy. The strong Al–Zr covalent bond in Al3 Zr is greater than the Al–Li covalent bond in Al3 Li will lead to Al3 Zr using as an inhomogeneous nucleus for Al3 Li, play a great role in grain refinement and improve the toughness of alloy. The calculated results of atomic bonding of Al–Li–Zr segregation cell in Table 6 indicate that the Al–Li bond and Li–Zr covalent bond are much stronger, therefore it is easily to form an Al3 Zr core and Al3 Li shell, or to say, a composite phase Al3 Li/Al3 Zr. In recent years, the investigation [17–18] has shown that the coherent composite Al3 Li/Al3 Zr phase precipitates in matrix and enhances the thermal stability, and strengthens the alloy. These can be well explained by the calculated atomic bonding in this paper.
Table 5 Atomic bonding of Al3 Zr cell Bond
I˛
Dn˛ (nm)
¯ n˛ (nm) D
nA
D (nm)
Al–Zr DnA Al–Al DnB Al–Al DnC Zr–Zr DnD
24 24 18 6
0.28638 0.28638 0.40500 0.40500
0.28554 0.28554 0.40417 0.40417
0.30611 0.16128 0.00170 0.00679
0.0003 0.0003 0.0003 0.0003
a = 0.4050 nm; Al: Al = 5, nc = 2.8970, R(1) = 0.119 nm; Zr: Zr = 13, nc = 2.9893, R(1) = 0.13705 nm.
Table 6 Atomic bonding of Al–Li–Zr cell Table 2 Atomic bonding of Al–Li cell Bond
I˛
Dn˛ (nm)
¯ n˛ (nm) D
nA
D(nm)
Al–Al DnA Al–Li DnB Al–Al DnD Li–Li DnC
24 24 18 6
0.28591 0.28591 0.40434 0.40434
0.28287 0.28287 0.40130 0.40130
0.2262 0.1714 0.0050 0.0013
0.0003 0.0003 0.0003 0.0003
a = 0.40434 nm; Al: R(1) = 0.11046 nm.
Al = 6,
nc = 3,
R(1) = 0.119 nm;
Li:
Li = 3,
nc = 0.6511,
Bond
I˛
Dn␣ (nm)
¯ n˛ (nm) D
nA
D (nm)
Al–Zr DnA Al–Al DnB Li–Zr DnC Al–Li DnD Al–Al DnE Zr–Zr DnG
16 8 8 16 12 6
0.28591 0.28591 0.28591 0.28591 0.40434 0.40434
0.28571 0.28571 0.28571 0.28571 0.40414 0.40414
0.2683 0.2113 0.2034 0.1602 0.0014 0.0051
0.00020 0.00020 0.00020 0.00020 0.00020 0.00020
a = 0.40434 nm; Al: Al = 5, nc = 2.8970, R(1) = 0.119 nm; Li: Li = 3, nc = 0.6511, R(1) = 0.11046 nm; Zr: Zr = B13, nc = 3.8430, R(1) = 0.12637 nm.
302
Y.J. Gao et al. / Materials Science and Engineering A 499 (2009) 299–303
In aging process of Al–Li–Zr alloy, a lot of tiny secondary Al3 Zr particles which structure are coherence with matrix, also precipitate in solid solution matrix due to the stronger interaction between Al and Zr atoms. Because the Al–Zr covalent bond network is much stronger in Al3 Zr particles, which are not easy cut by dislocations. It can repress the grain boundary moving when grains grow up, and hence increase the thermal stability of alloy. Because the particles Al3 Zr and Al3 (Li, Zr) are coherently existed in matrix, the mismatch of lattice constant at interphase boundary are so small that to form a strong covalence bond network in matrix. The stronger covalence Al–Zr bond in Al3 Zr is not easily to be broken down by dislocation cutting and can be hinder dislocation moving. For this reason, the matrix strengthening of alloy is owing to the stronger Al–Zr covalent bond network. The precipitation of composite particle Al3 (Li, Zr) can reduce the mismatch of coherence interface atoms and decrease interface energy, make the interface of phases with uniform atomic bonding distribution, and lead to the uniform atomic bonding network in between the matrix and precipitations. It is also the reason that strength and toughness of the alloy can be enhanced by better match of covalent bonding on interface among Al, Li and Zr atoms and the higher modulus are obtained for Al–Li alloy. 5. Conclusion (i) Because the combination incline of Al and Zr is the strongest, there is quite a lot of Zr and Al atoms to form the primary Al3 Zr particles before Al–Li–Zr alloy solidifies. The strong Al–Zr covalent bond in Al3 Zr is greater than the Al–Li covalent bond in Al3 Li will lead to Al3 Zr using as an inhomogeneous nucleus for Al3 Li, play a great role in grain refinement and improve the toughness of alloy. (ii) The stronger covalence Al–Zr bond in Al3 Zr is not easily to be broken down by dislocation cutting and can be hinder dislocation moving. For this reason, the matrix strengthening of alloy is owing to the stronger Al–Zr covalent bond network. (iii) The precipitation of composite particle Al3 (Li, Zr) reducing the mismatch of coherence interface atoms and decreasing interface energy, are owing to uniform atomic bonding distribution on interface between matrix and precipitations. It is also the reason that strength and toughness of the alloy can be enhanced by better match of covalent bonding on interface among Al, Li and Zr atoms. Acknowledgments This work was financially supported by the Natural Science Foundation of China under Project numbers 50061001 and 50661001; supported by the Science Foundation of Guangxi Province under Project numbers 0823029 and 0639004. Appendix A. Appendix Fundamental hypothesis in the empirical electron theory (EET) of solid and molecule:
means the valence electron, which is in the space enclosed by two or more atoms in a solid system formed by many atoms. A.2. Fundamental hypothesis 2 Normally, the hybridization of states is not continuous. Let Ct represent the content of the t states in the hybrid state, and Ch represent that of the h state. In most structures, Ct and Ch can be given with the following formulas: Ct = k=
1 , 1 + k2
Ct + Ch = 1
l + n + m l + n + m
(A1)
√ √ l + n + m l ± 5n ± 3m √ √ l + n + m l ± 5n ± 3m
(A2)
which l, n, m and l , n , m represent the sum of covalent electron number and lattice electron number of s, p, d electron of the h and t states, respectively. The terms and are parameters for the h and t states, respectively, and value 1 when the s electron is covalent electron or 0 when the s electron is lattice electron. The terms k = ∞ and k = 0 represent the h and t states, respectively, the number of various k values is called hybrid level number. The character parameters describing the atom state at hybrid level can be given with the following formulas: nT = nTh Ch + nTt Ct = (l + m + n)Ch + (l + m + n )Ct
nl = nlh Ch + nlt Ct = (1 − )lCh + (1 − )l Ct
(A3) (A4)
nc = nch Ch + nct Ct = (l + m + n)Ch + ( l + m + n )Ct
(A5)
R(1) = R(1)h Ch + R(1)t Ct
(A6)
where Ch and Ct represent the contents of the h and t states at hybrid level, respectively; R(1)h and R(1)t represent the bond radii of them, respectively. A.3. Fundamental hypothesis 3 Except in some special conditions, there will always be covalent electrons pairs between two adjacent atoms u and v. The number of this covalent electron pair is represented by n˛ and the distance between these two atoms is called covalent bond length, which is uv , according to Pauling‘s research, the following represented by Dn˛ uv , Ru (1), Rv (1) and n : relation exists among Dn˛ ˛ uv = Ru (1) + Rv (1) − ˇ ln n˛ Dn˛
(A7)
where u and v can be the same kind or different kind of atoms; n˛ can be an integer or a fraction; ˛ = A, B, C, . . ., N represents all bonds that cannot be neglected in a structure. The bond cannot be neglected means the bond with so long a bond length that the n˛ calculated with formula (A7) cannot be neglected compared with the possible of the largest nM ˛ in the structure. The value of ˇ should observe the following rule: ˇ=
0.710,
M when nM ˛ < 0.25 or n˛ > 0.75 0.600, when 0.300 ≤ nM ˛ ≤ 0.700 M 0.710−2.2ε, when nM ˛ = 0.25+ε, or, n˛ = 0.750−ε, where 0 < ε < 0.050.
(A8)
A.1. Fundamental hypothesis 1 In solids and molecules, an atom is normally formed by the hybridization of two atom states. These two states are called the h (head) state and the t (tail) states. At least one of them is the ground state or the excited state that is nearest to the ground state. Both of the states have their own total valence electron numbers nT , covalent electron numbers nc , lattice electron numbers nl , and bond radius R(1). Here, lattice electron is a new concept, which
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