Materials Transactions, Vol. 46, No. 6 (2005) pp. 1148 to 1153
Special Issue on Computer Modeling of Materials and Processes
#2005 The Japan Institute of Metals
Atomic Bonding and Properties of Al–Mg–Sc Alloy
Gao Yingjun*1 , Huang Chuanggao, Hou Xianhua*2 , Mo Qifeng*2 and Liu Hui*2
College of Physical Science and Engineering, Guangxi University, Nanning 530004, P. R. China
Atomic bonding of Al–Mg alloy with minor Sc is calculated according to the ‘‘Empirical electron theory in solid’’ (EET). The results show
that the particles Al3 Sc precipitate firstly in melting during solidification is owing to the strong interaction of Al with Sc atom. The strong
interaction of Al with Sc atom can also cause to form the Al–Sc and Mg–Sc segregation regions in solid solution matrix of Al–Mg alloy in
further process of solidification. So in following homogenization treatment, finer dispersed Al3 Sc second-particles owing to its strong Al–Sc
covalent bonds are easily precipitated in these segregation regions. These secondary tiny Al3 Sc particles coherent with the matrix can hinder the
recrystallization of the alloy under high temperature by pinning the grain boundary. The reason, that the large Al3 Sc particles can improve the
strength and hardness of the alloy, is attributed to the strong Al–Sc covalent bond net in larger Al3 Sc particles which are difficultly sheared by
dislocation.
(Received December 20, 2004; Accepted March 8, 2005; Published June 15, 2005)
Keywords: aluminum-marnesium alloy, scandium, covalent bond, grain refinement, thermal stability
1.
Introduction
The aluminum alloys containing minor Sc with multiple
excellent properties, such as high toughness, strength, good
weld performance, and corrosion resistance, are used widely
in space flight, automobile, and naval vessel as a new
structural material. With the increase of requirement of space
flight and aviation materials, modifying and developing of
the aluminum alloys containing Sc on microstructure level
have become a focus that materials scientists pay close
attention1–3) to at present.
In recent years Al–Mg–Sc alloy has been studied systematically3–6) by many research groups. The results show that,
when different Sc contents are added, the effect of grain
refinement of casting Al–Mg alloy is different remarkably.
Adding Sc (0:3 mass%) improves the strength, hardness,
and other mechanical properties of casting Al–Mg alloy, and
raises the thermal-stability of the alloy at the same time.7) It is
suggested that the effect of minor Sc addition on property of
alloy has an inherent relation to the strong interaction
between Sc and other atoms in matrix, especially in inner
relation between bond characteristics of Sc and Al atoms.
The Empirical Electronic Theory in Solid (EET),8) which
is established on basis of Pauling’s valence theory9) and the
energy band theory, offers a simple, direct and practical
empirical method-bond length difference (BLD) method to
deal with valence electron structures of complicated system
and has been used in segregation and design in alloys
successfully.10,11) Further more it enables the researcher to
trace the macrophysics properties of the alloy back to its
source on the valence electron structure level and offers
theory guide in deep level in alloy design.12,13) The purpose
of this paper is to reveal the microstructure mechanism of
minor Sc addition to property improvement of the casting Al–
5 mass%Mg alloys on valence electron structure level from
the bonding between the atoms.
*1Corresponding
*2Graduate
aurhor, E-mail: [email protected]
Student, Guangxi University
nA
nD
nC
nE n
B
(a)
(b)
(c)
Fig. 1 Atom arrangement in (a) the pure Al cell, (b) the Al cell with Mg
and (c) the pure Mg cell, where ; Al atom, ; Mg atom. nA , nB , nC , nD
and nE represent the bond of shortest length, second shortest length . . . . . .
in the pure Mg cell, respectively.
2.
Structure Model of Cell
The -Al matrix is a face-center cubic structure in Al–Mg
alloy. After Mg atom is dissolved in Al matrix, Mg
substitutes for Al atoms, and forms an Al–Mg solid solution.
As reference13) points out, when the content of Mg is not very
high, Al–5 mass%Mg solid solution is made up of two kinds
of structure unit cell. One is the Al cell without Mg, the other
is the Al cell containing Mg, as shown in Fig. 1. The pure Mg
metal is the h.c.p. structure with lattice constant14) a ¼
0:32094 nm and c ¼ 0:52056 nm.
The lattice constant a of Al crystal is 0.40496 nm given in
reference.15) Because lattice constant increases 0.0005 nm
when the addition of Mg increases 1%, and the lattice
constant of Al–Mg solid solution containing 5% Mg is
0.40745 nm, thus the lattice constant of the Al cell containing
Mg become 0.41632 nm.13)
When minor Sc are added in Al–Mg alloy, because of the
radius of Sc atom near to that of Mg atom, Sc atom will
substitute for Al atom and lie in face center of Al cell.
According to reference,10) besides pure Al and Al–Mg unit
cell, Al–Mg–Sc solid solution alloy should include other two
kinds of unit cells such as Al–Sc and Al–Mg–Sc unit cells as
well, so the solid solution is composed of these four possible
unit cells by mixing. These cell structures are shown in Fig. 1
and Fig. 2. The Sc atom replaces Al atom in face-center of
cell and make the lattice constant of Al–Mg–Sc cell change,
Atomic Bonding and Properties of Al–Mg–Sc Alloy
1149
Table 1
(a)
(b)
(c)
Fig. 2 Atom arrangement in (a) the Al–Sc cell, (b) the Al–Mg–Sc cell and
(c) the Al3 Sc cell. Where ; Al atom, ; Mg atom, ; Sc atom.
which will bring difficulty to use BLD analysis. Since
contents of Sc in the alloy are little, so here lattice constant of
Al–Sc cell is replaced by that of pure Al cell, also the lattice
constant of Al–Mg–Sc cell is replaced by that of Al–Mg cell
when such problem is handled. The influence of Sc on lattice
constant, that is to say the change of bond lengths caused by
Sc substituting for Al atom, can be reflected by the change of
atom-hybrid-state of Sc.10)
When Sc is added in Al–Mg alloy, the primary Al3 Sc
particles precipitate firstly in higher temperature before Al–
Mg alloy solidifies. Besides Al–Mg and Al–Mg–Sc cell, the
solid solution should include the Al3 Sc particles with L12
structure shown in Fig. 2. The lattice constant of the Al3 Sc
particle is 0.4106 nm.16)
3.
Calculation method
On the basis of the combination of the energy band theory
and Pauling’s metal bond theory, Yu8) established the EET
theory and its BLD method. The electron structures are
calculated with EET and BLD in this article, so here we
briefly introduce the fundamental hypotheses of EET in
Appendix.
3.1 Atom states of Al atoms
According to the fundamental hypothesis 1 in Appendix,
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
some conditions, the s electron is considered to be the lattice
electron to form the h state from the ground state, while the p
electron is considered to be the electron to form covalence
bond in order to ensure the one-covalence state. Let one
lattice electron be denoted by , the covalent electron by ,
and the empty orbit by ; then, the h state can be represented
as follows:
s
p
s
p
h, state: 3s2 3p1 !
t, state: 3s1 3p2 !
l
2
l0
1
n
1
n0
2
m
0
m0
0
0
0
1
where
represents two lattice electrons. The t state is
selected 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. According to fundamental hypothesis 2 in Appendix, substituting the parameters of the h and t
states of l ¼ 2, n ¼ 1, m ¼ 0, ¼ 0; l0 ¼ 1, n0 ¼ 2, m0 ¼ 0,
Hybridization Table of Al atom.
1
2
3
4
5
6
Ch
1
0.9835
0.9135
0.2352
0.0515
0
Ct
0
0.0165
0.0867
0.7648
0.9485
1
nT
3
3
3
3
3
3
nl
nc
2
1
1.9670
1.0330
1.8266
1.1734
0.4704
2.5296
0.1030
2.8970
0
3
0.1190
0.1190
0.1190
0.1190
0.1190
0.1190
R ð1Þ(nm)
0 ¼ 1 into formulas (A1) and (A2), we obtain six different k
values, which means there are six levels ().
The C1 values of these levels are 0, 0.0165, 0.0867,
0.7648, 0.9485, and 1, respectively. The nT , nc , and nl of all can be calculated with formulas (A3) through (A5). For s–p
hybridization, Rð1Þh ¼ Rð1Þt , so the Rð1Þ of all hybrid levels
is constant. Yu has given the bond radii of the elements in the
IVA group as follows: Rð1ÞB ¼ 0:0768 nm, Rð1ÞAl ¼
0:1190 nm, Rð1ÞGa ¼ 0:1226 nm, Rð1ÞIn ¼ 0:1442 nm, and
Rð1ÞTl ¼ 0:1460 nm. Collecting all the parameters of Al
atom, we obtain the hybridization table of Al, shown in
Table 1.
3.2 Electron structure of Al metal
The valence electron structure of the metal means the
states of the atoms that form the alloy and the electron
distribution of the covalent bonds formed by these atoms.
To any structure that is formed by covalent combination,
let the bond order of the shortest bond ¼ A, and the bond
orders of the other bonds 0 ¼ B, C, D . . . N. According the
equation Dn uv ¼ Ru ð1Þ þ Rv ð1Þ lg n given in fundamental hypothesis 3 in Appendix, the equations of all valence
bonds can be obtained.
DnA uv ¼ Ru ð1Þ þ Rv ð1Þ lg nA
Dn0 uv ¼ Ru ð1Þ þ Rv ð1Þ lg n0
ð1Þ
ð2Þ
Take the BLD treatment:
st
u
v
s
0
Duv
nA Dn0 ¼ R ð1Þ þ R ð1Þ R ð1Þ R ð1Þ þ lg
Let lg r0 ¼ lgðn0 =nA Þ; then,
n 0
st
s
¼ lg r0 ¼ ½Dnv
lg
nA Dn0 þ R ð1Þ
nA
þ Rl ð1Þ R00 ð1Þ Rv ð1Þ=
n 0
ð3Þ
nA
ð4Þ
Because all bond lengths between atoms connected by
covalent bond should obey the bond-length equation, the
ratio between the number of covalent electron pairs on
various bonds, r0 , can be obtained by the BLD. Furthermore,
if the atom states given according to the fundamental
hypothesis 1 in Appendix are correct, all covalent bond
lengths obtained from the atom state parameter Rð1Þ should
accord with the corresponding experimental lengths. Therefore, r0 can be calculated from the experiment bond lengths
and state parameters Rð1Þ .
Consulting Fig. 1, we calculate the experimental bond
lengths of the bonds in the Al lattice according to its lattice
constant, here we only consider the nearest and second
nearest neighbor atom bonds.
1150
G. Yingjun, H. Chuanggao, H. Xianhua, M. Qifeng and L. Hui
DAl{Al
nA
pffiffiffi
2
a and
¼
2
Table 2 Atomic bonding in pure Al metal.
DAl{Al
nB
¼ a:
ð5Þ
The following bong-length equations are given according
to the fundamental hypothesis 3 in Appendix:
DAl{Al
¼ RAl ð1Þ þ RAl ð1Þ lg nA
nA
DAl{Al
¼ RAl ð1Þ þ RAl ð1Þ lg nB
nB
ð6Þ
Treating the equation set (6) according to eq. (4) and
substituting the Rð1Þ of all levels in Table 1 and the
experimental bond lengths in equation set (5) into it, we
obtain rA ¼ 1:0000, rB ¼ 0:02157. Up to now, n0 ¼ nA r0
has been calculated, where 0 ¼ B; C; D . . . N.
Because the considered crystal structure unit is electric
neutral, the total covalentPelectron number provided by all
atoms in a structure unit, j nc j , should equal the sum of the
number
on all covalent bonds in the structure unit,
P
n
I
r
A ; i.e.,
X
X
X
nc j ¼
n A I r ¼ nA
I r
ð7Þ
j
Where I represents the equivalent bond number of a bond
with the bond order of ; its value can be calculated as
follows:
I ¼ IM IS IK
ð8Þ
Where IM represents the reference atom number in a
structure; IS represents the equivalent bond number for a
reference atom to form bonds with order; IK is a parameter,
which equals 1 when the two atoms that form bond are the
same or 2 when the two atoms are different.
For Al metal, there are
IA ¼
A
IM
ISA
IKA
ð4 3mÞorder
24
1¼1
1 ¼ 12
¼1
ðmÞorder
2
B
IB ¼ IM
ISB IKB ¼ 1 ð4 3mÞorder
24
1¼1
1¼6
ð2mÞorder
4
Substituting these parameters into eq. (7), we obtain
X
nc j
nA ¼
j
12 þ 6rB
ð9Þ
As obtained from the fundamental hypothesis 1 in
Appendix and shown in Table 1, the Al atom has six hybrid
levels, so nc has six different values. Substituting them into
eq. (9) we obtain the nA and n0 r0 of Al at all six levels.
For a given structure, if the calculated atom states of the
consisting atoms are correct, the theoretical bond lengths D n
determined by atom state parameters nc and Rð1Þ should all
accord with the experimental bond lengths Dn . Therefore,
D n by comparing the theoretical values calculated from
certain atom state with the experimental one of all covalent
bonds in a structure unit, we can determine if the given atom
state accord with the actual states. To determine if the
theoretical bond lengths D n , accord with the experimental
one Dn , quantitatively, Yu suggested that the absolute value
of their different should be less than 0.005 nm;
Al a ¼ 0:40496 nm Rð1Þ ¼ 0:1190 nm ¼ 0:0710 ¼ 4
nc ¼ 2:5296
bond
I
Dn
(nm)
D n
(nm)
n
jDj
(nm)
DMg{Mg
nA
12
0.28635
0.28633
0.20857
0.00002
DMg{Mg
nB
6
0.40496
0.40494
0.00445
0.00002
Dn ¼ jD n Dn j < 0:005 nm
ð10Þ
Substituting the nA and n0 obtained previously and all Rð1Þ in
Table 1 into equation set (6) and comparing the calculated
D n with Dn in eq. (5), we find that there is one atom state of
Al in Al metal satisfying eq. (10), i.e., is at the fourth level.
The electron structure parameters at the state are shown in
Table 2.
3.3 Electron structure of Mg metal
According to the fundamental hypothesis 1 in Appendix,
the electron configuration relating to ground state of element
Mg in the IIA group is 3s2 3p0 , which is a two-s electron state.
Because of the metallic properties of the elements under
some conditions, the s electron is considered to be the lattice
electron to form the h state in the ground state. Let two lattice
electron be denoted by , and the empty orbit by ; then, the
h state can be represented as follows:
s
p
s
p
l
2
l0
1
h state: 3s2 3p0 !
t state: 3s1 3p1 !
n
0
n0
1
m
0
m0
0
0
0
1
The t state is selected considering the two-covalence state,
which often appears in solids and molecules. The related
electron configuration is 3s1 3p1 . To ensure the two-covalence state, all of them should be covalent electrons. So the t
state can be represented as above. Following the procedure
stated above, we can obtain hybridization table of Mg and
atomic bonding of Mg in metal, shown in Table 3 and
Table 4.
3.4 Electronic structures of Al–Mg–Sc alloys
Following above the procedure, the electronic structure of
the Al–Mg, Al–Sc, Al–Mg–Sc, Al3 Sc cells can be calculated
one by one with BLD-method8) and BLD-criterion,8) and the
detail procedure has been shown in the references.17,18) The
results of the atomic bonding of each cell show thorough
Table 3
Hybridization Table of Mg atom.
1
2
3
4
k2
1
2.73205
0.73205
0
Ch
Ct
1
0
0.88185
0.11815
0.3489
0.6511
0
1
nT
2
2
2
2
nl
2
1.7637
0.6978
0
nc
0
0.2363
1.3022
2
0.12758
0.12730
0.12580
0.12521
R ð1Þ(nm)
Atomic Bonding and Properties of Al–Mg–Sc Alloy
Table 4 Atomic bonding in pure Mg metal.
Table 8
1151
Atomic bonding in Al–Mg–Sc cell.
Mg: a ¼ 0:32094 nm c ¼ 0:52056 nm Rð1Þ ¼ 0:12580 nm ¼ 0:0710
Al–Mg–Sc a ¼ 0:41632 nm ¼ 0:0710
¼ 3 nc ¼ 1:3022
Al: Rð1Þ ¼ 0:1190 nm Al ¼ 5 nc ¼ 2:897, Mg: Rð1Þ ¼ 0:12580 nm
Mg ¼ 3 nc ¼ 1:3022, Sc: Rð1Þ ¼ 0:12785 nm Sc ¼ 2 nc ¼ 2:0365,
bond
DMg{Mg
nA
DMg{Mg
nB
DMg{Mg
nC
DMg{Mg
nD
DMg{Mg
nE
Dn
(nm)
D (nm)
n
6
0.319 50
0.319 59
0.110 25
0.0009
6
0.320 94
0.321 03
0.105 21
0.0009
DMg{Sc
NA
8
0.29438
0.29408
0.24016
0.0030
6
2
0.452 86
0.520 56
0.452 95
0.520 65
0.001 45
0.000 16
0.0009
0.0009
DAl{Sc
NB
DAl{Mg
NC
16
16
0.29438
0.29438
0.29408
0.2.9408
0.19263
0.18024
0.0030
0.0030
6
0.555 89
0.555 98
0.000 05
0.0009
DAl-Al
ND
8
0.29438
0.29408
0.14457
0.0030
DSc{Sc
NE
6
0.41632
0.41602
0.00492
0.0030
DMg{Mg
Nf
6
0.41632
0.41602
0.00277
0.0030
DAl{Al
Ng
12
0.41632
0.41602
0.00430
0.0030
I
Table 5
jDj
(nm)
Atomic bonding in Al–Mg cell.
bond
I
D n
(nm)
Dn
(nm)
jDj
(nm)
n
Al–Mg a ¼ 0:41632 nm ¼ 0:0710
Al: Rð1Þ ¼ 0:1190 nm Al ¼ 4 nc ¼ 2:5296 Mg: Rð1Þ ¼ 0:12580 nm
Mg ¼ 3 nc ¼ 1:3022
D n
(nm)
n
jDj
(nm)
bond
I
Dn
(nm)
DAl{Mg
nA
24
0.29438
0.29387
0.20359
0.00050
Table 9 The Strongest covalent bonds of Al, Al–Sc, Al–Mg, Al–Mg–Sc
and Al3 Sc cells in Al–Mg–Sc alloy.18Þ
Unit cell
Atomic hybridization state
Al
Mg
Sc
Strongest
covalent bond
nA
jDj
(nm)
DAl{Al
nB
24
0.29438
0.29387
0.16330
0.00050
DAl{Mg
nC
Al
4
/
/
Al–Al
6
0.41632
0.41581
0.00486
0.00050
DAl{Al
nD
Al–Mg
4
3
/
Al–Mg
0.20359 0.00050
18
0.41632
0.41581
0.00313
0.00050
Al–Sc
5
/
2
Al–Sc
0.25258 0.00290
Al–Mg–Sc
Al3 Sc
5
5
3
/
2
3
Mg–Sc
Al–Sc
0.24016 0.00320
0.26309 0.00025
Table 6
Atomic bonding in Al–Sc cell.
0.20857 0.00002
Al–Sc a ¼ 0:40496 nm ¼ 0:0710
Al: Rð1Þ ¼ 0:1190 nm Al ¼ 5 nc ¼ 2:8970 Sc: Rð1Þ ¼ 0:12785 nm
Sc ¼ 2 nc ¼ 2:0365
bond
I
Dn
(nm)
D n
(nm)
n
jDj
(nm)
DAl{Sc
nA
24
0.28635
0.28928
0.25258
0.00293
DAl{Al
nB
DSc{Sc
nC
24
6
0.28635
0.40496
0.28928
0.40789
0.18956
0.00718
0.00293
0.00293
DAl{Al
nD
18
0.40496
0.40789
0.00404
0.00293
Table 7
Atomic bonding in Al3 Sc cell.
Al3 Sc a ¼ 0:4106 nm ¼ 0:0710
Al: Rð1Þ ¼ 0:1190 nm Al ¼ 5 nc ¼ 2:8970 Sc: Rð1Þ ¼ 0:13042 nm
Sc ¼ 3 nc ¼ 2:0958
bond
I
Dn
(nm)
D n
(nm)
n
jDj
(nm)
DAl{Sc
nA
DAl{Al
nB
24
24
0.29034
0.29034
0.29059
0.29059
0.26309
0.18166
0.00025
0.00025
DSc{Sc
nC
6
0.41060
0.41085
0.00771
0.00025
DAl{Al
nD
18
0.41060
0.41085
0.00367
0.00025
Table 5 to Table 8. The is determined according to the
eq. (A·8) in Appendix. The results of all the strongest bonds
in each cell with different f.c.c structure from Table 5 to
Table 8 are collected in Table 9.
4.
Analysis and Discussion
4.1 Effect of Sc added in the Al–Mg alloy
The calculated results in Table 9 show that the Al–Sc bond
and Mg–Sc bond of each cell with Sc is stronger than the Al–
Al bond and Al–Mg bond of each cell without Sc, and the
strongest bond nA among five different f.c.c. unit cells in
Table 9 is the Al–Sc bond in Al3 Sc, which value nA is
0.26309. This indicates that coalescent incline of Al and Sc is
strongest, and covalent bond with p-d electron between Al
and Sc atoms is easily formed. The Al–Sc phase diagram16)
shows the precipitated temperature of Al3 Sc in an Al–Sc
alloy is 930 K, which is much higher than the solidification
temperature (830 K) of the Al–Mg alloy. Therefore, some Al
and Sc atoms are combined to form primary Al3 Sc particles
before the Al–Mg alloy solidifies. These primary Al3 Sc
particles with L12 crystal structure are coherent with matrix,
act as the heterogeneous nucleation, are easily formed in Al–
Mg alloy during solidification because interface energy19)
between Al3 Sc and matrix is very low. The limited solid
solubility16) of Sc in rich Al alloy is 0.38 mass% Sc at the
eutectic temperature (933 K) and so the number of the
primary Al3 Sc particles precipitated from the melting are
very small when the contents of Sc are less than 0.38 mass%.
In this case, the Sc atoms mainly exist in Al–Mg matrix in the
form of solid solution, not in that form of Al3 Sc, so the effect
of grain refinement of Sc in Al–Mg alloy is not notable. On
the contrary, when the contents of Sc are more than
0.38 mass%, because of the strong interaction between Al
and Sc, quite a lot of Sc atoms are firstly precipitated in the
form of Al3 Sc, then grow up to coarse primary Al3 Sc
particles in the melting. These great amount of primary Al3 Sc
particles become the heterogeneous nucleation of Al–Mg
alloy during solidification, and play a great role in grain
refinement of alloy. The experimental results of grain
refinement in reference4,5) can be well explained in terms
1152
G. Yingjun, H. Chuanggao, H. Xianhua, M. Qifeng and L. Hui
of stronger Al–Sc and Mg–Sc bonds (seeing in Table 9) and
the Al–Sc or Mg–Sc segregation regions in the early stages of
Al3 Sc precipitation. When the Sc contents in solid solution
are rich enough to form the Al–Sc and Mg–Sc segregation
regions with the strong Al–Sc and Mg–Sc bonds in which
precipitate great amount of Al3 Sc clusters (particles) and
Al3 Sc clusters with Mg in its center, to act as the heterogeneous nucleation, then the grain refinement in matrix is more
notable.
4.2 The bond distribution and properties
The strongest covalent bond is the Al–Sc bond in Al–Sc
cell with nA ¼ 0:25258, known from Table 9 in which the
strongest covalent bonds of each segregation cell containing
Sc are given. The strongest covalent bond in Al–Mg–Sc cell
is Mg–Sc bond with nA ¼ 0:24016. The above two strongest
covalent bonds, Al–Sc bond and Mg–Sc bond, are much
stronger than the covalent bonds in Al cell and Al–Mg cell.
Therefore in the view of the microstructure of solid solution,
the Al cell containing Sc and the Al–Mg cell containing Sc
don’t disperse in the form of single cell in the solid solution,
but do in the form of Al–Sc and Mg–Sc segregation regions
larger than some cell scales, and how large the scale of the
segregation regions depends on the strong Al–Sc bond and
Mg–Sc bond in the matrix. These Al–Sc and Mg–Sc
segregation regions with stronger Al–Sc bond and Mg–Sc
bond are the Al–Sc and Al–Mg–Sc atom clusters, while is
probably the embryo structure20) of Al3 Sc or Al3 Sc with Mg
in its center.21) That is why the Mg–Sc bond plays an
important role21) in early stage of Al3 Sc precipitation.
Some Sc atoms precipitate as coarse primary Al3 Sc
particles from the melting during solidification, the other
Sc atoms exist mainly in the form of solid solution in the
matrix of Al–Mg–Sc alloy. The saturate solubility of Sc
decreases when the solidification temperature of the alloy
declines, which make the alloy become supersaturated Sc
solid solution. In the following-up course of homogeneous
treatment, tiny second Al3 Sc particles are easily precipitated
homogeneously and dispersed in the Al–Sc segregation
region. Because these tiny Al3 Sc particles are completely
coherent with matrix and its grain boundary have excellent
lattice matching relation with matrix, so in the bond of view
the covalent bond net of Al3 Sc and the covalent bond net of
matrix (with pure Al cell) are well combinative in interphase
boundary. Likewise, these Al3 Sc particles which has higher
melting point and better thermal-stability, attributing to its
strong Al–Sc covalent bond, can make the benefit of stability
for grain boundary, restrain the transfer of grain boundary
under high temperature, hinder the recrystallization of the
alloy. That is why the strongest Al–Sc covalent bond in tiny
Al3 Sc particle plays an important role in pinning of grain
boundary20) to prevent recrystallization till higher temperature. When the dispersed tiny Al3 Sc particle grow up in
Orowan’s coarse stage, and become large Al3 Sc particles
dispersed in matrix with greater regions of strong Al–Sc bond
net, then the Al3 Sc particles are more difficultly sheared by
dislocation. Therefore the large Al3 Sc particle can improve
strength and hardness of the alloy. Further investigation
shows when Sc and Zr added simultaneously in Al–Mg alloy
to form Al3 (Sc, Zr) particles, the recrystallization temper-
ature of the alloy is enhanced by more stronger Al–Zr
bond.18)
5.
Conclusions
(1) Because the Al and Sc atoms are easier to combine and
form the strong Al–Sc bond, the primary Al3 Sc
particles are formed firstly in the casting Al–Mg–Sc
alloy when solidifying.
(2) Because Al–Sc bond and Mg–Sc bond were much
stronger, Al–Sc and Mg–Sc segregation regions are
easy to be formed, and become the embryo of secondary
Al3 Sc particles or Al3 Sc with Mg in its center, during
the following-up homogeneous treatment course, such
as aging.
(3) Since the secondary tiny Al3 Sc particles with strong
Al–Sc covalent bonds are coherent with the matrix, the
strength of covalent bond in the matrix would be
improved by the precipitation, and the recrystallization
of the alloy are hindered under high temperature.
(4) The reason that the Al3 Sc particle can improve strength
and hardness of the alloy is the larger particles with the
great regions of strong Al–Sc bond net are difficultly
sheared by dislocation.
ACKNOWLEDGMENT
This work was financially supported by the Natural
Science Foundation of China under Project number
50061001.
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1153
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
given with the follow formulas:
nT ¼ nTh Ch þ nTt Ct
Appendix
¼ ðl þ m þ nÞCh þ ðl0 þ m0 þ n0 ÞCt
Fundamental hypothesis in the empirical electron theory
of solid and molecule
nl ¼ nlh Ch þ nlt Ct
A. 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 number nT , covalent electron number nc ,
lattice electron number nl , and bond radius Rð1Þ. Here, lattice
electron is a new concept, which means the valence electron,
which is in the space enclosed by two or more atoms in a
solid system formed by many atoms.
nc ¼ nch Ch þ nct Ct
B. Fundamental hypothesis 2
Normally, the hybridization of states is not continuous. Let
Ct represent the content of the t states in the hybrid state Ct
and Ch represent that of the h state. In most structures, Ct and
Ch can be given with the following formulas:
Ct ¼
1
;
1 þ k2
Ct þ Ch ¼ 1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l0 þ n0 þ m0
l 0 þ n0 þ m 0
k¼
l þ n þ m
lþnþm
pffiffiffiffiffi pffiffiffiffiffiffi
l 5n 3m
pffiffiffiffiffiffi pffiffiffiffiffiffiffi :
l0 5n0 3m0
ðA:1Þ
ðA:2Þ
Which l; n; m and l0 ; n0 ; m0 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 0
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 ¼ 1 and k ¼ 0
¼ ð1 ÞlCh þ ð1 þ 0 Þl0 Ct
¼ ðl þ m þ nÞCh þ ð0 l0 þ m0 þ n0 ÞCt
Rð1Þ ¼ Rð1Þh Ch þ Rð1Þt Ct
ðA:3Þ
ðA:4Þ
ðA:5Þ
ðA:6Þ
Where Ch and Ct represent the contents of the h and t states
at hybrid level, respectively; Rð1Þh and Rð1Þt represent the
bond radii of them, respectively.
C. 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 represented by Duv
n , according to
Pauling’s research, the following relation exists among Duv
n ,
Ru ð1Þ, Rv ð1Þ and n :
u
v
Duv
n ¼ R ð1Þ þ R ð1Þ lg n
ðA:7Þ
Where u and v can be the same kind or different kind of
atoms; n can be an integer or a fraction; and ¼
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 of
the largest nM
in the structure. The value of should observe
the following rule:
8
< 0710
¼ 0:600
ðA:8Þ
when 0:300 nM
0:700
:
0:710 2:2"
where 0 < " < 0:050.