In this article we will discuss about:- 1. Introduction to Phase Diagram 2. Phase Rule 3. Prediction 4. Uses 5. Limitations.

Introduction to Phase Diagram:

As the properties of a material depends to a large extent on the number, amount, type and the form (shape and the size) of the phases present, and can be altered by changing these quantities, thus, the study of phase relationship plays an important role in the understanding of the properties of the materials.

Actually, the state of a material depends on the variables/conditions like composition, temperature and pressure. The best way to understand these effects is through phase diagram, or equilibrium diagram. The equilibrium diagram is a map which gives relationship between phases in equilibrium in a system as a function of temperature, pressure and composition.

It gives a “blue print” of the alloy system from which we can anticipate at what compositions the alloys are likely to have useful properties, what treatments must be given to alloys to develop their properties to the best effects, and what treatments are likely to be harmful and must be avoided.


Equilibrium diagrams also called constitutional diagrams illustrate the stable states of a metal, or alloys, i.e., those phases which have lowest free energy under given conditions, which means phases obtained under very slow cooling, or heating rate. When the graphical representation deals with phases which are in equilibrium with the surroundings, it is called an equilibrium diagram, otherwise, it is called a phase diagram. In the iron-carbon system, stable phases are solid solution of carbon in iron and graphite.

However, under normal conditions, the solid solution and the compound, Fe3C (cementite) are formed. The graphical representation of Fe-Fe3C system is a phase diagram (or a metastable diagram), while that of iron- graphite is an equilibrium diagram.

The study of microstructure of a material may reveal phases which may, or may not correspond to those given in equilibrium diagram, and in fact, the heat treatment given to a material more often results in phases other than those in the diagram. Inspite of that, the study of phase diagrams is an important beginning to feel and understand insight in the processes of control of the microstructure.

Phase Rule:

The changes in the number of phases in an alloy under equilibrium conditions are expressed by the Gibb’s phase rule. It is the most informative law for alloy behaviour. This law is the basis for checking the accuracy of constitutional diagrams, and also for rationalising the behaviour of alloys for which diagrams are not available.


Thermodynamic reasoning has been used to derive this law. However, it is important to realise that there are no exceptions to the phase rule, and any conclusion which actually violates it, has been proved to be incorrect.

The phase rule is stated by a very simple equation:

F = C – P + 2 … (3.1)

where, F is the number of degrees of freedom possessed by the system, C is the number of components present in the system, and P is the number of phases present in the system. It is essential to understand the meaning of each term used here.


1. System:

A system may consist of a substance, or a number of substances, which are completely isolated from the surroundings, i.e., the substances present in it are free to react only with each other. The system under study is subjected to changes in overall composition, temperature, pressure, or total volume.

A system may consist of gases, liquids, or solids, or any combination of these, and it may contain only one chemical element, or compound, or any number of elements (metal or non- metal, or both), or compounds together.

2. Equilibrium:


It is the state of minimum free energy under any fixed set of conditions of composition, temperature, pressure and total volume. Under one set of conditions, even the slightest change in the amount, composition, volume, or physical state of any substance present within it, causes an increase in its free energy, unless the system is put under some other conditions.

Phase rule truly applies only to systems which are under conditions of true equilibrium. Often, wide departure from equilibrium conditions under industrial practice, intentional, or unintentional result in microstructures and thus, the properties, which at equilibrium, would be impossible. Phase rule does not apply then.

3. Components:

The components of a system refer to the smallest number of stable individual substances which describe completely the chemical composition of the system at a temperature or pressure that may be of interest.


Thus, the components of a system may be elements, ions, or compounds. In the ice-water-steam system, the only component is H2O. In NaCl-KCl system, as both the compounds are completely stable over the common range of temperatures and pressures, there are two components, inspite of the fact that three chemical elements are present in it.

Among metallic alloys, some truly stable compounds (FeO, Cu2O, MnS etc.,) are formed and thus, each of these is considered a component of the system in which it exists. The intermetallic compounds are not counted as components as these invariably, decompose into their component elements. In metallic systems, the number of components is simply the number of different chemical elements present, even if intermetallic compounds exist at certain specific compositions.

In Cu-Zn system, the elements Cu and Zn are the components (two components or binary alloy system). In the Fe-C system, iron and graphite can be the components, but it may be often convenient to choose iron and iron carbide (cementite) as the components. The Fe-C-Mn is three component, or ternary system; Fe-C-Cr-Ni is four component or quaternary system; Al-Cu-Si-Mn-Mg is a five component or quinary system.

4. Phases:

In the classic definition, a phase is a physically-distinct, chemically-homogeneous and mechanically-separable region of a system. There should be some detectable difference in composition, physical state, crystal structure, or properties to differentiate one phase from another.

The three common states of matter constitute three phases. For example, ice, water and steam are three different phases because they differ in physical state, although, they are identical in composition (H2O). Alpha-iron and gamma-iron are two difference phases because alpha is BCC, and gamma-iron is FCC in crystal structure, although both are solid, pure iron.

All gases are completely miscible in each other in all proportions, and thus, gaseous state is taken as one phase. A liquid solution, such as common salt in water, is also a single phase, but a liquid-mixture, for example of oil and water, essentially insoluble, forms two separate phases. A solid solution, where atoms are mixed within the unit cells, is a single phase. In solid alloys, each different type of crystal present in the system is a separate phase. Several phases may be present, in them.

5. Degrees of Freedom:

In general, it is the number of independent variables whose values must be stated in order to define exactly the conditions of the system. The degree of freedom refers to the number of variables which can be changed independently without bringing out the disappearance of a phase, or the formation of a new phase.

Thus, it represents the number of external and internal variables such as temperature, pressure and concentration of the components, which can be changed without changing the number of phases in the system. Of these variables, two are external parameters, temperature and pressure.

The internal variables are to specify the compositions of the phases present. To clarify this fact further, it must be borne in mind that the overall composition of a system is fixed when that system is isolated, i.e., it is not a degree of freedom for it. But, when two or more phases are present in a system, the concentration (composition) of various phases becomes another degree of freedom, which could be more than one.

The number of possible degrees of freedom represented by concentration varies with the type of system, but is always one less than number of components present in it (explained later). That is why, in one component system, there is no degree of freedom in concentration.

As the compositions are normally expressed as weight, or atomic percent, the number of variables needed to know the composition of a phase is C – 1, where C is the number of components in the system (the composition of the last component is obtained by subtracting the sum of C – 1 from 100).

If the system has P phases, then the total number of composition variables is P (C – 1). Thus, the total number of variables including the two external variables (temperature and pressure) is P(C – 1) + 2. Out of these, the number of independent variables gives the degrees of freedom, F, i.e., the degree of freedom cannot become more than the total number of variables, i.e.,

F = C – P + 2 ≤ P (C – 1) + 2

For a single phase system (putting P = 1),

F = C + 1 ≤ C + 1

the degrees of freedom and the total variables are equal. The degree of freedom decreases as the number of phases increase, but the degree of freedom cannot be less than zero. This puts a limit to the possible maximum number of phases that can exist in equilibrium in a given system.

If the number of degrees of freedom is zero (it is then called invariant), then the variables (temperature, pressure, concentration) cannot be changed without disturbing the equilibrium, or without changing the number of phases.


Enumerate the degrees of freedom of a three- component system with various numbers of possible phases.


In phase rule, C = 3 and as the minimum number of phases can be one, thus for different values of P starting from P = 1, the various degrees of freedom are calculated

A three-component system cannot have more than 5 phases in equilibrium.


A binary phase diagram has been drawn at constant one atmospheric pressure. What is the maximum number of phases that can coexist for at least one degree of freedom.


As the pressure variable has been kept constant, thus, the modified phase rule to be used is:

F = C – P + 1

where C = 2.

For at least one or more degrees of freedom:

The degrees of freedom (with pressure already fixed) cannot be more than two.


Find the number of degree of freedom for a system having equal number of components and phases.


Putting C = P in the phase rule,

F = C – P + 2 = 2


If four distinct phases are observed in a laboratory specimen of a binary alloy made at one atmospheric pressure. Is such an observation possible? Explain.


Using modified phase rule (as pressure is already constant)

F = C – P + 1

= 2 – 4 + 1

= – 1

As the degrees of freedom cannot be less than zero, such an observation is not possible under equilibrium conditions. The alloy can show four phases under non-equilibrium conditions for which phase rule is not valid.


What can be the maximum number of phases coexisting in equilibrium in a binary system?


To maximise the number of phases, the degrees of freedom should be minimum, which can be, F = 0, thus,

F = C – P + 2

O = 2 – P + 2

P = 4


What can be the maximum possible degrees of freedom in a binary system?


To maximise the degree of freedom, the number of phases should be minimum, which can be, P = 1, and thus,

F = C – P + 2

= 2 – 1 + 2

= 3

Prediction of Phases Diagram:

A phase diagram can be used to obtain again the original data used in drawing it.

That means, for a definite composition of an alloy and at a definite temperature, it is possible, under equilibrium conditions to know:

(i) The phases present there,

(ii) Chemical composition of phases and

(iii) The amount of each phase.

Prediction of Type of Phases:

The equilibrium diagram can be used to know the type of phases present, under equilibrium conditions only, if a temperature and the composition of the alloy are given. Such a pair of values (temperature, and composition) locates a point in the diagram. Such points are very commonly used to understand the phase diagrams.

For example, the state of the alloy of composition 80 Ni, 20 Cu can be found out with reference to a certain temperature, that is, this alloy at temperature 1500°C is indicated by point Z, but at 1100°C is indicated by point X in Fig. 3.6. Once such a point has been located in the diagram, it is easy to know which phase or phases are present.

The alloy has phases corresponding to the phase field in which this point lies. For example, the 80 Ni 20 Cu alloy at 1500°C as indicated by point Z, has only one phase, the liquid solution, whereas it at 1100°C as indicated by point X, has one ;phase, the solid solution. This alloy at 1400°C as indicated by point D consists of a mixture of liquid solution and solid solution. Normally, a point on the liquidus, or solidus line is assumed to have two phases. .

Prediction of Chemical Compositions of Phases:

When in an alloy, only a single phase is present, as at points X and Z in Fig. 3.6, the composition of the phase is tire same as that of the alloy. The alloy (80 Ni, 20 Cu) at point Z is completely liquid, and thus, the chemical composition of the liquid solution is 80 Ni, 20 Cu.

This alloy at X is completely solid solution, which has a chemical composition of 80 Ni 20 Cu. (It is unnecessary to draw a tie line when the alloy composition-temperature point lies in a single-phase field, which is needed for a point lying in two-phase field).

Another alloy (40 Ni 60 Cu) at temperature 1250°C is indicated by point Y in Fig. 3.6. At Y, the alloy has two phases present, and each phase has a different composition, and neither of these compositions is same as the original composition of the alloy. If the two phases are to be in equilibrium, they must be at the same temperature.

Thus, the points representing these phases must lie at the temperature of original point (i.e., 1250°C). The exact points characterizing the liquid and the solid phases are given by the intersections of the temperature-horizontal line with the liquidus and solidus lines, respectively. Thus, the composition of the liquid phase in the 40Ni60Cu alloy at 1250°C is given by point 1 (32 Ni, 68 Cu), while that of solid phase is given by point 2 (48 Ni 52 Cu).

The composition of a given phase is read on the composition axis (x-axis) as indicated by the vertical dashed line from point 1 to x-axis for liquid. The horizontal (constant temperature) line connecting the points that represent the two different phases is called a tie line.

Actually for knowing the compositions of the two phases, a horizontal line (representing that temperature) is drawn through the alloy composition-temperature point up to the boundary lines of the two-phase fields. This is called tie line. The two intersection points respectively give the compositions of the two phases at the end of the tie line (after dropping these points on the composition axis and the compositions are read directly).

Prediction of Amounts of Phases- Lever Rule:

If an alloy has only a single phase, such as at point z in Fig. 3.6, weight of the phase is equal to the weight of the alloy. This alloy at z has 100% liquid phase. If the weight of the alloy is 100 grams, the amount of liquid phase at z point is 100 grams. If the weight of the alloy is 5 grams, even then the alloy is 100% liquid phase having a weight of 5 grams.

If the alloy has two phases, and if the total amount of alloy is known, then it is possible to determine the amount of each phase that is present at equilibrium at a given temperature. If the total weight of alloy is 100 grams, then the numerical calculations are simple. However, similar convenience in calculations can still be had for any weight of the alloy by expressing the amount of each phase as a percentage of the weight of the alloy, or in terms of fraction of the alloy.

Lever rule is used to find the amount of phases present in only the two-phase regions of the binary equilibrium diagrams. This rule can be derived based on the principle of conservation of mass. The relative amounts of the two phases in an alloy depend on their chemical compositions relative to the composition of the alloy. This is because, the total weight of one of the metals, and say metal B in the alloy must be equal to its combined weight in the liquid and in the solid phases.

Let Wα and WL be the fractional amounts by weight of solid and liquid, respectively. Then, from conservation of mass, the sum of the fractional amounts must be equal to unity,

Lever rule states that the relative amount of a given phase is proportional to the length of the lever arm on the opposite side of the alloy point of the lever, as expressed above. The weights of the two phases are such that they would balance as illustrated in Fig. 3.8, i.e., amount of solid x its lever arm = Amount of liquid x its lever arm. Assumed weight of alloy = 100 gram. It is true that 68.75 x 5 = 31.25 x 11.

Lever rule can be used now to know the amount of phases in an alloy (27 Ni, 73 Cu) at 1200°C as illustrated in Fig. 3.7. For the given composition and the temperature, locate the point ‘O’, which lies in two-phase field containing liquid and solid. Through ‘O’, draw a horizontal line for temperature 1200°C. This line called the tie-line extends only up to the two phase boundaries. The tie- line ‘pon’ is also called the lever, having the fulcrum at ‘O’, the composition of the alloy.

Both the parts of lever are called the lever arms. As the composition of two points p (22% Ni) and n (38% Ni) can be read from the x-axis of the diagram, thus, the length ‘po’ (= a) can be calculated as 27 – 22 = 5 units, and of ‘on’ (= b) as 38 – 27 = 11 units. The length of lever is 38 – 22 = 16 units. Thus, applying lever rule,

Lever rule can also be applied when two solid phases are present under equilibrium conditions. The application of lever rule can be summarised. For the given composition of alloy and its temperature, locate the point. If it is in two phase region, draw a tie-line (lever) through this point.

The intersections of the tie-line with the boundaries of the two-phase region determine the composition of respective phases (the intersection point for solidus indicates the composition of the solid, and with liquidus indicates the composition of the liquid. The same applies for other phases).

Determine the three distances, a, b and l (in units of percent composition) as illustrated in Fig. 3.8. The fraction of phase corresponding to point p (liquid here) is given by b/l, while corresponding to point n (solid here) is given by a/l. It is clear that the amount of a phase is proportional to the length of the kever arm on the opposite side of the fulcrum (the alloy composition).

Lever rule cannot be applied at an invariant temperature as three phases are in equilibrium. It can be applied above or below the invariant temperature. (Fraction of eutectic mixture can be calculated when the end of lever arm must end at composition of eutectic).

Equilibrium Cooling of a Solid Solution Alloy:

Equilibrium cooling (even heating) means a very slow rate of change of temperature so that at all times, equilibrium conditions are maintained in the system. In the Cu-Ni isomorphous system, suppose we have an alloy containing 70 Ni, 30 Cu. A vertical line is drawn at this composition as in Fig. 3.9. This alloy above point X to 0 has single-phase liquid solution (70Ni, 30Cu), and below point Y3 has single phase solid solution (70Ni, 30Cu), but between points X and Y3 contains two phases-one liquid and other solid.

This alloy has homogeneous liquid solution at point O. When cooled, no change occurs till its temperature becomes T (its liquidus temperature), where solidification begins with nuclei formation. A temperature horizontal through X and T cuts the solidus at point Y, which is the composition of the first solid (nuclei) formed (the composition of point Y can be read by dropping a vertical line from Y to x-axis).

As the first solid nuclei have higher nickel content (as given by point Y ≈ 90Ni) than the content in liquid alloy, the remaining liquid gets depleted of nickel and correspondingly gets richer in copper as more solid forms. As the alloy cools further below temperature T, more solid forms taking the shape of dendrites (tree like structure).

This alloy at temperature T1 (given by point O) with tie-line X1 OY1 drawn (through O at temperature T1 ≈ 1365°C) has homogeneous liquid having chemical composition given by point X1 (59 Ni 41 Cu) and homogeneous solid (as dendrites) having chemical composition given by point Y1 (85 Ni 15 Cu) under equilibrium conditions.

Having obtained the chemical composition of the solid as well as liquid phase, lever rule could be used to know the weights of these phases, by using the same lever X1 OY1 with its two arms X1O and OY1:

As the alloy is cooled slowly further, more solid forms till temperature T2 is attained. Under equilibrium conditions the chemical composition of homogenous liquid phase is given by point X2 and of homogeneous solid phase by point Y2. The chemical composition of liquid phase, as given by X2 is 46 Ni 54 Cu, and of solid phase as given by Y2 is 78 Ni 22 Cu.

Their relative amounts are:

As the cooling continues further under equilibrium conditions, more solid forms till the solidification is completed at T3 to result in number of grains of homogeneous solid solution as illustrated schematically in Fig. 3.9, having composition of the alloy (70% Ni, 30% Cu). The last traces of liquid which solidified had the composition given by point X3.

Few conclusions are:

(i) The solid solution alloy freezes over a range of temperature (as also illustrated in Fig. 3.4). As the temperature decreases in this range, the amount of solid increases but the amount of liquid decreases.

(ii) As the temperature drops from T to T3, the composition of the liquid varies along the liquidus, i.e. changes from X to X3. The composition of the solid varies along the solidus, i.e. changes from Y to Y3. The composition of both the phases shifts in the some direction, i.e., both liquid and solid become richer in copper.

This happens because with the fall of temperature, there is also a corresponding change in the amount of phases, so that the amount of copper (also nickel) in liquid and the solid phase added together is equal to the amount of copper (also nickel) in the original alloy.

(iii) As the temperature drops, more and more solid forms, there must be a continuous change of composition taking place in the already formed solid, because at temperature, T1, the uniform composition of solid is 85% Ni and 15% Cu, but the homogeneous composition at temperature T2 becomes 78% Ni and 22% Cu.

This can occur if diffusion takes place continuously, where nickel atoms move outwards from the centre of the dendrites to be replaced by copper atoms which move inwards. The equilibrium compositions are attained by lot of such diffusion of atoms. The importance of diffusion of atoms becomes more clear, when the alloy attains the temperature T3, when the last traces of liquid (of composition X3 = 38 Ni, 62 Cu) solidify at the crystal boundaries and are immediately absorbed by diffusion so that the solid as a whole becomes homogeneous and of composition Y3 (70% Ni, 30% Cu) which is also the composition of the original liquid alloy.

The initial liquid phase and the final solid phase have same uniform composition because of principle of conservation of matter. Thus, the equilibrium cooling must be at a very slow rate so that at each step of temperature enough diffu­sion occurs to result in homogeneous solid.

The copper-nickel phase-diagram is a typical solid-solution system. Similar alloy systems (such as Ag-Au) have similar pattern of solidification behaviour, and characteristic varia­tions in mechanical and physical properties appear with changes in chemical composition. An understanding of such patterns is useful in making rough predictions of the properties of alloys of various compositions even in complex phase diagrams.

Fig. 3.10 illustrates variations in properties of industrial copper-nickel alloys as the chemical compositions of the solid solutions changes from 100% copper to 100% nickel. Properties like lattice constant, specific heat, specific volume, and thermal expansion vary almost linearly with composition. But, properties like tensile strength, hardness, yield strength, electrical resistivity show a maximum, while ductility shows a minimum.

Coring in Solid Solutions:

The equilibrium solidification of a solid solution alloy assumes that the rate of cooling is ‘infinitely slow’, so that complete equilibrium by means of convection (in the liquid) and diffusion (in the solid) could be achieved at each stage of the process. Under actual conditions of industrial casting practice, however, the rate of solidification is much faster than the rate of diffusion, and thus, the solid phase is unable to attain uniformity and equilibrium.

The core (centre) of each dendrite (or a grain) contains higher amount of high melting metal than the surface of the dendrite (or inter-dendritic region). In copper-nickel solid solution (for example 70 Ni 30 Cu alloy), the core has more nickel content (composition of point Y in Fig. 3.9) than the equilibrium amount, so that outer fringes (inter-dendritic spaces) contain correspondingly more copper than the equilibrium content as schematically illustrated in Fig. 3.11.

This variation of composition from the core of the dendrite to the interdendritic space (content of nickel decreases continuously from the core to the centre of the interdendritic space) is called dendritic-segregation.

Dendritic-segregation on a microscopic scale is called coring. Since the rate of chemical attack with an etchant varies with the composition, proper etching of polished surface usually reveals coring as illustrated in Fig. 3.12 (a).

Let us examine the solidification of 70% Ni, 30% Cu alloy when cooled at a faster rate than equilibrium cooling. Under equilibrium conditions, the composition of the liquid phase varies continuously along the liquidus and that of solid along the solidus. Even under non-equilibrium conditions, there is not often much undercooling of the liquid, and diffusion and convection currents within the liquid are usually rapid enough to maintain it at very nearly the composition indicated by liquidus curve XX1X3 (Fig. 3.13).

The solidification of this alloy begins at T, freezing a solid solution of composition Y. As the temperature drops, then at T1 the liquid have composition of X1 and the solid solution now freezing is of composition Y1.

Since the diffusion is too slow (cooling rate is fast) to keep pace with the crystal growth, not enough time is being allowed to achieve uniformity in the solid, thus its average composition may be between Y and Y1, say Y1.

As the temperature decreases, the average composition of the solid solution departs still further from equilibrium conditions. It appears that the composition of the solid follows a ‘non-equilibrium’ solidus line YY2 K5 Y4 shown dotted in Fig. 3.13. Under equilibrium cooling, the solidification should be complete at however, since the average composition of the solid solution at this temperature is and has not become equal to the average composition of the alloy, some liquid must be present in the alloy (which can be calculated by using lever rule with lever length of X3 Y5 and fulcrum at Y3).

The solidification is completed at T3 as at this temperature the composition of the solid solution Y4 coincides with the alloy com­position (X). The last traces of liquid to solidify have the composition X4, much richer in copper than obtained under equi­librium conditions.

Faster the rate of cooling, the greater is the composition range in the solidified alloy, i.e., there is an increased temperature range over which liquid and solid are present, and last freezing occurs at a much lower temperature than predicted by phase diagram. This solidi­fying liquid is much richer in low melting metal (here copper). The real microstructure of the alloy may differ substantially from the equilibrium structure.

The degree of coring in an alloy system depends on the diffusivity of the two unlike atoms in the solid solution and the time available for diffusion. The latter in turn depends on the solidification rate, being short for chill-casting (more intense coring), and relatively long for sand-castings because of lower heat transfer rates of sands.

Coring takes place not only in an isomorphorous system like Cu-Ni having complete solid solubility, but almost in all equilibrium diagrams exhibiting certain ranges of solidification. Greater is the gap between liquidus and solidus, more pronounced is the coring, which becomes still intense if the temperatures of freezing of alloys are low.

Though diffusivity of both types of atoms at such temperatures is important, but normally it is very slow. The process of diffusion at low temperatures is very slow, more so because the number of vacancies, which aid diffusion, decrease exponentially with the drop of temperature. That is why, cast tin-bronzes are invariably found cored even the commercial castings of brasses and stainless steels are cored.

Coring frequently results in weakness and brittleness of castings, because of different composition near the grain boundaries. More serious (in many cases) is the susceptibility to corrosion of such cored structures to cause intergranular corrosion. In general, cored castings lack uniformity of mechanical and physical properties.

The problem of coring can be solved by one of the following methods:

1. Use of Slow Cooling Rates during Solidification of the Alloys:

Slow, or almost equilibrium cooling rates prevent the formation of coring because then, enough time is available for the diffusion to occur to homogenise the solid phase. But this method is normally not adopted commercially, because slow cooling rates produce coarse-grained castings, which develop inferior properties.

As more time is needed, the productivity decreases. Moreover slow cooling rates may not be practically feasible if a casting has been designed to be produced, say by chill-casting method. Even sand-casting method is fast enough to give coring.

2. Homogenisation of Cored Castings:

Homogenisation is a prolonged annealing treatment at a temperature high enough so that diffusion within the alloy is relatively rapid, but still safely below the depressed solidus, so that burning of the alloy does not occur. For example, a 70% Ni 30% Cu alloy casting must be heated below T3 temperature (Fig. 3.13), below the depressed solidus.

If the temperature of homogenising is fixed (looking at solidus temperature in the equilibrium diagram), below T2 and above T3, the liquation of the grain boundaries occurs, called ‘burning’, impairing the shape and physical properties of the casting.

A burnt alloy is permanently damaged, because oxidation occurs at the grain boundaries. The alloy cannot be salvaged by any heat or and mechanical treatment. However, over-heated alloy, due to heating below but very close to depressed solidus, which shows decreased ductility and toughness, can be salvaged by proper heat, or and mechanical treatment.

The time required to homogenise a given cored alloy varies with grain size, extent of coring, temperature of diffusion, diffusion rate at this temperature in the alloy. The homogenising time can in some cases be reduced by first cold-working and then doing homogenising annealing.

As recrystallisation occurs during annealing and recrystallisation increases diffusion rates. More important is that cold working reduces interdendritic distance, that is, the distance through which atoms have to diffuse. Thus, homogenisation is faster. Most hot-worked, or cold-worked and annealed products have homogeneity which approaches the equilibrium state. Fig. 3.12 (b) illustrates homogenised casting.

Size Factor Effect:

The four Hume Rothery rules for the formation of primary substitutional solid solutions are crystal structure effect, the electro-chemical effect, relative valency effect and atomic size factor. When the first three rules are observed, but as the size difference between the atoms of two component metals A and B increases and approaches 15 percent, the equilibrium diagram changes from that of the copper-nickel type to one of the eutectic system with limited primary solid solubility as illustrated in Fig. 3.14.

This is because with the increasing atomic size difference, distortions produced in the parent lattice increases to produce a miscibility gap, and to ultimately the partial solid-solubility type of eutectic phase diagram.

Some alloy systems show a minimum, or may be a maxima in the phase diagram (Fig. 3.14 and 3.15), i.e. liquidus and solidus curves are tangent to each other and at a constant temperature line at the point of intersection. Such points are called congruent points. Alloy of such a composition is called a congruent- melting alloy.

A congruent-melting alloy freezes at a constant tempera­ture. Its cooling curve resembles that of a pure metal, except that the resulting solid is not a pure metal but is a solid solution. Gold-nickel (Fig. 3.15 a) is one typical case showing congruent point at 950°C with 82% Au, 18% Ni as congruent composition.

When the atomic size difference is about 15% as in Au-Ni system, a two-solid-phase field appears in the phase diagram at lower temperature and is called a miscibility gap. Au-Ni system shows formations of solid solutions in all proportions. But below 812°C and inside the curve (Fig. 3.15 a), two phases, α1 and α2 are stable. Here, α1 is a solid solution of nickel in the gold lattice, and α2 is a solid solution of solubility of gold in nickel lattice.

As the atomic size difference is large, little distortions in the parent lattice of the other metal, which restricts the solid solubilities. Both α1 as well as α2 are FCC, but have difference in lattice parameters, colour, etc. Separate crystals of the two phases are formed with high angle grain boundaries between them.

Fig. 3.15 (b) illustrates a complete solid solution forming system having a maximum in the curve. Lithium-magnesium system illustrates such a maximum at 601°C at 13% lithium.

It is good to now compare Fig. 3.14 (c) and (d). As atomic size difference becomes more (Fig. 3.14 d), the miscibility gap and the solidus line intersect, i.e. minima of curve of Fig. 3.14 (c) gets lowered to intersect the raised miscibility curve in a horizontal temperature line to result in eutectic type diagram (explained below).

The eutectic point, thus, is equivalent to the minimum point. In Au-Ni system, a congruent alloy (alloy with composition of minima in the curve) on cooling first solidifies as a single homogeneous solid solution at 950°C, and then, on further cooling, decomposes into two solid phases as it crosses the miscibility curve.

Whereas, a eutectic alloy in Fig. 3.14 (d) freezes directly from liquid into two-phase mixture. In a eutectic system, the component metals need not have same crystal structure, or similar chemical nature.

Uses of Phase Diagrams:

Phase diagrams are of immense importance to a metallurgist.

Most of the purposes, as summarised below:

1. To Predict the Temperature at which Freezing, or Melting Begins, or Ends for any Specific Alloy Composition in an Alloy System:

Draw a vertical line representing the composition of the specific alloy, say X in Fig. 3.39. Its intersections with the solidus (T2) and liquidus (T1) indicate the temperature below which, at equilibrium, the alloy is completely solid, and above which, it is completely liquid respectively.

On heating, melting begins at T2 and is completely liquid above T1 temperature. If an alloy is to be cast, then, the temperature of the molten alloy has to be higher than T1. In order to fill the mould completely before freezing blocks of any thin section in it, the alloy should be at least 50°C higher than its liquidus temperature, T1.

2. To Predict the Safe Temperature of Working, or Heat Treatment:

The temperature of hot working, or heat treatment should be less than (at least 30°C) its solidus temperature to give allowances for the impurities present in it and for the temperature fluctuations in the furnace. Heating the alloy above T2 temperature, causes partial melting, called burning of the alloy.

The ‘sweat out’ molten metal leaves behind voids whose interior surface gets oxidised (which cannot be welded during hot working). A burnt alloy is good only for scrap. If the alloy is in cored state, then a homogenising temperature above T3, shall again cause ‘burning’ of the alloy. A safer temperature is T4.

If for age hardening, the solutionising temperature T6 is chosen instead of T6, grain growth of α-phase occurs which cannot be refined later on.

3. To Determine the Number of Phases, Types of Phases, Composition of Phases Present in any given Alloy at a Specific Temperature:

The primary function of an equilibrium diagram is to graphically show the extent and boundaries of composi­tion-temperature regions within which an alloy exists as a single phase, or as two-phases. Thus, the fields of the diagram are so labelled that the number and the general nature of the phases present, under equilibrium conditions, in a specific alloy at a particular temperature are indicated on the diagram.

If the point with coordinates of a specific composition and a particular temperature lies in single-phase field, then the alloy is either unsaturated homogeneous liquid, or solid solution with composition as of the alloy.

Such a point in a two-phase field indicates both the phases to be saturated solutions, which could be liquid solutions, solid solutions, or a liquid and a solid solution. If the composition of the alloy is changed at the same temperature but still being within the two-phase field, the number, type, or the composition of the phases do not change but their relative amounts get changed.

A horizontal line in a binary phase diagram indicates a particular temperature and a range of alloy compositions at which three-phases can coexist at equilibrium. Such a line separates either a two-phase field from some other two-phase field which has only one phase in common with it, or a two-phase field from a one-phase field that is different from either of these two phases.

4. To Calculate the -Relative Amounts of the Phases Present in a Two-Phase Alloy:

Lever rule has been used to calculate the amounts of the two phases present in a two-phase field.

5. To Describe the Freezing or Melting of an Alloy:

Cooling of a number of alloys from the molten state to room temperature is described with the help of the equilibrium diagram. During slow heating of the alloy, the changes would be exactly reversed.

6. To Predict the Microstructure of an Alloy at any given Temperature:

It is a major advantage of equilibrium diagrams to make usefully accurate predictions of the microstructure developed in an alloy at a specific temperature, or after an actual or proposed heat treatment. It is more important to a metallurgist as the microstructure controls the properties of an alloy.

For example, the shape of the pure element which is separating with the fall of temperature is quite important. Zinc has HCP crystal structure and strongly anisotropic. When zinc crystals are growing freely in liquid, growth tends to be more along one or another of its close packed directions.

Thus, there develops acicular (needle-shaped) crystals. Zinc in a eutectic mixture does not find the same opportunity for free growth. Thus, it is present as finely distributed phase with another phase in eutectic mixture.

7. To Predict the Possible Heat Treatment which can be Given:

The presence of a solvus line in a phase diagram, which if shows decrease of solid solubility (‘OD’ in Fig. 3.39) with the decrease of temperature indicates the chance of giving precipitation-hardening heat treatment to the alloys in the system. The presence of eutectoid reaction in a phase diagram helps to predict possible heat treatments like annealing, normalising, or hardening.

It is possible to predict what heat treatments are likely to be harmful and must be avoided. For example, if the shape of an alloy is not to be changed by mechanical working, and if no phase transformation occurs during heating, or cooling, then heating such an alloy to high temperatures shall coarsen the grains with resulting inferior properties developed in it. The coarse grains cannot be refined again.

8. To Choose the Composition to Develop Best Properties:

The composition of the alloy can be chosen which gives the best properties. An alloy having maximum solute content indicated by the solubility limit by the solvus line can develop maximum hardness by precipitation hardening. If the solute content is more (or less), the maximum hardness attained is less.

The maximum tensile and yield strengths are developed if a slowly cooled alloy (if alloy is to be used in annealed state) has 100% eutectic structure, or 100% eutectoid structure, if one of the two reactions occurs in the phase diagram, but the ductility is low. Good ductility is present if the amount of these mixtures are minimum, or it should be a single phase alloy. The castability in an alloy system is best at the eutectic point of the system.

Limitations of Phases Diagrams:

Phase diagrams play an extremely useful role in the interpretation of the microstructures developed in alloys, but have several restrictions:

1. Phase diagrams show only equilibrium state of alloys (i.e., under very slow cooling rates), but alloys under normal industrial practice are rarely cooled under equilibrium conditions.

2. Phase diagrams do not indicate whether a high temperature phase can be retained at room temperature by say fast quenching.

3. Phase diagrams do not indicate whether a particular transformation (for example eutectoid transformation) can be suppressed, and what should be the rate of cooling of the alloy to-avoid the transformation.

4. These diagrams do not indicate the phases produced by fast rates of cooling such as martensite is not shown in Fe-Fe3C phase diagram. These, thus, do not indicate the temperature of start of such transformations (Ms, etc.) and their kinetics of formation.

5. Even under equilibrium conditions, the diagrams do not indicate the character of the transformations. These do not indicate the rate at which the equilibrium shall be attained.

6. The most important limitation is that the diagram gives information only on the constitution of alloys, like the number of phases present at a point, but does not give information about structural distribution of the phases, i.e., does not indicate the size, shape, distribution of the phases, which effect the mechanical properties of the alloys.

The structural distributions of phases are effected by the surface energy between phases and the strain energy produced by the transformation. For example, if the beta phase, in a mixture of alpha and beta, is in small amounts, and is entirely distributed with alpha grains, the mechanical properties shall be largely governed by the alpha phase, but if beta is present around the grain boundaries of alpha, then the strength and ductility of the alloy is largely dictated by properties of beta-phase.