3D computer model of the Co-Cu-CoS-Cu 2 S subsystem T-x-y diagram above 800  C

The three-dimensional computer model of the Co-Cu-CoS-Cu 2 S subsystem T-x-y diagram at temperatures above 800 o C is represented. It is shown that the liquid immiscibility in the binary subsystem Cu-Cu 2 S is transformed within the ternary system with Co into the wide two-phase region of two immiscible melts, which interrupts the univariant curve of the Co and Cu 2 S co-crystallization. The special features of the structure of the solidus surface of cobalt, caused by liquid-phase immiscibility are considered.

the multiphase compositions, which are formed during the crystallization of the quaternary melts.

Initial experimental data
Binary systems are well studied and reference books data of their phase diagrams can be confidently used [32][33][34] (Figure 1).
Investigation of the Co-Cu-S system (the Co-Cu-CoS-Cu2S subsystem) is limited in the current research in concentration -by the quasi-binary section CoS-Cu2S and in temperature -by 800 o C ( Figure 2).
The binary Co-Cu (A-B) system is the usual peritectic system with one transformation L+CoCu (pAB: L+AB) at 1112 o C (Figure 1a). It is characterized by the variable solubility of Cu in -Co. It is observed a "retrograde" view of the solidus in the temperature range 1112-1495 o C, when the solubility of copper in the solid solution first grows with an increase of temperature, and then it decreases and has a maximum at 1367 o C and the concentration of copper 20.8 % weight.
The compound Co1±xS (R1) is melted congruently at the temperature 11821 o C and is characterized by the region of the homogeneity in the range of concentrations 35.5-40 % weight of sulfur.
The ternary Co-Cu-CoS-Cu2S (A-B-R1-R2) subsystem is characterized by the wide region of liquid-phase immiscibility ( Figure 2). The critical point K of the immiscibility region has the temperature 1334 o C. Tie-lines (K is their degeneration) on the border of the immiscibility region are isotherms, which form two ruled surfaces as the boundaries of three-phase monotectic reactions with participation of either Co (L1L2+Co and directing lines KN and KM) or the solid solution on the base of Cu2S (L1L2+Cu2S and directing lines nN and mM). The thermal analysis showed the temperature 1076 o C of the four-phase invariant monotectic transformation L1L2+Co+Cu2S (M(N): L1L2+A+R2) [18]. Besides, the subsystem includes two invariant transformations of quasi-peritectic type L+CoSCu2S+Co4S3 (Q1: L+R1R2+R3) at 875C, L+CoCu+Cu2S (Q2: L+AB+R2) at 1070 o C and one invariant transformation of the eutectic type LCo+Cu2S+Co4S3 (E: LA+R2+R3) at 8372 o C. Since the reaction Q2 temperature is not experimentally determined and is indicated only in the temperature interval 1067-1076 o C [18], that it is accepted equal to 1070 o C in the 3D model.
Concentration coordinates of the apexes of complexes, which correspond to quasiperitectic Q1 and Q2 reactions, are cited in [18] such, that the points Q1 and Q2, corresponding to liquid concentrations, occur within R1Q1R2Q1R3Q1 and AQ2BQ2R2Q2 triangles, and they correspond to not quasi-peritectic Q1: L+R1R2+R3 and Q2: L+AB+R2 reactions, but to eutectic LR1+R2+R3 and LA+B+R2+R3 ones, accordingly. Therefore for the 3D computer model coordinates of the R3Q1 and BQ2 points were corrected in such a way that the Q1 and Q2 points would not appear inside the tie-triangle (Table 1).  The lowest temperatures here is 800 o C (below the eutectic invariant point E at 837 o C); 3D model is constructed down to this temperature and corresponding points are noted by the superscript "0" b Concentration coordinates of the R3Q1 and BQ2 points were corrected in such a way that the Q1 and Q2 points would not fall inside the R1Q1R2Q1R3Q1 and AQ2BQ2R2Q2 tie-triangles, accordingly. Coordinates of R3Q1 and BQ2 points, which correspond to data [18], are indicated in the brackets

3D simulation
3D simulation is carried out into several steps [31]. The scheme of uni-and invariant states ( Table 2) is formed firstly. It is the description of uni-and invariant reactions and, correspondingly, of three-and four-phase regions (degenerated into a planar complex).
Further, it is transformed into the three-dimensional form, in which the corresponding ruled surfaces and the isothermal planes, which play the role of the complexes and correspond to invariant reactions, graphically are presented. Then the necessary unruled surfaces (q -liquidus, s -solidus, vsolvus, etc in the 3D model and in the Table 3) are added to these ruled surfaces and planes.
As a result, the prototype of T-x-y diagram is obtained, which is transformed into the real system T-x-y diagram 3D model after the input of the coordinates of all base points and correction of the curvature of lines and surfaces ( Figure 2). It is possible to divide the 3D image of the T-x-y diagram ( Figure 2) into its constituent ("exploded" [23]) phase regions (Figure 3;  The analysis of the T-x-y diagram geometric structure is presented as the scheme of uniand invariant states ( Table 2). Four M(N), Q2, Q1 and E invariant transformations of the Co-Cu-CoS-Cu2S (A-B-R1-R2) subsystem, in accordance with the temperature row, are written to the scheme, where every three-phase reaction has the initial and final points of the interacting phases concentrations. Contours of the unruled surfaces are combined from these lines. For example, eR1R2Q1, R1R2R1Q1, R2R1R2Q1 lines correspond to concentrations change in the liquid (L) and solid (R1 and R2) phases in the LR1+R2 reaction.
The eR1R2Q1 curve is one of the contour lines of CoS (R1) on the border of two liquidus surfaces, while the R1R2R1Q1 curve forms its solidus contour. The R2R1R2Q1 curve participates in the shaping of R2 solidus contour. Thus, the scheme makes it possible to determine the types (ruled, unruled, plane, etc) and to calculate number of all surfaces, to indicate their contours and to describe borders of all phase regions. As a result, the Co-Cu-CoS-Cu2S T-x-y diagram consists of 82 surfaces (Table 3) and 33 phase regions (Table 4; Figure 3; Figure 4).
The 3D model has been constructed on the base of experimental liquidus and solidus surfaces. It consists of the immiscibility cupola, six liquidus and six solidus surfaces of Co (A), Cu (B), CoS (R1), Cu2S (R2), Co4S3 (R3) (where the liquidus and solidus surfaces of Cu2S (R2) are more complex: both surfaces are divided into two parts -"up" and "down" (qR2_up, qR2_down , sR2_up, sR2_down in the Table 3).    Earlier the reference book of computer models of T-x-y and T-x-y-z diagrams of the basic topological types has been created [35]. It includes >200 3D models of T-x-y and 7 4D models of T-x-y-z diagrams. Each computer model is the prototype of the phase diagram with corresponding topology. In the case of the Co-Cu-CoS-Cu2S T-x-y diagram the construction of its 3D model incorporates the templates of the system with the binary incongruently melting compound and of the system with the univariant monotectic immiscibility.
In contrast to the simple surfaces of primary crystallization of copper and R1, R3, the liquidus surface of Co (A) has the complicate contour. It is formed by eight points A, eAR3, E, N, K, M, Q2, pAB (the point "max" denotes the curvature of the NE line and it is not considered in this case as the liquidus surface contour) (Figure 2b).
Since the contour of the Co (A) liquidus is formed by eight points, then the corresponding to it solidus must be given by eight points. However, they are only seven: A, AR3, AE, AK, AM(N), AQ2, AB. Therefore the basic innovation in this type computer models consists of the presence of the AKAM(N) pseudo-fold on this surface of solidus [28][29][30]. It is the directing curve of two ruled surfaces and it does not influence the smoothness of the Co (A) solidus surface. Its upper AK point has the critical K point temperature in the intersection of the immiscibility cupola with this liquidus surface. Thanks to the fold, the liquidus and solidus become topologically equivalent surfaces, because the AKAM(N) curve is the conjugate one with two parts of the arc NKM. Because of the liquid immiscibility to L1 and L2 under the cupola, which intersects the Q2E line of co-crystallization of A and R2 crystals, it is necessary in this, would seem routine, work to distinguish with what precisely liquid, L1 or L2, the Co (A), Cu (B), Cu2S (R2) crystals coexist. Since the monotectic reaction in the binary Cu-Cu2S (B-R2) subsystem is carried out in the form L1L2+R2, then it must be the two-phase region L2+R2 in the lower part (between the mM and pABQ2 lines under the qR2_up liquidus surface) and L1+R2 region (under the qR2_down liquidus surface) in the upper part of the Co-Cu-CoS-Cu2S (A-B-R1-R2) subsystem. Corresponding surfaces of liquidus and solidus (borders of Cu2S (R2)) primary crystallization beginning and finishing) are also divided into two parts.
The T-x-y diagram, besides the immiscibility cupola, six liquidus and six solidus surfaces, includes 39 ruled surfaces and 16 planes (simplexes of four complexes), corresponding to Q1, Q2, M(N) and E reactions.
After transformation of the scheme of uni-and invariant states from the tabular form into the graphic one, when the horizontal (isothermal) planes, corresponding to the M(N), Q1, Q2, E invariant reactions, and the ruled surfaces are designed, the T-x-y diagram prototype is obtained. It is converted into the real system T-x-y diagram 3D model, when concentrations and temperatures of base points (Table 1) are introduced, and the curvature of lines and surfaces is specified. Comparison of the 3D model ( Figure 5a) and experimental [18] (Figure 5b) isothermal lines on the liquidus surfaces shows that the model line 1335 o C adjoins the critical point K, and this is correct, because the temperature of this point is equal to 1334 o C, however, experimental line passes far from this point K.
There are considerable disagreements according to the form of the isothermal line 1080 o C on the liquidus surface of Cu2S (R2). The 3D model of this surface includes two lines at 1080 o C. The first one is located around the N point, since the temperature of the N point is equal to 1076 o C (Figure 5a). The second line at 1080 o C is located between the NmaxE (in the temperature interval of 1076-1091-837 o C) and Cu2S-eR1R2 (1131-941 o C) lines. As for the experimental version the isothermal line (1080 o C), it is located around the point of maximum with 1091 o C (Figure 5b). But it is impossible, as the temperature 1080 o C is on the Cu2S-eR1R2 line.
A main difference in the 3D model ( Figure 6a) and experimental (Figure 6b) isothermal lines of the solidus surfaces and connected with them ruled surfaces is caused by the already discussed above contradictions in the concentration coordinates of the apexes of the complexes, which correspond to the Q1 and Q2 quasi-peritectic reactions ( Table 1).
As a whole it is possible to consider that the 3D computer model adequately reproduces experiment, and therefore it can be used in the practical work, in particular, for designing any isopleths. For instance, two isopleths have been designed: parallel to the sides Co-Cu (A-B) S1   Table 1. Base points (weight portions), according to which the 3D model of the subsystem Co-Cu-CoS-Cu2S (A-B-R1-R2) has been designed Table 2. Scheme of uni-and invariant states of the subsystem Co-Cu-CoS-Cu2S (A-B-R1-R2) (Figure 2)*, k>A>K>R1>R2>pAB>m(n)>B>M(N)>Q2>eBR2>eR1R2>pR1R3>eAR3>Q1>E** Table 3. Surfaces of the T-x-y diagram Table 4. Phase regions (Figure 3; Figure 4)