Int. J. Communications, Network and System Sciences, 2013, 6, 415-442
http://dx.doi.org/10.4236/ijcns.2013.610045 Published Online October 2013 (http://www.scirp.org/journal/ijcns)
Copyright © 2013 SciRes. IJCNS
Alloy Gene Gibbs Energy Partition Function and
Equilibrium Holographic Network Phase Diagrams of
AuCu-Type Sublattice System
Youqing Xie1,2,3*, Xiaobo Li4, Xinbi Liu1,2,3, Yaozhuang Nie5, Hongjian Peng6
1School of Materials Science and Engineering, Central South University, Changsha, China
2Powder Metallurgy Research Institute, Central South University, Changsha, China
3State Key Laboratory of Powder Metallurgy, Central South University, Changsha, China
4College of Mechanical Engineering, Xiangtan University, Xiangtan, China
5School of Physical Science and Technique, Central South University, Changsha, China
6School of Chemistry and Chemical Engineering, Central South University, Changsha, China
Email: *xieyouq8088@163.com
Received August 1, 2013; revised September 3, 2013; accepted September 14, 2013
Copyright © 2013 Youqing Xie et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Taking AuCu-sublattice system as an example, we present two discoveries and a method. First, the alloy gene se-
quences are the central characteristic atom sequences in the basic coordination cluster sequences. Second, the transmis-
sion mode of the information about structures and properties of the alloy genes is described by the alloy gene Gibbs
energy partition function. The most valuable method in the system sciences is “the whole obtained from a few parts”.
We have established the alloy gene database and holographic alloy positioning system of the Au-Cu system, as well as
alloy gene Gibbs energy partition function and equilibrium holographic network phase diagrams of the AuCu-type
sublattice system. It means that a standard way for researchers to share predictive algorithms and computational meth-
ods may be produced during designing advanced alloys.
Keywords: Alloy Gene Gibbs Energy Partition Function; Holographic Alloy Positioning System; Equilibrium
Holographic Network Phase Diagrams; Systematic Metal Materials Science; Au-Cu System
1. Introduction
There are 81 kinds of metal elements accounting for 79%
in the Element Periodic Table, of which several alloy
system groups can be composed: 2
81 3240C kinds of
binary alloy systems, 3
81 85320C kinds of ternary al-
loy systems, 4
81 1663740C kinds of quaternary alloy
systems, and so on. In order to quickly and efficiently
discover advanced alloys, we have to establish the Sys-
tematic Metal Materials Science (SMMS) by new think-
ing modes and methods of system sciences [1] (Supple-
mentary Section 1).
“A diversity of structures of a system is attributed to
combination and arrangement of structural units in the
basic structure unit sequences”. It is the first philosophic
proposition of system sciences. Now, we deliver an ex-
tensive definition of the gene sequence: the gene se-
quence is a basic structure unit sequence carrying a set of
transmission information about structures and properties
for determining diversity of structures and properties of a
system, which may be a biologic or non-biologic system.
Therefore, it is our first task to seek alloy gene (AG) se-
quences for establishing the SMMS framework.
We have discovered that the AG-sequences are the
central characteristic atom sequences in the basic coor-
dination cluster sequences (Supplementary Figure S1(c)).
In the fcc-based lattice Au-Cu system, the basic coordi-
nation clusters

Au AuAu Cu
ii
BA Iii 


and
Cu CuAuCu
ii
BA Iii 
consist of the central cha-
racteristic atoms Au
i
A
and Cu
i
A
respectively, and the
nearest coordinative configuration

Au CuIi i


.
Here, I is the coordinative number and equal to 12; i is
the number of Cu-atoms and can change from 0 to 12;
(I-i) is the number of Au-atoms. Therefore, the AG-se-
quences of the Au-Cu system are the central characteris-
tic atom Au
i
A
- and Cu
i
A
-sequences respectively in the
Au
i
B- and Cu
i
B-sequences. They may be used to replace
*Corresponding author.
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
416
atomic pair-sequence [2] and atomic cluster-sequences
[3,4], as well as other structural unit sequences [5-7] in
the current alloy theories, because they have more ana-
logous characters to biologic gene sequences: (1) The
AG Au
i
A
- and Cu
i
A
-sequences carry a set of transmis-
sion information: coordinative configurations, electronic
structures, physical and thermodynamic properties, which
have been obtained by the AG-theory (Supplementary
Section 1). (2) Based on the AG-information, a Holo-
graphic Alloy Positioning (HAP) system of the Au-Cu
system has been established by “the Whole Obtained
from a few Parts” (WOP) method (Section 2). (3) AG-
information may be transmitted to alloy phases by AG-
Gibbs energy partition function (Section 3). The Equilib-
rium Holographic Network Phase (EHNP) diagrams of
the AuCu-type sublattice system have been obtained by
the HAP system (Section 4). (4) Alloy genes may be in-
terconvertible, the essence of the orderdisorder transi-
tion can be explained by the AG-concentrations distribu-
tions (Figure 4). (5) Alloy genes will be used to design
alloys, which may be called as the AG-arranging design.
2. Holographic Alloy Positioning System
The main objective of the HAP system of an alloy sys-
tem is to provide a set of EHNP diagrams for designing
advanced alloys. The HAP system of the Au-Cu system
consists of AG-theory, AG-Gibbs energy partition func-
tions of Au3Cu-, AuCu- and AuCu3-type sublattice sys-
tems and balance theory between sublattice systems
(Figure 1). Its performing procedures are as follows: (1)
The first step is to establish AG-database obtained by the
AG-theory, only using experimental mixed enthalpies
Figure 1. Holographic alloy positioning system of Au-Cu
system.
exp
m
H
T and mixed volumes


exp
m
VT of the
AuCu and AuCu3 compounds [8,9] (Supplementary
Figures S2 and S5). (2) The second step is respectively
to establish EHNP diagrams of the Au3Cu-, AuCu- and
AuCu3-type sublattice systems, according to the AG-
Gibbs energy partition functions of Au3Cu-, AuCu- and
AuCu3-type sublattice systems and the Bragg-Williams
sublattice models [10,11]. (3) The third step is to es-
tablish EHNP diagrams of the Au-Cu system based on
the balance theory between sublattice systems. So much
information is obtained by computing technique and a
few experimental data, i.e., WOP method (Supplemen-
tary Section 1.5).
3. AG-Gibbs Energy Partition Function
“A diversity of properties of a system is attributed to
contents and transmission mode of the information of
basic structure unit sequences”. It is the second philoso-
phic proposition of system sciences. The transmission
mode of AG-information is described by AG-Gibbs en-
ergy
,
x
T-partition function, which consists of the
AG-Gibbs energy transmission

*,GxT-function and
AG-arranging structure
 

Au Cu
,, ,
ii
g
xxTxxT-func-
tion (Supplementary Section 2).
An equilibrium orderdisorder transition of the
AG-arranging structure of an alloy is defined as that “the
AG-Gibbs energy levels
 
Au Cu
,
ii
GTGT and AG-
concentrations

Au Cu
,, ,
ii
x
xT xxT occupied at the
Au
i
GT- and
Cu
i
GT-energy levels can respond im-
mediately and change synchronously with each small
variation in temperature”. It has following general be-
haviors: (1) The order disorder transition upon heating
and disorder order transition upon cooling proceed
along the same minimal mixed Gibbs energy min
m
GT
path, i.e., there exists no a so-called hysteresis pheno-
menon between both transitions. (2) The min
m
GT
path is continuous and has no jumping phenomenon.
4. EHNP-Diagrams of AuCu-Type Sublattice
System
“Properties are determined by structures; properties should
be suitable for environments; environments change struc-
tures”. It is the fourth philosophic proposition of system
sciences (the third one is presented in Supplementary
Section 1.2). The man’s knowledge of relationships of
structures, properties and temperature for alloys has been
changed from single causality to systematic correlativity,
due to discoveries of alloy gene sequences and their in-
formation transmission mode, and establishments of the
AG-Gibbs energy partition function and HAP system.
This systematic correlativity may be described by a set of
HNP diagrams.
An EHNP diagram consists of curves linked by the
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
417
network

,,qxT -points as functions of the state q,
composition x and temperature T. Each position inside
the phase region and on the phase boundary (PB) curve
may be marked by the

,,qxT -point, where the alloy
state q denotes the AG-concentrations (Au
i
x
and Cu
i
x
),
mixed Gibbs energy m
G, order degree
, configura-
tional entropy c
S, mixed characteristic Gibbs energy
*m
G, mixed enthalpy m
H
, mixed potential energy
m
E, mixed volume m
V, generalized vibration free
energy v
X
, generalized vibration energy v
U, general-
ized vibration entropy v
S, mixed heat capacity m
p
C,
mixed thermal expansion coefficient m
and activi-
ties (Au
a and Cu
a). Each kind of the q-EHNP diagram
includes four diagrams: three-dimension qxT phase
diagram, two-dimension x
qT
, q
Tx and T
qx
path phase diagrams. In the text the
,
m
GxT,

,
x
T
,

,
c
SxT,

Au ,
i
x
xT and

Cu ,
i
x
xT EHNP
diagrams are presented. The other EHNP diagrams are
shown in Supplementary Figure 6. The calculations of
all diagrams are performed by compositional step x
=
0.5at.% and temperature step 1KT
. These diagrams
are interconnected. Therefore, a set of information about
structures, properties and their variations with tempera-
ture of a designed alloy in the equilibrium state can be pre-
dicated by interlinking method and without calculation.
4.1. Mixed Gibbs Energy and Order Degree
EHNP Diagrams
Using the minimal mixed Gibbs energy path method, we
have obtained the three-dimension m
GxT and
x
T
 EHNP diagrams, and two-dimension
m
x
GT
, xT
, m
G
Tx
, Tx
, m
T
Gx
and
T
x
path phase diagrams. From these diagrams, we
have obtained following main understandings: (1) From
Figures 2(a) and (a’), it can be known that there are or-
dered phase region (denoted by the symbol “O”), disor-
dered phase region (denoted by the symbol “D”), AuCu
compound consisted of Au
8
A
and Cu
4
A
genes (denoted
by the symbol “C”), and the

,
m
PB
GxT- and

,
PB
x
T
-phase boundary curves in the each diagram.
(2) From Figures 2(b) and (b’ ), it can be known that the
Au47.5%Cu52.5% alloy has the highest critical temperature
836 K
c
T, which is higher than the critical temperature
(833K) of the Au50%Cu50% alloy , that agrees with the
experimental phenomenon (see Figure 2(c’)). (3) From
Figures 2(c) and (c’), it can be known that the phase
diagrams can be respectively divided into the several
regions by some m
G
Tx
and Tx
curves. It is very
useful to enrich the EHNP diagrams. (4) From Figure
2(c’), we have discovered that all experimental jumping
temperatures

j
T, which are denoted by symbols
”[12] and “” [13], approach to the iso-order
0.65
Tx
curve. The experimental j
Tx curve is
very analogous to the equilibrium PB
Tx curve, i.e.,
c
Tx
curve. However, the experimental jumping long-
range order degree
L
R
j
of the AuCu compound is
about 0.8, and the corresponding short-range order de-
grees
SR
is about 0.40. These phenomena show that
the experimental order disorder transition belongs in
metastable and non-equilibrium. (5) The isocomposi-
tional m
x
GT
and xT
path diagrams (Figures
2(d) and (d’)) will be used to establish EHNP diagrams
of the Au-Cu system.
4.2. Configuration Entropy and AG-
Concentration EHNP Diagrams
The configuration entropy


,
c
SxT and AG-con-
centration
Au Cu
,, ,
ii
x
xT xxT EHNP diagrams have
been established, based on the AG-arranging structure
Au Cu
,, ,
ii
g
xxTxxT-function in the

,
x
T-func-
tion. From Figures 3 and 4, it can be known that each
kind of q-EHNP diagram includes not only the 4-type
diagrams indicated above, but also other–type diagrams.
From Figure 3, we have obtained following main under-
standings: (1) The configuration entropy of each ordered
alloy can change continually from the configuration en-
tropy of the maximum order degree max
state to one of
the ideal disordered state, that need not to induct any
parameter. It means that we should take the ideal disor-
dered state as the standard. (2) The structural units used
for calculating configuration entropy should be in agree-
ment with the structural units used for calculating corre-
sponding energy levels. These are two rules to establish
partition function. However, these rules are often ne-
glected in the currently used thermodynamic models of
alloy phases [14].
From Figure 4, we have obtained following main un-
derstandings: (1) The AG-concentration EHNP diagram
may be described by two modes: the Au
i
x
xT and
Cu
i
x
xT
EHNP diagrams (Figures 4(a) and (a’)) in
the AG-arranging crystallography [15], where the Au
i
x
and Cu
i
x are the probabilities occupied at the lattice
points; the Au
i
x
xi
and Cu
i
xi EHNP diagrams
(Figures 4(a), (a’), (c) and (c’)) in the AG-arranging
band theory, where the Au
i
x
and Cu
i
x
are the probabili-
ties occupied at the
Au
i
GT and

Cu
i
GT energy lev-
els. (2) There exists a probability of AG-arranging struc-
tures in the alloy phases, which is described by the AG-
concentration

Au Cu
,, ,
ii
x
xT xxT functions. This
probability leads to atom-scale heterogeneity of proper-
ties of alloys. (3) There exists an emergent phenomenon
of AG-arranging structures in the ordered alloy phases,
which is defined as that some AG-concentrations of the
alloy in ordered state are larger than one in the disor-
dered state, such as Au
9
x
, Au
8
x
, Au
7
x
, Cu
3
x
, Cu
4
x
and
Cu
5
x
(Figures 4(b)-(c’)). (4) Alloy genes may be inter-
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
418
(a) (a’)
(b) (b’)
(c) (c’)
(d) (d’)
Figure 2. Mixed Gibbs energy and order degree EHNP diagrams. (a), (a’), Three-dimension m
GxT
and xT
diagrams with phase boundary curves
m
PB
GxT, and
PB xT
,; (b), (b’), Two-dimension isocompositional m
x
GT
and xT
path diagrams with phase boundary curves
m
PB
GT and
PB T
; (c), (c’), Two-dimension iso-mixed Gibbs
energy m
G
Tx
and iso-order degree Tx
diagrams with phase boundary curves
PB
Tx
; (d), (d’), Two-dimension
isothermal m
T
Gx and Tx
diagrams with phase boundary curves
m
PB
Gx and
PB x
.
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
419
(a) (b)
(c) (d)
Figure 3. Configuration entropy EHNP diagrams of the AuCu-type sublattice system. a) Three-dimension c
xT phase
diagram with phase boundary curve
c
PB
S
xT,; b) Three-dimension c
S
xT
.Au phase diagram with phase boundary
curve

.,
cAu
PB
S
xT of the Au-component; c) Three-dimension .cCu
S
xT
diagram with phase boundary curve
.,
cCu
PB
S
xT
of the Cu-component; d) Three-dimension

..cAu cCu
S
SxT
 combinational phase diagrams with phase boundary curves

.,
cAu
PB
S
xT and

.,
cCu
PB
S
xT of the Au- and Cu-components.
convertible. The essence of the order disorder tran-
sition of the stoichiometric AuCu compound consisting
of Au
8
A
- and Cu
4
A
-alloy genes is that the Au
8
A
- and
Cu
4
A
-alloy genes are split into the Au
i
A
- and Cu
i
A
-se-
quences, respectively; otherwise the essence of the dis-
order order transition of the stoichiometric disordered
AuCu alloy is that the Au
i
A
- and Cu
i
A
-sequences are
degenerated into the Au
8
A
- and Cu
4
A
-alloy genes (Fig-
ures 4(d) and (d’)).
5. Conclusions
In this paper, these creative achievements are of broad
significances: (1) Four philosophic propositions of sys-
tem sciences will prove stimulating to researchers work-
ing on general system sciences, matter systems and social
systems, and who may well find value in using them as a
philosophic thinking logic for establishing various sys-
tem science frameworks. (2) the AG-sequence instead of
compositional atoms, compositional atom pairs and
compositional atom clusters sucessively proposed during
last one hundred years will prove stimulating to re-
searchers working on physics, chemistry and material
science, and who may well find value in using it as basic
structure unit sequence for establish SMMS framework.
(3) The AG-Gibbs energy partition function composed of
AG-Gibbs energy level model of chemical potential and
AG-statistic model of entropy replaces two current func-
tions: one is composed of atom pair energy level model
Y. Q. XIE ET AL.
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420
Figure 4. AG-concentration EHNP diagrams. (a), (a’), Three-dimension i
xxT
Au and i
xxT
Cu diagrams; (b), (b’),
Three-dimension i
xxi
Au and i
xxi
Cu diagrams of the ordered Au(1-x)Cux alloys with maximum order degree
max
at 0K; (c), (c’), Three-dimension i
xxi
Au and i
xxi
Cu diagrams of the perfectly disordered Au(1-x)Cux alloys; (d), (d’),
Three-dimension i
xTi
Au and i
xTi
Cu path diagrams of disordering for the stoichiometric AuCu compound.
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
421
of chemical potential and compositional atom statistic
model of entropy; another one is composed of electron
energy level model of chemical potential and composi-
tional atom cluster model of entropy. That will prove
stimulating to researchers working on physics, chemistry
and materials science, and who may well find value in
using it as a theoretic foundation for establishing alloy
system sciences of binary and ternary alloy systems. (4)
The HAP system based on the AG-Gibbs energy parti-
tion function and the WOP method will prove stimulat-
ing to researchers working on physics, chemistry and
materials science, and who may well find value in using
it as a platform for establishing EHNP diagrams of alloy
systems. (5) The great holographic alloy database con-
sisting of AG-database and EHNP diagrams of alloy sys-
tems will prove stimulating to others working on materi-
als engineering, and who may well find value in using it
as a tool for materials discovery, design, manufacture
and application, which may be called as “AG-Arranging
(AGA) Engineering”. The investigation in AuCu-type
sublattice system may be analogous to Mendel’s investi-
gation in genetic characters of peas.
6. Acknowledgements
The work was supported by the National Natural Science
Foundation of China (51071181); and by the Natural
Science Foundation of Hunan province (2012FJ4044),
China.
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Transactions of Nonferrous Metals Society of China, Vol.
21, No. 8, 2011, pp. 1801-1807.
http://dx.doi.org/10.1016/S1003-6326(11)60934-5
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
423
Supplementary Information
1. Alloy Gene (AG) Theory of Au-Cu System
1.1. New Thinking Modes of System Sciences
‘A diversity of structures of a system is attributed to
combination and arrangement of structure units in the
basic structure unit sequence’. It is the first philosophic
proposition of system sciences proposed by us. For ex-
amples, the diversity of species in the biological system
is attributed to splices of some biological genes in the
biological gene sequences; the diversity of free atom
states in the free atom system is attributed to arrangement
of electrons in the electronic orbital (state) sequence de-
scribed by four quantum numbers; the diversity of crys-
talline species in the geometric crystallography is attrib-
uted to combination of symmetric elements in the sym-
metric element sequence; the diversity of music scores in
the music system is attributed to combination and ar-
rangement of notes in the music note sequence, and so on.
This philosophic proposition guides us to seek the most
important structure units for designing advanced alloys.
They are the central characteristic atoms in the basic co-
ordination clusters, and called as alloy genes (AG).
“A diversity of properties of a system is attributed to
the contents and transmission mode of the information of
the basic structure unit sequences”. It is the second phi-
losophic proposition of system sciences proposed by us.
It is also essential condition to establish a systematic
science theory of a certain system. It is a protracted and
tortuous history to research structures, properties, evolu-
tions and functions of the biologic gene sequences for
establishing biosystematics. It is another good example
to research structures, properties, evolutions and func-
tions of electronic state sequences for establishing sys-
tematic science of free atom system. Through a long-
term exploring we have proposed a fictitious structure
unit (Supplementary Section 1.3), which is the character-
istic crystal consisted of the same alloy genes with
known potential energy and volume, and with the same
based lattice to the alloy system. Then, we have estab-
lished the valence bond theory and thermodynamics of
characteristic crystals. By both these theories, we can
calculate the valence electron structures of characteristic
crystals, which may represent ones of alloy genes, and
the physical and thermodynamic properties of charac-
teristic crystals, which may represent contributions of
alloy genes to properties of alloys. Now, we have dis-
covered that the transmission mode of the AG-in-forma-
tion about structures and properties can be described by
the AG-Gibbs energy partition function.
1.2. Structural Levels of Alloy Systems
“The complexity and entirety of a complex system are
attributed to the multilevel of the structures and proper-
ties and to the various correlatives between different
structural levels, between different properties and be-
tween the structures and properties.” It is the third phi-
losophic proposition of system sciences proposed by us.
The SMMS framework contains three levels: the elec-
tron-structures of atoms, atom- and electron (valence
bond)-structures of phases and phase-structures of or-
ganizations (Supplementary Figure S1(a)), which are
simplified as atom-, phase- and organization-levels, re-
spectively. The traditional crystallography and thermo-
dynamics of alloys contain phase-level and organization-
level. The first-principles electron theory of alloys be-
longs in the electron-structures of phases (Supplemen-
tary Figure S1(b)). Therefore, traditional alloy theories
have no atom-level theory, i.e., AG-theory.
1.3. Structural Unit Sequences of the Au-Cu Al-
loy System
In order to explain diversities of structures and properties
of alloy phases, we proposed three structural unit se-
quences.
Basic coordination cluster B-sequences:
Au Au Au
012
i
BBA ; Cu Cu Cu
012
i
BBB . They are formed
by the central characteristic atoms

Au Cu
,
ii
AA and the
nearest coordinative configuration

12 iAu iCu

.
The alloy phase is formed by the Basic Coordination
Cluster Overlapping (Simplified as BCO model) (Sup-
plementary Figure S1(c)).
Alloy genes A-sequences: Au Au Au
012
i
A
AA;
Cu Cu Cu
012
i
A
AA. They are the central characteristic
atoms of basic coordination clusters. The alloy phase is
formed by the Alloy Gene Arranging (Simplified as
AGA model) (Supplementary Figure S1(c)).
Characteristic crystal C-sequences: Au Au Au
012
i
CCC ;
Cu Cu Cu
012
i
CCC . Each fictitious characteristic crystal
consists of the same characteristic atoms. The alloy phase
is formed by the Characteristic Crystal Mixing (Simpli-
fied as CCM model) (see Supplementary Figure S1(d),
Supplementary Section 1.6).
Recently, we have discovered that the alloy gene se-
quences are very useful to design advanced alloys.
1.4. The Separated Theory of Potential Energies
and Volumes of Alloy Genes
The AG-theory consists of three parts (Supplementary
Figure S2(a)): separated theory of potential energies and
volumes of alloy genes [16,17], valence bond theory
[18,19] and thermodynamics of characteristic crystals
(see Equations (9) to (24)).
The extensive properties

,qxT ,

,
A
qxT,
,
B
qxT functions of a given alloy phase and its com-
ponents can be obtained by a transmission law of the
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
424
Figure S1. Theoretic levels and three structural unit sequences. a) Theoretic levels in SMMS framework; b) Theoretic levels
in the traditional alloy theory; c) Basic coordination cluster sequences and AG-sequences: BBB
Au Au Au
0812
 ,
BBB
Cu CuCu
12 40
 and AAA
Au Au Au
0812
 , AAA
Cu CuCu
12 40
 ; d) The characteristic crystal-sequences: CCC
Au Au Au
0812
 ,
CCC
Cu CuCu
12 40
 .
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
425
Figure S2. AG-theory and AG-properties. a) AG-theory based on the experimental techniques or the first-principles electron
theory of alloys; b), c) AG-potential energiesi
Au - and i
Cu -sequencesi
vAu -and i
vCu -sequences; d), e) Potential energy
curves of alloy genes:
WrWrWr
Au AuAu
0812
 ,
WrWrWr
Cu CuCu
12 40
 ; f), g) Gibbs energy curves:
GT GTGT
Au Au Au
0812
 ,
 
i
GT GT GT
Cu CuCu
12 0
 .
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
426
extensive q-properties of alloy genes, which may be sim-
plified as AGT law:
 
 
 
00
0
0
,, ,
,,
,,
II
AA BB
iiii
ii
IAA
AiiA
i
IBB
BiiB
i
qxTxxTq TxxTq T
qxTxxTqTx
qxTxxTqTx



(1)
where, the q denotes potential energy (E) and volume (v),
which are the functions as temperature (T); the ),(TxxA
i
and

,
B
i
x
xT are the concentrations of the alloy genes;
which are the functions of the composition (x) and tem-
perature (T); A
x
and
B
x
denote compositions of the A
and B components, respectively.
In order to make the Equation (1) become the simple,
applicable and separable

,,qxT
function, three type
relations of A
i
q and
B
i
q with i have been designed:
(i)Type I of linear relation,



00
0
AA AA
iI
B
BBB
iI I
qq iIqq
qq IiIqq
 
 

(2)
(ii)Type II of concave parabola relation,




2
00
2
0
AA AA
iI
B
BBB
iI I
qq qIqq
qq IiIqq
 
 


(3)
(iii)Type III of convex parabola relation,








2
00 0
0
2
0
2
2
AA AAAA
iI I
BB BB
iI I
BB
I
qqiI qqiIqq
qq IiIqq
IiIq q
 
 




(4)
where, the 0
A
q and
B
I
qdenote, respectively, q-proper-
ties of the primary state alloy genes; the A
I
q and 0
B
q
denote, respectively, q-properties of the terminal state
alloy genes.
By combining Equations (2)-(4) and substituting them
into Equation (1), nine
,qxT functions can be ob-
tained. In Supplementary Table S1 the nine general
,qxT -functions of alloy phases can be used to com-
pounds, ordered and disordered alloy phases. In Sup-
plementary Table S2, the nine

,qxT -functions can
be used for the disordered alloy phases only.
1.5. Methodology
The most valuable method in the systematic science is
“the whole obtained from a few parts”, which may be
simplified as the WOP method. It means that a system
can be reproduced from it’s a few parts. Such as the
whole of a tree can be reproduced from a seed, a branch
or even a leaf of this tree. It means also that the whole
information of a system can be obtained from the infor-
Table S1. Nine general
qxT,-functions of alloy phases at T K*.
No. Type
A B Functions
1 I I
  

 

000
00
,
II
AB AAABBB
ABIiIi I
ii
iIi
qxTxqTxqTxqT qTxqT qT
II

 

 
 

2 I
  

 

2
000
00
,
II
AB AAABBB
ABIiIi I
ii
iIi
qxTxqTxqTxqT qTxqT qT
II


 
 

 
 



3
 

  

2
000
2
00
2
,
II
AB AAABBB
ABI iIiI
ii
II iI i
i
qxTxqTxqTxqT qTxqT qT
II


 

 


 


4
  

 

2
000
00
,
II
AB AAABBB
ABIiIi I
ii
iIi
qxTxqTxqTxqT qTxqT qT
II


 
 

 
 



5
  

 

22
000
00
,
II
AB AAABBB
ABIiIi I
ii
iIi
qxTxqTxqTxqT qTxqT qT
II

 
 
 
 
 
 
 
 

6
  

  

2
2
000
2
00
2
,
II
AB AAABBB
ABI iIiI
ii
II iI i
i
qxTxqT xqTxqTqTx qTqT
II


  

 




 
 

7
  

 

2
000
2
00
2
,
II
ABAAA BBB
ABIiIi I
ii
Ii iIi
qxTxqT xqTxqT qTxqT qT
II


 
 

 


8
  

 

2
2
000
2
00
2
,
II
AB AAABBB
ABIiIi I
ii
Ii iIi
qxTxqTxqTxqT qTx qT qT
II



 
 


 
 


9
 

 

2
2
000
22
0
2
2
,
I
AB AAABBB
ABI iIiI
i
II iI i
Ii i
qxTxqTxqTxqT qTx qT qT
II

 

 


 


*: Here, the

,
AA
ii
x
xxT and
,
BB
ii
x
xxT.
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
427
Table S2.
qxT,-functions of disordered alloy phases at T K.
No. Type
A B Functions
1 I
 

 

000
,ABAA BB
ABIABI I
qxTxqTxqTxxqTqTqTqT 
2 ⅠⅡ
 

 

2
000
1
,A BAAABABBB
ABIABII
Ixxxx
qxTxqTxqTxxqTqTqTqT
I

 
3
ⅠⅢ
 

 

2
000
1
,ABAAABAB BB
ABIABII
IxxIxx
qxTxqTxqTxxqTqTqTqT
I

 
4
ⅡⅠ
 
 

 

2
000
1
,AB ABABAA BB
ABIIAB I
Ixxxx
qxTxqTxqTqTqTxxqTqT
I


5
 
 

 

22
000
11
,ABAB ABAAAB ABBB
ABI II
Ixxxx Ixxxx
qxTxqTxqTqTqTqTqT
II
 
  
6
 
 

 

22
000
11
,ABABAB AAABABBB
ABI II
Ixxxx IxxIxx
qxTxq Txq Tq Tq Tq Tq T
II
 
 
7
 
 

 

2
000
1
,ABABAB AABB
ABIIABI
IxxIxx
qxTxqTxqTqTqTxxqTqT
I

 
8
 
 

 

22
000
11
,ABABAB AAABAB BB
ABI II
IxxIxx Ixxxx
qxTxq Txq Tq Tq Tq Tq T
II
 
 
9
 
 

 

2 2
000
11
,ABABABAAAB ABBB
ABI II
I xxIxxIxxIxx
qxTxqTxqTqTqTqTqT
II
 

mation of it’s a few parts.
Due to the discovery of alloy genes and the establish-
ments of holographic alloy positioning (HAP) system
and a big database consisting of the AG-database and the
holographic network phase diagrams of alloy systems, a
standard method for researchers to share predictive algo-
rithms and computational methods may be produced. The
whole information of an alloy system and a designed
alloy may be obtained. This study will create a new way
of intelligent design advanced alloys and lead to funda-
mental variations in the metallic materials science.
1.6. The Choices of Potential Energy E-function
and Volume V-function in the Au-Cu System
The 5th E-function in Supplementary Table S1 has been
chosen for describing Au-Cu system, according to the
calculated values of the mixed enthalpies

max
,0,
m
Hx
, excess potential energies

max
,0,
ex
Ex
and the lowest critical temperatures

c
Tx of AuCu- and AuCu3-type sublattice ordered
Au(1-x)Cux alloys with maximum order degrees, based on
potential energies of the Au
i
A
- and Cu
i
A
-sequences
separated from mixed enthalpies of the AuCu and AuCu3
compounds respectively by nine E-functions at room
temperature ( Supplementary Figure S3).
The 6th V-function in Supplementary Table S1 has
been chosen for describing Au-Cu system, according to
the calculated values of the mixed volumes

max
,,
m
r
vxT
and excess volumes
max
,,
ex
r
vxT
of the AuCu- and
AuCu3-type sublattice ordered Au(1-x)Cux alloys with
maximum order degrees based on volumes of the Au
i
A
-
and Cu
i
A
-sequences separated from experimental vol-
umes of AuCu and AuCu3 compounds respectively by
the 2nd-9th v-functions at room temperature (Supple-
mentary Figure S4).
In the Au-Cu system, the correlative function between
potential energies of alloy genes is,




2
AuAuAu Au
00
2
CuCuCu Cu
0.
iI
iI I
iI
IiI
 
 
 
 


(5)
The potential energy function of alloys is,

 

 

Au Cu
Au 0Cu
2
Au AuAu
0
0
2
Cu CuCu
0
0
,I
I
iI
i
I
iI
i
xTxTxT
ixTT
I
Ii
x
TT
I




















(6)
The correlative function between volumes of alloy
genes is:






2
AuAuAuAu
00
CuCuCu Cu
0
2Cu Cu
0
2
iI
iI I
I
vv iIvv
vv IiIvv
IiI vv
 
 

 


(7)
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
428
Figure S3. Energetic properties and critical temperatures of AuCu- and AuCu3-type sublattice ordered Au(1-x)Cux alloys with
maximal order degrees calculated by nine E-functions. a), b) Mixed enthalpy
m
HxAuCu and

m
Hx3
AuCu , here the ex-
perimental values are denoted by symbol “”; c), d) Excess potential energy differences

ex
E
xAuCu and
ex
E
x3
AuCu rela-
tive to disordered Au(1-x)Cux alloys; e), f) Critical temperature
m
c
Tx
,AuCu and
m
c
Tx
3
,AuCu, here the experimental values are
denoted by symbol “”.
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
429
Figure S4. Volumetric properties of AuCu- and AuCu3-type sublattice ordered Au(1-x)Cux alloys with maximal order degrees
calculated by nine V-functions. a), b) Mixed volumes of formation
m
VxAuCu and
m
Vx3
AuCu ; c), d) Excess volume differ-
ences
ex
VxAuCu and
ex
Vx3
AuCu relative to disordered Au(1-x)Cux alloys.
The volume function of alloys is,
 
 

 

Au Cu
Au 0Cu
2
Au AuAu
0
0
2
Cu Cu
0
2
0
,
2
I
I
iI
i
IB
iI
i
vxTx vTx vT
ixvTvT
I
II iIi
x
vTvT
I










 




(8)
where, Au
03.8140 eV
 , Au4.0555 eV
I
 ,
Cu
03.6488 eV
 , Cu 3.4824 eV
I
 ;

Au3 3
016.958110nmv
,

Au3 3
15.5213 10nm
I
v
,

Cu3 3
012.6828 10nmv
,

Cu33
11.8124 10nm
I
v
, I =
12.
Their features are as follows (Supplementary Figures
2(b) and (c), Supplementary Table S2):
The Au
i
decreases with increasing number i of Cu-
atoms in the coordinative configuration:



2
Au AuAu Au
00iI
iI
 
 .
The Cu
i
decreases with increasing number (I-i) of
Au-atoms in the coordinative configuration:

2
Cu CuCu Cu
0.
iI I
IiI
 



The Au
i
v decreases with increasing number i of Cu-
atoms in the coordinative configuration:

2
Au AuAu Au
00iI
vv iIvv .
The Cu
i
v increase with increasing number (I-i) of Au-
atoms in the coordinative configuration:



Cu CuCu Cu
0
2Cu Cu
0
2
iI I
I
vv IiIvv
IiI vv
 


 


.
1.7. Valence Electron Structures of
AG-Sequences
It is a very-very difficult problem to obtain electronic
structures, physical and thermodynamic properties of
alloy genes. After knowing the potential energies and
volumes of alloy genes, a fictitious ‘characteristic crys-
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
430
tal’ has been proposed in order to resolve this difficult
problem. It is an fcc-based consisting of the same alloy
genes with known potential energy and volume in the
fcc-based lattice Au-Cu system. Therefore, there are
Au
i
C- and Cu
i
C-sequences of characteristic crystals cor-
responding to Au
i
A
- and Cu
i
A
-sequences. The Au
0
C and
Cu
12
C are called as primary state characteristic crystals,
i.e., Au- and Cu- pure metals. The Au
12
C and Cu
0
C are
called as terminal state characteristic crystals.
After knowing the potential energies and volumes of
alloy genes, the valence electron structures, single bond
radii and potential energy curves of alloy genes can be
calculated based on the valence bond theory of charac-
teristic crystals and using inverse calculation method;
and then, the Debye temperatures and bulk moduli of
characteristic crystals can be obtained. These results are
shown in Supplementary Figures S2(d) and (e), and
listed in Supplementary Table S3. The valence electron
Table S3. The valence electron structures (the number of free electrons (sf), covalent electrons (sc, dc) and non-valent elec-
trons (dn)), volumes(v), potential energies (ε) and single bond radii (R) of alloy genes, cohesive energy (Ec), Debye tempera-
tures (θ) and bulk moduli (B) of characteristic crystals.
valence electrons in outer shell v ε R Ec θ B
i
C
dn d
c s
c s
f 103 nm3 (eV/at.) 101 nm(J/mol) (K) (GPa)
Au characteristic crystals
Au
0
C 3.4013 5.8506 0 1.7481 16.9581 3.8130 1.3428 368000 165.00 170.99
Au
1
C 3.4016 5.8523 0 1.7461 16.9524 3.8146 1.3426 368162 165.06 171.17
Au
2
C 3.4007 5.8578 0 1.7415 16.9353 3.8197 1.3423 368647 165.22 171.69
Au
3
C 3.3995 5.8669 0 1.7336 16.9067 3.8281 1.3417 369456 165.49 172.58
Au
4
C 3.399 5.8793 0 1.7216 16.8667 3.8398 1.3409 370589 165.87 173.83
Au
5
C 3.3994 5.8951 0 1.7055 16.8152 3.8549 1.3397 372046 166.37 175.45
Au
6
C 3.3998 5.9143 0 1.6859 16.7524 3.8733 1.3383 373826 166.98 177.46
Au
7
C 3.4005 5.9368 0 1.6627 16.6780 3.8951 1.3367 375929 167.70 179.86
Au
8
C 3.4014 5.9626 0 1.636 16.5923 3.9203 1.3348 378357 168.52 182.68
Au
9
C 3.404 5.9912 0 1.6048 16.4951 3.9488 1.3326 381108 169.47 185.94
Au
10
C 3.4074 6.0229 0 1.5698 16.3865 3.9806 1.3302 384182 170.52 189.67
Au
11
C 3.4138 6.0569 0 1.5293 16.2665 4.0159 1.3274 387581 171.70 193.89
Au
12
C 3.4218 6.0936 0 1.4847 16.1351 4.0544 1.3244 391303 172.99 198.65
Cu characteristic crystals
Cu
0
C 4.2877 5.0495 0 1.6628 12.3631 3.6478 1.1793 352055 345.81 134.68
Cu
1
C 4.3485 5.0119 0 1.6396 12.3593 3.6212 1.1782 349491 344.58 133.76
Cu
2
C 4.411 4.9762 0 1.6129 12.3478 3.5969 1.1768 347150 343.53 133.05
Cu
3
C 4.4737 4.9426 0 1.5837 12.3287 3.5750 1.1753 345031 342.66 132.56
Cu
4
C 4.5366 4.9111 0 1.5523 12.3019 3.5553 1.1735 343136 341.96 132.27
Cu
5
C 4.6006 4.8815 0 1.5179 12.2675 3.5380 1.1715 341463 341.45 132.20
Cu
6
C 4.6648 4.854 0 1.4812 12.2254 3.5230 1.1693 340014 341.10 132.34
Cu
7
C 4.7309 4.8282 0 1.4409 12.1757 3.5103 1.1669 338787 340.96 132.70
Cu
8
C 4.7982 4.8042 0 1.3976 12.1183 3.4999 1.1642 337784 340.99 133.28
Cu
9
C 4.866 4.7821 0 1.3519 12.0533 3.4918 1.1612 337003 341.21 134.09
Cu
10
C 4.9345 4.7618 0 1.3037 11.9807 3.4860 1.1581 336446 341.61 135.15
Cu
11
C 5.0053 4.7429 0 1.2519 11.9003 3.4826 1.1547 336111 342.20 136.44
Cu
12
C 5.0778 4.7254 0 1.1968 11.8124 3.4814 1.1511 336000 343.00 138.00
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
431
structures of characteristic crystals may represent ones of
alloy genes and their physical properties may represent
the contributions of alloy genes to properties of alloys.
In the free atom system, the electron structures of at-
oms are described by four quantum numbers. The out-
shell electron structures of the Au- and Cu- free atoms at
the basic states are respectively 10 1
56ds and 10 1
34ds.
The electronic structure units of the alloy genes are
valence electrons described by four quantum numbers:
free electron

f
s
, covalent electron

,,,
cccc
s
pd f,
magnetic electron

,
mm
df and non-valence electron

,,
nnn
pd f. In the Au-Cu system, the valence electron
structures of AG-sequences are listed in Supplementary
Table S3. The valence electron structures of the primary
states

Au Cu
012
,

and terminal states

Au Cu
12 0
,

of
alloy genes are as follows:
Au3.4013 5.85061.7481
0556
ncf
dds
;
Au3.4218 6.0936 1.4847
12 556
ncf
dd s
.
Cu5.0778 4.7254 1.1968
12 334
nc f
dds
;
Cu4.2877 5.04954 1.6628
033 4
nc f
dd s
.
The essence of properties of alloy genes can be ex-
plained by their valence electron structures.
The essence for forming fcc-based lattice: As changing
from Au
0
to Au
12
the n
d-electrons slightly increase
(3.4012/atom 3.4218/atomand occupy at the
d-energy band
g
e state; and the dc-electrons obviously
increase (5.8506/atom 6.0936/atomand occupy at the
d-energy band g
t2 state. Therefore, each Au
i
A
-atom
possesses 12 covalent bonds to form fcc-based lattice.
As changing from Cu
12
to Cu
0
the dn-electrons
obviously decrease (5.0778/atom 4.2877/atom), few
of them occupy at the d-energy band 2
g
t state, besides
filling the
g
e state fully; the dc-electrons obviously in-
crease (4.7254/atom 5.0495/atomand occupy at the
d-energy band 2
g
t state. Each Cu
i
A
-atom possesses 12
covalent bonds to form fcc-based lattice.
For these reasons, the intermetallics, ordered and dis-
ordered alloys formed by Au
i
A
- and Cu
i
A
-sequences
possess fcc-based lattice.
The essence for variations in potential energies and
volumes of alloy genes:
As changing from Au
0
to Au
12
, the
f
s
-electrons
decrease: 1.7481/atom 1.4847/atom, and the de-
creased
f
s
-electrons are translated into dc-electrons. It
results in falling Au
12
potential energy: 3.813 eV/atom
4.0544eV/atom, shorting single bond radus: 1.3428
× 101 nm 1.3244 × 101 nm, and reducing Au
12
v
volume: 16.9581 × 103 nm3 16.1351 × 103 nm3.
Therefore, The Au
i
-potential energies decrease and the
Au
i
v-volumes reduce with increasing number i of Cu-
atoms in the coordinative configurations.
As changing from Cu
12
to Cu
0
, the
f
s
-electrons
increase: 1.1968/atom 1.6628/atomand the dc- elec-
trons increase: 4.7254/atom 5.0495/atom. It results in
falling Cu
0
potential energy: 3.4814 eV/atom
3.6478 eV/atom, lengthening single bond radius: 1.1511
× 101 nm 1.1793 × 101 nm, and expanding Cu
0
v
volume: 11.8124 × 103 nm3 12.3631 × 103 nm3.
Therefore, the Cu
i
-potential energies decrease and
Cu
i
v-volumes increases with increasing number (I-i) of
Au-atoms in the coordinative configurations, that is con-
trary to the expected behavior.
1.8. The Thermodynamics Properties of
AG-Sequences
In the SMMS framework, the Gibbs energy

i
GT of
each characteristic crystal (or ally gene) may be split into
two parts: a temperature-independent contribution of po-
tential energy
0E, of which the variation with tem-
peratures has been accounted in the attaching vibration
energy, and a temperature-dependent contribution of
generalized vibration free energy


v
X
T, but both are
configuration(i)-dependent (Equation (9)). The enthalpy
i
H
T of each characteristic crystal may be also split
into two parts: a temperature-independent contribution of
potential energy
0E and a temperature-dependent
contribution of generalized vibration energy
v
UT
(Equation (10)). The generalized vibration free energy
includes the generalized vibration energies
v
UT ,
which include Debye vibration energies

D
UT and
attaching vibration energies


E
UT, and the contribu-
tion of the generalized vibration entropies
v
ST ,
which include Debye vibration entropies

D
ST and
attaching vibration entropies

E
STthe generalized
vibration heat capacity

v
p
CT, which include Debye
vibration heat capacity

D
p
CT, attaching vibration
heat capacity
E
p
CT. The attaching vibration energy
includes contributions of electron excitation, energy of
formation of holes, variation of potential energy with
temperature, and so on. Therefore, the multi-level ener-
getic functions of characteristic crystals at T K as fol-
lows:
 
 
AuAuAu.AuAu.
CuCu Cu.CuCu.
0
0
vv
iiii i
vv
iiii i
GT EXTHTTST
GT EXTHTTST
 
 
(9)
 
 
AuAu Au.
CuCuCu.
0
0
v
iii
v
iii
H
TEU T
H
TEUT


(10)
 
 
Au. Au.Au.
Cu. Cu.Cu.
vv v
ii i
vv v
ii i
X
TUTTST
X
TUTTS T


(11)
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
432
 
 
Au. Au.Au.
Cu. Cu.Cu.
DD D
ii i
DD D
ii i
X
TU TTS T
X
TUTTS T


(12)
 
 
Au. Au.Au.
Cu. Cu.Cu.
EE E
ii i
EE E
ii i
X
TUTTS T
X
TU TTST


(13)
 

Au. Au. Au.
Cu. Cu.Cu.
vDE
iii
vDE
iii
UTU TUT
UTUTU T


(14)
 
 
Au. Au.
.
0
Cu. Cu.
.
0
d
d
T
DD
ipi
T
DD
ipi
UTCTT
UTCTT
(15)

 
Au. Au.
.
0
Cu. Cu.
.
0
() d
d
T
EE
ipi
T
EE
ipi
UTCTT
UTCTT
(16)
 
 
Au. Au.Au.
Cu. Cu.Cu.
vDE
iii
vDE
iii
STSTS T
STSTS T


(17)

 
Au.
.
Au.
0
Cu.
.
Cu.
0
()d
d
D
T
pi
D
i
D
T
pi
D
i
CT
ST T
T
CT
ST T
T
(18)
 
 
Au.
.
Au.
0
Cu.
.
Cu.
0
d
d
E
T
pi
E
i
E
T
pi
E
i
CT
STT
T
CT
ST T
T
(19)
 


 


 


 


23 23
Au AuAuAuAuAu
000
23 23
Cu CuCuCuCuCu
12 12
00 00
0000
iii
iI ii
EVEV
EV EV




(20)




Au
Cu
34
Au.
.Au 2
0
34
Cu.
.Cu 2
0
e
9d
e1
e
9d
e1
i
i
x
T
D
pi x
i
x
T
D
pi x
i
Tx
CTR x
Tx
CTR x






(21)


Au.106 2
.0
Au.46 2
.12
1.858993 104.22258410
2.763134 105.481502 10
E
p
E
p
CTTT
CTTT




(22)


Cu.6 2
.12
Cu.62
.0
0.001680352.826912 10
0.005646143.632353 10
E
p
E
p
CT TT
CT TT


(23)
where, Au
0165 K
, Au
12 175.21 K
, Cu
12 343 K
,
Cu
0343.07 K
.
It can be demonstrated that the general correlative
functions between other energetic properties (q) of char-
acteristic crystals (ally genes) at T K are as follows
  
  
2
AuAuAuAu
00
2
CuCuCuCu
0I
iI
iI
qT qTiI qTqT
qT qTIiIqTqT

 






(24)
The AG-thermodynamic properties of characteristic
crystals (ally genes) of Au-Cu system are shown in Sup-
plementary Figure S5.
2. Alloy Gene Gibbs Energy Partition
Function
In this section we present the AG-Gibbs energy partition
function of AuCu-type sublattice alloy system, which
includes AG-Gibbs energy transmission law, and AG-
arranging function, as well as other functions of thermo-
dynamic properties.
Characteristic Gibbs energy function:
The characteristic Gibbs energy

*,GxT function
can be obtained by the AGT law (S1):
 

12 AuAu CuCu
0
,, ,
iiii
i
GxTxxTGTxxT GT

(25)
Partition function of the alloy:

Au Cu
*
,,,,
exp ,
ii
x
TgxxTxxT
GxT kT


(26)
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
433
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
434
Figure S5. Curves of AG-thermodynamic properties in Au-Cu system. a), b) Generalized vibration heat capacity curves:
vvv
ppip
CCC
Au..Au .Au
.0 12
 , vvv
ppip
CCC
Cu. Cu. Cu.
.0 ..12
 ; c), d) Generalized vibration energy curves: vvv
i
UUU
Au. Au. Au.
012
 ,
vvv
i
UUU
Cu. Cu. Cu.
012
 ; e), f) Generalized vibration entropies curves: vvv
i
SSS
Au. Au. Au.
012
 , vvv
i
SSS
Cu. Cu.Cu.
012
 ; g), h) General-
ized vibration free energy curves: vvv
i
XXX
Au. Au. Au.
012
 , vvv
i
XXX
Cu. Cu.Cu.
012
 ; i), j) Enthalpy of formation curves:
i
HHH
Au Au Au
012
 i
HHH
Cu Cu Cu
012
 ; k), l) Gibbs energy curves: i
GGG
Au Au Au
012
 i
GGG
Cu CuCu
12 0
 .
AG-arranging function extending to all Au
i
G- and
Cu
i
G-energy levels:


 

 
 

 
Au Cu
Au 1Cu 1Au 2Cu2
00
Au 1Cu 1Au2Cu 2
0000
,, ,
!!
!! !!
ii
II
iii i
ii
III I
iii i
iiii
gxxTxxT
Nx xNxx
Nx NxNxNx

 
 

 
 





 
(27)
Concentrations of the Au
i
A
and Cu
i
A
alloy genes at
the

Au
i
GT- and

Cu
i
GT-energy levels::










Au 1Au2
Au
Au 1Au2
Au
,,,
,,,
ii i
ii i
x
xTxxTxxT
x
xTxxTxxT


(28)
Concentrations of the Au
i
A
and Cu
i
A
alloy genes oc-
cupied at the (1) and (2) sublattice points:


  


  


  


  
Au 1111
Au
Cu 1111
Cu
Au 2222
Au
Cu 2222
Cu
,
,
,
,
ii
ii
ii
ii
xxTvP
xxTvP
xxTvP
xxTvP
(29)
Probabilities of Au and Cu atoms occupied at (1) and
(2) sublattice points:
 
 

 
 

11
Au Au
11 1
Cu AuAu
21
Au Au
22 1
Cu AuAu
1,
11 1
,
11 .
Px v
PPx v
Pxv
PP xv





 
(30)
Fractions of the (1) and (2) sublattice points:
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
435
 
 
11
22
12
12
vNN
vNN


(31)
Definition of order degree:




 
  
12 Au1Cu2
Au AuCuCuAuCu
1212 121221
11
ii
ii
PxPxxxxx
vvvvwvvvwv

 
 (32)
The probability

1
i
of the coordinative cluster
Au,CuIii
surrounding the (1)-sublattice point:
 
 

 

 

 


84
1
122 211
8AuCu4AuCu
Kiiii ii
ii
i
ii ii
ik
CPPC PP





 


(33)
In the

1
i
equation, the probability

2
i
of co-
ordinative cluster

8Au,Cuii



occupied at the
(2)-sublattice points, and the probability


2
ii
of co-
ordinative cluster

4Au,Cuii ii


occupied
at the (1)-sublattice points:
 

 



  

 


8
222
8Au Cu
4
111
4AuCu
Kii
i
i
ik
Kiiii
ii
ii
ik
CP P
CP P

 





(34)
where, the i is the number of the Cu atoms occupied at
the (2)-sublattice points.
The probability

2
i
of coordinative cluster
Au,CuIii
surrounding the (2)-sublattice point:
 
 

 

  

 


84
2
211122
8AuCu4 AuCu
Kiiii ii
ii
i
ii
ii
ik
CPPCPP






 


(35)
where, the i is the number of the Cu atoms occupied at
the (1)-sublattice points.
In the

2
i
equation, the probability

1
i
of co-
ordinative cluster

8Au,Cuii



occupied at the (1)-
sublattice points, and the probability


2
ii
of coordina-
tive cluster

4Au,Cuii ii

 

occupied at the
(2)-sublattice points
 

 



  

 


8
111
8Au Cu
'
4
222
4Au Cu
'
Kii
i
i
ik
Kiiii
ii
ii
ik
CPP
CP P


 







(36)
In the Equations (32) to (35), k and K take their values,
according to
0,,if 4;
4,,if 48;
4,8,if 812.
kKii
ki Kii
ki Ki
 
 
 
(37)
The maximum order degree max
as function of com-
position Au
x
:
For the stoichiometric alloy, due to

1
Au 1P,
max 1
;
For the alloys with

1
Au
x
v, due to

2
Au 0P
,


1
max Au
x
xv
;
For the alloys with

1
Au
x
v, due to

1
Au 1P,



1
max Au
11
x
xv
 .
The methods for calculating

1
i
and

2
i
are listed
in Supplementary Tables S4 and S5.
The configurational entropies of alloy and compo-
nents:



 


 



 



 



Au CuAu.Cu.
Au Cu
Au 111Au222
Au.
Au Au
000 0
Au
Cu 111Cu2
Cu.
Cu
00
Cu
,ln,, ,
,ln,ln
,ln,
ccc
Bii
III I
c
iiii
iii i
II
c
iii
iii
SxTkgx xxSxTxSxT
R
SxxTPxxTP
x
R
Sx xTPxxT
x

 




 





 




 

22
Cu
00
ln
II
i
i
P








(38)
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
436
Table S4. The methods for calculating the probability

i
1 of coordinative cluster
Ii iAu, Cu surrounding the (1)-
sublattice point in AuCu-type order alloys.

2
i


1
ii
i
i 8
i
C


 


8
22
ii
AB
PP
ii

4
ii
C


 


4
11
ii ii
AB
PP

 


0 0 1




80
22
AB
PP 0 1




40
11
AB
PP
1 0 1




80
22
AB
PP 1 4




31
11
AB
PP
1 1 8




71
22
AB
PP 0 1




40
11
AB
PP
2 0 1




80
22
AB
PP 2 6




22
11
AB
PP
2 1 8




71
22
AB
PP 1 4




31
11
AB
PP
2 2 28




62
22
AB
PP 0 1




40
11
AB
PP
3 0 1




80
22
AB
PP 3 4




13
11
AB
PP
3 1 8




71
22
AB
PP 2 6




22
11
AB
PP
3 2 28




62
22
AB
PP 1 4




31
11
AB
PP
3 3 56




53
22
AB
PP 0 1




40
11
AB
PP
4 0 1




80
22
AB
PP 4 1




04
11
AB
PP
4 1 8




71
22
AB
PP 3 4




13
11
AB
PP
4 2 28




62
22
AB
PP 2 6




22
11
AB
PP
4 3 56




53
22
AB
PP 1 4




31
11
AB
PP
04i
4 4 70




44
22
AB
PP 0 1




40
11
AB
PP
5 1 8




71
22
AB
PP 4 1




04
11
AB
PP
5 2 28




62
22
AB
PP 3 4




13
11
AB
PP
5 3 56




53
22
AB
PP 2 6




22
11
AB
PP
5 4 70




44
22
AB
PP 1 4




31
11
AB
PP
5 5 56




35
22
AB
PP 0 1




40
11
AB
PP
6 2 28




62
22
AB
PP 4 1




04
11
AB
PP
6 3 56




53
22
AB
PP 3 4




13
11
AB
PP
6 4 70




44
22
AB
PP 2 6




22
11
AB
PP
6 5 56




35
22
AB
PP 1 4




31
11
AB
PP
6 6 28




26
22
AB
PP 0 1




40
11
AB
PP
7 3 56




53
22
AB
PP 4 1




04
11
AB
PP
48i
7 4 70




44
22
AB
PP 3 4




13
11
AB
PP
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
437
Continued
7 5 56




35
22
AB
PP 2 6




22
11
AB
PP
7 6 28




26
22
AB
PP 1 4




31
11
AB
PP
7 7 8




17
22
AB
PP 0 1




40
11
AB
PP
8 4 70




44
22
AB
PP 4 1




04
11
AB
PP
8 5 56




35
22
AB
PP 3 4




13
11
AB
PP
8 6 28




26
22
AB
PP 2 6




22
11
AB
PP
8 7 8




17
22
AB
PP 1 4




31
11
AB
PP
8 8 1




08
22
AB
PP 0 1




40
11
AB
PP
9 5 56




35
22
AB
PP 4 1




04
11
AB
PP
9 6 28




26
22
AB
PP 3 4




13
11
AB
PP
9 7 8




17
22
AB
PP 2 6




22
11
AB
PP
9 8 1




08
22
AB
PP 1 4




31
11
AB
PP
10 6 28




26
22
AB
PP 4 1




04
11
AB
PP
10 7 8




17
22
AB
PP 3 4




13
11
AB
PP
10 8 1 8)2(0)2( )()( BAPP 2 6




22
11
AB
PP
11 7 8




17
22
AB
PP 4 1




04
11
AB
PP
11 8 1




08
22
AB
PP 3 4




13
11
AB
PP
812i
12 8 1




08
22
AB
PP 4 1




04
11
AB
PP
The mole potential energies of the alloy and compo-
nents:
 


 


 
Au Cu
Au Cu
12
AuAu Au
Au
0
12
CuCuCu
Cu
0
,,,
,1, 0
,1, 0
I
ii
i
I
ii
i
ExTxExTx ExT
ExT xxxTE
ExTxxxTE

(39)
The Debye vibration energies of alloy and compo-
nents:
 


 


 
Au. Cu.
Au Cu
12
Au.Au Au.
Au ,
0
0
12
Cu.CuCu.
Cu ,
0
0
,,,
,1 ,d
,1 ,d
DDD
IT
DD
ipi
i
IT
DD
ipi
i
UxTxUxTxXxT
UxT xxxTCTT
UxTxxxTCTT

(40)
The attaching vibration energies of alloy and compo-
nents:
 


 


 
Au. Cu.
Au Cu
12
Au.Au Au.
Au ,
0
0
12
Cu.Cu Cu.
Cu ,
0
0
,,,
,1 ,d
,1 ,d
EEE
IT
EE
ipi
i
IT
EE
ipi
i
UxTxUxT xXxT
UxT xxxTCTT
UxTxxxTCTT

(41)
The generalized vibration energies of the alloy and
components:
 


 


 
Au. Cu.
Au Cu
12
Au.Au Au.
Au
0
12
Cu.Cu Cu.
Cu
0
,,,
,1,
,1 ,
vvv
I
vv
ii
i
I
vv
ii
i
UxTxUxT xUxT
UxT xxxTUT
UxTxxxTUT

(42)
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
438
Table S5. The methods for calculating the probability

i
2 of coordinative cluster
Ii iAu, Cu surrounding the (2)-
sublattice point in AuCu-type order alloys.

2
i


1
ii
i
i 8
i
C


 


8
11
ii
AB
PP
ii

4
ii
C


 


4
22
ii ii
AB
PP

 


0 0 1




80
11
AB
PP 0 1




40
22
AB
PP
1 0 1




80
11
AB
PP 1 4




31
22
AB
PP
1 1 8




71
11
AB
PP 0 1




40
22
AB
PP
2 0 1




80
11
AB
PP 2 6




22
22
AB
PP
2 1 8




71
11
AB
PP 1 4




31
22
AB
PP
2 2 28




62
11
AB
PP 0 1




40
22
AB
PP
3 0 1




80
11
AB
PP 3 4




13
22
AB
PP
3 1 8




71
11
AB
PP 2 6




22
22
AB
PP
3 2 28




62
11
AB
PP 1 4




31
22
AB
PP
3 3 56




53
11
AB
PP 0 1




40
22
AB
PP
4 0 1




80
11
AB
PP 4 1




04
22
AB
PP
4 1 8




71
11
AB
PP 3 4




13
22
AB
PP
4 2 28




62
11
AB
PP 2 6




22
22
AB
PP
4 3 56




53
11
AB
PP 1 4




31
22
AB
PP
04i
4 4 70




44
11
AB
PP 0 1




40
22
AB
PP
5 1 8




71
11
AB
PP 4 1




04
22
AB
PP
5 2 28




62
11
AB
PP 3 4




13
22
AB
PP
5 3 56




53
11
AB
PP 2 6




22
22
AB
PP
5 4 70




44
11
AB
PP 1 4




31
22
AB
PP
5 5 56




35
11
AB
PP 0 1




40
22
AB
PP
6 2 28




62
11
AB
PP 4 1




04
22
AB
PP
6 3 56




53
11
AB
PP 3 4




13
22
AB
PP
6 4 70




44
11
AB
PP 2 6




22
22
AB
PP
6 5 56




35
11
AB
PP 1 4




31
22
AB
PP
6 6 28




26
11
AB
PP 0 1




40
22
AB
PP
7 3 56




53
11
AB
PP 4 1




04
22
AB
PP
7 4 70




44
11
AB
PP 3 4




13
22
AB
PP
7 5 56




35
11
AB
PP 2 6




22
22
AB
PP
48i
7 6 28




26
11
AB
PP 1 4




31
22
AB
PP
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
439
Continued
7 7 8




17
11
AB
PP 0 1




40
22
AB
PP
8 4 70




44
11
AB
PP 4 1




04
22
AB
PP
8 5 56




35
11
AB
PP 3 4




13
22
AB
PP
8 6 28




26
11
AB
PP 2 6




22
22
AB
PP
8 7 8




17
11
AB
PP 1 4




31
22
AB
PP
8 8 1




08
22
AB
PP 0 1




40
22
AB
PP
9 5 56




35
11
AB
PP 4 1




04
22
AB
PP
9 6 28




26
11
AB
PP 3 4




13
22
AB
PP
9 7 8




17
11
AB
PP 2 6




22
22
AB
PP
9 8 1




08
22
AB
PP 1 4




31
22
AB
PP
10 6 28




26
11
AB
PP 4 1




04
22
AB
PP
10 7 8




17
11
AB
PP 3 4




13
22
AB
PP
10 8 1




08
22
AB
PP 2 6




22
22
AB
PP
11 7 8




17
11
AB
PP 4 1




04
22
AB
PP
11 8 1




08
22
AB
PP 3 4




13
22
AB
PP
812i
12 8 1




08
22
AB
PP 4 1




04
22
AB
PP
The Debye vibration entropies of the alloy and com-
ponents:




 


 
Au.
Au
Cu.
Cu
Au.
12 ,
Au. Au
Au 0
0
Cu.
12 ,
Cu. Cu
Cu 0
0
,,
,
,1 ,d
,1 ,d
DD
D
D
ITpi
D
i
i
D
ITpi
D
i
i
SxT xSxT
xS xT
CT
SxT xxxTT
T
CT
SxTxxxTT
T
(43)
The attaching vibration entropies of the alloy and
components:
 



 


 
Au.
Au
Cu.
Cu
Au.
12 ,
Au. Au
Au 0
0
Cu.
12 ,
Cu. Cu
Cu 0
0
,,
,
,1 ,d
,1 ,d
EE
E
E
ITpi
E
i
i
E
ITpi
E
i
i
SxTxSxT
xS xT
CT
SxTxxxTT
T
CT
SxT xxxTT
T
(44)
The generalized vibration entropies of the alloy and
components:
 


 


 
Au. Cu.
Au Cu
12
Au.AuAu.
Au
0
12
Cu.Cu Cu.
Cu
0
,,,
,1,
,1 ,
vvv
I
vv
ii
i
I
vv
ii
i
SxTxSxTxSxT
SxTxxxTST
SxT xxxTST

(45)
The generalized vibration free energies of the alloy
and components:



 


 
Au. Cu.
Au Cu
12
Au.AuAu.
Au
0
12
Cu.Cu Cu.
Cu
0
,,,
,1 ,
,1 ,
vvv
I
vv
ii
i
I
vv
ii
i
X
xTxXxTx XxT
X
xTxxxTXT
X
xTxxxT XT

(46)
The enthalpies of the alloy and components:
 


 



 

Au Cu
Au Cu
12
AuAuAu Au.
Au
0
12
CuCuCuCu.
Cu
0
,,,
,1, 0
,1, 0
Iv
iii
i
Iv
iii
i
HxTx HxTx HxT
H
xTxxxT EUT
H
xTxxxTEUT



(47)
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
440
The characteristic Gibbs energies of the alloy and
components:
 


 


 
Au Cu
Au Cu
12
AuAu Au
Au
0
12
CuCuCu
Cu
0
,,,
,1,
,1 ,
I
ii
i
I
ii
i
GxTxGxT xGxT
GxT xxxTGT
GxT xxxTGT
 

(48)
The Gibbs energies of the alloy and components:
 


 



 

Au Cu
Au Cu
Au*Au Au.
Au
Cu*CuCu.
Cu
,,,
,1 , ,
,1 ,,
c
c
GxTxGxTx GxT
GxTxGxTTSxT
GxTx G xTTSxT



(49)
The mixed heat capacity of the alloy:
 
  

 

   

  

..
.AuAu.Au.
,,0
0
CuCu. Cu.
,,12
0
Au
.AuAu
0
0
Cu
Cu Cu
12
0
,,,
,,
,
,d
,d
,d
d
mmvms
pp p
I
mvv v
pipip
i
Ivv
ipip
i
Ii
ms
pi
i
Ii
i
i
CxTC xTC xT
CxTxxTC TCT
xxTCTC T
xxT
CxT HTHT
T
xxT
H
THT
T
 
 

 

(50)
The activities of the components:
 

 



 



 

*
AuAu Au
*
CuCu Cu
12
*AuAuAu Au
Au 0
0
12
*CuCuCu Cu
Cu
0
,exp,
,exp,
,1 ,
,1 ,
m
m
I
ii
i
I
iiI
i
axT xGxTRT
axT xGxTRT
GxT xxxTGTGT
GxT xxxTGTGT
 


 
(51)
The volumes of the alloy and components:







Au Cu
Au Cu
12
AuAu Au
Au
0
12
CuCu Cu
Cu
0
,,,
,1,
,1 ,
I
ii
i
I
ii
i
vxTx vxTxvxT
vxT xxxTvT
ExTxxxTvT

(52)
The mixed volume thermal expansion coefficient of
the alloy:



 

 
  


  

..
.AuAu. Au.CuCu. Cu.
012
00
Au Cu
.AuAuCuCu
012
Au Cu
0 0
,,,
,,,
,,
1d 1d
,dd
mmvms
II
mvv vv v
ii ii
ii
I I
i i
ms
i i
i i
i i
xTxT xT
xTxxTTTx xTTT
xxT xxT
x
T vTvTvTvT
TT
vT vT





 
 

 

  
 
 

 


(53)
It should be emphasized that the new

,
x
T-func-
tion can be used to describe both long-distance order and
short-distance order states of alloy phases, because our
model includes the sublattice model and coordinative
model (Equation (32)), and that, the thermodynamic
properties of alloy components need no representatives
by the partial molar properties. We have proved that the
partial molar properties cannot represent the average
properties of the corresponding components [20].
3. EHNP Diagrams of Thermodynamic
Properties of AuCu-type Sublattice Alloy
System
According to three-dimension m
GxT and
x
T
EHNP diagrams, we have obtained other
qxT
EHNP diagrams of the AuCu-type sublattice
system in Supplementary Figure S6.
It should be emphasized that from each three-dimen-
sion qxT
EHNP diagram we can obtain isocompo-
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
441
Y. Q. XIE ET AL.
Copyright © 2013 SciRes. IJCNS
442
Figure S6. The
q
xT EHNP diagrams of the AuCu-type sublattice system. a) Mixed Gibbs energy m
GxT
three-dimension EHNP diagram; b) Mixed characteristic Gibbs energy m
GxT

* three-dimension EHNP diagram,
without including configuration entropy; c) Mixed enthalpy m
HxT
 three-dimension EHNP diagram; d) Mixed po-
tential energy m
E
xT three-dimension EHNP diagram, without including variations in AG-potential energies with
temperatures; e) Mixed volume m
VxT three-dimension EHNP diagram; f) Generalized mixed vibration free energy
v
X
xT three-dimension EHNP diagram; g) Generalized mixed vibration energy v
UxT
 three-dimension EHNP
diagram, including variations in AG-potential energies with temperatures; h) Generalized mixed vibration entropy
v
xT three-dimension EHNP diagram; i) Heat capacity m
p
CxT
three-dimension EHNP diagram; j) Thermal
expansion coefficient xT
three-dimension EHNP diagram; k) axT
Au three-dimension diagram; l) axT
Cu
three-dimension diagram.
sitional x
qT, isoproperty q
Tx, and isothermal
T
qx path phase diagrams. These diagrams are inter-
connected to form a big database about structures, prop-
erties and their variations with temperature of alloy sys-
tems. Therefore, the man’s knowledge of relationships of
structures, properties and environments for alloy systems
has been changed from single causality to systematic
correlativity.