Introduction

A complete solubility or complete insolubility of the components of a binary alloy system are only special cases. In general, the components are neither completely immiscible nor completely miscible (only complete solubility exists with a complete solid solution series).

In reality, an alloying element B can always be dissolved to a certain degree in the host material A and vice versa. In general, therefore, a limited solubility of the components in the solid state is always obtained.

When the components of an alloy are partially soluble, the microstructure ultimately forms a mixture of solid solutions. Up to a certain percentage of B atoms can then be stored in the crystal of substance A. Conversely, this also applies to the crystal of substance B, in whose lattice atoms of type A can be dissolved to a certain degree.

How to read the phase diagram

In the following sections, a fictitious AB alloy system is described in more detail whose components A and B are only partially miscible in the solid state. The creation of the corresponding phase diagram based on selected cooling curves is not described in detail here. Basic information can be found in the chapter on solid solutions.

A typical phase diagram of an alloy system with limited solubility is shown in the figure below. This will be discussed in more detail.

Solid solution based microstructure with precipitations

If the solubility limits of the components in the respective other lattice are not exceeded, then the alloy will initially solidify in the same way as a solid solution whose components dissolve completely into one another. After all, it is perfectly miscible below the solubility limit. The cooling curve of such an alloy first shows the known solidification range (flattening cooling curve) within the alloy solidifies completely. In the following, the alloy AB10, which consists of 10 % B atoms, will be considered as an example for this case.

When the AB10 alloy solidifies, the phase diagram first passes through the known two phase region consisting of melt and solid solution. In this two phase region, the chemical composition of the phases can again be determined by drawing a perpendicular line from the liquidus line (composition of the melt) or solidus line (composition of the solid solution) onto the concentration axis. The phase fractions at a certain temperature can also be determined using the lever rule.

Immediately after solidification, component B is completely dissolved in the lattice of the host substance A at 10 % due to its relatively low concentration. However, solubility generally decreases with decreasing temperature. This can be explained by the decreasing lattice vibrations at decreasing temperature. Because with decreasing lattice vibrations, the space between the atoms also decreases.

Atoms embedded in the “shrinking” lattice can then no longer be held and are “pressed out” of the microstructure, so to speak. These atoms form precipitations (precipitates). The formation of such precipitations due to the solubility limit is also called precipitation or segregation. The line in the phase diagram showing the solubility limit is also referred to as solvus line (Solvus).

Segregation or is the formation of a phase from an existing microstructure due to the solubility limit! The formed segregations are also called precipitations or precipates!

The solvus lines for the respective components are shown in the phase diagram in purple and green. They therefore reflect the maximum solubility of the stored component.

As the temperature drops, the solubility of the components in the other lattice decreases!

The solvus line shows that the host substance A has its maximum solubility at a temperature of 700 °C and can store 20 % B atoms. At this temperature, the entire B atoms of the alloying element remain dissolved in the lattice structure of the host material A.

However, as the temperature drops, the solubility decreases more and more. At a temperature of 600 °C, for example, only a maximum of 15 % and at 500 °C only a maximum of 12 % of B atoms can be dissolved in the lattice of host substance A.

Finally, at 400 °C exactly the existing concentration of 10 % B atoms is reached. At this temperature, the lattice structure of the host component A can just exactly dissolve the entire B atoms. This is also known as a saturated \(\alpha\) solid solution.

\(\alpha\) solid solution is a solid solution consisting mainly of the host component and having dissolved relatively small amounts of alloying elements therein!

If this state is further cooled down, then obviously fewer B atoms can be dissolved in the A-lattice than are still present in the solid solution. So far, the solid solution still has 10 % B atoms, but at 300 °C only a maximum of 9 % and at 200 °C only 8 % can be dissolved. The “too many” B atoms that can no longer be dissolved are therefore precipitated from the A-lattice. The \(\alpha\) solid solution thus always remains saturated to the maximum solubility.

However, no pure B crystals will form during precipitation. Rather, B-rich \(\beta\) solid solutions will form or separate from the lattice (also called \(\beta_{pre}\)-solid solutions). The chemical composition of the \(\beta_{pre}\) can be determined by drawing a perpendicular line from the corresponding solvus line onto the concentration axis (here: 78 % B; 22 % A). Obviously, the \(\beta_{pre}\) always contains a certain amount of A atoms, depending on the temperature!

\(\beta\) solid solution is a solid solution consisting mainly of the alloying element and having dissolved relatively small amounts of the host material in it!

The microstructure fraction of \(\beta_{pre}\) in total can be determined at a given temperature by the lever rule. At a temperature of 200 °C, a microstructure fraction of about 4 % is obtained for the \(\beta_{pre}\) solid solution. Correspondingly, this leads to a microstructure fraction of about 96 % for the \(\alpha\) solid solution.

\begin{align}
\underline{\beta_{pre}} &= \frac{10-7}{78-7} \cdot 100 \text{ %} = \underline{4 \text{ %}} \\[5px]
\underline{\alpha} &= \frac{(78-10)}{(78-7)} \cdot 100 \text{ %} = \underline{96 \text{ %}} \\[5px]
\end{align}

When the solubility limit is undershot, the precipitated solid solution is formed by diffusion processes. Due to the low temperature and the crystal structure already present, these processes take place relatively slow. The precipitations are therefore often limited to very small regions and take place preferably at energetically favourable locations such as grain boundaries, dislocations, inclusions, vacancies, etc.

As the cooling process continues, the solubility of the B atoms in the lattice of component A decreases more and more. B atoms in the form of B-rich \(\beta\) solid solutions are therefore further precipitated, whereby their chemical composition also changes permanently as can be read off from the solvus line.

Note that all precipitation processes take place in an already completely solidified state.

Eutectic microstructure

In addition to the lens-shaped two phase region typical of solid solutions, the phase diagram also contains liquidus lines which meet at a common point. This is the already known eutectic point that one obtains in case of an alloy systems whose components are completely insoluble in one another. The formation of the microstructure of the eutectic AB40 alloy is therefore similar. In turn, such an alloy does not solidify in a temperature range but at a certain temperature with a respective thermal arrest.

The only difference is that no pure crystals of atomic type A or B are formed during the solidification process, but \(\alpha\) solid solutions and \(\beta\) solid solutions. The respective components contain the other type of atom in their lattice structures according to their solubility. Immediately after solidification, a chemical composition of 80 % A and 20 % B can be read from the phase diagram for the \(\alpha\) solid solution (see figure above). For the \(\beta\) solid solution a chemical composition of 30 % A and 70 % B results according to its solubility. As is usual with eutectic alloys, a very fine-grained or rather fine-lamellar microstructure is present due to the strong supercooling.

In the further course of cooling, the composition of the two different solid solutions changes by diffusion processes. The \(\alpha\) solid solution at room temperature (~0 °C) consists of 95 % A and 5% B and the \(\beta\) solid solution of 20 % A and 80 % B. Finally, the microstructure fractions can also be determined using the lever rule. In this case, the \(\alpha\) solid solution has a microstructure fraction of about 53 % and the \(\beta\) solid solution has a microstructure fraction of about 47 %. For further basic information on the determination of the microstructure fractions, please read the article Complete insolubility of components in solid state.

Hypoeutectic based microstructure with precipitations

In the following, the hypoeutectic alloy AB25 is considered. This alloy will crystallize in a solidification range between 960 °C and 700 °C. When the liquidus line is reached, \(\alpha\) solid solutions will first precipitate. Their composition can be determined by drawing a perpendicular line from the solidus line onto the concentration axis. For example, the \(\alpha\) solid solutions consist of 13% B atoms (or 87% A atoms) at a temperature of 800°C.

The precipitation of the A-rich \(\alpha\) solid solutions causes the melt to become poor in A atoms. Consequently, the concentration of B atoms in the melt increases. The increase in concentration for the melt can be determined by drawing a perpendicular line from the solidus line onto the concentration axis again.

At 800 °C, the residual melt has a concentration of 34 % B. It becomes apparent that as the temperature decreases, the melt increasingly approaches the eutectic composition according to the liquidus line. Finally, the eutectic composition in the residual melt is reached at 40 % B at 700 °C. The remaining melt now behaves like a eutectic AB40 alloy and solidifies in a thermal arrest to the eutectic (finely distributed \(\alpha\) and \(\beta\) solid solution).

Immediately after complete solidification of the microstructure, the \(\alpha\) solid solutions precipitated from the melt have a composition corresponding to point P (20 % B). However, the solubility of the B atoms in this \(\alpha\) solid solution now decreases with decreasing temperature. B atoms in the form of \(\beta_{pre}\) solid solutions are therefore precipitated at the grain boundaries during further cooling. Their composition can be read off again at the corresponding solvus line (green line).

Thus, the \(\beta_{pre}\) solid solutions at a temperature of 200 °C have a concentration of B atoms of about 79%, while the \(\alpha\) solid solutions contain only 7% B atoms. The latter concentration applies both to the \(\alpha\)-solid solution precipitated from the melt and to the \(\alpha\)-solid solution of the eutectic formed from the melt!

At room temperature there is thus a eutectic based microstructure, consisting of finely distributed \(\alpha\) and \(\beta\) solid solutions. The primarily formed  \(\alpha\) solid solutions during solidification of the melt are embedded therein. In addition, there are \(\beta_{pre}\) solid solution crystals precipitated at the grain boundaries due to decreasing solubility during cooling.

Hypereutectic based microstructure with precipitations

The processes for a hypereutectic alloy are similar to those for a hypoeutectic alloy. The only difference is that \(\beta\) solid solutions are precipitated from the melt after reaching the liquidus line. The concentration of B atoms will decrease during cooling acording to the liquidus line. Once the eutectic composition is finally reached in the residual melt, it will transform into the eutectic.

If the solubility limit of the \(\beta\) solid solution primarily precipitated from the melt is undershot, then this time A-rich \(\alpha\) solid solution is precipitated at the grain boundaries. The microstructure then consists of a eutectic based microstructure. The primarily formed  \(\beta\) solid solutions during solidification of the melt are embedded therein as well as \(\alpha_{pre}\) solid solutions precipitated at the grain boundaries.

Solid solution microstructure without precipitations

For alloy concentrations below 5 % B, the solubility limit down to room temperature is always above the existing alloy concentration. In this case, neither eutectic solidification (the melt is completely solidified before it could have reached the eutectic composition) nor precipitation takes place, since the entire B atoms can always be dissolved in the host lattice. In this case, the microstructure consists of a single solid solution phase (\(\alpha\) solid solution).

In principle, the same situation applies to alloys with an alloy concentration of more than 80 % B atoms according to the phase diagram. In this case, the microstructure consists of only one \(\beta\) solid solution phase.

Microstructure diagram

To determine the microstructure fractions at room temperature relatively easily, it is helpful to create an microstructure diagram. In principle, the creation of such a diagram is carried out in the same way as in the case of an alloy whose components are completely immiscible – by applying the lever rule in the phase diagram at room temperature (~ 0 °C).

Initially, the microstructure consists entirely of \(\alpha\) solid solutions up to an alloying element content of 5 %. At higher concentrations, \(\beta\) solid solutions are increasingly precipitated due to the limited solubility (\(\beta_{pre}\)). Most \(\beta_{pre}\) solid solutions will form when the microstructure is fully saturated immediately after solidification, so that \(\beta_{pre}\) solid solutions will precipitate immediately upon further cooling. This is the case for an alloy concentration of 20 % B. Due to the linear displacement of the imaginary fulcrum when applying the lever rule, the microstructure fraction of the \(\beta_{pre}\) solid solution also increases linearly up to the already said maximum.

At higher concentrations of B atoms, additional eutectic components (\(Eu\)) are formed, while the fraction of precipitated \(\beta_{pre}\) solid solutions decreases again. The eutectic microstructural components increase linearly up to the eutectic composition and finally reach 100 %.

Hypereutectic alloys precipitate \(\beta\) solid solutions from the melt before the eutectic forms. As the alloy concentration in the hypereutectic range increases, the eutectic microstructures decrease in favor of the \(\beta\) solid solutions. In addition, the fraction of \(\alpha_{pre}\) solid solutions precipitated increases due to the limited solubility. The maximum fraction of \(\alpha_{pre}\) precipitations is finally obtained at 70 % B, since the \(\beta\) solid solution formed there is saturated to the maximum immediately after solidification and thus \(\alpha_{pre}\) solid solution is precipitated directly. In this case, the microstructure no longer contains any eutectic, as the microstructure is solidified before the eutectic composition could have reached in the melt.

When the alloy concentration is further increased, the fraction of \(\alpha_{pre}\) solid solution decreases to the maximum solubility limit of 80 % B and finally reaches zero. From that point on, the microstructure consists only of \(\beta\) solid solutions.

Note: The concentration range between the pure solid solution microstructures, within which several phases occur in the microstructure, is also referred to as the miscibility gap.

Aging

The precipitation processes that occur through diffusion processes when the solubility limit is undershot always take certain time. However, this time can be taken away by rapid cooling (known as quenching). As a result, some of the normally insoluble alloying element atoms remain forcibly dissolved in the host lattice, as the time is taken for them to diffuse out. This leads to correspondingly strong lattice distortions and can lead to enormous increases in strength, as the dislocation movement is made more difficult by the lattice distortions and the forcely dissolved atoms.

The solid solutions are thus in an oversaturated state when rapidly cooled to room temperature and are not in thermodynamic equilibrium. In principle, however, each system strives for a thermodynamic equilibrium. For this reason, precipitates from the supersaturated solid solution will form over time, even at room temperature. However, due to the low diffusibility at such low temperatures, this process takes much longer. This can be a question of days, weeks, months or even years. By striving for a thermodynamic equilibrium state or the subsequent formation of precipitations, the original properties of the material naturally change over time. This process is also known as the aging.

Aging is the (usually negative) change in the properties of a material due to precipitation processes!

The aging process increases with rising temperature, since diffusion processes are correspondingly faster at higher temperatures. Thus, a material can be artificially aged by heating in order to investigate the effects of aging in economic times. Ageing phenomena in steels are primarily due to nitrogen and carbon. Over time, these are separated from the supersaturated ferrite lattice in the form of iron nitride and iron carbide and cause the steel to become unwantedly brittle.

However, aging processes do not always have to have a negative influence on material properties. In some cases, the desired properties only occur with the formation of precipitates and thus with aging (see article on precipitation hardening).

This is the case, for example, with so-called precipitation hardenable aluminium alloys. An important material representative in this context is duralumin, an alloy of aluminum and copper as well as other alloying elements such as iron and magnesium. Immediately after quenching, this alloy is relatively soft and can be easily formed or processed. Only through the subsequent aging process of one to two days does the alloy achieve its strength and hardness. In this case, the desired aging process is also referred to as artificial aging.

Artificial aging is the desired aging process of a material in order to maintain its wanted properties!

Introduction

If the two components of a binary alloy system are completely insoluble in one another in the solid state, one speaks of a crystal mixture. In this case, each of the two components forms its own crystal lattice structure. The individual crystals are only composed of the atoms of the corresponding substance, i.e. they are pure crystals. Hence the term “crystal mixture”.

The bismuth/cadmium alloy system exhibits such a complete insolubility of the components in the solid state over almost the entire mixing range. Therefore, this alloy system will be explained in more detail below.

A complete insolubility of the components in the solid state will lead to a mixture of pure crystals!

Cooling curves

The cooling curves of insoluble components (accounts only for the solid state, not in the liquid state!) show a flattening during solidification as well as a thermal arrest. While the beginning of the flattening indicates the start of solidification, the solidification ends after the thermal arrest. Depending on the concentration of the alloying element, the solidification starts at different temperatures. The end of solidification, on the other hand, always takes place in a thermal arrest at the same temperature, irrespective of the alloy concentration!

Alloys, whose components are completely insoluble in one another, solidify in a temperature range, whereby the solidification is always finished at the same temperature (thermal arrest)!

Starting with pure bismuth, the start of solidification is shifted more and more towards lower temperatures as the cadmium content increases. At an alloy concentration of 40 % cadmium, the start of solidification has fallen so far that it even coincides with the thermal arrest. This special alloy solidifies like a pure substance with only a thermal arrest (eutectic alloy). If the cadmium concentration is increased further, the start of solidification shifts back to higher temperatures and finally ends at the solidification temperature of the pure cadmium.

In order to clearly show the different solidification ranges as a function of the alloy composition, a phase diagram can be generated at this point. Its creation will be discussed in more detail in the next section.

Phase diagram

How to build up a phase diagram

The phase diagram in this case is generated analogous to that of a solid solution. For this purpose, the start of solidification and the end of solidification are transferred from selected cooling curves into a concentration-temperature diagram. One gets again two characteristic lines. The upper polylines mark the beginning of solidification (liquidus line) and the lower horizontal line the end of solidification (solidus line).

It can be seen that the liquidi start to fall off from the respective pure substances. Obviously, the start of solidification is shifted to low temperatures by the presence of the other substance. The falling lines finally meet at a common point. This particular point is also known as the eutectic point.

The Solidus, however, with the exception of the pure substances, is always at the same temperature regardless of the alloy concentration. Even if a different start of solidification results for each alloy, the end of solidification for all alloys is always at the same temperature (here: 146 °C). The Solidus is often referred to as the eutectic line.

The actual crystallisation takes place between the liquidus and the solidus line. In this region, parts of the alloy are already solidified, while others are still in a liquid state (“mushy” state). These regions are therefore again two phase regions. Note that there is a two phase region to the right and to the left of the eutectic point.

Apparently, different atomic processes take place during the crystallization, which initially lead to a flattened cooling curve and finally end in a thermal arrest. The processes that lead to such behavior are explained in more detail in the following sections.

Real phase diagram (limited solubility)

The phase diagram above gives the impression that adding even the smallest concentration of cadmium or bismuth causes an immediate shift of the solidification end to 146 °C (see Solidus). Although the two substances are in principle not soluble in each other in the solid state, in reality small concentrations of one substance can be dissolved in the other substance.

Therefore, the Solidus starting from the pure substances does not actually fall off immediately, but gradually (see dashed lines). Note that for a pure substance the liquidus line and solidus line must in principle coincide, since the start of solidification and the end of solidification are at a common temperature (thermal arrest). Therefore, the solidus line must converge to the liquidus line in the marginal areas.

Such a state diagram is already a preliminary step to the phase diagram of an alloy system with limited solubility of the components in the solid state. This type of alloy is described in more detail in a separate article.

How to read the phase diagram

Eutectic alloy

First, the formation of the microstructure is examined using the example of the eutectic alloy BiCd40. For better orientation, the considered alloy is drawn as a vertical line in the phase diagram.

Starting from the liquid state, the cooling initially leads to a reduction in the kinetic energy of the bismuth atoms and cadmium atoms. Due to the very different chemical properties, the substances initially prevent each other from crystallizing. The solidification temperature of the alloy is therefore below that of the pure substances. In the case of the eutectic alloy, the start of solidification (liquidus line) has even been reduced to such an extent that it coincides in the phase diagram with the end of crystallization (solidus line). Eutectic alloys solidify like pure substances in a thermal arrest.

Eutectic alloys solidify at a constant temperature and have the lowest solidification temperature of the entire alloy system!

Once the temperature has finally dropped to the eutectic point, the undercooling has progressed so far (cf. solidification temperature of the pure substances of 271 °C or 321 °C vs. local temperature of 146 °C) that each substance start to crystallize for their own at constant temperature.

Due to the strong supercooling, many nuclei have formed. Around these nuclei, the respective atoms now attach themselves more and more and the microstructure begins to form. Each growing crystal consists of either pure bismuth atoms or pure cadmium atoms. Note, that there is insolubility in the solid state, i.e. the different atoms cannot be mixed in a common lattice structure. They are therefore forced to form their own pure crystals.

The individual crystallites continue to grow over time as further atoms attach to the lattice structure. The individual grains finally begin to collide towards the end of solidification. After the remaining melt has been completely consumed, the crystallization process is completed.

Due to the strong undercooling, a very fine-grained structure has developed, which is also called eutectic. As a structural component, the eutectic is thus a crystal mixture or phase mixture consisting of finely divided bismuth crystals and cadmium crystals. A further cooling only leads to a reduction of the temperature, the microstructure does not change any further. Hence, the formation of the microstructure is complete.

The melt is severely supercooled during crystallization. Diffusion processes in the melt and in the crystal are thus strongly inhibited. So movements of particles are relative slow and thus can cover only short distances. This usually leads to a fine lamellar crystal growth. Therefore, the grains of a eutectic alloy often have a lamellar structure rather than a roundish shape!

The eutectic is a finely distributed phase mixture (usually lamellar) which forms at a constant temperature out of the melt in the eutectic point!

Due to their very fine structure, eutectic alloys generally have very good strength and toughness values. This is due to the difficult dislocation movement over the fine lamellar structure. The lamellas serve as “obstacles” for the dislocations, so to speak.

Furthermore, eutectic alloys have the lowest melting point in the entire alloy system. This is precisely why they are ideal as casting materials or solders. The lower the processing temperature of a casting material, the lower the requirements on the respective casting moulds.

In addition, eutectic alloys do not solidify in a wide temperature range but in a temperature point, so that the shrinkage can be kept correspondingly low. Shrinkage is the reduction in volume during solidification.

Eutectic alloys have good strength and toughness properties and are often used as casting materials or solders!

Hypoeutectic alloy

In the following section, the microstructure formation using the example of the bismuth-cadmium alloy BiCd15 is described in more detail. Such an alloy, which is to the left of the eutectic point, is also called a hypoeutectic alloy.

Hypoeutectic alloys are alloys to the left of the eutectic point!

Hypoeutectic alloys always show both a flattening section as well as a and horizontal section (thermal arrest) within the cooling curve. Obviously different atomic processes take place during solidification, which are explained in more detail below.

The mutual influence of the components initially has an effect on the start of solidification. Due to the presence of cadmium atoms in the melt, the formation of bismuth nuclei is initially suppressed. The solidification temperature of the alloy is therefore below the pure substance bismuth. After all, at some point supercooling has progressed so far that the cadmium atoms can no longer stop the start of crystallization of the bismuth atoms. The liquidus line is reached and the solidification process begins.

If the temperature falls below the liquidus line, individual nuclei are initially formed, which consist solely of bismuth atoms. The formation of cadmium nuclei is initially completely suppressed by the bismuth atoms due to the very different chemical properties!

With a hypoeutectic alloy, only crystals consisting of the host material (primary crystals) form during solidification in the two-phase region!

The bismuth nuclei grow into larger crystals as they cool further, as more and more bismuth atoms accumulate from the melt. These crystallites consist only of bismuth atoms, since cadmium cannot be dissolved in the bismuth crystal structure due to the insolubility of the components. Since bismuth is obviously the first to crystallize out of the melt during this solidification process, the resulting bismuth crystals are also called primary bismuth crystals (Bi_pc).

With further cooling, more and more bismuth atoms from the residual melt attach themselves to the crystals already present. As a result, the concentration of bismuth atoms in the residual melt decreases or rather the cadmium content of the melt increases permanently.

The cadmium concentration in the residual melt (L) can be read off – as with solid solutions – by drawing a perpendicular line onto the concentration axis at the corresponding temperature (here: ~20 % at ~200 °C).

The composition of the primary bismuth crystals (Bi_pc) can also be determined in the same way. Obviously one get a cadmium content of 0%; because it is a bismuth crystal, consisting of 100% bismuth. Note that the corresponding phase boundary must always be approached for drawing the perpendicular line – in this case, the phase boundary of the primary crystal is 0% cadmium and thus 100% bismuth.

By drawing a perpendicular line during further cooling, the increase in the cadmium content due to the increased excretion of bismuth in the residual melt becomes directly visible in the phase diagram. The cadmium content of the residual melt at 175 °C has already risen to approx. 28 %.

The cadmium content in the residual melt is thus increasingly approaching the eutectic composition with 40 % cadmium. After all, the residual melt at 146 °C has just reached the eutectic composition of 40 % cadmium. The residual melt now behaves like the solidification of the eutectic alloy. The eutectic melt thus begins to solidify into the eutectic at a constant temperature (finely distributed bismuth and cadmium crystals).

After the eutectic melt has solidified into eutectic, the crystallization process is completed. In the solidified state, the microstructure finally consists of the previously precipitated primary bismuth crystals and the eutectic formed at last. A further cooling only leads to a reduction of the temperature, the microstructure does not change any further.

The microstructure of a hypoeutectic alloy consists of primary precipitated crystals of the base material which are embedded in a eutectic matrix!

Due to their special casting properties, mixed crystal alloys are also referred to as casting alloys. However, since the microstructure generally contains primary precipitated crystals and is therefore relatively heterogeneous, these casting alloys are only partially suitable for forming. Solid solution alloys, on the other hand, generally have a homogeneous structure, since the components – unlike crystal mixtures – are completely soluble in one another. Solid solution alloys therefore generally exhibit better forming properties and are also referred to as wrought alloys.

Hypereutectic alloy

The microstructure of a so-called hypereutectic alloy (e.g. BiCd80) is formed in an analogous way to a hypoeutectic alloy.

Hypereutectic alloys are alloys to the right of the eutectic point!

The only difference in the solidification process is that cadmium primary crystals (Cd_pc) precipitate from the melt when the liquidus line is reached. This leads to a corresponding depletion of cadmium atoms in the residual melt. Finally, the concentration of bismuth in the melt at 146 °C will have fallen to the eutectic concentration of 40 %. Now the eutectic melt begins to solidify at a constant temperature to form the eutectic.

In the solidified microstructure of a hypereutectic bismuth-cadmium alloy, the primary precipitated cadmium crystals (Cd_pc) are embedded in the eutectic.

The microstructure of a hypereutectic alloy consists of primary precipitated crystals of the alloying element which are embedded in a eutectic matrix!

Determination of phase fraction and microstructure fraction

In principle, the terms phase component and microstructural component must be strictly separated. A phase is a state of a substance which is characterized by a uniform chemical and physical structure (for further information see here)! Microscopically delimitable areas with certain physical properties, e.g. individual grains, are called microstructural components.

The eutectic, for example, is a microstructural component (microscopically visible with certain physical properties), but has no uniform chemical structure. Therefore, the eutectic is not a single phase but a phase mixture consisting of finely distributed cadmium crystals (1st phase) and bismuth crystals (2nd phase). The precipitated primary crystals, on the other hand, are both a microstructural component (microscopically visible with certain physical properties) and a phase (uniform chemical structure).

Phase fractions

In the two phase regions, the phases melt and primary crystals are present. As with solid solution alloys, the phase fractions can also be determined using the corresponding lever rule.

The hypoeutectic alloy BiCd15 at a temperature of 175 °C is taken as an example. Starting from the state point, the corresponding phase boundaries are first approached with a horizontal line (tie line). In the case of the melt this is the liquidus line and in the case of primary bismuth crystals the alloy concentration of 0 % cadmium (100 % bismuth).

To determine the phase fractions of primary crystals (Bi_pc) and residual melt (L), the lever rule is now applied (“opposite lever arm divided by tie line length”). In the present case, the opposite lever arm of the primary bismuth crystals has a length of b=13 and the opposite lever arm of the melt a length of a=15. Thus the tie line has a total length of a+b=28. These values can now be used to determine the phase fraction of the primary crystals and the melt:

\begin{align}
\underline{Bi_{pc}} = \frac{b}{a+b} \cdot 100 \text{ %} = \frac{13}{28} \cdot 100 \text{ %} = \underline{46.4 \text{ %}} \\[5px]
\underline{L} = \frac{a}{a+b} \cdot 100 \text{ %} = \frac{15}{28} \cdot 100 \text{ %} = \underline{53.6 \text{ %}} \\[5px]
\end{align}

For the present BiCd15 alloy, the phase fraction of solidified primary crystals at 175 °C is 46,4 %. The remaining 53,6 % is accounted for the liquid phase melt.

Microstructure fraction

If the BiCd15 alloy is completely solidified, then the phases bismuth and cadmium are present in the microstructure. While the primary excreted bismuth crystals form their own microstructural component, the eutectic matrix consists of a phase mixture (of cadmium and bismuth) as a further microstructural component.

In order to determine the fractions of the individual microstructural components, the lever rule can be applied again. Any state point below the solidus line is now considered. The exact temperature chosen is irrelevant, as the microstructure below the eutectic line does not change anyway. Analogous to drawing a line to the phase boundaries, the lever arms are now drawn up to the corresponding microstructural boundaries.

In the case of primary bismuth crystals, the microstructure boundary is identical to the phase boundary at 0%, since the primary crystals are both a structural component and a phase. However, as a phase mixture, the eutectic has a cadmium concentration of 40 %. This is why the microstructural boundary of the eutectic is also located there. The lever arm must therefore be drawn to the eutectic concentration to calculate the microstructural component.

Note: This procedure also results directly from the fact that the calculation of the phase fractions in the two phase region in the limiting case lies on the solidus line and from the residual melt finally the eutectic emerges. Therefore, for the calculation of the microstructural components, one can simply imagine a state immediately before the formation of the eutectic. In this state point, the phase melt then simply becomes the microstructural component eutectic.

For the determination of the microstructure of primary crystals (Bi_pc) and eutectic (E) the lever rule is applied again. In the present case, the opposite lever arm of the primary bismuth crystals results to b=25 and the opposite lever length of the eutectic results to a=15. Thus the tie line has a total length of a+b=40. These values can now be used to determine the microstructure fraction of the primary crystals and the eutectic:

\begin{align}
\underline{Bi_{pc}} = \frac{b}{a+b} \cdot 100 \text{ %} = \frac{25}{40} \cdot 100 \text{ %} = \underline{62.5 \text{ %}}   \\[5px]
\underline{E} = \frac{a}{a+b} \cdot 100 \text{ %} = \frac{15}{40} \cdot 100 \text{ %} = \underline{37.5 \text{ %}}   \\[5px]
\end{align}

This results in a primary crystal content of 62.5% for the alloy BiCd15 at room temperature. The remaining 37.5% is accounted for the eutectic.

Excursus: Aluminium-silicon alloy for engine blocks

The determination of the primary crystals present in the microstructure is of great importance for many applications. A hypereutectic aluminium-silicon alloy is often used to cast an engine block (the eutectic composition is at 12.6% Si).

Due to the primary precipitated and very hard and thus wear-resistant silicon crystals, it is thus possible that the piston slides in the cylinder without additional reinforcements. For this it is however necessary that the microstructure consists of at least 5 % primary crystals.

With the understanding of the determination of the microstructure fractions, the required alloy concentration can then be determined. In this case, the alloy requires 17.0 % silicon (AlSi17).

Microstructure diagram

The phase diagram below schematically shows the individual phases of the bismuth/cadmium alloy system. Although the microstructure in the solidified state basically consists of a crystal mixture of cadmium and bismuth crystals, the microstructure can be classified more precisely depending on the alloy composition.

In eutectic alloys, the microstructure consists exclusively of eutectic, i.e. a very fine crystal mixture of both components. In the case of hypoeutectic alloys, the crystals of the host element, which were previously primary precipitated, also occur as microstructural component. In hypereutectic alloys, the primary precipitated crystals of the alloying element result as further microstructural components in addition to the eutectic.

Note, that an alloy concentration close to the eutectic composition reduces the grain size of the primary precipitated crystals. After all, the alloy concentration is already close to the eutectic point. Thus, only a small proportion of primary crystals is excreted in order to achieve the eutectic composition for complete solidification.

This also becomes clear quickly with the help of the lever rule for the determination of the microstructure fractions. The microstructure components shift linearly with the change of the fulcrum of the imaginary balance scale and thus the alloy concentration.

For example, the “pivot point” of a 20 % cadmium alloy is located exactly in the middle of the tie line. The solidified microstructure thus consists in equal mass fractions of primary bismuth crystals and eutectic. With increasing cadmium content, the balance shifts more and more to a larger fractions of eutectic. In 40 % cadmium, the microstructure consists of 100% eutectic. Conversely, a reduction in the cadmium content leads to a shift in the equilibrium towards a larger primary crystal content. At 0% cadmium, the structure is no longer eutectic but consists of 100% bismuth crystals. This results in a linear relationship between the cadmium content and the corresponding microstructural components.

The fractions of the microstructure components is shown below the phase diagram in the figure above. In this way, the fractions of the respective microstructural components can be determined easily on the basis of the alloy concentration. Such a representation of the microstructural fractions as a function of the alloy concentration is also referred to as a microstructure diagram.

The 30 % cadmium alloy is considered as an example. According to the microstructure diagram, 75 % of the microstructure consists of eutectic and 25 % of primary bismuth crystals.

Conversely, the required alloy concentration can also be determined from the microstructure diagram for a desired primary crystal content. If, for example, 25 % of the solidified structure is to consist of primary cadmium crystals (correspondingly 75 % of eutectic), the diagram shows the required alloy concentration of 55 % cadmium.

Introduction

If the two components of an alloy (binary system) are completely soluble in each other in the solid state, one speaks of a solid solution alloy. In this case, the two components form a common crystal lattice, which is made up of both the atoms of the host material and the atoms of the alloying element (for more information, see article types of alloys).

The copper/nickel alloy system exhibits such perfect miscibility over the entire concentration range (complete solid solution series). In the following, therefore, the copper/nickel alloy system will be explained in more detail as an example of an alloy with complete solubility of the components in the solid state.

A complete solubility of the components in the solid state is also called solid solution!

Cooling curves

As already explained in the chapter heat of crystallization, a pure substance solidifies at a so-called thermal arrest, i.e. the temperature remains constant during crystallization. In contrast to this, however, substance mixtures generally no longer have a thermal arrest.

The temperature decrease is no longer stopped completely during crystallization but only slowed down. As shown below, this can be seen in the bent temperature curves for the copper-nickel alloys .

Alloys have no longer a solidification point but a solidification range!

In addition, the cooling curves show that the solidification process extends over different temperature ranges, depending on the nickel content. Starting from the relatively low melting point of pure copper (1085 °C), the solidification range shifts with increasing nickel content to higher values and finally culminates in the solidification point of pure nickel (1455 °C).

For a nickel concentration of 20 %, the alloy solidifies in the temperature range between approx. 1190 °C and 1160 °C, while at a nickel content of 40 %, solidification starts at around 1280 °C and is already finished at around 1230 °C.

The reason for the different solidification ranges is that the presence of the alloying element also changes the chemical properties – such as the solidification temperature.

In principle, alloys can be produced in countless concentration variants, depending on how much of the alloying component is added to the corresponding host material. Accordingly, there is also an innumerable variation in different solidification ranges. To clearly show the solidification ranges across an entire alloy system (i.e. for all conceivable alloy concentrations), so-called phase diagrams are required. Their preparation will be discussed in more detail below.

Phase diagram

How to build up a phase diagram

In order to clearly show the solidification range of alloys as a function of the concentration, a diagram is chosen in which the alloy content is plotted on the horizontal axis in mass percent (here: nickel content) and the temperature is plotted on the vertical axis. Then, for selected cooling curves, the respective start of solidification and the end of solidification are entered in the diagram. In this way, two characteristic lines are obtained which mark the beginning of solidification (liquidus line) and the end of solidification (solidus line) of the entire alloy system.

Above the liquidus line the alloy is completely liquid (melt), while below the solidus line the alloy is completely solidified (solid solution)!

Note that the limiting case of a nickel content of 0 % ultimately corresponds to the solidification point of pure copper and a nickel content of 100 % corresponds to that of pure nickel. Since the solidification of these pure substances takes place at a constant temperature, the liquidus line and solidus line coincide there.

Since the thermodynamic states of an alloy become clear in this diagram form, such a concentration-temperature-diagram is also called a state diagram. In addition, the different states are also referred to as phases and the state diagram is therefore often referred to as a phase diagram.

A phase is a state of a substance which is characterized by a uniform chemical and physical structure!

Note that this definition includes not only the different states of matter of a substance, but also its atomic structure. Although diamond and graphite each consist of carbon atoms and both are in the same solid state, the chemical structure is different for diamond and graphite (diamond: diamond lattice, graphite: hexagonal lattice). In this sense, diamond and graphite also represent different phases.

Note that in the region between the liquidus line and the solidus line, two phases coexist – the liquid phase (melt) and the solidified phase (solid solution). This intermediate region is therefore also referred to as a two phase region.

The region between the liquidus line and the solidus line is also referred to as a “two phase region”!

How to read a phase diagram

Using a copper-nickel alloy consisting of 55 % nickel (CuNi55 alloy) as an example, the interpretation of the phase diagram at different temperatures is explained in more detail below.

For better orientation, the alloy is first entered into the phase diagram as a line of state (yellow solid line). To locate the corresponding states of the alloy for different temperatures, a horizontal line at the respective temperature (yellow dashed line) is used. The intersection of both lines finally represents the state point, i.e. the state of the alloy at this temperature.

At a temperature of e.g. 1350 °C, the state point of the CuNi55 alloy is above the liquidus line. The alloy is thus completely in a liquid state (melt).

At a temperature of about 1330 °C, the state point is on the liquidus line. The crystallization process starts from existing nuclei and the microstructure begins to form (start of solidification).

At a temperature of 1325 °C, the state point is just below the liquidus line in the two phase region. The solidification process has only just begun and a few crystals have formed, but the majority of the alloy is still liquid.

At a temperature of 1295 °C, the state point is just above the solidus line within the two phase region. The crystallization process is therefore almost complete, so that the crystallites have grown at the expense of the melt, i.e. the microstructure is almost completely solidified.

At a temperature of around 1290 °C the state point of the specified copper-nickel alloy is directly on the solidus line and the microstructure is thus completely solidified.

With further cooling to e.g. 1275 °C the state point is below the solidus line. The microstructure does not change any further. The micrograph shows the typical grains with their grain boundaries.

Scope of validity

Note that the two phases in the two phase region – melt and solid solution – exist stable side by side, i.e. it is a thermodynamically stable state. Although areas that have already solidified may partially liquefy again, at the same time liquid areas crystallize to the same extent to solid components (“mushy state”). This is therefore a thermodynamic equilibrium in which recrystallization processes take place on a microscopic level, but nothing changes on a macroscopic scale.

Since setting a state of equilibrium takes time (by diffusion processes), the cooling curves needed for creating the phase diagram must in principle be recorded at (infinitely) slow cooling. Only then is it always guaranteed that the system is always in thermodynamic equilibrium with the environment. If the cooling rate is too fast, the equilibrium state cannot be reached and the solidification ranges (i.e. liquidus line and solidus line) shift! Strictly speaking, phase diagrams therefore only apply to (infinitely) slow cooling processes. In this respect, the state diagram only shows states in thermodynamic equilibrium (equilibrium diagram).

How to determine the phase composition

During the solidification of an alloy, the chemical composition of both the liquid (melt) and the solid solution changes constantly. The chemical composition of these individual phases is also called phase composition.

The phase composition is the chemical composition of the individual phases (liquid and solid solution)!

One can imagine that the melt is “sieved” from the solid solution crystals that have already solidified. The liquid and the solid are then examined for their chemical composition in separate containers.

Using the example of the already considered CuNi55 alloy, the determination of the phase composition is explained using the state diagram.

The state of the alloy at a temperature of 1300 °C is considered. This state point lies in the two phase region, so that parts of the alloy are present as melt and solid solution. Both the melt and the solid solution show different chemical compositions. These can be determined as follows.

Starting from the state point, a horizontal temperature line (black dashed line) is drawn until the respective phase boundary is reached (liquidus line or solidus line). This isothermal line between the two phases in the two phase region is also referred to as a tie line.

A tie line is a isothermal line between two phases in a two phase region!

The tie line and the solidus line or liquidus line each mark an intersection. By drawing a vertical line based on the intersection points to the concentration axis, the corresponding chemical composition of the melt and the solid solution can now be read off directly. In this case, the solid solution consists of 58 % Ni (or 42 % Cu) and the remaining liquid melt of 46 % Ni (or 54 % Cu).

Note that the chemical concentrations obtained do not contradict the total alloy concentration of 55% nickel. Although the solid solutions have a higher nickel content (58%) than the alloy itself, the melt has a lower nickel concentration (46%).

For the determination of the total nickel content, the average of both concentrations must not simply be calculated at this point. This consideration would only be justified if the melt and solid solutions were present in equal parts.

However, more solid solution crystals than melt are present in the considered state point. Consequently, the higher nickel concentration of the solid solution with 58 % nickel has a higher weight than the nickel content of the melt. Weighted proportionally, the nickel concentration across the entire alloy would finally return to the required 55%!

How to determine the phase fractions (lever rule)

In the previous section, the determination of the chemical composition of melt and solid solution was explained using the phase diagram. This section deals with the question what percentage of the total alloy is still liquid at a certain temperature (i.e. present as melt) and what percentage is already solidified (i.e. present as solid solution).

For a better understanding, one can imagine that the partially solidified alloy is screened off from the melt. Now the screened solid solution and the remaining melt are weighed separately and their respective mass fractions of the total are determined.

In this context, one also speaks of phase fraction, i.e. the mass fraction between the phases of melt and solid solution. The phase fraction can also be determined using the phase diagram, as explained below.

The phase fraction is the percentage of the respective phases of the total alloy!

Since the converted crystallisation heat is coupled to the respective mass of the phase, the phase fractions can be determined with the lever rule. To explain this, the CuNi55 alloy at a temperature of 1300 °C is again considered.

Starting from the considered state point, the corresponding distances to the liquidus line an solidus line form the lever arms of an imaginary balance scale. The fulcrum of this balance scale is in the state point.

Now the respective containers with the sieved phases are attached to the ends of the levers in thought. The container with the solidified solid solutions is suspended at the phase boundary to the solid region (solidus line). The melt container is attached to the liquidus line adjacent to the liquid region.

With the condition that the scale has to be balanced, the respective phase fractions can be determined. In a descriptive presentation it will be assumed that the considered alloy have a total mass of 100 kg. In this context, the question arises how much of this total mass is allocated to the respective lever arms in order to be in equilibrium.

If the mass of the liquid melt is denoted by \(L\) and the mass of the solid solution by \(S\), the lever rule first provides the following relationship with the corresponding lever arms \(a\) and \(b\):

\begin{align}
\label{hebelgesetz}
& L \cdot a = S \cdot b  \\[5px]
\end{align}

A further condition is that the mass fraction of melt and solid solution must add up to 100 % (in the descriptive presentation this corresponds to the use of 100 kg alloy mass):

\begin{align}
\label{massengesetz}
& L + S = 100 \text{ %}  \\[5px]
\end{align}

The two equations can now be inserted into each other and solved for the values \(L\) or \(S\):

\begin{align}
\underline{L\cdot a  = S \cdot b} ~~~\text{and}~~~\underline{S = 100 \text{ %} -L} ~~~\text{:} \\[5px]
\end{align}

\begin{align}
L \cdot a &= (100 \text{ %} – L) \cdot b  \\[5px]
L \cdot a &= 100 \text{ %} \cdot b – L \cdot b  \\[5px]
L \cdot a + L \cdot b &= 100 \text{ %} \cdot b   \\[5px]
L \cdot (a+b) &= 100 \text{ %} \cdot b   \\[5px]
\end{align}

\begin{align}
\label{Sm}
\boxed{L = \frac{b}{a+b} \cdot 100 \text{ %}}   \\[5px]
\end{align}

The phase fraction of the solid solution is obtained in the same way:

\begin{align}
\underline{L \cdot a  = S \cdot b} ~~~\text{and}~~~\underline{L = 100 \text{ %} -S} ~~~\text{:} \\[5px]
\end{align}

\begin{align}
(100 \text{ %} – S) \cdot a &= S \cdot b  \\[5px]
100 \text{ %} \cdot a – S \cdot a &= S \cdot b  \\[5px]
100 \text{ %} \cdot a  &= S \cdot b + S \cdot a  \\[5px]
100 \text{ %} \cdot a  &= S \cdot (a+b) \\[5px]
\end{align}

\begin{align}
\label{Mk}
\boxed{S = \frac{a}{a+b} \cdot 100 \text{ %}}   \\[5px]
\end{align}

Finally, the results can be interpreted as follows. The fraction of a phase is always determined by the length of the opposite lever arm divided by the total length of the tie line!

The phase fraction is calculated by the length of the opposite lever arm divided by the total length of the tie line (lever rule)!

This is much more memorable than the corresponding formulas, especially since the names of the lever arms \(a\) and \(b\) were chosen arbitrarily and the respective formulas would also change if they were mixed up.

In the present case, the opposite lever arm of the solid solution container has \(a=9\) and the opposite lever length of the melt container \(b=3\) (where the lever lengths refer to the distance of the respective concentration values). This results in a total length of \(a+b=12\). These values can now be used to determine the phase fraction of the melt and solid solution:

\begin{align}
\underline{L} = \frac{b}{a+b} \cdot 100 \text{ %} = \frac{3}{12} \cdot 100 \text{ %} = \underline{25 \text{ %}}   \\[5px]
\underline{S} = \frac{a}{a+b} \cdot 100 \text{ %} = \frac{9}{12} \cdot 100 \text{ %} = \underline{75 \text{ %}}   \\[5px]
\end{align}

Accordingly 25 % of the total alloy mass is present as melt, while the remaining 75 % of the alloy is already solidified and thus present as solid solution. Of course, the sum of the phase fractions must add up to 100 %. Note that as the temperature drops, the balance continuously shifts towards a larger fraction of solid solution, which also shows that the microstructure is solidifying more and more.

In order to make the figures calculated somewhat clearer, a total alloy mass of 100 kg is assumed. Then 75 kg of the alloy are already solidified, while 25 kg are still in liquid form. As already determined in the section of phase composition, the 75 kg of solid solution consist on average of 58 % nickel, while the 25 kg of melt has a nickel content of 45 %.

The solid solution thus contains a nickel mass of 43.5 kg (=75 kg x 0.58) and in the melt a nickel mass of 11.5 kg (=25 kg x 0.46). In total, the alloy thus contains a nickel mass of 55 kg (=43.5 kg + 11.5 kg). This corresponds exactly to the mass of nickel needed for a CuNi55 alloy.

Crystal segregation

If a copper-nickel alloy is solidified, a subsequent material test often reveals that the nickel content within a grain can differ greatly from region to region. While the grain center is characterized by a relatively high nickel content, the edge areas of the grain, for example, are rather low in nickel. Such concentration differences within a grain are called crystal segregations or microsegregation.

Crystal Segregations are differences in the chemical composition within a grain!

However, since the alloy concentration has a decisive influence on the material properties, the grain centre thus shows a different property (e.g. harder) than the edge area (e.g. softer). Such different properties within a grain are generally not desired, as this can lead to unpredictable material failure.

The formation of crystal segregations is relatively easy to understand with the help of the rules explained in the section of phase composition. In order to explain the formation of crystal segregations, a CuNi55 alloy with 55 % nickel is considered.

The phase diagram shows that during solidification, the nickel concentration within the resulting solid solution decreases over time. The first solid solution formed, has a nickel content of around 66 % immediately upon the onset of solidification at the time \(t_0\).

As the cooling progresses, additional atoms from the melt accumulate around the crystals that have already formed. At a later point in time \(t_1\), a new layer has finally accumulated, which according to the phase diagram leads to an average nickel content in the entire grain of only 63.5 %. Since this is an average value within the entire solid solution, the deposited layer must obviously have a lower nickel content than the previously solidified layer. This is the only way to explain the drop in the average nickel content.

Finally, as solidification progresses, the mean nickel content in the solid solution decreases further (\(t_1\)→\(t_2\)→\(t_3\)). If the solidification process is completed at the time \(t_4\), the nickel content has finally dropped to 55 % and is now completely in the solidified microstructure.

In reality, the atomic layers are not deposited gradually but continuously around the solidified crystals. Consequently, a continuous decrease of nickel concentration is obtained starting from the grain centre (start of solidification) up to the corresponding edge areas (end of solidification). This is how the formation of crystal segregations can be explained.

Note that crystal segregation is not a phenomenon of copper-nickel alloys, but a fundamental phenomenon of solid solutions!

In principle, crystal segregation could be avoided by a sufficiently slow cooling, as the resulting differences in concentration can always be compensated by diffusion processes. In reality, however, cooling processes cannot take place infinitely slowly, which inevitably results in differences in concentration in the microstructure. In such cases, the resulting differences in concentration can be partially compensated by subsequent heat treatment at sufficiently high temperatures (called diffusion annealing).

Introduction

In many technical fields, high demands are placed on the materials used, for example in aeronautical engineering. In some combustion chambers the temperature can exceed 2000 °C. The materials must therefore not only withstand high mechanical loads but also thermal stresses.

Simple metals often do not meet these requirements. For this reason, several metals are usually melted together in order to obtain completely new properties after solidification. Such mixtures of two or more metals are also known as alloys. On a chemical level, alloys are therefore characterized by their metallic bonds.

Mixtures of substances with metallic character are called alloys!

In order to specifically influence the desired material properties, profound knowledge of alloying is required. The basics of alloying will therefore be explained in this article. Due to their complexity, only alloys which are consist of two components are considered (also referred to as binary systems). The total of the possible mixture concentrations of a two-component alloy is also referred to as alloying system.

Binary systems are alloy systems consisting of two components!

In general, alloys are obtained by melting, mixturing and subsequent solidification. For this purpose, a certain amount of an alloying element B (solute) is added to a host material A (solvent) in the liquid state. In the liquid state, the atoms of the substances involved are only weakly bound to each other. In general, the substances can therefore be mixed relatively well.

During solidification, this solubility can then either be retained completely (complete solid solution) or be completely lost (mixture of pure crystals). Partial solubilities of the substances can also occur during solidification (mixture of solid solutions).

Depending on the solubility of the two components A and B in the solid state, alloys can thus be divided into three different types.

Solid solution

If the two components A and B of an alloy remain completely soluble in one another in the solid state, the atoms of the alloying component B are incorporated in the host crystal (host matrix) of the host component A. The atoms A and B then form a common lattice structure. Such a crystal structure of mixed atoms in a common lattice is called solid solution.

A complete solubility of the alloy components in the solid state is referred to as a solid solution!

Figuratively, the components of a solid solution behave like a mixture of water and alcohol, in which the alcohol particles can also be completely dissolved in the water.

In a solid solution, the alloying atoms B can accumulate in the host lattice of the host substance A in two different ways during crystallization. Accordingly, a distinction can be made between substitutional solid solutions and interstitial solid solutions. These are explained in more detail in the corresponding chapters.

Substitutional solid solution

If the atoms of the alloying element B occupy regular positions in the host lattice of the element A during crystallization, this is referred to as a substitutional solid solution. A comparison between the host lattice before melting down and the common crystal lattice after solidification shows that individual A atoms were simply substituted by B atoms.

In general, the two alloy components have different atom radii and chemical properties. Therefore, in reality, there is a lattice distortion within the crystalls. The lattice distortion increases with the number of substituted atoms and finally leads to the fact that the host atoms cannot be replaced to an unlimited extent by alloying atoms. The solubility of the alloying element in the host material is therefore generally limited (partial solubility of the components in the solid state).

Only under the conditions that the alloying component B, compared to the host component A, has

• the same lattice structure,
• similar atomic radii (differ less than 15 %) and
• similar chemical properties,

the alloying atoms B can occupy regular lattice sites of the host crystal A “unnoticed” over the entire mixing range. Any alloy concentration can ultimately be produced without a so-called miscibility gap by means of such a substitutional solid solution.

A solubility over the entire mixing range is also referred to as a complete solid solution series, i.e. a perfect solubility of the components in the solid state. In principle, the copper-nickel alloy system forms such a complete solid solution series.

An alloy system which shows a complete solubility of the components in the solid state over the entire concentration range is also referred to as a “complete solid solution series”!

The conditions mentioned above to form a solid solution are also called Hume-Rothery rules.

Interstitial solid solution

If the atoms B of the alloying element are relatively small compared to the atoms A of the host material (maximum diameter ratio 0.4), there is another possibility of atomic arrangement in the lattice. Due to their small size, the alloying atoms B can then also be placed in interstitial lattices sites of the host crystal structure.

Such a mixture of atoms at intermediate lattice sites is also known as an interstitial solid solution. Due to their dominant role, the regular lattice sites are reserved exclusively for the atoms of the host lattice.

Since only the intermediate lattice sites are available in an interstitial solid solution, a complete solubility of the components only occurs within a strongly limited concentration range (usually only a few percent). If alloying is carried out beyond this solubility limit, the “excess” of alloying atoms B will precipitate and form its own crystal in the microstructure. This crystal can in turn partly contain atoms of the host component A. Therefore, only a partial solubility of the components in the solid state is obtained for interstitial solid solutions.

Mixture of pure crystals

If the conditions for the formation of a solid solution are not met, the alloying atoms B may not be able to occupy regular lattice sites or intermediate lattice sites as well. This is the case when the alloying component B, in comparison to the host component A, has

• another lattice structure or
• has very different chemical properties.

The atoms are then practically displaced from the other lattice structure during solidification and are forced to form their own (“pure”) crystals. Each type of atom then forms its own crystal structure, so that no alloying atoms can be found in the host lattice and no host atoms ca be found in the alloying lattice (complete insolubility of the components in the solid state). The microstructure consists of a mixture of completely different crystallites (grains).

If the components of an alloy are not soluble, each component will form its own crystal structure!

Figuratively, the components of such a alloy behave like a mixture of water and oil, whose components cannot be mixed either – neither the water particles in oil nor the oil particles in water.

Note that just because the atoms of a mixed crystal alloy cannot be mixed in a common lattice structure, this does not mean that the alloy is less stable than a solid solution alloy! Very high interatomic forces also act between the pure crystals of such an alloy to ensure cohesion.

A mixture of pure crystallites will not be found in reality, since the components can always be mixed to a certain degree, even if the solubility is very low. But in the bismuth-cadmium alloy system, for example, the solubility is so low that a complete insolubility of the components can be assumed for simplicity.

Mixture of solid solutions

Perfect solubility (solid solution) or complete insolubility of the components (crystal mixture) are only special cases. In general, the components are neither completely miscible nor immiscible.

In reality, an alloying component B can always be dissolved to a certain degree in the host component A and vice versa. In general, therefore, a limited solubility of the components in the solid state is always obtained.

Figuratively, partial solubility can be compared with a water-sugar mixture in which the solubility of sugar in water is also limited. The water can only dissolve the sugar in it to a certain extent, the undissolved sugar will eventually settle.

In the lattice structure of the host component A (solvent), B atoms (solute) will also be found to a certain extent. Depending on their chemical properties, the B atoms can either occupy regular lattice sites in the host lattice A or be stored at intermediate lattice sites. It is then either a substitutional solid solution or an interstitial solid solution.

Such a solid solution, which primarily consists of the host element A (in particular of the lattice structure of component A) and contains only small amounts of alloying element B, is also referred to as \(\alpha\)-solid solution (alpha solid solution).

Conversely, at very high concentrations of B atoms, the crystallites consist mainly of the lattice structure of component B (solvent), while small amounts of atoms A (solute) will be deposited therein. In such a case one speaks of a \(\beta\)-solid solution (beta solid solution).