Note: DoITPoMS Teaching and Learning Packages are intended to be used interactively at a computer! This print-friendly version of the TLP is provided for convenience, but does not display all the content of the TLP. For example, any video clips and answers to questions are missing. The formatting (page breaks, etc) of the printed version is unpredictable and highly dependent on your browser.

Contents

Aims

On completion of this teaching and learning package, you should:

• Understand conventions used in representing ternary phase diagrams
• Be able to read off and plot compositions on a ternary phase diagram
• Understand the advantages and disadvantages of different sections taken from a ternary phase diagram
• Be able to determine the composition of phases in two and three phase regions from tie lines and tie triangles
• Be able to apply the Gibbs phase rule to ternary systems
• Understand the solidification process for alloys in different ternary systems
• Know of a few industrially useful ternary alloys

Before you start

This TLP builds on many of the ideas introduced in the Phase Diagrams and Solidification TLP.

You should be familiar with the concepts of a binary phase diagram, phase fields, the Gibbs phase rule, tie lines, the lever rule, congruent and incongruent melting, eutectic reactions and peritectic reactions.

Introduction

Phase diagrams provide thermodynamic information for a system – the equilibrium phase or phases for a given set of conditions (usually temperature, pressure, composition). From phase diagrams, the proportions and compositions of the phase or phases present can also be inferred. This information is incredibly valuable when considering the behaviour of a system during heating and cooling.

Cooling and heating regimes (initial cooling rate, heat treatments, the reactions the alloy experiences or bypasses, etc.) determine the microstructure of an alloy, which in turn strongly influences its mechanical properties. Phase diagrams are a vital tool in optimising the microstructure of an alloy for a given purpose.

Many industrially important systems can be sufficiently described by binary phase diagrams but as alloys become more complex and incorporate more components, more complex analysis is needed.

Ternary phase diagrams are used to analyse ternary systems (alloys composed of three components). They can also be used in the simplified analysis of higher order systems (quaternary, quinary, etc).

The complexity of phase diagrams increases with the number of components involved (for example binary systems being more complex than unary systems) so the creation, analysis and interpretation of ternary phase diagrams is more involved than it is for binary phase diagrams.

A typical, simple ternary phase diagram is shown below:

Figure 1: A relatively simple ternary phase diagram showing a single eutectic valley. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

Most real ternary systems are very complex and so simpler, schematic ternary systems are used throughout this TLP to convey key points. These schematic systems have generic components (A, B and C) and up to three solid phases (\( \alpha \) - an A-rich phase, \( \beta \) - a B-rich phase and \( \gamma \) - a C-rich phase). The liquid phase is denoted by L in all schematic systems.

While very powerful tools, ternary phase diagrams suffer the limitations of all phase diagrams. They show only thermodynamic information with no consideration for kinetics or non-equilibrium systems.

Representing ternary phase diagrams

Ternary phase diagrams contain three components. This extra component relative to binary phase diagrams makes representing ternary phase diagrams more difficult but analogies can be made between the two kinds of phase diagram.

Representing Composition

The composition of binary alloys can be plotted as a point on a line. As there are only two components and the sum of the fractions of each component must equal one, defining the fraction of component defines the composition of the whole system.

Similarly, in ternary systems, defining the fractions of two of the three components inherently defines the fraction of the third component and the overall composition of the system.

Binary compositions can be represented as a point on a line. As the system can be defined by the fraction of just one component, the composition can be represented on a one-dimensional object.

Each end of the line represents a pure component. The fractional distance of the point along the line can be related to the fraction of each pure component by the lever rule.

As ternary compositions can be defined by defining two of the three components, they require a two-dimensional representation. Ternary compositions are conventionally shown as a point on an equilateral triangle. Analogously to the binary case, each corner of the triangle represents a pure component.

This triangle is referred to as the composition triangle.

This representation of composition is more complex than the binary counterpart and reading off the overall composition is a more involved process:

Similarly, plotting compositions onto a ternary phase diagram is more difficult than for a binary phase diagram but involves many of the same principles required to read off a composition:

Representing the whole phase diagram

Now that the standard representation of composition has been established, it is time to build up the rest of the phase diagram.

Phase diagrams are useful as they give information on the reactions in a system and how the equilibrium phases vary. To see how the equilibrium phases change we need to add an axis for the variable that is being changed.

In most cases, ternary phase diagrams are plotted as temperature-composition diagrams for a fixed pressure but they can be plotted as pressure-composition diagrams for a fixed temperature just as easily.

Adding a temperature axis perpendicular to the plane of the composition triangle creates a triangular prism:

Figure 2: A temperature composition diagram for a ternary system. Image source: https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

This is the temperature-composition diagram for a ternary system. Each vertical edge of the prism shows how the pure components vary with temperature. The faces of the prism joining a pair of components (A and B or B and C for example) are binary phase diagrams. The triangular face is the composition triangle. The region within the prism is the ternary space.

This representation of a ternary phase diagram is known as a space model.

A few examples of space models:

Figure 3: Two space models: once showing complete solid solubility (left) and one showing a ternary eutectic reaction (right). Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

Sections through ternary phase diagrams

Why take sections through Ternary Phase Diagrams?

Space models contain all the temperature-composition information of a system. By looking at just one section of the space model, information is lost.

Some systems are simple enough (like those on the previous page) that interpretation can be done from the space model alone. However, more complex systems are difficult to accurately represent and interpret this way.

It is in these more complex systems that taking sections becomes a vital tool in the interpretation of ternary systems. While there is information loss in taking the section, the simplification of the representation and ease with which sections can be manipulated vastly outweighs this drawback.

Different Sections Taken through Ternary Phase Diagrams

There are a few different kinds of sections commonly used to analyse ternary systems.

Each kind of section provides different information and suffers from different information loss. Interpreting ternary (and higher order systems) often involves a trade-off of information for simplicity.

Isothermal Sections

Isothermal (or horizontal) sections are sections through the space model taken at a constant temperature. They are triangular sections that are parallel to the composition triangle.

Isothermal sections are the most common way to represent information about ternary systems. They show which phase (or phases) are in equilibrium for varying compositions at the given temperature.

Isothermal sections can provide compositional information: the limits of solid solubility, ternary eutectic compositions, compositions of phases in two and three phase regions and more.

However, isothermal sections contain no information about the system at any other temperature. Information about the temperatures at which reactions occur or which phases are stable at different temperatures must be found on different sections.

The animation below shows isothermal sections at various temperatures for three different kinds of ternary system:

Vertical Sections

Vertical sections (sometimes known as isopleths) are sections taken through a ternary phase diagram that are perpendicular to the plane of the composition triangle. They are rectangular with a horizontal composition axis (showing a series of compositions that plot on a straight line in ternary composition space) and a vertical temperature axis.

Vertical sections show the equilibrium phase (or phases) for a restricted set of compositions at varying temperatures.

Vertical sections provide information on the temperatures at which reactions occur. They provide information on the solidus and liquidus temperatures and eutectic and peritectic temperatures.

However, vertical sections do not contain compositional information. Despite looking similar to binary phase diagrams, the composition of phases cannot be read off vertical sections and the lever rule cannot be applied (for more information see Tie Lines and Tie Triangles)

Vertical sections are often taken parallel to one of the binaries (so it corresponds to a constant proportion of one component while the other two vary) or from one component to a point on the opposite binary (so it corresponds to a constant ratio of the other two components). These are simply the most common geometries though; vertical sections can be taken along any line in compositional space.

The animation below shows a vertical section of each type through three different kinds of ternary system:

Increasingly complex ternary systems (such as those containing class II and class III reactions – see Applying the Gibbs Phase Rule to Ternary Systems) give rise to vertical sections that are increasingly difficult to determine and interpret. For real ternary systems (which often have multiple reactions), vertical sections quickly become unwieldy. This is why isothermal sections are preferred for analysis of ternary systems.

Liquidus Projections

A liquidus projection is a two-dimensional projection of the three-dimensional liquidus surface onto the composition triangle.

This is often done using contours - lines linking points on the liquidus surface that are at the same temperature – in the same way that maps use contours to show how the elevation of the area being mapped varies.

The positions of equilibria involving the liquid phase (such are eutectic valleys along which the liquid decomposes into two solids or invariant points where liquids can undergo congruent solidification) are also included on liquidus projections.

Liquidus projections also include labels to indicate which solid is in equilibrium with the liquid in different regions (e.g., \( \rm{L + \alpha} \) and \( \rm{L+\beta} \) regions are labelled).

Liquidus projections are incredibly useful as they contain information on the changes an alloy undergoes upon solidification. It is possible to determine which solid phases form (and in which order) and which reactions occur during the cooling of an alloy. Conversely, it is possible to determine the changes of the alloy upon melting.

hile vital in understanding cooling histories of alloys, liquidus projections have their limitations. They provide no information on solid state reactions or any changes in solid solubility which occur below the liquidus surface.

ther projections (such as solidus and solid solubility surface projections) can be used to provide the information that liquidus projections lack. They are constructed in very similar ways with features of the surface being plotted as well as temperature contours.

The animation below shows the construction of the liquidus projection for a variety of different ternary systems:

Tie lines and tie triangles

Tie Lines

Tie lines are isothermal and isobaric lines which span two phase regions.

Each end of a tie line represents the composition of a phase. Any point along the line can be described as a proportion of each of the endmember phases.

In binary systems, tie lines are horizontal lines spanning the area that represents the two-phase region. Each tie line is associated with a specific temperature, and they run from one bounding curve of the two-phase region to the other bounding curve.

In ternary systems, tie lines are horizontal lines (they lie in the planes of isothermal sections) and run from one bounding curve of the two-phase region to the other bounding curve. There are multiple tie lines for a given temperature.

Figure 4: Isothermal section from a system with complete solid solubility. The two-phase region has tie lines plotted.

The figure above is of an isothermal section through a ternary phase diagram showing complete solid solubility. This isothermal section has the tie lines for the two-phase region plotted onto it. The whole section is taken for a single temperature but there are many tie lines.

The positions of tie lines on an isothermal section cannot be inferred from the section alone. Often, tie lines are plotted onto isothermal sections but if they are not then the tie lines cannot be guessed.

Restrictions on Tie Lines in Ternary Systems

Two tie lines at a given temperature can never cross.

Consider an isothermal section of a two-phase region in ternary space which has two tie lines that cross. Consider the alloy with the composition of this crossing point. When this alloy reaches the temperature of the crossing tie lines there is no deterministic outcome.

If the Gibbs free energy associated with each pair of phases (a tie line defines each pair) is the same, then the system would decompose into four phases. This violates the Gibbs phase rule and is impossible.

As the Gibbs free energy associated with each pair of phases cannot be the same, it must be lower for one pair (for one tie line) than the other. The system will then decompose into two phases given by the tie line with lower Gibbs free energy. This makes one of the tie lines redundant.

Crossing tie lines, therefore, cannot exist at a given temperature.

It follows then, that if a two-phase region spans from one binary to another binary (as it does in the figure above) then the tie lines fan out between the two binaries. This allows the tie lines at the edge of the ternary space to be parallel to their respective binaries and for no tie lines to overlap. The tie lines rotate smoothly from being parallel to one binary at one edge of the two-phase region to being parallel to the other binary at the other edge of the region.

Moving down temperature, tie lines rotate in the direction of decreasing melting point of the endmember components (For example, in the above system A has the highest melting point, then B, then C. As temperature decreases the tie lines will rotate from A to B and then round to C). This is known as Konovalov’s Rule.

The rotation of tie lines limits the use of vertical sections. It is unlikely that the alloy of interest lies on a tie line that is also in the plane of vertical section at one temperature, let alone a range of temperatures. In nearly every case the ends of the relevant tie line (which represent the compositions of the two phases) do not lie in the plane of the vertical sections. Subsequently, a horizontal (isothermal) line drawn joining the edges of a two-phase region is not a tie line; it does not link the compositions of the two phases that form.

There are cases in which vertical sections can be treated like binary phase diagrams. If a congruently melting compound occurs in one of the binaries making up the ternary phase diagram, then a section from the third component to this compound can be used to read off compositions of phases. These sections are often known as pseudo binary sections.

There is an example of a pseudobinary in the Au-Pb-Sn system. AuSn is an intermetallic phase which melts congruently. The vertical section taken from the Pb corner to this compound is a true binary. Isothermal lines on this diagram are tie lines. This vertical section is shown below:

Figure 5: AuSn to Pb phase diagram - a pseudo binary from the Au-Sn-Pb system. Adapted from figure 1. https://www.sciencedirect.com/science/article/pii/002250886790094X

Tie Triangles

Tie triangles are isothermal, isobaric triangles which span three phase regions.

Each corner of a tie triangle represents the composition of a phase. Any point within the triangle can be described in terms of proportions of these phases - in a similar way that any point within the composition triangle can be described in terms of proportions of each component, though tie triangles are rarely equilateral.

For tie lines, the proportions of each phase could be determined by simply using the lever rule. The process for determining the proportion of each phase in a three-phase region is slightly more involved but quite similar to determining the proportions of each component in a ternary alloy.

Similar to finding the proportions of two components in an alloy, finding the proportions of two phases allows the third to be calculated without doing a construction on the tie triangle (as the percentages of the three phases must sum to 100%). However, it is good practice to do a construction for all three phases and check that they sum to 100% to ensure that no mistakes have been made.

Unlike tie lines in ternary systems, each three-phase equilibrium only has one tie triangle at any given temperature.

On an isothermal section, a three-phase region is represented by a tie triangle which is associated with three two-phase regions generally along the edges of the triangle) and three single phase regions (at each vertex of the triangle).

Usually, the equilibrium composition of the phases involved in a tie triangle vary with temperature. Subsequently, when comparing isothermal sections for multiple temperatures, tie triangles will appear to move across the composition triangle.

Three phase regions in ternary systems are associated with a three-phase reaction (eutectic or peritectic for example) in one of their binary systems. The binary reaction is invariant (occurs at a specific temperature and composition), but the three-phase region is univariant (exists over a range of temperatures and compositions - see Applying the Gibbs Phase Rule to Ternary Systems).

Applying the Gibbs phase rule to ternary systems

In a ternary system the number of components is three, c = 3. The total number of degrees of freedom (the number of variables - composition variable, temperature or pressure - that must be specified to define the state of a system) and phases (homogenous regions of the system) is five.

As ternary systems are often considered as temperature-composition diagrams, pressure is fixed. This reduces the number of degrees of freedom by one.

For a constant pressure (or equally a constant temperature), the total number of degrees of freedom and phases in a ternary system is four.

There can be:

• One phase and three degrees of freedom
• Two phases and two degrees of freedom
• Three phases and one degree of freedom
• Four phases and no degrees of freedom

Consider a system with a ternary eutectic reaction:

Figure 6: A space model for a ternary system with a ternary eutectic point. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

Single Phase Regions

Single phase regions are volumes where the alloy can lie on any point in the volume.

Single phase regions have three degrees of freedom. They are trivariant equilibria.

This means that to define the equilibrium (the point in ternary space the alloy lies on), three variables must be specified and that three variables can be altered without moving out of the single-phase region.

Temperature and the proportion of two of the endmembers must be specified to define the position of the alloy.

Temperature and the proportion of two of the endmembers (which implicitly alters the proportion of the third endmember – see Representing Ternary Phase Diagrams for more about defining a ternary composition) can be varied without leaving the single-phase region.

Two Phase Regions

Two phase regions are volumes. They can be considered as a stack of isothermal sections; the edges of these sections define the surfaces bounding the two-phase region. Each isothermal section contains a stack of tie lines which span the area. An alloy can lie on any tie line in the volume.

Two phase regions have two degrees of freedom. They are bivariant equilibria.

To define which tie line the alloy lies on, a temperature and the proportion of one component in one of the two phases must be specified or two composition variables can be specified (the proportion of two components in one of the two phases or the proportion of a component in each phase).

Once the tie line is specified, the temperature and the composition of both phases can be found, and the state of the system is defined.

Three Phase Regions

Three phase regions are volumes. They can be thought of as a stack of isothermal tie triangles (each corner of the triangle gives the composition of a phase – see Tie Lines and Tie Triangles for more information). The edges of these tie triangles define the surfaces bounding the three-phase region. An alloy can lie on any of these isothermal tie triangles.

Three phase regions have one degree of freedom. They are univariant (or monovariant) equilibria.

To define which tie triangle the alloy lies on, either the temperature or the proportion of one component in one of the constituent phases must be specified.

Four Phase Regions

Four phase equilibria have no degrees of freedom. They are invariant equilibria and are represented by points in ternary space at a fixed temperature and composition.

By specifying that four phases are in equilibrium, the state of the system is defined to be at the invariant point.

Changing any variables requires the system to move off this point and to cease being in a four-phase equilibrium.

Invariant Reactions

Reactions are often associated with heating and cooling through invariant points as there is a drastic change in the phases present across the point.

There are three main classes of ternary invariant reactions:

Class I – ternary eutectic reactions which are analogous to binary eutectic reactions. They have a generalised equation of:

\[ \rm{L \rightleftharpoons \alpha + \beta + \gamma} \]

Directly above the invariant point there is a single-phase region and directly below it there is a three-phase region.

Class II – this ternary invariant reaction has no direct binary equivalent. It is sometimes referred to as a peritectic type, but this can cause confusion with Class III reactions. To avoid confusion, this type of reaction is sometimes called an ubergang reaction, from the German for ‘crossing’. They have a generalised equation of:

\[ \rm{L + \alpha \rightleftharpoons \beta + \gamma} \]

Directly above the invariant point there is a two-phase region and directly below it there is a different two-phase region.

Class III – ternary peritectic reactions which are analogous to binary peritectic reactions. They have a generalised equation of:

\[ \rm{L + \alpha + \beta \rightleftharpoons \gamma} \]

Directly above the invariant point there is a three-phase region and directly below it there is a single-phase region.

There is more information on each of these reactions on the pages about solidification of ternary alloys.

There are other kinds of ternary invariant reactions (for example solid state reactions and reactions involving gases) but they can all be placed into one of the three classes outlined above.

Solidification of ternary alloys – complete solid solution

The ternary system below shows complete solid solubility. Each of its bounding binaries show complete solid solubility.

Figure 7: Space model for a ternary system with complete solid solubility. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

This system has a single-phase liquid region (at high temperatures); a single-phase solid region (at low temperatures, spanning the whole composition triangle) and a two-phase (liquid + solid) region.

All alloys in this system cool through all three regions.

Initially they cool through the liquid phase field. When they reach the liquidus, alloys enter the two-phase field where the solid phase begins to form and the proportion of liquid in the system decreases. Once all the liquid has been exhausted, the alloy passes across the solidus into the single-phase solid field.

When in the single-phase fields (the liquid at high temperatures and the solid at low temperatures), the alloy is homogenous, and its composition is the bulk composition.

When the alloy is in the two-phase field, it is heterogenous. The compositions of the solid and liquid at each temperature are given by tie lines which span the two-phase region. These tie lines change with decreasing temperature so the equilibrium compositions of the solid and liquid also change with decreasing temperature.

Consider alloy W in the diagram below:

Figure 8: The composition of alloy W is plotted in a ternary system showing complete solid solubility. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

The solidification behaviour of alloy W is described below:

Solidification of ternary alloys – one eutectic valley

The ternary system below has one eutectic valley. Two of its bounding binaries have a eutectic point and the third shows complete solid solution.

Figure 9: Space model of a ternary system with one eutectic valley. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

This system has a single-phase liquid region (at high temperatures); two solid single-phase fields (\( \beta \) in the B rich corner and \( \rm{\alpha} \) which spans the AC join); two 'liquid + solid' two-phase regions \( \rm{(L + \alpha} \) and \( \rm{L +\beta} \)); a two-phase solid region \( \rm{\alpha +\beta} \)) and a three-phase field (\( \rm{L +\alpha + \beta} \)).

There are two solidification routes worth considering in this system:

• Alloys that do not pass through the three-phase region on cooling
• Alloys that do pass through the three-phase region on cooling

Solidification of Alloys that Do Not Pass Through the Three-Phase Region

Alloys whose compositions lie within the limit of maximum solubility of B in \( \rm{ \alpha} \) or A/C in \( \rm{\beta} \) do not experience the eutectic reaction. They do not pass through the three-phase region.

Alloys X and Y in the diagram below are both alloys that do not pass through the three-phase region.

Figure 10: Compositions of alloys X and Y plotted onto a ternary system with a single eutectic valley. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

As Alloy X cools from the liquid phase field, it passes across the liquidus into the \( \rm{L + \alpha} \) region. At this point, solid \( \rm{\alpha} \) begins to form. In a similar way to the complete solid solubility case, the alloy cools through the two-phase region with the composition of the solid and liquid varying according to the ends of rotating tie lines. The proportion of the liquid in the system decreases as the alloy cools through this region until there is no liquid remaining. At this point, the alloy crosses the solidus into the single phase \( \rm{\alpha} \) region. The entire system is solid and single phase.

The solidification route for alloy Y is very similar to that for alloy X except that this alloy lies in the \( \rm{L + \beta} \) phase field. The alloys cools across the liquidus from the liquid phase field into the \( \rm{ L + \beta} \) two-phase field. The solidification of \( \rm{ \beta} \) occurs. As the alloy cools through the two-phase region, the compositions of the solid and liquid change and the proportion of remaining liquid decreases. The liquid is exhausted as the alloy crosses the solidus into the single phase \( \rm{ \beta} \) region. At low temperatures the alloy crosses into the two-phase \( \rm{ \alpha + \beta} \) region. If the alloy is cooled slowly enough, \( \rm{ \alpha} \) will precipitate at triple points and grain boundaries until equilibrium proportions of each phase are obtained.

Solidification of Alloys that Do Pass Through the Three-Phase Region

Any alloy that passes through the three-phase region experiences the eutectic reaction. These alloys lie outside the limits of maximum solubility of B in \( \rm{ \alpha} \) or A/C in \( \rm{ \beta} \).

Alloy Z in the diagram below is an alloy that passes through the three-phase region.

Figure 11: Composition of alloy Z plotted on a ternary system with a single eutectic valley. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

The solidification of alloys passing through the three-phase region is more complex than that of alloys that do not:

Solidification of ternary alloys – Class I reactions

The ternary system below has a ternary eutectic invariant point. Associated with this point is a ternary eutectic (or class I) reaction:

\[ \rm{ L \rightleftharpoons \alpha + \beta + \gamma} \]

Each of the bounding binaries has a binary eutectic point. A eutectic valley runs from each binary eutectic point to the invariant point.

>

Figure 12: A ternary system with a ternary eutectic point. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

This system has three single-phase regions (\( \rm{\alpha, \beta} \) and \( \rm{\gamma} \)), three two-phase solid regions (\( \rm{\alpha + \beta}, \rm{\alpha + \gamma}, \rm{\beta + \gamma} \)), one three-phase solid region (\( \rm{\alpha + \beta + \gamma} \)), three two-phase solid and liquid regions (\(\rm{ L + \alpha, L + \beta, L + \gamma} \)), three three-phase regions (\(\rm{ L + \alpha + \beta, L + \beta + \gamma, L + \alpha + \gamma} \)) and a single phase liquid region.

There are three kinds of solidification routes worth considering in this system:

• Alloys that do not pass through a three-phase region on cooling
• Alloys that pass through a three-phase region but not the invariant point on cooling
• Alloys that pass through the invariant point

The kind of solidification route an alloy falls into can be determined by observing which phase field its composition lies in at the invariant temperature.

An example isothermal section from a system with a class I reaction at the invariant temperature is shown below:

Figure 13: The invariant reaction plane for a ternary system with a class I reaction.

This is the invariant reaction plane. It is analogous to the invariant reaction line seen in some binary systems. This section shows four phases in equilibrium. This is shown by a tie triangle (\(\rm{\alpha + \beta + \gamma} \)) with the composition of the liquid plotted at the invariant point inside the triangle.

The position of the liquid within the tie triangle linking the other three phases is characteristic of class I reactions. The relationship of the liquid composition to the tie triangle linking the other three phases is different for each class of reaction and can be used to identify reaction types.

Any point within the tie triangle can be described in terms of proportions of the phases at the vertices. The liquid composition can be described as proportions of each solid phase. If the liquid composition lay outside the triangle this would not be possible.

For the liquid to decompose into the three solid phases, its composition must lie in the tie triangle for matter to be conserved. If the liquid lay outside the tie triangle, for example if it was poorer in C and lay closer to the AB join, then all three solid phases would be richer in C than the liquid. There would simply not be enough C in the liquid to form the three solid phases. The eutectic reaction could not occur.

At the invariant temperature, maximum solubility of unlike atoms in single phase solids occurs. If an alloy plots in a single-phase field at this temperature, it will cool without passing through a three-phase region. This cooling is analogous to an alloy cooling through a two-phase solid + liquid region in a system with complete solid solubility. This case has been discussed on previous pages in more detail.

If an alloy plots in a four-phase region at this temperature, it will undergo a binary eutectic reaction and pass through the corresponding three-phase region. This alloy will not cool through the invariant point as the system is completely solid at the invariant temperature and the invariant reaction requires a liquid phase.

The solidification of this kind of alloy is therefore analogous to the case discussed on the previous page of a system with a single eutectic valley. The alloy will cool through the single-phase liquid field into the two-phase field where a primary solid will form. The alloy then cools into and through the three-phase region with the proportions of the phase present changing (liquid decreasing while the two solid phases increase) as the tie triangle sweeps past the bulk composition. Once the tie triangle is such that the bulk composition lies on the side joining the two solid phases, the system is fully solid and the alloy cools through the two-phase solid region. This is discussed in more detail on the previous page.

If an alloy plots in the three-phase region at the invariant temperature, it undergoes the invariant reaction.

Alloy P in the diagram below undergoes the ternary eutectic reaction:

Figure 14: Composition of alloy P plotted onto a ternary system with a class I reaction. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

The solidification of alloys undergoing the ternary eutectic reaction is more complex than that of other alloys in this system:

Solidification of ternary alloys – Class II reactions

The ternary system below has a class II invariant point. Associated with this point is an Ubergang or (class II) reaction:

\[ \rm{L + \alpha \rightleftharpoons \beta + \gamma} \]

Two of the bounding binaries for this system have binary eutectic points. The third has a binary peritectic point. A eutectic valley and a peritectic valley run from their respective binaries to the invariant point. A second eutectic valley runs from the invariant point to the lowest temperature binary eutectic point.

Figure 15: A ternary system with a class II invariant point. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

This system has three single-phase regions (\( \rm{\alpha, \beta } \) and \( \rm{\gamma} \)), three two-phase solid regions (\( \rm{\alpha + \beta, \alpha + \gamma, \beta + \gamma} \)), one three-phase solid region (\(\rm{ \alpha + \beta + \gamma} \)), three two-phase solid and liquid regions (\(\rm{ L + \alpha, L + \beta, L + \gamma} \)), three three-phase regions (\(\rm{ L + \alpha + \beta, L + \beta + \gamma, L + \alpha + \gamma} \)) and a single phase liquid region.

A notable difference between this system and one with a ternary eutectic point is that two of the three phase regions lie below the invariant point (\(\rm{ \alpha + \beta + \gamma} \) and \(\rm{ L + \beta + \gamma} \)) rather than just one (as in the ternary eutectic case).

There are five kinds of solidification routes worth considering in this system:

• Alloys that do not pass through a three-phase region on cooling
• Alloys that pass through a three-phase region above the invariant point but not through the invariant point
• Alloys that pass through a three-phase region below the invariant point but not through the invariant point
• Alloys that go through the invariant reaction and are fully solid below the invariant point
• Alloys that go through the invariant reaction and are not fully solid below the invariant point

The kind of solidification route an alloy takes can be determined by observing where the composition lies in at the invariant temperature.

An example isothermal section from a system with a class II reaction at the invariant temperature is shown below:

Figure 16: The invariant reaction plane for a ternary system with a class II reaction.

This is the invariant reaction plane. It shows four phases in equilibrium. This can be shown by either the \(\rm{ L + \alpha + \beta} \) and the \(\rm{ L + \alpha + \gamma} \) tie triangles or by the \(\rm{ \alpha + \beta + \gamma} \) and the \(\rm{ L + \beta + \gamma} \) tie triangles. The invariant plane is a trapezium which forms from two tie triangles on cooling into the invariant temperature and it decomposes into two tie triangles cooling out of the invariant temperature.

The invariant point is the composition of the liquid in this four-phase equilibrium. Note that the liquid composition lies outside of the \( \rm{\alpha + \beta + \gamma} \) tie triangle to form a trapezium. This is characteristic of class II reactions. The liquid can combine with the solid opposite it to form the other two solids. Any other reactions involving these four phases, of these compositions would violate the conservation of mass.

Figure 17: Isothermal section at the invariant temperature for a system with a class II reaction showing only single-phase solid fields.

At the invariant temperature, maximum solubility of unlike atoms in single phase solids occurs. If an alloy plots in a single-phase field at this temperature, it will cool without passing through a three-phase region. This cooling is analogous to an alloy cooling through a two-phase solid + liquid region in a system with complete solid solubility. This case has been discussed on previous pages.

There are two cases where an alloy passes through a three-phase region above the invariant point but not through the invariant reaction.

Figure 18: Isothermal section at the invariant temperature for a system with a class II reaction showing only the \( \rm{\alpha + \beta } \) field.

If an alloy plots in the \(\rm{ \alpha + \beta} \) region it will undergo the monovariant peritectic reaction. First, \(\rm{ \alpha} \) will be precipitated until the composition of the liquid lies on the peritectic valley. The peritectic reaction will occur (in an analogous way to a monovariant eutectic with the relevant tie triangle sweeping past the bulk composition) and the a will react with the liquid to form \( \rm{\beta} \) until the liquid is fully exhausted and the system is fully solid.

Figure 19: Isothermal section at the invariant temperature for a system with a class II reaction showing only the \( \rm{\alpha + \gamma} \) field.

If an alloy plots in the \( \rm{\alpha + \gamma} \) region, then it will undergo a monovariant eutectic reaction where either \( \rm{\alpha} \) or \(\rm{ \gamma} \) are precipitated until the liquid composition lies on the \( \rm{\alpha + \gamma} \) eutectic valley. At this point both \(\rm{ \alpha} \) and \( \rm{\gamma} \) are precipitated until the system is fully solid.

Figure 20: Isothermal section at the invariant temperature for a system with a class II reaction showing only the \(\rm{ L + \beta , L + \gamma} \) and \(\rm{ L} \) fields.

In the case where an alloy passes through a three-phase region below the invariant point, this alloy plots in the \(\rm{ L + \beta , L + \gamma} \) or \(\rm{ L} \) region at the invariant temperature. Either \( \rm{\beta} \) or \( \rm{\gamma} \) precipitates initially until the composition of the liquid lies on the \(\rm{\beta + \gamma} \) eutectic valley. Then both \(\rm{ \beta} \) and \(\rm{ \gamma} \) precipitate until the system is full solid.

If an alloy plots in the four-phase trapezium it will undergo the invariant reaction. As there are two reactants in a class II reaction (\(\rm{ L } \) and \(\rm{\alpha} \)) the reaction can be terminated by either of them being exhausted.

Figure 21: Isothermal section at the invariant temperature for a system with a class II reaction showing only the \(\rm{ \alpha + \beta + \gamma} \) tie triangle.

In the case where the liquid is exhausted, the class II reaction is the terminal solidification event for the system as the system is fully solid when the reaction is over. If an alloy plots in the \(\rm{ \alpha + \beta + \gamma} \) tie triangle, then the liquid will be exhausted before \(\rm{ \alpha} \) in the invariant reaction and solidification will cease there.

Figure 22: Isothermal section at the invariant temperature for a system with a class II reaction showing only the \( \rm{L + \beta + \gamma} \) tie triangle.

In the case that \( \rm{\alpha} \) is exhausted and there is remaining liquid, the system is not fully solid at the end of the reaction and will continue to solidify by the \( \rm{\beta + \gamma} \) monovariant eutectic reaction. The composition of the liquid will move away from the invariant point and down the eutectic valley, precipitating \( \rm{\beta} \) and \( \rm{\gamma} \) until the system is completely solid. If an alloy plots in the \(\rm{ L + \beta + \gamma} \) tie triangle, then \(\rm{ \alpha} \) will be exhausted before the liquid in the invariant reaction and this alloy will finish solidifying by the \(\rm{ \beta + \gamma} \) eutectic reaction.

Alloy Q in the diagram below will fully solidify by the class II reaction:

Figure 23: Composition of alloy Q plotted onto a ternary system with a class II reaction. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

The solidification of alloy Q is described below:

Solidification of ternary alloys – Class III reactions

The ternary system below has a ternary peritectic invariant point. Associated with this point is a ternary peritectic (or class III) reaction:

\[\rm{ L + \alpha + \beta \rightleftharpoons \gamma} \]

Two of the bounding binaries have binary peritectic points while the third has a binary eutectic point. A eutectic valley runs from the eutectic point to the invariant point and two peritectic valleys run from the invariant point to the binary peritectic points.

Figure 24: A ternary system with a ternary peritectic point. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

This system has three single-phase regions (\(\rm{ \alpha, \beta} \) and \( \rm{\gamma} \)), three two-phase solid regions (\( \rm{\alpha + \beta, \alpha + \gamma, \beta + \gamma} \)), one three-phase solid region (\(\rm{ \alpha + \beta + \gamma} \)), three two-phase solid and liquid regions (\(\rm{ L + \alpha, L + \beta, L + \gamma} \)), three three-phase regions (\(\rm{ L + \alpha + \beta, L + \beta + \gamma, L + \alpha + \gamma} \)) and a single phase liquid region.

Notably, three of the four three-phase regions lie below the invariant point (\( \rm{\alpha + \beta + \gamma, L + \beta + \gamma} \) and \( \rm{L + \alpha + \gamma} \)). This is distinct from the class I and class II cases where one and two three-phase regions lie below the invariant point, respectively.

There are five kinds of solidification routes worth considering in this system:

• Alloys that do not pass through a three-phase region on cooling
• Alloys that pass through a three-phase region above the invariant point but not through the invariant point
• Alloys that pass through a three-phase region below the invariant point but not through the invariant point
• Alloys that pass through the invariant point and are fully solid below the invariant point
• Alloys that go through the invariant reaction and are not fully solid below the invariant point.

The kind of solidification route an alloy takes can be determined by observing where the composition lies at the invariant temperature.

An example isothermal section from a system with a class III reaction at the invariant temperature is shown below:

Figure 25: The invariant reaction plane for a ternary system with a class III reaction.

The invariant reaction plane shows four phases in equilibrium. This is shown as a tie triangle (\(\rm{ L + \alpha + \beta} \)) with the composition of \(\rm{ \gamma} \) plotted inside the triangle. The invariant point is at the composition of the liquid at the invariant temperature.

The position of the composition of \( \rm{\gamma} \) (the solid phase formed in the peritectic reaction) within the \(\rm{ L + \alpha + \beta} \) tie triangle (the reactants) is characteristic of class III reactions.

Any point within the \(\rm{ L + \alpha + \beta} \) tie triangle (including the composition of \(\rm{ \gamma} \)) can be described as proportions of each phase. Combining \(\rm{ L, \alpha} \) and \( \rm{\beta} \) in the reaction can make \( \rm{\gamma} \) without violating the conservation of matter.

Figure 26: Isothermal section at the invariant temperature for a system with a class II reaction showing only single-phase solid fields.

At the invariant temperature, maximum solubility of unlike atoms in single phase solids occurs for \( \rm{\alpha} \) and \( \rm{\beta} \). If an alloy plots in a single-phase field at this temperature, it will cool without passing through a three-phase region. This cooling is analogous to an alloy cooling through a two-phase solid + liquid region in a system with complete solid solubility. Compositions plotting in the single-phase liquid region will also solidify in this way, passing into the \(\rm{ L + \gamma} \) region then into the \(\rm{ \gamma} \) region. This case has been discussed on the previous two pages in more detail.

Figure 27: Isothermal section at the invariant temperature for a system with a class III reaction showing only the \( \rm{\alpha + \gamma} \) field.

If an alloy plots in the \( \rm{\alpha + \beta} \) field, then it will pass through a three-phase region above the invariant temperature (due to the monovariant eutectic reaction) but not the invariant point. These alloys will precipitate out either \(\rm{ \alpha} \) or \(\rm{ \beta} \) until the composition of the liquid lies on the eutectic valley, at which point coprecipitation of \(\rm{ \alpha} \) and \( \rm{\beta} \) will occur until the system is fully solid.

Figure 28: Isothermal section at the invariant temperature for a system with a class III reaction showing only the solid + liquid fields.

If an alloy plots in either of the liquid + solid regions, it will undergo one of the monovariant peritectic reactions. These alloys will pass through one of the three-phase regions below the invariant point. Their solidification will begin with the precipitation of \(\rm{ \alpha} \) or \( \rm{\beta} \) (depending on which primary solid field the composition lies in). The alloys will then undergo one of the monovariant peritectic reactions where the primary solid reacts with the liquid to produce \( \rm{\gamma} \) until the liquid is exhausted and the system is fully solid – a mixture of the primary solid (\(\rm{ \alpha} \) or \(\rm{ \beta} \)) and \( \rm{\gamma} \).

If an alloy plots in the four-phase region it will undergo the ternary peritectic reaction. As there are three reactants in this reaction, it can terminate when any of these reactants are exhausted.

Figure 29: Isothermal section at the invariant temperature for a system with a class III reaction showing only the \( \rm{\alpha + \beta + \gamma} \) tie triangle.

If the alloy plots in the \(\rm{ \alpha + \beta + \gamma} \) tie triangle, the liquid will be exhausted first, and the system will be fully solid once the invariant is complete.

Figure 30: Isothermal section at the invariant temperature for a system with a class III reaction showing only the \( \rm{L + \beta + \gamma} \) tie triangle.

If the alloy plots in the \(\rm{ L + \beta + \gamma} \) tie triangle, \(\rm{ \alpha} \) will be exhausted first and there will be liquid remaining after the invariant reaction. The system will solidify via the \(\rm{ L + \beta} \) peritectic reaction (as there is no \(\rm{ \alpha} \) present for the \(\rm{ L + \alpha} \) peritectic).

Figure 31: Isothermal section at the invariant temperature for a system with a class III reaction showing only the \( \rm{L + \alpha + \gamma} \) tie triangle.

If the alloy plots in the \(\rm{ L + \alpha + \gamma} \) tie triangle, \(\rm{ \beta} \) will be exhausted first and there will be liquid remaining after the invariant reaction. The system will solidify via the \(\rm{ L + \alpha} \) peritectic reaction.

Alloy R in the diagram below will undergo the ternary peritectic reaction but will complete solidification by the \(\rm{ L + \beta \rightarrow \gamma} \) reaction:

Figure 32: Composition of alloy R plotted onto a ternary system with a class III reaction. Adapted from https://commons.wikimedia.org/wiki/File:Space_diagram_of_a_three-component_system.jpg

The solidification of alloy R is described below:

Case Study: Sn-Ag-Cu system

Lead-based solders have been used for centuries due to their low cost and low melting points (the lead-tin eutectic point is 183°C).

However, due to the negative health impacts of lead, using lead solder to join pipes carrying water for drinking, cooking or bathing became illegal in the UK in 1999.

The EU also placed restrictions on the concentration of lead allowed in electronic devices in 2003 due to concerns about lead leeching into the environment when they are disposed of.

While lead-tin solders are still used in some applications, there has been a significant move towards lead-free solders.

One of the most promising lead-free solders is from the tin, silver, copper (Sn-Ag-Cu or SAC) system.

Figure 33: The liquidus projection for the Cu-Ag-Sn system. Adapted from https://matdata.asminternational.org/apd/viewPicture.aspx?dbKey=grantami_apd&id=10704740&revision=394995

This system has a ternary eutectic point in the tin rich corner.

Figure 34: The Sn rich corner of the Cu-Ag-Sn system showing a ternary eutectic reaction. Adapted from https://matdata.asminternational.org/apd/viewPicture.aspx?dbKey=grantami_apd&id=10704728&revision=394989

The composition of the eutectic point is 3.5 wt.% Ag 0.9 wt.% Cu and balance Sn. It occurs at 217°C.

The commonly used commercial composition of SAC solder is 3.5 wt.% Ag 0.5 wt.% Cu balance Sn. This is a just off-eutectic composition so melts close to the eutectic temperature.

Solidification of Lead-Free Solder

The liquidus projection for the Sn-Ag-Cu system can be used to predict the solidification behaviour and microstructure of lead-free solder.

There are two key assumptions made when inferring this cooling history:

• That the primary solid is pure Sn
• That equilibrium is maintained

For the first, the very low solubilities of Ag and Cu in Sn make this assumption reasonable. It is rarely the case that this assumption is appropriate, and care must be taken with systems that show any significant solid solubility.

While the solder was cooled relatively rapidly in air, the assumption that equilibrium is maintained is reasonable (again due to specific features of this system. This is not a universally applicable assumption).

Often, the equilibrium composition of solid phases varies with temperature. To maintain equilibrium, diffusion of solute in and out of solid phases must occur. Diffusion is temperature-dependent and takes time so this limits the cooling rates at which equilibrium can be maintained.

Due to the low solid solubility in this system and the stoichiometric nature of the phases, there is little change in equilibrium composition of the solid phases with decreasing temperature. Hence, very little slow solid-state diffusion is required to maintain equilibrium. This allows for equilibrium to be maintained at faster cooling rates, such as cooling in air.

Micrographs

A sample of lead-free solder (of the typical commercial composition) was melted down and mounted. The sample was ground and polished and etched with iron(III) chloride (ferric chloride) to reveal the microstructure. This micrograph was taken on an optical microscope using reflected light.

Figure 35: A reflected light micrograph of an alloy of composition 3.5 wt.% Ag 0.5 wt.% Cu balance Sn. Some primary solid and some eutectic regions can be seen.

The expected microstructure for this sample is primary Sn growth followed by an intergrowth of Sn and Ag₃Sn and finally a eutectic intergrowth of Sn, Ag₃Sn and Cu₆Sn₅. As cooling was rapid, the sample is expected to be fairly fine grained.

The proportion of solid Sn formed before the liquid reaches the eutectic valley is very small so the volume fraction of primary solid is expected to be very small. The Sn that does form is approximately spherical.

Due to the rapid cooling rate and intermetallic nature of Ag₃Sn, the reaction along the eutectic valley does not yield the expected, stripey eutectic microstructure. Instead, a fine dispersion of particle-like Ag₃Sn forms in a Sn matrix.

The Cu₆Sn₅ phase is also intermetallic. This causes the microstructure formed by the ternary eutectic (which is already a complex question) to be somewhat abnormal.

Due to complexities of this system and the morphologies of its phases, it is difficult to see from an optical micrograph alone whether the prediction of solidification history from the phase diagram is accurate or not.

A compositional map of the sample was taken using an SEM.

This is a backscattered image of the sample:

Figure 36: Backscatter SEM image of the lead-free solder. The darker regions correspond to Cu rich phases.

Elements with higher atomic numbers can deflect incident electrons more strongly resulting in a higher yield of backscattered electrons appearing brighter. Tin has an atomic number of 50, silver of 47 and copper of 19.

Due to the similar atomic numbers of silver and tin, it is difficult to distinguish these two elements in a backscattered image. Copper has a lower atomic number so copper-rich phases (Cu₆Sn₅) appear darker in the image.

To help differentiate between tin and silver, a compositional map of the sample was made using energy dispersive spectroscopy. High energy electrons incident on the sample can knock low energy electrons out of atoms, leaving a vacancy which a higher energy electron in the atom can fall into. The energy lost by the falling electron is released as an X-ray with an energy equal to the energy gap between the electron knocked out and the falling electron. These energy gaps are unique to specific elements so the X-rays produced by a sample can be used to identify the elements present.

A map of each element was made:

Figure 37: Composition map showing distribution of silver in the lead-free solder sample

Figure 38: Composition map showing distribution of tin in the lead-free solder sample

Figure 39: Composition map showing distribution of copper in the lead-free solder sample. The dark regions in the backscatter SEM image match up with the more copper rich regions of this image

A map of the whole sample with each element overlayed was made:

Figure 40: All three composition maps collated into a single image. The Cu rich regions become quite apparent

It can be seen from the composition map that the darker regions seen in the backscattered image are copper-rich, as expected (although the phase is Cu₆Sn₅ - nearly equilmolar in tin so there is significant tin in these areas too).The rest of the micrograph appears to have both silver and tin fairly evenly distributed throughout (though Sn, being much more abundant in the bulk composition, is more densely distributed in the sample). This makes sense due the particulate nature of the Ag₃Sn phase and the rapid cooling which results in a fine grain size and intergrowth.

The phase diagram (along with knowledge of the morphology of phases present) allows the solidification history and microstructure of an alloy to be predicted.

Uses of ternary alloys

There are many industrially important ternary alloys. The ability to have more than two components in an alloy opens up possibilities to improve mechanical properties beyond the limits of binary alloys: reduced melting points for solder, maximising solid solution strengthening without changing phase, extending the temperature range of desirable phases. This section provides a few examples of useful or interesting ternary systems.

Ti–Al–V

Titanium exists in a hcp structure (\( \rm{\alpha} \)) at high temperatures (above 882°C) and a body centred cubic structure (\(\rm{ \beta} \)) at low temperature (below 882°C).

Alloying additions will often act to stabilise one of these fields – extending its stable temperature range (e.g., \( \rm{\alpha} \) being stable to lower temperatures). Additions can sometimes act to form a eutectoid too.

One of the most important titanium alloys is 6 wt.% Al 4 wt.% V and balance Ti. Aluminium is an \( \rm{\alpha} \) stabiliser and vanadium is a \( \rm{\beta} \) stabiliser. This alloy lies in the \(\rm{ \alpha + \beta} \) two phase field at room temperature.

The exact microstructure will depend on the extent of hot working and heat treatments but slow cooled Ti–6Al–4V will crystallise as \( \rm{\beta} \) initially and then on further cooling \( \rm{\alpha} \) will nucleate and grow to give a Widmanstätten microstructure (laths of \( \rm{\alpha} \) in a \( \rm{\beta} \) matrix).

Figure 41: Reflected light micrograph of a Ti-6Al-4V alloy showing both \( \rm{\alpha} \) and \( \rm{\beta} \) phases. Image from DoITPoMS micrograph library 54, a mixed α+β Ti alloy with a Widmanstätten microstructure

This microstructure has good strength, toughness and fatigue resistance. This alloy is light weight and biocompatible, and has been used in applications such as prosthetics, in the aerospace industry and for hydraulics systems.

Ti–Ni–Nb

Near equimolar binary alloys of Ti and Ni (also known as Nitinol) show exceptional shape memory behaviour (see Superelasticity and Shape Memory Alloys TLP). It can be deformed at low temperature and when it is heated it can recover the deformation and take on its original shape again. This property is possible because of an austenite/martensite like transition that occurs in this alloy.

The phase change is associated with a change in shape but not a change in volume. The dimension of a sample will show a hysteresis loop upon heating and cooling.

Nitinol and related alloys have had significant use in the medical field (notably as stents as they can be fitted in a less invasive procedure than traditional stainless-steel stents). They’ve also been used in electrical switches, aircrafts and as vibration dampeners.

Adding a third element to NiTi alloys can alter the mechanical properties of the alloy but they can also alter the temperature at which the phase transformation (which mediates the shape memory behaviour) occurs.

The addition of Nb reduces the temperature at which the austenite phase begins to convert to the martensitic phase. This increases the width of the hysteresis loop and allows medical devices made from these allows to be stored over a wider range of temperatures.

The addition of Nb pushes the martensite start temperature and the austenite start temperature apart. This increases the width of the hysteresis loop and allows these alloys to be stored over a wider range of temperatures.

A micrograph of an alloy with compositions 47 at. % Ni 44 at. % Ti 9 at. % Nb is shown below in its as-cast state. This alloy is commonly used in pipe coupling. This image was taken using an SEM in backscattered mode so the elements with the highest atomic numbers (Nb: 41, Ni: 28, Ti: 22) will appear brightest.

Figure 42: Backscatter SEM image of a 47 at. % Ni 44 at. % Ti 9 at. % Nb alloy showing primary dendrites and interdendritic eutectic. Image by Oliver Reed

This composition lies in the two-phase TiNi + (Ti, Nb) field. Nb and Ti show complete solid solubility when Ti is in the \( \rm{\beta} \) phase. (Ti, Nb) refers to this phase which spans the Nb-Ti side of the ternary.

The microstructure is dominated by the eutectic reaction involving these two phases. Prior to the reaction there is extensive growth of TiNi dendrites as the primary solid. The interdendritic space is then filled by eutectic intergrowth of TiNi and (Ti, Nb) showing typical lamellar eutectic microstructure.

Au–Pb–Sn

While there has been a move away from lead-based solders now, there was a period where Pb-Sn solders were the gold standard for soldering in electronic devices.

One problem encountered was the diffusion of gold from plating diffusing into the solder. The addition of gold to these alloys could cause embrittlement, reducing the efficacy of the join.

The Au–Pb–Sn system was studied to learn more about the impact of gold on solder.

It is quite an interesting ternary system as it contains a pseudobinary. The intermediate phase AuSn melts congruently. A vertical section drawn from the Pb corner to the AuSn composition is a true binary (all the tie lines for alloys along the section lie in the plane of the section).

The pseudo-binary is a thermal divide. Alloys that lie on one side of the thermal divide cannot evolve to produce melts or solids that lie on the other side of the thermal divide. This allows each side of the diagram to be considered in isolation.

The gold-poor half of the diagram is more commonly studied as it is rare for the concentration of gold in solder joints to exceed 50 wt.%.

The gold-poor half of the diagram is relatively simple, with two class II (ubergang, U) reactions and a terminal class I (eutectic, E) reaction.

The diagram below shows the liquidus projection of the gold-poor side of the ternary phase diagram.

Figure 43: Au-poor side of the Au-Pb-Sn ternary system – separated from the rest of the system by a thermal divide. This partial ternary system shows three invariant reactions. Adapted from https://matdata.asminternational.org/apd/viewPicture.aspx?dbKey=grantami_apd&id=10759796&revision=422525

A micrograph with composition 5.2 wt.% Au 10 wt.% Pb balance Sn in its as cast state is shown below. This composition is just off the composition of the terminal eutectic.

The phases involved in the terminal eutectic reaction are Sn, AuSn₄ and Pb.

The composition of this sample lies in the Sn + L field so the primary solid is dendritic Sn. This is followed by co-crystallisation of Sn and AuSn₄ as the composition of the remaining liquid reaches the eutectic valley. The composition of the liquid moves down temperature to the eutectic point where the liquid is exhausted in the ternary eutectic reaction:

\[ \rm{ L \rightleftharpoons Sn + Pb + AuSn_4} \]

The system is fully solid.  The Sn dendrites, secondary AuSn₄  and ternary eutectic intergrowth can all be seen in the micrograph.

AuSn₄ is an intermetallic phase which forms needle-like crystals rather than dendrites or spherical precipitates as would be expected for a metallic phase.

Cu–Ni–Sn

Alloys from the Cu–Ni–Sn system have become increasingly important in the aerospace and electronics industries. They have excellent thermal and electrical conductivity, high strength and good wear resistance. They are ideal alternatives to Cu–Be alloys, whose production creates toxic dust.

This system has a lot of potential applications as a bearing material.

A micrograph of an alloy with composition 12 at.% Ni 3 at.% Sn balance Cu is shown below. This image was taken using an SEM in backscattered mode. The elements with the highest atomic numbers (Sn: 50, Cu: 29, Ni: 28) appear brightest.

Figure 44: Backscatter SEM image of an alloy of composition 12 at. % Ni 3 at. % Sn balance Cu. Image by James Hogg

The Cu-rich corner of the ternary phase diagram features a class II (ubergang) reaction. The alloy lies in the (Cu, Ni) phase field so an fcc solid with Cu and Ni forms first. The composition of the liquid moves away from the bulk composition towards the (Cu, Ni) + Cu_0.85Sn_0.15 eutectic valley. Once the liquid reaches the eutectic valley (Cu, Ni) and Cu_0.85Sn_0.15 solidify.

The composition of the liquid moves down temperature towards the invariant class II point. When it reaches the point the quasi-peritectic reaction occurs:

\[ \rm{L + Cu, Ni \rightleftharpoons (Cu, Ni)_3Sn + Cu_{0.85}Sn_{0.15}} \]

The (Cu, Ni) dendritic phase reacts with the liquid to produce the two more tin-rich phases. There is an intergrowth of the two tin bearing phases in the interdendritic region.

Duplex stainless steels

Stainless steels are a family of steel alloys that are based on the Fe – Ni – Cr ternary system, although they will often contain other elements (such as Mo and N). Cr stabilises the (\( \rm{\alpha} \)) ferrite phase while Ni stabilises the (\( \rm{\gamma} \)) austenite phase. There are four main types of stainless steels: ferritic, austenitic, martensitic (like ferritic but with higher carbon content) and duplex (contains ferrite and austenite).

These alloys are useful because of their good mechanical properties, cleanability (ability to remove contaminants from the surface and corrosion resistance.

Their corrosion resistance comes from the chromium present in alloy, which forms a coherent, chromium oxide layer on the surface of the alloy, which protects it from further oxidation.

Duplex stainless steels are stronger than fully ferritic stainless steels (though at the expense of some ductility).

These alloys solidify as fully ferritic. Nucleation and growth of the austenite phase occurs during subsequent heat treatments.

Below is an optical micrograph of a relatively fine-grained sample of duplex stainless steel.

Figure 45: Reflected light micrograph of a coarse-grained duplex stainless steel

It is clear from the micrograph that there are two phases present as there are grains of two distinct colours. The phases cannot be differentiated from optical microscopy alone. The application of ferrofluid (FeO) which clusters on the magnetic ferrite grains would allow the two phases to be distinguished.

Olivine–clinopyroxene-plagioclase

Ternary phase diagrams can also be used in the study of ceramic systems.

The olivine–clinopyroxene–plagioclase system is of vital importance to the study of Earth sciences as it describes the melting, evolution and solidification of the Earth’s mantle.

Each endmember in this system is not a single element as they have been in previously discussed systems. Each corner represents the composition of a specific mineral. Typically, these are forsterite (the magnesium-rich endmember of olivine, Mg₂SiO₄), diopside (a magnesium and calcium bearing clinopyroxene, CaMgSi₂O₆) and anorthite (calcium endmember of the plagioclase feldspars, CaAl₂Si₂O₈).

Another vital difference between this ternary system and the metallic ternary systems discussed above is that this system is incredibly simplified.

The number of components in natural systems such as the mantle is much greater than three. For example, 11 major oxides are required to describe a mudstone. This classification of major oxides does not account for minor contributions from a vast array of elements which can have huge effects on the kinetics of the system (and consequently the phases that have time to form).

The effective number of components in these complex systems must be reduced for them to be analysed with ternary phase diagrams. This can be done by assuming that a component is in excess. Once this assumption is made, that component can be projected away from (consider a tetrahedron projected from one corner becoming a triangle) reducing the number of components in the system. The number of components can also be reduced by combining components. Both approaches, while simplifying, come with a loss of information.

The olivine – clinopyroxene – plagioclase ternary system is only a section of the basalt tetrahedron. It is simplified by projecting from the mineral quartz.

Figure 46: Liquidus projection of the Anorthite (An)– Diopside (Di) – Forsterite (Fo) ternary system. Image source: figure 8.55 from https://opengeology.org/petrology/8-igneous-phase-diagrams-and-phase-equilibria/?print=print

Below is a micrograph of a thin section of a peridotite from the Island of Rum. It is a 30 μm section of the rock (most minerals become translucent at this thickness) viewed in plan polarised light in a transmission optical microscope.

Figure 47: A transmission microscopy image in plane polarised light of a thin section from a peridotite on Rum. The sample is from the Igneous Petrology reference series, Department of Earth Sciences, University of Cambridge

The same thin section is viewed between crossed polars as the interaction of polarised light with minerals is useful in identifying them. The width of the field of view is 1mm.

Figure 48: A transmission microscopy image of the sample between crossed polars. the sample is from the Igneous Petrology reference series, Department of Earth Sciences, University of Cambridge

These micrographs show grains with very different morphologies to those seen in metallic micrographs. This is due to the minerals present having strongly preferred growth directions and non-cubic crystal systems.

Melts from the mantle nearly always lie in the olivine + liquid phase field. Olivine is crystallised first; this is often followed by olivine + plagioclase crystallisation then by the olivine + clinopyroxene + plagioclase eutectic reaction.

This reaction is not an invariant reaction but a univariant reaction, even though it includes four phases (liquid, olivine, clinopyroxene and plagioclase). This is because the system considered here is quaternary (four components) and according to the Gibbs phase rule, there is an additional degree of freedom relative to true ternary systems.

There is evidence for this solidification history in the micrograph. The most prominent crystal (labelled ‘olivine’) is likely a primary olivine while the smaller, equally brightly coloured crystals are secondary olivine crystals. There are plagioclase crystals into the largest olivine and between smaller olivine crystals. This implies that the olivine came before the plagioclase. The clinopyroxene is the last phase to appear in the solidification history. It fills whatever space is left by the olivine and plagioclase crystals. Many of the olivine crystals in the left of the micrograph are enveloped in a single clinopyroxene grain (appears blue in crossed polars). These olivine crystals must predate the clinopyroxene growing around them.

Summary

This has been a brief introduction to the rich and interesting field of ternary phase diagrams.

Conventions for representing information on ternary systems (both as the whole space model and as sections taken through it) have been discussed. The trade-off of simplicity and information loss in taking sections through a complete space has been considered and the limitations of each kind of section should be appreciated.

The various kinds of equilibria (two-phase, three-phase, etc.) and how to determine the compositions and proportions of phases present using tie lines and tie triangles should be understood.

There should be an appreciation of how alloys solidify in a variety of simple ternary systems and the different kinds of invariant reactions ternary alloys can undergo:

Class I    \(\rm{ L \rightleftharpoons \alpha + \beta + \gamma} \)

Class II   \(\rm{ L + \alpha \rightleftharpoons \beta + \gamma} \)

Class III  \(\rm{ L + \alpha + \beta \rightleftharpoons \gamma} \)

Examples of industrially and scientifically important ternary systems have been provided to demonstrate both the applications of ternary phase diagrams and the complexities of real systems.

Questions

Quick questions

You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!

1. In a ternary system ABC at a specific temperature there is a three-phase equilibrium of α, β and γ such that the compositions of the phases (by weight) are:
   α  75% A  15% B  10% C
   β  20% A  50% B  30% C
   γ  5% A  30% B  65% C
   An alloy lies in this three-phase region. It consists of equal proportions of α, β and γ. What is its composition?

   	a	33% A  32% B  35% C
   	b	27% A  34% B  39% C
   	c	49% A  26% B  25% C
   	d	21% A  43% B  36% C

2. A two-phase region exists in a ternary system. Which set (or sets – you may choose more than one) of variables can be used to define the system?

   	a	Temperature
   	b	Proportion of one component in one phase
   	c	Temperature and the proportion of one component in one phase
   	d	The proportion of two components in either of the two phases (or one of each)

Deeper questions

The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.

3. What is the composition of alloy X in the diagram below?
   
   (a) 70% A 20% B 10% C
   (b) 15% A 75% B 10% C
   (c) 40% A 55% B 20% C
   (d) 20% A 65% B 15% C

4. Which of the following statements about vertical sections is incorrect?

   	a	Vertical sections show the solidus and liquidus temperatures
   	b	Compositions can be read off most vertical sections using horizontal tie lines
   	c	Vertical sections are perpendicular to the composition triangle
   	d	Vertical sections are pseudo binaries if they link two congruently melting solids

5. Which of the following diagrams has possible tie lines shown?

   	a
   	b
   	c
   	d

6. In the diagram below, which two phase fields are separated by the shaded surface?
   

   	a	L and L + β
   	b	L + α and α
   	c	α + β and α + β + γ
   	d	L and L + α

7. During a univariant eutectic reaction (where a binary eutectic reaction is extended into the ternary space, shown by a eutectic valley) the proportions of phases present change as the reaction proceeds (the proportion of L decreases to zero and the proportions of the two solid phases increase). Why is this?

   	a	The bulk composition moves past the tie triangle on the composition triangle
   	b	The tie triangle and the bulk composition are both moving but at different rates so there is relative motion
   	c	The tie triangle moves past the stationary bulk composition
   	d	The tie triangle disappears when it reaches the bulk composition leaving a two-phase region.

8. Which of the following is associated with a class II reaction?

   	a
   	b	L + α + β → γ
   	c
   	d

9. Consider a ternary phase diagram where one of the bounding binaries has a eutectic point and the other two show complete solid solubility. The diagram below shows such a system (the ternary space is left intentionally blank):
   
   Sketch what an isothermal section for this system might look like a) above the eutectic temperature but below the melting point of B b) below the eutectic temperature and c) below the melting point of C.

Going further

Books

D. R. F. West and N. Saunders, Ternary Phase Diagrams in Materials Science, CRC Press, Third Edition, 2002

F. N. Rhines, Phase Diagrams in Metallurgy, McGraw-Hill Book Company, Inc., Metallurgy and Metallurgical Engineering Series, 1956.

A. Prince, Alloy Phase Equilibria, Elsevier, 1966.
Note: Alloy Phase Equilibria is out of print but can be accessed at https://www.msiport.com/msi-research/free-tools/a-prince-alloy-phase-equilibria/ for free.

Websites

Other resources

Academic consultant: Howard Stone, James Hogg (University of Cambridge)
Content development: Toni Renz
Photography and video: Toni Renz
Web development: David Brook, Lianne Sallows

This DoITPoMS TLP was funded by the Peter Mason Fund in conjunction with the Worshipful Company of Armourers and Brasiers and the Department of Materials Science and Metallurgy, University of Cambridge.