In this article we will discuss about how to construct ternary diagram for solid solubility.

Many useful metallic alloys (stainless steels, aluminium-based age-hardenable alloys contain not just two important components, but three or more to obtain much greater improvements in the properties. Metals for high temperature services may contain ten important elements.

Thus, most commercial alloys are more complex than binary alloys. But it is difficult to represent phase relations in such multicomponent systems. That is why, relatively few three-component diagrams and almost no higher-component diagrams exist.

To simplify the representation of three components, pressure is fixed at one atmosphere (i.e., pressure is made constant), as three independent variables can only be specified, i.e., two to define the composition and the third to define the temperature (in ternary systems, % of the third component, say C = 100 – % A – % B).

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Consequently, a 3-dimensional model is required, with two of the three axes show changes in the compositions of the two components, and third axis shows changes in temperature. A two dimensional figure is required to plot the two independent compositions in a three component system, while the temperature axis is made perpendicular to it.

The most convenient figure is an equilateral triangle, Fig. 3.55, also called as Gibbs triangle. Each vertex represents a pure metal. Each side of the Gibbs triangle can be divided into 100 equal parts, representing 100% on the binary composition scale.

As vertex A represents pure metal A, B pure metal B and C represents pure metal C, the composition of alloy I (indicated point O) is given by, % A = PC, % B = RP and % C = RA. The base of the triangle (CB) opposite the metal A vertex represents 0% A (similarly for other components B and C). Thus, the composition of alloy I is 30% A, 40% B and 30% C. Lines parallel to the base line of the triangle indicate varying percentage of A, but one fixed % for one line. This is true for all lines.** **

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**Thus, in the Gibbs triangle: **

1. The vertex of the triangle represents one pure component A, B or C.

2. Points of binary alloys lie on the sides of the triangle.

3. A point inside the triangle represents a ternary alloy.

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4. Alloys represented by the points of a line parallel to one of the sides of the triangle have a constant composition of the component present at the opposite corner of the triangle.

5. Alloys represented by the points on the line passing through a corner of the triangle have constant proportion of the other two components.

6. Ternary diagrams also obey the Gibbs phase rule. Lever rule could be used in a three dimensional ternary diagram, but is normally very difficult to interpret. But the lever rule cannot be used in vertical-section diagrams called the isopleths.

However, it can be used in isothermal diagrams (horizontal sections cut through the three-dimensional temperature- composition diagram at a series of temperatures). One such diagram at the constant room temperature is illustrated in Fig. 3.55.

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The line, SL (which lies in a two phase region having liquid and solid), may be seen to represent a condition of an alloy of composition D, which has two phases present at the temperature under consideration. If S represents point on solidus and L on liquidus, their chemical compositions can be read from the Gibbs triangle (S = 20% A, 10% B, 70 % C; L = 40% A, 30% B, 30% C).

**The line SL can be used as a tie line to know the amounts of these phases such as: **

**Types of Derived Two Dimensional Diagrams from Ternary Diagrams****: **

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Three dimensional diagrams is not convenient for making detailed phase analysis.

**But, the information can be presented in two dimensions by any of the several methods: **

**(i) Liquidus Plot: **

It is a projection of the ‘plan’ of three- dimensional diagram, Fig. 3.58. Normally, liquidus temperatures can be plotted as isothermal contours, Fig. 3.61 (b). Thus, freezing temperature of the material can be predicted. Liquidus plot also gives the identity of the primary phase that forms during solidification for any given composition. Tie-lines cannot be used.

**(ii) Isothermal Plot: **

Simple composition triangles at constant temperature (pressure is already constant) are important in determining the phase constitution of alloy, Fig. 3.56 (d). Thus, isothermal (horizontal) sections cut through the three dimensional composition-temperature diagram at a series of temperatures can be obtained.

It is useful in predicting the phases, their amounts and compositions at that temperature. These diagrams generally provide the most satisfactory means for recording ternary equilibrium in two dimensions. These plots are thus more often used,

**(iii)** Vertical Sections (Isopleths), either from one corner of the space diagram or parallel to one side of the space diagram, Fig. 3.56 (c), though resemble binary diagrams (composition v temperature), but lack an important characteristic.

The tie lines do not usually lie in the plane of a vertical section, and thus cannot be used to obtain amounts and compositions at that temperature. However, vertical sections are quite valuable in showing the phases that are present in an alloy during equilibrium cooling and heating. The vertical sections also show the temperatures at which the various phase changes occur.

Fig. 3.56 illustrates the ternary diagram for a system with complete solid solubility with three methods of representing in two dimensions.

It is a system in which all three metals are completely insoluble in each other in solid state. Each two metals form binary eutectic, being illustrated on each side of the triangle, but all three form a ternary eutectic. Fig. 3.57 (a) illustrates the three dimensional view of the ternary diagram, whereas Fig. 3.58 shows the developed view, i.e., projection of liquidus on the base.

The liquidus of the ternary eutectic forms three separate smooth liquidus surfaces, extending inwards from the melting point of each pure metal and sloping downwards to form three valleys which meet at E, the ternary eutectic point. A projection of the liquidus surface on to the base is illustrated in Fig. 3.58.

A ternary eutectic forms in such a system but at a constant temperature, i.e., here L gives a mixture of A + B + C at a constant temperature (F = 3 – 4 + 1 = 0). A ternary alloy can start solidifying by freezing first, one of the components, [say here A-cooling curve 3.57 (b)] over a range of temperature, and then, from the liquid a binary eutectic forms over a range of temperatures (F = 3 – 3 + 1 = 1), and then in the end at a constant temperature, the liquid completes the solidification by forming a ternary eutectic as illustrated in cooling curve in Fig. 3.57 (b).

In Fig. 3.58, E_{1}A is the projection of a part of the liquidus line of binary diagram between A and B, whereas E_{3}A is of A and C. ‘A’ is common in them. Point E_{1} is the projection of the eutectic point of A and B binary diagram. E_{1}E illustrates the effect of element C on the eutectic point of A and B, whereas E_{3}E is due to metal B on eutectic between A and C.

As metal A is the first solid which forms along liquidus line E_{1}A in binary diagram of A & B, and along liquidus line E_{3}A in binary diagram of A & C, thus, the metal ‘A’ is also the first solid formed for all the alloys whose composition lies within the area E_{1}AE_{3}E. Thus, the surface E_{1}AE_{3}E (Fig. 3.57 a) is the liquidus surface for the primary solidification of A. Also, surface BE_{1}EE_{2} is the liquidus surface for primary freezing of B; while CE_{2}EE_{3} is for freezing of C.

Consider the solidification of alloy ‘O’ (Fig. 3.58). The freezing begins at temperature T_{1} with the solidification of pure A, and thus the liquid becomes richer in B and C. As more ‘A’ solidifies, the ratio of B to C in the liquid remains constant till the temperature falls to when the liquid composition is given by point P, in the valley between two liquidus surfaces, i.e., the composition of-the liquid varies between O to P (extrapolation of straight line AO) as temperature drops from T_{1} to T_{3}. Lever rule gives the amounts of phases at just attained temperature, T_{3} or the point ‘P’.

Point P is on the eutectic line E_{1}E (line E_{1}E is the line of solidification of binary eutectic (A + B). Thus, at the moment when the continuous freezing of A reaches the points ‘P’, binary eutectic (of pure A and of pure B) begins to solidify. As the solidification of this binary eutectic continues, the composition of the liquid now varies along PE (i.e., concentration of C continuously increases in liquid), until it reaches point ‘E’. The liquid of composition E at temperature t’ (Fig. 3.57 a) solidifies as a ternary eutectic of A + B + C.

Join point E and A by a straight line (Fig. 3.58). Any alloy in area EAE_{1}P freezes first with the formation of pure metal A, then the solidification occurs of binary eutectic of A and B, and then at E (temperature t’), remaining liquid solidifies as ternary eutectic of A + B + C. If the composition of the alloy lies in area EAE_{3}, then the sequence of freezing from liquid is first pure A, then eutectic between A and C, and then the ternary eutectic of (A + B + C) at t’ temperature.

Similar behaviour for B-rich, or C-rich alloys takes place, and the liquid reaches either the valley E_{2}E or E_{3}E. And ultimately the liquid is in equilibrium with solids A, B and C (F = 3 – 4 + 1 = 0) at t’ where liquid freezes into three phase mixture eutectic. Every ternary alloy ultimately produces the ternary eutectic at constant temperature, t’. Thus, the solidus, or ternary surface is a horizontal plane across the diagram (Fig. 3.57 a). Table 3.3 gives regions and the resulting structures (solidified).

**Few simple conclusions are: **

1. No ternary eutectic forms if it is a binary alloy composition.

2. No primary pure crystal forms if the composition lies on the lines of binary eutectics (E_{1}E, E_{2}E, and E_{3}E), i.e., solidification in these alloys starts directly with solidification of binary eutectic.

3. Alloys having compositions which fall on the lines connecting the ternary eutectic point, E, with the corners of the triangle, do not show solidification of binary eutectic at all, i.e., after the solidification of pure component, ternary eutectic forms directly to complete the solidification.

4. Alloy of composition of ternary eutectic (E) is the lowest melting alloy and solidification occurs at a constant temperature.

The best way, probably, for the analysis of ternary diagrams is to use horizontal section for a fixed temperature, cut through a three-dimensional ternary diagram. Five such diagrams at five different temperatures of the diagram shown in Fig. 3.57 (a) are illustrated in Fig. 3.59. Now analysis is made at a single temperature. Consider for analysis the alloy ‘X’ in the liquid region at temperature T_{1} as shown in isothermal diagram, 3.59 (a). It is evident that the alloy point lies in the liquid region. Thus,

Alloy ‘X’ at temperature T_{1}

Phase — liquid

Amount — 100%

Composition — given by point X (i.e. of the original alloy)

The other phase fields that exist at temperature T_{1} are not involved in this analysis. Let us take section diagram at temperature T_{2} (Fig. 3.59 b). The alloy X is still 100% liquid. Another alloy whose composition is given by point X_{1} is in two phase field, liquid and solid pure A. Tie- lines have been drawn.

The first tie line on each edge of L + A region is the boundary line of the figure at temperature T_{2}. The directions of the tie lines lying within a region vary ‘fan wise’, so that there is a gradual change from the direction of one boundary tie line to that of the other. No two tie-lines at the same temperature may ever cross.

**They run from liquidus to solidus. Tie-line is used for analysis of amount of phases present at this temperature T _{2} in the alloy X_{1}: **

Other diagrams can be used in the same way for analysis. Fig. 3.59 (e) gives phases present at room temperature under equilibrium conditions.

A single isothermal diagram does not tell the changes occurring with the fall of temperature, and thus, vertical sections are used. Fig. 3.60 illustrates one such vertical section of diagram 3.57 (a). An alloy of composition X_{2} is liquid above temperature T_{5}. Freezing of pure ‘A’ starts at T_{5} and continues, till T_{6} temperature is reached, where eutectic of (A + B) starts to solidify.

As the decrease of temperature takes place, and more of eutectic (binary) solidifies till at T_{e}, the remaining liquid solidifies as ternary eutectic of A + B + C. Tie-lines cannot be used here to calculate the amount of phases.

**Ternary Diagram Showing Partial Solid Solubilities****: **

The Fig. 3.61 (a) is a ternary three dimensional phase diagram of metals, each having partial solid solubility in the other. Fig. 3.61 (b) is the liquidus plot with temperature contours, while (c) shows the isothermal section at room temperature.

**Problem: **

Using diagrams of 3.61, find the following in each of ternary alloys-

1. 80% B, 10% A, 10% C,

2. 10% A, 30% B, 60% C,

3. 40% A, 20% B, 40% C

(a) Liquidus temperature

(b) Primary phase that solidifies

(c) Phases at room temperature.

**Solution: **

**For Ternary Alloy 1: **

Located point is at I and liquidus isotherm give its liquidus temperature ≈ 400°C. The primary phase as indicated in diagram is β. This alloy at room temperature (Fig. 3.61 c) has 100% β.

**For Ternary Alloy 2: **

Located point for this composition is II (Fig. b), and liquidus temperature is ≈ 275°C. The primary phase is ϒ. At room temperature, the alloy has structure consisting of β + ϒ.

**For Ternary Alloy 3: **

Located point for this composition is III, and thus the liquidus temperature is 350°C and the primary phase is α. The structure at room temperature consists of α + β + ϒ.

The three dimensional ternary phase diagram is shown in Fig. 3.62 (a). Four isothermal section at T_{1}, T_{2}, T_{3}, T_{4} (Fig. 3.62 a) temperatures are shown in Fig. 3.62 (b, c, d and e). Let us do phase analysis of alloy X with the help of these isothermal plots.

At temperature T_{1}, the alloy X (A- 60% B, 20% C) (Fig. 3.62 b)

Phase = liquid

Amount = 100%

Composition of phase = 20%A, 60%B, 20%C

Looking at diagram in Fig. 3.62 (a) when the alloy X (60% B, 20% C) reaches its liquidus temperature, freezing starts of solid solution beta, which is a solid solution of A and C in B. Thus,

**At temperature T _{2} (Fig. 3.62 c), the alloy X (60% B, 20% C): **

As the temperature decreases further, freezing of eutectic of beta and alpha begins. Thus, At T_{3} temperature, the alloy X (60% B, 20% C) (Fig. 3.62 c)

As the temperature drops further, liquid freezes completely, and thus, the alloy is left with two phases α and β. Thus,

**At temperature T _{4}, the alloy X (60% B, 20% C) (Fig. 3.62 d):**

We have discussed very simple ternary, diagrams. The drawing and analysis of most metallurgical important ternary diagrams is extremely difficult. Understanding of ternary diagrams through the isothermal sections is probably the easiest method to solve most problems.