Medical, Pharma, Engineering, Science, Technology and Business

^{1}Institute of Physical Materials Science, Siberian Branch of Russian Academy of Sciences, Russia

^{2}Buryat State University Ulan-Ude, Russia

- *Corresponding Author:
- Lutsyk VI,

Institute of Physical Materials Science

Siberian Branch of Russian Academy of Sciences, Russia

**Tel:**+79247514406

**E-mail:**[email protected]

**Received date:** February 28, 2017; **Accepted date:** March 10, 2017; **Published date:** March 15, 2017

**Citation: **Lutsyk VI, Vorob’eva (2017) Verification of Phase Diagrams by Three-Dimension Computer Models. Mod Chem Appl 5:215. doi: 10.4172/2329-6798.1000215

**Copyright:** © 2017 Lutsyk VI, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Present paper is the survey of the works, dedicated to the elimination of contradictions in the publications, which describe the calculated and/or experimental results of investigations of the three-component systems phase diagrams. Special approach to the construction of phase diagrams in the form of their assembling from the surfaces and the phase regions into the three-dimensional (3D) computer model as the effective tool of the detection of the incorrect interpretation of the obtained experiment or of errors in the thermodynamic calculations of the phase diagrams fragments, caused by a deficiency in the initial information, is proposed. 3D computer models of Au-Ge-Sn, Au-Ge-Sb, Ag-Au-Bi, Ag-Sb-Sn, Au-Bi-Sb T-x-y diagram are considered.

Phasediagrams; Computer simulation; Threedimensional Visualization

Such known programs as Lukas Program, ThermoCalc, ChemSage, FACTSage, MTDATA, PanEngine, PANDAT are created for the calculations of phase equilibria. Thanks to them it became possible the use of more realistic models of the thermodynamic properties of phases, the calculation of phase diagrams in the complex two-component systems and the systems with the large number of components. The CALPHAD-method, which makes possible to generalize and to refine within the framework information about the phase equilibria and the thermodynamics of phases for one model, is most claimed today. It is effective means for decreasing the volumes of the experiments, necessary for understanding of phase transformations in the alloys and the ceramics. The CALPHAD ideology became the powerful means of theoretical studies and obtaining of adequate information about the phase equilibria. Thermodynamic properties and phase diagrams for the technologically important multi-component materials can be predicted with its use. The reliable thermodynamic descriptions of two-component systems are the basis of the data bases with such characteristics. However, the CALPHAD application is limited by a deficiency in reliable thermodynamic data and the weak possibilities of the visualization of three-dimensional objects. In addition to this, using of thermodynamic methods of the states diagram calculating is hindered by the need of evaluating the thermodynamic properties of phases (in the absence of experimental data) and the agreement of experimental data of phase equilibria with the thermodynamic models. The innovation technology of assembling the space models of multidimensional phase diagrams from the entire totality of the geometric images corresponding to them is proposed: “To decode the diagrams topology the schemes of uni- and invariant states had been elaborated. This sort of schemes with phase’s routes designations makes possible to calculate the number of phase regions, surfaces and to know a type of every surface (plane, ruled or unruled surface). Detailed analysis of T-x-y diagrams geometrical constructions had been carried out with their aid, and their computer models had been designed” [1].

**3D models of T-x-y diagrams: Approaches, the principles of
the construction**

Basic principle of the design of the three-dimensional (3D)
computer model of the ternary system T-x-y diagram is the assembling
of three-dimensional objects of its surfaces and phase regions [2]. The
3D model constructing is fulfilled in several stages: 1) the twodimensional
(2D) table and then the 3D scheme of uni- and invariant
states (planes of invariant reactions and ruled surfaces), 2) the prototype (unruled surfaces), 3) transformation of the prototype into the real
system model. It should be noted that, as many geometrically simple
diagrams are already described in the known monographs (for instance,
[3,4], therefore the Reference book of 3D and 4D models of virtual
T-x-y and T-x-y-z diagrams of basic topological types is created (look
about it, for instance, in [5]. And to construct the real system T-x-y
diagram of a simple topology it suffices to take from the Reference book
the finished model (or the combination of two-three simple models).
The 3D model is obtained after the input of the concrete coordinates of
base points (corresponding to invariant transformations in the binary
and ternary system) and correction of the curvature of T-x-y diagram
lines and surfaces. For instance, the liquidus of the Au-Ge-Sn=A-B-C
phase diagram with six binary compounds (incongruently melting R1,
R2 of the variable and R4=AuSn_{2}, R5=AuSn_{4} of the constant
concentration (R5 is decomposed below 49.8°C), R3=AuSn is the
stoicheometry congruently melting compound, R6=Au_{5}Sn exists below
179.3°C) is the result of triangulation by the quasi-binary Ge-AuSn
section, and each subsystem (Au-Ge-AuSn and AuSn-Ge-Sn) is the
double combination of the classical topological type of the T-x-y
diagram with a incongruently melting binary compound. But both
subsystems are formed by two incongruently compounds (**Figure 1**).
And the sub-solidus is complicated by the allotropy of tin
(transformation of 2 modifications at low, ∼13°C, temperatures). First
step of the 3D model design is the analysis of the T-x-y diagram
geometric structure using the uni- and invariant states scheme. This is
the usual Sheil’ phase reactions scheme [6], the trajectories of a change
in the concentrations of the interacting phases in which are written
below each three-phase reaction [2] (**Tables 1** and **2**). The first (with
highest temperature) and second (with lowest one) points are accepted
as the base points for each trajectory. They obtain designations (for
instance, Table 1 and Figure 1): binary eutectic “e” and peritectic “p”
with the subscripts e_{AB}, e_{BC}, p_{AR1}, etc; concentrations of appropriate
solid phases “A_{B}” and “B_{A}” as participants of, for instance, the eutectic
reaction L→A+B; liquid concentrations in ternary eutectic “E”, quasiperitectic “Q”, peritectic “P”, etc invariant reactions; concentrations of
appropriate solid phases “A_{Q1}”, “B_{Q1}”, “R1_{Q1}” (corresponding, for
instance, to the reaction Q1: L+A→B+R1). Superscript “0” is assigned to
based points of the temperature scale (on the lower face of the trigonal
prism of the T-x-y diagram). These trajectories together with the
appropriate lines of binary systems form the contours of the diagram
surfaces [1]: curves e_{AB}Q_{1}, p_{AR1}Q_{1}, corresponding to the liquid L, belong
to inner contours of the liquidus surfaces (liquidus – q_{A}, q_{B}, q_{R1}, etc),
and appropriate them curves A_{B}A_{Q1} and A_{R1}A_{Q1} belong to the solidus
surfaces (solidus - s_{A}, s_{B}, s_{R1}, etc). The binary combinations of such lines
are the directing curves for three ruled surfaces (ruled - superscript “r”)
on the boundaries of the three-phase region. Since these curves belong
to the contours of liquidus (q) or solidus (s), than ruled surfaces on the
boundaries of the region L+A+B are designated as q^{r}_{AB}, q^{r}_{BA}, s^{r}_{AB}. If both
the directing curves of a ruled surface are located on the contour of
solidus surface, such ruled surface of “solidus type” is designated by the
letter “s”. For instance, the solidus type ruled surface s^{r}_{AB} has directing
curves A_{B}A_{Q1} and A_{R1}A_{Q1}, which belong to contours of solidus surfaces
s_{A} and s_{B}. But if one of the directing curve (e_{AB}Q1) belongs to the
liquidus surface (q_{A}), then the ruled surface is named as “the liquidus
type” (q^{r}_{AB}). As a result, according to the uni- and invariant states
schema of the Au-Ge-Sn system (**Table 1**), one should expect besides six
invariant four-phase transformations (two – eutectic E1, E2 and four
quasi-peritectic Q_{1}, Q_{2}, Q_{3}, Q_{4}), indicated in the quasi-peritectictoid Q_{5}:
R_{2}+R_{3}→B+R_{6} and the eutectoid E_{3}: R_{5}→B+C+R_{4} interactions, and also
the polymorphous transformation E_{4}: C→C_{1}+B+R_{4} in the sub-solidus.
These “derived” from the uni- and invariant states scheme solid-phase
invariant transformations Q_{5}, E_{3}, E_{4} are confirmed by the isopleths [7].
Thus, according to the uni- and invariant states scheme, the Au-Ge-
Sn=A-B-C T-x-y diagram consists of 173 surfaces (8 - liquidus and 8 -
solidus, 38 - solvus, 2 - transus, 81 ruled surfaces and 9 horizontal
(isothermal) planes-complexes, corresponding to invariant reactions in
the ternary system, and each of them is divided into 4 simplexes.
Surfaces are borders of 65 phase regions (8 two-phase L+I with liquid
(I=A, B, C, R1-R5), 10 one-phase (I=A, B, C, C1, R1-R6), 20 two-phase
I+J without liquid, 13 three-phase L+I+J with liquid, 14 three-phase
I+J+K without liquid). Further the tabular (2D) scheme is transferred
into the graphic 3D form. Since the horizontal (isothermal) plane or the
complex, which consists of four simplexes, corresponds to each four-phase transformation, then, it is possible to construct, for instance, for
the Au-Ge-Sn system the quadrangle A_{Q1}B_{Q1}Q_{1}R_{1}Q_{1} and other 8
complexes of different types for reactions Q_{2}-Q_{5}, E_{1}-E_{4} (**Table 1**). Then it
makes possible to draw three lines e_{AB}Q_{1}, A_{B}A_{Q1}, B_{A}B_{Q1} as the directing
ones and to obtain the prototype of the three-phase region L+A+B. The
preliminary contours of other three-phase regions are depicted
analogously. The 3D uni- and invariant states scheme is so constructed.
Next step is to obtain the prototype of the T-x-y diagram by designing
of the liquidus, solidus, solvus, transus surfaces. And last step includes
the refinement of the curvature of the directing curves of ruled surfaces,
closing the contours of unruled surfaces by curves of binary systems
and correcting their isothermal lines. In this stage the 3D model of the
real system T-x-y diagram is formed. Finished T-x-y diagram 3D model
allows to construct any arbitrarily assigned sections and to calculate
mass balances of the coexisting phases in all stages of the crystallization for any arbitrarily assigned concentration [8]. Furthermore, the option
of the determination of conditions for changing the type of three-phase
reaction in any three-phase region is provided in 3D models [9].
Moreover 3D computer models of phase diagrams are an effective tool
for the verification of those experimentally constructed isothermal
sections and isopleths for checking the correctness of the interpretation
of data, obtained from the experiment and the thermodynamic
calculation [10,11]. The quality of each model depends on completeness
and authenticity of initial data. But, independent of the initial
information, even primitive initial model is capable to carry out such
very important function as searching of contradictions in the
description of phase diagrams geometric structures or data, obtained
from the different publications. Such possibilities of 3D models can be
seen on the examples of the using of the metal systems T-x-y diagrams
- the bases of the creation of the materials, promising as the lead-free
solders. Initial data for these 3D models are taken from the special
atlases of phase diagrams for Pb-free solders [7,12]. Their authors
carried out for each system thorough selection and agreement of
experimental data, which was being accompanied by the necessary
thermodynamic calculations in the CALPHAD-technology. The
additional experiment was fulfilled in the absence or the doubtfulness
of data. Final result for each phase diagram of ternary system was
represented in the form of the table of invariant reactions, x-y projection
of liquidus and two-three isothermal sections and isopleths.
Nevertheless, in spite of so scrupulous selection of represented data,
they are not always deprived of contradictions. It should be noted that
the 3D computer models construction is accompanied by the redesignation
of components and compounds. For instance, Au, Ge, Sn,
low-temperature modification of tin, six binary compounds are denoted
in the Au-Ge-Sn T-x-y diagram 3D model, correspondingly, as A, B, C,
C_{1}, R_{1}, R_{2}, etc. This makes it possible to avoid confusion in the
designations, accepted in the CALPHAD, where it is taken into
consideration for the designation of phases their crystal structure. For
instance, compounds AuSn4 and AuSn are denoted in atlases of phase
diagrams for lead-free solders in the volume, devoted to binary systems
as AUSN_{4} и AU1SN, correspondingly, but in other volume of the same
series as PTSN4_TYPE and NIAS_TYPE. Both germanium and lowtemperature
modification of tin are designated by the same symbol
DIAMOND_A4 (of their structure prototype).

**Au-Ge-Sb:** “Template” for the Au-Ge-Sb T-x-y diagram can
be taken from the Reference book and, after input the coordinates
(concentration-temperature) of base points, after the correction of the
curvature of lines and surfaces, to obtain the 3D model of the real Au-
Ge-Sb T-x-y diagram with the compound AuSb_{2} (**Figure 2**). In this case
the comparison of the 3D model isopleths 17 at % Ge (**Figure 3b**) with
that published (**Figure 3a**) makes it possible to reveal immediately the error of the latter: the intersection of the liquidus curve and the ruled
surface section on the border of the regions L+Ge and L+Ge+Sb. It is
worthwhile to note that the analogous section, parallel to side Au-Sb,
but at other distance from Ge (not 17 at %, but 15 at %), in other paper,
devoted to this system [13], is also constructed. The sections of liquidus
and ruled surface are passed closely, but they do not intersect each
other in this section.

**Ag-Au-Bi:** Despite the fact that the information, placed into the
atlas, was thoroughly checked (according to its authors), the description
of the Ag-Au-Bi system is contradictory: in the binary system Au-Bi,
cited according to the data [14-19], the compound Au_{2}Bi decomposes
at the temperature 110°C, whereas the solid solution on its basis in
the ternary system isopleths exists also at 0°C (**Figures 4c** and **4d**). To
understand these contradictions, it is necessary to discuss the papers
[14-18], the information from which in the form of an isothermal
section and four isopleths were included into the atlas. The Ag-Au-
Bi=A-B-C T-x-y diagram has a simple geometric construction. It is
formed by binary systems: with continuous series of solid solutions
(Ag-Au=A-B), an eutectic (Ag-Bi=A-C), and an eutectic-peritectic (Au-Bi=B-C) with the binary incongruently melting compound Au_{2}Bi=R.
Liquidus consists of three fields of primary crystallization of: bismuth,
solid solution based on the binary compound Au_{2}Bi=R and the solid
solution Ag(Au)=A(B). Liquidus surfaces intersect in the curves,
connecting the binary eutectics e_{AC} and e_{CR} and the peritectic p_{BR} (**Figure 4**) with the point Q, corresponding to the invariant quasi-peritectic
reaction L+Ag(Au)→Bi+Au_{2}Bi or L+A(B)→C+R. The compound
Au_{2}Bi exists up to 50°C according to the 2005 year paper [14] (**Figure 5a**). This compound decomposes at 110°C (**Figure 5b**) according to
the 2006 year work [16], and these data, as the most reliable, were
chosen for the atlas [12]. However, the solid solution based on Au2Bi
occurs in isopleths of below by temperatures from 450K [17] and to
0°C [12], in the phase regions with the lowest temperatures. But if this
compound decomposes in the binary system Au-Bi at 110°C, then the
solid solution on its basis must indeed decompose, also, in the Ag-
Au-Bi ternary system. This problem may be resolved by means of the
3D model too. In order to demonstrate this contradiction graphically,
two versions of the 3D computer model of the T-x-y diagram were
constructed: in one version the compound Au_{2}Bi decomposes in the binary system at 110°C and, correspondingly, the solid solution on its
basis in the ternary system does not exist below by this temperature
too (**Figure 4a**), in other version a decomposition is absent (**Figure 4b**). Both versions of the T-x-y diagram differ in the solvus surfaces
and the boundaries of the three-phase region A(B)+C+R. If the solid
solution on the basis of the compound R decomposes, then the process
R→A(B)+C is closed within the three-phase region with boundaries
given by the directing curves R_{Q}R, A(B)_{Q}B_{R}, C_{Q}C_{R} (**Figure 4a**). If the
solid solution based on the compound R in the ternary system does not
decompose, then the region A(B)+C+R below the horizontal plane Q is
bounded by three ruled surfaces with the directing curves R_{Q}R^{0}_{Q}, A(B)_{Q}A(B)^{0}_{Q}, C_{Q}C^{0} (**Figure 4b**). Both versions of the T-x-y diagram consists
of 28 surfaces (3 liquidus and 3 solidus, 6 solvus, 12 ruled surfaces
and 4 horizontal planes, corresponding to simplexes of the complex of
the invariant transformation Q: L+A(B)→C+R) and 14 phase regions.
Border solution on the compound R base has the linear region of the
homogeneity. Bordering it surfaces of two solvus v_{R_A(B)} (R_{B}RR_{Q} or R_{B}R_{Q}_{R}^{0}_{Q}R^{0}_{B}), v_{RC} (R_{C}RR_{Q} or R_{C}R_{Q}R^{0}_{Q}R^{0}_{C}) and solidus s_{R} (R_{B}R_{Q}R_{C}) have
a triangular shape (**Figure 4b**). A conclusion about the temperature
boundaries of the compound Au_{2}Bi existence cannot be made from the
phase reactions scheme of [14]. It corresponds to the uni- and invariant
states scheme, which does not consider its decomposition. Note also
that phase reactions are written in the scheme of [14] incorrectly: the
peritectic reaction L+Au→Au_{2}Bi and the eutectic one L→Au_{2}Bi+Bi are
changed by the places. Moreover the eutectic reaction is written as
L→(Ag)+(Bi). Isopleths 20 at% Ag (**Figure 5a**-**c**), 50 at% Bi, 85 at% Bi,
Ag:Au=1:4 and the isothermal section T=230°C are shown in papers
[12,17]. There is no contradictions of the 3D model and the published
sections, including the isothermal section at T<T^{Q} in both versions and
the high-temperature part of the T-x-y diagram (higher than reaction Q
at T^{Q}=251.9°C). If the compound Au^{2}Bi decomposes, the versions of the
20 at % Ag isopleth, published in [12] (**Figure 4c**) and [17,18] (**Figure 5d**-**f**), are reproduced by the 3D model, where the section S^{1}S^{2} of the
three-phase region, designated in [12] as FCC1_A1+RHOMBO_A7+AU2BI_
C15, is bounded by curves 6-8 and 7-9 (**Figure 5e**). However, in the case
of the based on R solid solution decomposition the vertical plane S_{1}S_{2}
intersects the ruled surfaces q^{rR}_{BC} and s^{rR}_{BC} (to notate the boundaries of the three-phase region with a phase reaction R→A(B)+C without liquid,
the superscript “R” is added) through curves 7-8 and 6-8. The last are the
boundaries of the three-phase region Ag(Au)+Bi+Au_{2}Bi=A(B)+C+R
(**Figure 5f**). Since the three-phase region Ag(Au)+Bi+Au_{2}Bi in the
published in [11,17] isopleths exists up to 0°C, then it is assumed that
the temperature of the Au_{2}Bi compound decomposition after addition
of the third component (Ag) reduces due to the formation of the solid
solution Ag(Au). However, the participation of the compound in the
invariant reaction L+Ag(Au)→Bi+Au_{2}Bi at 251.9°C and without the
decomposition of the compound Au_{2}Bi→Ag(Au)+Bi at 110°C suggests
that the three-phase region Ag(Au)+Bi+Au2Bi exists only within the
temperature interval 251.9-110°C. The two-phase region Ag(Au)+Bi
is lower than this region under the ruled surface A(B)_{Q}C_{Q}C_{B}B_{C} in all
concentration diapason. Because at ultralow temperatures, according to
the third law of thermodynamics, the continuous solid solution Ag(Au)
should undergo decomposition into components [20,21] this twophase
region should be replaced by the three-phase region Ag+Au+Bi.

**Ag-Sb-Sn: **Analogous contradictions are in the description of
the Ag-Sb-Sn T-x-y diagram. From one side, the binary compound
Sb_{2}Sn_{3} in the Ag-Sb-Sn T-x-y diagram, represented in [22-24], exists at
temperatures up to 0°C. With another side, this compound decomposes
at 242.4°C in [12]. Its existence is limited by the same temperature in
the analogous system Ni-Sb-Sn too. Consequently, the conditions
for the compound Sb2Sn3 existence require additional experimental
study. But now the 3D model is designed in two versions. The first one
corresponds to [22-24] with the polymorphous transformations in the
eutectoid reaction, and the 3D model consists of 99 surfaces and 62
phase regions. In the different version, constructed according to the data
of [12], one additional eutectoid reaction precedes the polymorphous
transformation, and the 3D model consists of 109 surfaces and 66 phase
regions.

**Au-Bi-Sb: **The results of experimental study and thermodynamic
correlation using the CALPHAD-technology of the Au-Bi-Sb=A-B-C
system with the compounds Au_{2}Bi=R1 and AuSb_{2}=R_{2}, are presented in
[25,26]. Earlier six isopleths had been shown in [27], which then were
also constructed in [26]. The authors of atlas [12] after the analysis of these publications preferred the results of the data of [26], but correcting
and obtaining “the better agreement in comparison with the
experimental results”. The region of the homogeneity of the solid
solution Bi(Sb)=B(C) is shown in [26] in the isothermal section 300°C.
Analogous section is used for the illustration of the phase diagrams
calculation results by the method of the convex hulls [28]. However, the
binary system Bi-Sb adjoins not with the solid solution B(C) region, but
with the two-phase region Bi(Sb)+AuSb_{2}=B(C)+R_{2}, containing the
compound with the third component (gold), in the same section,
published in the atlas and in [25]. So, versions of the Au-Bi-Sb T-x-y
diagram 3D computer model were also used for explaining this
contradiction in iso- and the polythermal sections, constructed by the
different authors. Each version is designed according to the data of
concrete publication and reproduces all given there sections and
projections. This makes it possible to compare the sections of diagrams,
obtained from different works. According to the atlas data, two invariant
reactions – quasi-peritectic Q: L+A→R_{1}+R_{2} at 296°C and eutectic E:
L→B(C)+R_{1}+R_{2} at 239.2°C with participation of the solid solution of
bismuth with antimony B(C) takes a place in the Au-Bi-Sb system.
However, the Au_{2}Bi=R1 compound decomposes in the binary system at
110°C and it is absent in the atlas sections at temperatures lower than
110°C. Consequently, it is necessary to add into the phase reactions
scheme the invariant decomposition Y: R_{1}→A+B(C)+R_{2} (**Table 2**).
Formal enumeration of surfaces and phase regions gives according to
the uni- and invariant states scheme (**Table 2**) 4 liquidus and 4 solidus
surfaces, 12 surfaces of solvus, the cupola of the disintegration of the
solid solution B(C), 3 complexes of the invariant reactions (Q at 296°C,
E at 239.2°C, Y at ∼110°C), divided each into 4 horizontal triangular
simplexes, 15 ruled surfaces or borders of 5 three-phase regions with
liquid and 12 ruled surfaces as borders of 4 three-phase regions without
liquid. Concentration coordinates (0, 0.948, 0.052) of the vertex B(C)_{E}
of the complex, which corresponds to the eutectic reaction in the
ternary system, are indicated in the atlas [12], as if this point belongs to
the Bi-Sb=B-C binary system. Since this is impossible, the point B(C)_{E}
was displaced in the computer model inside the prism to position
(0.010, 0.948, 0.042). Because of this the solidus s_{B(C)} surface ceased to
belong to edge B-C. Surfaces of solidus (s_{R1}) and solvus (v_{R1A}, v_{R1_B(C)},
v_{R1R2}), corresponding to the stoichiometric compound R_{1}, which does
not exist below 110°C, practically coincide with the vertical line R_{1}
within the temperature interval 377.5-110°C, where 377.5°C is the
temperature of the peritectic pAR1. Border solution based on the
compound R_{2} has the linear homogeneity region. The part of the solidus
s_{R2} (R_{2A}R_{2Q}R_{2E}R_{2C}) corresponding to it takes the triangular shape
R_{2C}R_{2A}R_{2Q}, and the line R_{2Q}R_{2E} coincides with the line R_{2C}R_{2E} (**Figure 6a**).
Analogously the surfaces of solvus v_{R2A} (R_{2A}R_{2Q}R_{2Y}R_{20}), v_{R2_B(C)}(R_{2C}R_{2E}R_{2Y}R_{20}), v_{R2R1} (R_{2Q}R_{2E}R_{2Y}) have in the same plane the
configurations with the triangular fragments vR2A (R2AR2QR2Y), vR2_B(C)
(R_{2C}R_{2E}R_{2Y}), v_{R2R1} (R_{2Q}R_{2E}R_{2Y}), corresponding to the solidus s_{R2}. But
since in the Au-Sb=A-C system the compound R_{2} is stoichiometric, and
points R_{2}°_{A} and R_{2}°_{C}, practically coincide with the vertical line in the
point R_{2}, they are supplemented in the ternary system by the point,denoted in the uni- and invariant states scheme (Table 2) as R20 Y, they all obtains the same designation R_{2} °, corresponding to the prism base (**Figure 6a**). So, the Au-Bi-Sb T-x-y diagram consists of 60 surfaces and 24 phase regions, including six surfaces (s_{R1}, v_{AB(C)}, v_{AR1}, v_{R1A}, v_{R1_B(C)},v_{R1R2}), which practically degenerated in the vertical lines A and R_{1} and four surfaces (s^{r}_{AR1}, q^{rR1}_{A_B(C)}, v^{r}_{AR1_(Q)}, v^{r}_{AR2_(Y)}) practically coincided with the A-B and A-C edges of the prism. Since this 3D model corresponds to the basic principles of geometric thermodynamics (the phase rule and the law of adjoining phase regions), then it makes it possible to explain the reasons for contradictions in the publications of different authors. The curve of the solid solution Bi(Sb) disintegration on Bi and Sb is depicted at temperatures lower than 200°C in the published in [12]
the Bi-Sb binary system phase diagram (**Figure 7**). This curve obviously
assigns the solid solution disintegration cupola in the Au-Bi-Sb ternary
system. Track of a section can be seen in the atlas [12] in the 20 at % Bi
isopleth (Figure 8a). However the two-phase region Bi+Sb is denoted in
this section as the three-phase : Bi+Sb+AuSb_{2}=B+C+R2 (or RHOMBO_
A7+RHOMBO_A7+AUSB_{2} in the designations of [12]). The twophase
region Bi(Sb)+AuSb_{2}=B(C)+R2 (RHOMBO_A7+AUSB2) is
denoted above it. Thus, the phase regions, which contain the compound
with gold, adjoin with the binary system, formed by bismuth and
antimony. Perhaps, the authors [12] were mistaken in the designations
of these phase regions, and the compound AuSb_{2}=R_{2} should be
removed, after renaming the two-phase region to the single-phase (the
solid solution of bismuth with antimony), and the three-phase to the
two-phase (disintegration of this solid solution). But the single-phase
region RHOMBO_A7 would then be adjacent to three-phase regions Bi
(Sb)+Au2Bi+AuSb_{2}=B(C)+R1+R2(RHOMBO_A7+AU2BI_
C15+AUSB2) and L+Bi(Sb)+AuSb_{2}=L+B(C)+R2 (L+RHOMBO_
A7+AUSB2is not designed in [12], violating the law of adjoining phase
regions. As was mentioned above, authors of [12] preferred the results
of the work [26]. It is necessary to note that the binary system Bi-Sb is
examined in [26] at temperatures above 500 K – above the binodal
curve and the temperature 400 K is the minimum in all isopleths. The
disintegration cupola of the solid solution Bi(Sb) is absent in the
sections, published in [26], and it is possible only to assume that it
would appear at temperatures below 400 К. However, there is no
contradiction in the designations of phase regions, because the singlephase
region of the solid solution Bi(Sb) adjoins directly to the Bi-Sb
system as, for instance, in the 20 at % Bi isopleth (**Figure 8c**). On the
other hand, the binodal curve assigns the region of the disintegration of
the solid solution in the Bi-Sb binary system, and it can be seen the
track of its section not only in isopleths in the atlas [12] (**Figure 8a**), but
also in the paper [25], moreover with the same contradictory
designations of phase regions near the Bi-Sb system. To explain the
reasons of the contradiction in determining phase regions and to
understand, whether the authors considered the decomposition of the
solid solution Bi(Sb)=B(C) in the binary system Bi-Sb, on the one hand
in the publications [12,25], and in [26,27] on the other hand, three
versions of the Au-Bi-Sb T-x-y diagram 3D model were designed
according to the data of [12] (**Figure 6**), [25] and [26]. In contrast to
[12], the compounds Au_{2}Bi=R1 and AuSb2=R2 are treated as
stoichiometric in [25], and there is no disintegration of the solid
solution Bi(Sb) in [26]. If “no”, then isopleths in [26] (**Figure 8c**) and
[27] are accurate. And they, like the corresponding 3D model isopleth
(**Figure 8d**), depicts the boundaries of the solid solution Bi(Sb)=B(C)
region as traces of sections of the surfaces of solidus and solvus (curves
5-6 and 6-18). They are very close to the temperature axis and are hard
to see, but they are present in the sections. If “yes”, and the solid solution
decomposes in the system Bi-Sb, then a curve corresponds to it in the
T-x diagram (**Figure 7**). It generates the disintegration cupola in the
ternary system, which is present in the isopleths of [12] (**Figure 8a**) and
[25]. In that case errors in the designations of phase regions in these
papers are connected with the fact that the sections of the solidus s_{B(C)}
and solvus v_{B(C)}_R_{2} surfaces - curves 5-6, 6-20 (**Figure 8b**) are lost in the
appropriate sections of the 3D models. At the same time, it should be
noted that the curve of solvus (the curve 6-18 in **Figure 8d**) of the model
section cannot be very close to the temperature axis, when the
disintegration cupola is present, which in this case must be assumed,
also, in the section, constructed in [26] **Figure 8c**. They should move
away from the axis in order “to give a place” for the trace of the section
of the disintegration cupola, as in Figure 8b. Thus, the final version of
the T-x-y diagram 3D computer model consists of 60 surfaces and 24 phase regions and reproduced the data of [12], but with the added
surfaces of solidus s_{B(C)} and solvus v_{B(C)}_R_{2}. Traces of these surfaces are
shown by dashed lines on the 20 at% Bi isopleth (**Figure 8b**).

**Figure 8:** Isopleth S_{1}(0, 0.2, 0.8)-S_{2}(0.8, 0.2, 0) of the Au-Bi-Sb: [12] (a): [26] (c): and 3D models according to data [12] (b): (the axis of temperature is marked in °C) and [26] (d): (sections of the added surfaces of solidus s_{B(C)} and solvus v_{B(C)}_R_{2}. are drawn with dashed lines).

It is convenient to use the 3D computer models of T-x-y diagrams,
designed according to the data of the different authors, for the agreement
of the sections and for searching of contradictions in calculations or
incorrect interpretation of experiment. Despite the fact that the binary
system Au-Bi is well studied [14,16,19] and it shown in the publications [16,19] that the solid solution on basis of the compound Au_{2}Bi does not
exist in the ternary system at temperatures below 110°C, the analysis
of isopleths, carried out with the aid of two versions of the Au-Ag-Bi
T-x-y diagram 3D computer model, constructed according to the data
[12,17], showed that the decomposition this solid solution was not
considered in [12] and [17]. Therefore the low-temperature part of
these sections must be corrected. The correct form of section is shown
in the **Figure 5f**. In order to be confident in the correctness of the Au-
Sb-Sn system description in [22-24], it is necessary to explain, actually
the compound Sb_{2}Sn_{3} is decomposed at 242.2°C as in [12], or the temperature boundaries of its existence are stretched to 0°C as it shown
in [22] Fragment of the 17 at % Ge isopleth is not correctly in **Figure 8** The analysis, carried out with the aid of three versions of the Au-
Bi-Sb T-x-y diagram 3D computer model for publications [12,25,26],
showed that: a) sections of the solidus sB(C) and solvus vB(C)_R2 surfaces
in isopleths of the 3D computer models by curves 5-6 and 6-20 (**Figure 8b**), 5-6 and 6-18 (**Figure 5d**), are missed in sections of the diagram
in [12] (**Figure 8a**) and in [25] b) phase regions Bi(Sb) and Bi+Sb are
missed in the same sections in [12] (**Figure 8a**) and in [25];c) the cupola
of the solid solution Bi(Sb) disintegration to bismuth and antimony
(**Figure 8c**), which is assigned by the binodal of the Bi-Sb binary
system [12], is lost in [26] (**Figure 6**). Most preferable is the version
of the T-x-y diagram 3D model, which is reconstructed according to
the data of [12], but with the added surfaces of solidus sB(C) and solvus v_{B(C)_}R_{2} (**Figure 6**). At the same time, it is possible only to speak, that the
structure of the ternary system phase regions near the binary system
Bi-Sb at temperatures below 150°C is contradictory and requires an
additional experimental study.

This work was been performed under the program of fundamental research SB RAS (Project 0336-2016-0006), it was partially supported by the RFBR (Projects 15-43-04304, 17-08-00875) and the RSF (Project 17-19-01171).

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