Hubbry Logo
Solid solutionSolid solutionMain
Open search
Solid solution
Community hub
Solid solution
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Contribute something
Solid solution
Solid solution
Solid solution
View on Wikipedia
from Wikipedia

A solid solution, a term popularly used for metals, is a homogeneous mixture of two compounds in solid state and having a single crystal structure.[1] Many examples can be found in metallurgy, geology, and solid-state chemistry. The word "solution" is used to describe the intimate mixing of components at the atomic level and distinguishes these homogeneous materials from physical mixtures of components. Two terms are mainly associated with solid solutions – solvents and solutes, depending on the relative abundance of the atomic species.

The solute may incorporate into the solvent crystal lattice substitutionally, by replacing a solvent particle in the lattice, or interstitially, by fitting into the space between solvent particles.[2]

Solid solutions consist of fractional composition of one or more of its constituent ions between pure, isostructural extremes, known as end members or parents. For example, parent compounds sodium chloride (NaCl) and potassium chloride (KCl) have the same cubic crystal structure, so it is possible to make a solid solution with any ratio of sodium to potassium (Na1-xKx)Cl, eg. by dissolving that ratio of NaCl and KCl in water and then removing the water by evaporation.

An example of a solid solution in this family is sold under the brand name Lo Salt which is (Na0.33K0.66)Cl, hence it contains 66% less sodium than pure NaCl.[3] Similarly, iodised salt is often composed of around 50-100 ppm of potassium iodide (KI) dissolved in a NaCl solvent. [4] In contrast, an example of a physical mixture is the mineral sylvinite - this contains separate, large chunks of NaCl and KCl, and is therefore inhomogenous and not a solid solution.

Because minerals are natural materials they are prone to large variations in composition. In many cases specimens are members for a solid solution family and geologists find it more helpful to discuss the composition of the family than an individual specimen. Olivine is described by the formula (Mg, Fe)2SiO4, which is equivalent to (Mg1−xFex)2SiO4. The ratio of magnesium to iron varies between the two endmembers of the solid solution series: forsterite (Mg-endmember: Mg2SiO4) and fayalite (Fe-endmember: Fe2SiO4)[5] but the ratio in olivine is not normally defined. With increasingly complex compositions the geological notation becomes significantly easier to manage than the chemical notation.

Nomenclature

[edit]

The IUPAC definition of a solid solution is a "solid in which components are compatible and form a unique phase".[6]

The definition "crystal containing a second constituent which fits into and is distributed in the lattice of the host crystal" given in refs.,[7][8] is not general and, thus, is not recommended.

The expression is to be used to describe a solid phase containing more than one substance when, for convenience, one (or more) of the substances, called the solvent, is treated differently from the other substances, called solutes.

One or several of the components can be macromolecules. Some of the other components can then act as plasticizers, i.e., as molecularly dispersed substances that decrease the glass-transition temperature at which the amorphous phase of a polymer is converted between glassy and rubbery states.

In pharmaceutical preparations, the concept of solid solution is often applied to the case of mixtures of drug and polymer.

The number of drug molecules that do behave as solvent (plasticizer) of polymers is small.[9]

Phase diagrams

[edit]
A binary phase diagram displaying solid solutions over the full range of relative concentrations

On a phase diagram a solid solution is represented by an area, often labeled with the structure type, which covers the compositional and temperature/pressure ranges. Where the end members are not isostructural there are likely to be two solid solution ranges with different structures dictated by the parents. In this case the ranges may overlap and the materials in this region can have either structure, or there may be a miscibility gap in solid state indicating that attempts to generate materials with this composition will result in mixtures. In areas on a phase diagram which are not covered by a solid solution there may be line phases, these are compounds with a known crystal structure and set stoichiometry. Where the crystalline phase consists of two (non-charged) organic molecules the line phase is commonly known as a cocrystal. In metallurgy alloys with a set composition are referred to as intermetallic compounds. A solid solution is likely to exist when the two elements (generally metals) involved are close together on the periodic table, an intermetallic compound generally results when two metals involved are not near each other on the periodic table.[10]

Details

[edit]

The solute may incorporate into the solvent crystal lattice substitutionally, by replacing a solvent particle in the lattice, or interstitially, by fitting into the space between solvent particles. Both of these types of solid solution affect the properties of the material by distorting the crystal lattice and disrupting the physical and electrical homogeneity of the solvent material.[11] Where the atomic radii of the solute atom is larger than the solvent atom it replaces the crystal structure (unit cell) often expands to accommodate it, this means that the composition of a material in a solid solution can be calculated from the unit cell volume a relationship known as Vegard's law.[12]

Some mixtures will readily form solid solutions over a range of concentrations, while other mixtures will not form solid solutions at all. The propensity for any two substances to form a solid solution is a complicated matter involving the chemical, crystallographic, and quantum properties of the substances in question. Substitutional solid solutions, in accordance with the Hume-Rothery rules, may form if the solute and solvent have:

a solid solution mixes with others to form a new solution

The phase diagram in the above diagram displays an alloy of two metals which forms a solid solution at all relative concentrations of the two species. In this case, the pure phase of each element is of the same crystal structure, and the similar properties of the two elements allow for unbiased substitution through the full range of relative concentrations. Solid solution of pseudo-binary systems in complex systems with three or more components may require a more involved representation of the phase diagram with more than one solvus curves drawn corresponding to different equilibrium chemical conditions.[13]

Solid solutions have important commercial and industrial applications, as such mixtures often have superior properties to pure materials. Many metal alloys are solid solutions. Even small amounts of solute can affect the electrical and physical properties of the solvent.

A binary phase diagram showing two solid solutions: and

The binary phase diagram in the above diagram shows the phases of a mixture of two substances in varying concentrations, and . The region labeled "" is a solid solution, with acting as the solute in a matrix of . On the other end of the concentration scale, the region labeled "" is also a solid solution, with acting as the solute in a matrix of . The large solid region in between the and solid solutions, labeled " + ", is not a solid solution. Instead, an examination of the microstructure of a mixture in this range would reveal two phases—solid solution -in- and solid solution -in- would form separate phases, perhaps lamella or grains.

Application

[edit]

In the phase diagram, at three different concentrations, the material will be solid until heated to its melting point, and then (after adding the heat of fusion) become liquid at that same temperature:

  • the unalloyed extreme left
  • the unalloyed extreme right
  • the dip in the center (the eutectic composition).

At other proportions, the material will enter a mushy or pasty phase until it warms up to being completely melted.

The mixture at the dip point of the diagram is called a eutectic alloy. Lead-tin mixtures formulated at that point (37/63 mixture) are useful when soldering electronic components, particularly if done manually, since the solid phase is quickly entered as the solder cools. In contrast, when lead-tin mixtures were used to solder seams in automobile bodies a pasty state enabled a shape to be formed with a wooden paddle or tool, so a 70–30 lead to tin ratio was used. (Lead is being removed from such applications owing to its toxicity and consequent difficulty in recycling devices and components that include lead.)

Exsolution

[edit]

When a solid solution becomes unstable—due to a lower temperature, for example—exsolution occurs and the two phases separate into distinct microscopic to megascopic lamellae. This is mainly caused by difference in cation size. Cations which have a large difference in radii are not likely to readily substitute.[14]

Alkali feldspar minerals, for example, have end members of albite, NaAlSi3O8 and microcline, KAlSi3O8. At high temperatures Na+ and K+ readily substitute for each other and so the minerals will form a solid solution, yet at low temperatures albite can only substitute a small amount of K+ and the same applies for Na+ in the microcline. This leads to exsolution where they will separate into two separate phases. In the case of the alkali feldspar minerals, thin white albite layers will alternate between typically pink microcline,[14] resulting in a perthite texture.

See also

[edit]

Notes

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Solid solution

View on Grokipedia
from Grokipedia
A solid solution is a homogeneous crystalline phase consisting of two or more in which the atoms or ions of one substance (the solute) are randomly dispersed within the lattice of another (the ), forming a single-phase material with uniform composition and properties at the . This phenomenon occurs in both metallic alloys and minerals, enabling compositional variability without under equilibrium conditions. Solid solutions are classified into two primary types based on the mechanism of incorporation: substitutional, where solute atoms replace host atoms in the lattice sites, and interstitial, where smaller solute atoms occupy voids between the host atoms. Substitutional solid solutions typically form between elements with similar atomic radii (differing by no more than 15%), crystal structures, electronegativities, and valences, as governed by the , which predict the extent of —unlimited in cases like copper-nickel alloys (Cu-Ni) and limited in others. Interstitial solid solutions, by contrast, involve small atoms such as , carbon, , or fitting into lattice interstices of larger host metals, often resulting in limited due to site availability, as seen in (iron-carbon systems). In , solid solutions manifest as compositional series where ions of similar size and charge substitute within crystal structures, leading to end-member compositions connected by continuous variation; examples include ( Mg₂SiO₄ to Fe₂SiO₄) and ( NaAlSi₃O₈ to CaAl₂Si₂O₈). Factors like , , and (typically within 15% difference for effective substitution) control the extent of , while coupled substitutions maintain charge balance when ions differ in valence. Upon cooling or changing conditions, exsolution can occur, producing lamellar textures such as in alkali feldspars. The formation and stability of solid solutions are depicted in phase diagrams, which illustrate regions of complete (isomorphous systems like Cu-Ni) versus limited , bounded by lines that define single-phase and two-phase equilibria during cooling or heating. These concepts are fundamental in materials engineering for designing alloys with tailored properties, such as enhanced strength in (copper-zinc substitutional solution), and in geosciences for interpreting compositions and petrogenetic histories.

Fundamentals

Definition and Nomenclature

A solid solution is a homogeneous crystalline phase formed when atoms of one or more substances (solutes) are incorporated into the crystal lattice of another substance (the ), resulting in a single phase without separation into distinct components. This incorporation maintains the overall crystal structure of the while allowing compositional variation across a range of solute concentrations. The concept of solid solutions has roots in ancient metallurgical practices, such as the creation of alloys, but the modern understanding developed through scientific inquiry in the 19th and early 20th centuries. Systematic studies began in the , particularly on metal alloys, with key advancements in phase equilibrium and crystal chemistry. Metallurgists like William Hume-Rothery contributed significantly in the 1920s by establishing empirical rules for predicting solid solution formation in metallic systems. In , the primary or majority component that provides the host lattice is termed the , while the added minor component is the solute. is described as complete if the solute can be incorporated across the entire composition range (from 0 to 100%), or partial if limited to a specific concentration range. Common notations include Greek letters to denote phases, such as the α-phase for face-centered cubic (FCC) solid solutions in metals like copper-based alloys. At the atomic level, solute atoms are incorporated by occupying either regular lattice sites of the solvent or interstitial positions between them, thereby preserving the host lattice's symmetry and structure while potentially altering properties like lattice parameters. Phase diagrams serve as visual representations of these solubility limits in binary systems.

Types of Solid Solutions

Solid solutions are broadly classified into two main types based on the mechanism by which solute atoms are incorporated into the lattice: substitutional and . In substitutional solid solutions, solute atoms replace atoms at lattice sites, requiring the solute and to have compatible atomic properties to minimize lattice strain. This type forms when the atomic radii of the solute and differ by less than 15%, they share the same , exhibit similar electronegativities, and have comparable valences, as outlined by the . These empirical guidelines, first proposed in the with key developments in , predict extensive solid in metallic systems by ensuring minimal distortion to the host lattice. For instance, the - system adheres to these rules, with atoms (radius 142 pm) substituting for atoms (radius 145 pm) in a face-centered cubic lattice, forming a substitutional solid solution (α-phase) with limited up to approximately 38 wt% Zn at 458 °C, as shown in phase diagrams. Interstitial solid solutions occur when small solute atoms occupy the interstitial voids between the larger atoms, without displacing them from lattice positions. This mechanism is limited by the size of the voids, typically accommodating solutes with atomic less than about 0.59 times that of the (based on the for tetrahedral sites), leading to higher lattice strain and restricted compared to substitutional types. A classic example is carbon in iron, where carbon atoms fit into the octahedral interstices of the body-centered cubic iron lattice in ferrite, with a maximum of about 0.022 wt% C at the eutectoid (727 °C), decreasing to approximately 0.008 wt% at (0 °C), beyond which precipitation occurs. These structural distinctions influence the properties of the resulting alloys, such as mechanical strength and ductility, with substitutional solutions often allowing broader compositional ranges than interstitial ones.

Thermodynamics and Stability

Thermodynamic Principles

The formation and stability of solid solutions are governed by the Gibbs free energy of mixing, ΔGmix\Delta G_{\text{mix}}, which determines whether the mixed phase is thermodynamically favored over the pure components. This is expressed as ΔGmix=ΔHmix−TΔSmix\Delta G_{\text{mix}} = \Delta H_{\text{mix}} - T \Delta S_{\text{mix}}, where ΔHmix\Delta H_{\text{mix}} is the enthalpy of mixing, ΔSmix\Delta S_{\text{mix}} is the entropy of mixing, and TT is the temperature; a negative ΔGmix\Delta G_{\text{mix}} drives spontaneous solution formation at constant temperature and pressure. In the ideal solution approximation, ΔHmix=0\Delta H_{\text{mix}} = 0, assuming no net energetic interactions between solute and atoms beyond random placement on the lattice, while the arises solely from configurational disorder. The configurational is given by ΔSmix=−[R](/page/Gasconstant)[xln⁡x+(1−x)ln⁡(1−x)]\Delta S_{\text{mix}} = -[R](/page/Gas_constant) [x \ln x + (1-x) \ln (1-x)], where RR is the and xx is the of the solute; this term is always positive and increases with , promoting . Vibrational contributions, stemming from changes in densities of states upon alloying, can further stabilize solutions but are typically smaller than configurational effects in metals and ceramics. For non-ideal behavior, the regular solution model accounts for enthalpic interactions via ΔHmix=Ωx(1−x)\Delta H_{\text{mix}} = \Omega x (1-x), where Ω\Omega is the temperature-independent interaction parameter reflecting pairwise atomic bonding energies; negative Ω\Omega (exothermic mixing) enhances solubility, while positive Ω\Omega (endothermic) limits it. In this model, the excess Gibbs free energy is primarily enthalpic, with entropy retaining the ideal form, allowing prediction of solution limits from measured Ω\Omega values. Phase stability in solid solutions is assessed by the curvature of ΔGmix\Delta G_{\text{mix}} versus composition; a emerges when the second derivative ∂2ΔGmix/∂x2<0\partial^2 \Delta G_{\text{mix}} / \partial x^2 < 0 in regions of positive ΔHmix\Delta H_{\text{mix}}, leading to solute clustering or decomposition into solute-rich and solvent-rich phases at lower temperatures. For regular solutions, this condition simplifies to Ω>2RT\Omega > 2RT, below the critical solution temperature where cannot overcome repulsive interactions.

Factors Influencing Solubility

The solubility of solute atoms in a host lattice generally increases with rising temperature, exhibiting an Arrhenius-type dependence driven by the thermal activation that overcomes the barriers to mixing. This behavior is evident in many metallic alloys, where higher temperatures expand the solvus boundaries in phase diagrams, allowing greater incorporation of the solute before occurs. In certain systems, such as some solid solutions, an upper consolute temperature (UCST) below which a forms, leading to decreased upon cooling due to energetic instabilities, while lower consolute temperatures can appear in systems with complex interactions. Pressure exerts a relatively minor influence on solid solubility under ambient conditions due to the incompressibility of most solids, but it becomes significant in high-pressure environments, such as those used for synthesizing materials like . In these processes, elevated pressures (typically 5-6 GPa) enhance the solubility of carbon in molten metal catalysts like or iron by stabilizing the denser phase over , facilitating its from the solution. Similarly, in systems like MgO-Y₂O₃ nanocomposites, applied pressures up to several GPa can shift phase equilibria and increase solid solubility by altering volume-dependent free energy terms. In ternary alloy systems, the introduction of a third element can significantly modify the solubility windows of the primary binary components by influencing lattice parameters, electronic structure, or phase stability. For instance, additions like magnesium to aluminum-cerium alloys extend the solid solubility of in the aluminum matrix through synergistic interactions that reduce tendencies, thereby enhancing overall mechanical properties. Impurities or alloying elements with differing valencies or sizes can either widen solubility limits by compensating strain or narrow them by promoting secondary phase formation, as observed in zirconium-based systems where third-element additions adjust the solubility of aluminum-stabilized precipitates. Empirical observations indicate that substitutional solid solubility is often limited to approximately 10-20 at.% when atomic size mismatches exceed thresholds, as excessive lattice strain destabilizes the uniform solution structure. The highlight that differences in greater than 15% generate prohibitive strain energies, restricting extensive mixing. For example, the Ni-Cr system exhibits complete mutual solid across all compositions due to their similar face-centered cubic structures and atomic s (differing by less than 2%), enabling stable austenitic phases. In contrast, the Cu-Ag system shows limited —around 5 at.% Ag in Cu and 0.1 at.% Cu in Ag at the eutectic temperature (779°C)—owing to a 12% mismatch and differing electronegativities that favor .

Phase Diagrams

Representation in Binary Systems

In binary phase diagrams, the graphical representation of solid solutions for two-component systems plots on the vertical axis against composition (typically mole or weight of one component) on the horizontal axis, at constant pressure, to delineate phase equilibria. The solvus line marks the boundary of solubility limits within the solid phase, separating single-phase solid solution regions from two-phase solid regions, while the solidus line defines the boundary between the single-phase solid solution and the solid-plus-liquid region. For isomorphous systems exhibiting complete solid solubility across all compositions, such as the copper-nickel (Cu-Ni) alloy, the phase diagram features a lens-shaped two-phase (liquid + solid) region bounded by the liquidus line (separating liquid from liquid + solid) and the solidus line (separating solid solution from liquid + solid), with a broad single-phase solid solution region extending below the solidus to room temperature. In this configuration, the solid solution phase, denoted as α, accommodates any ratio of Cu and Ni atoms on the same crystal lattice due to their similar atomic sizes and crystal structures. In eutectic systems with limited solid solubility, the diagram includes separate α and β solid solution fields for each end-member, connected by a two-phase solid region and separated by a solvus line that indicates the decreasing mutual with falling temperature. A classic example is the lead-tin (Pb-Sn) system, where the α phase (Pb-rich solid solution) and β phase (Sn-rich solid solution) exhibit narrow solubility ranges, flanked by lines that converge at the eutectic point—the lowest melting temperature where liquid coexists with both solid solutions. To quantify phase fractions in two-phase regions adjacent to solid solutions, such as the α + β solid region below the solvus, the is applied along a horizontal tie line at a given : the fraction of the β phase equals the length of the segment from the overall composition to the α solvus boundary divided by the total tie line , and vice versa for the α phase. This method, derived from , enables calculation of relative amounts without direct measurement, as in determining the proportions of α and β in a hypoeutectic Pb-Sn cooled into the two-phase field.

Key Features and Interpretation

In binary phase diagrams, key features such as boundary lines and invariant reactions provide critical insights into the stability and behavior of solid solutions, enabling predictions of phase transformations under varying and composition. The solvus line delineates the -composition boundary below which a solid solution decomposes into two distinct solid phases, marking the limit of in the system. Crossing this line upon cooling leads to of a secondary phase, as the decreases with . This boundary is particularly relevant in interpreting , where a solid solution phase melts directly to a of identical composition without , versus incongruent melting, in which the solid transforms to a and a different solid phase at the peritectic . Peritectic reactions involve a phase and one solid solution reacting to form a new solid phase at a fixed temperature, often appearing as a horizontal line in the diagram connecting the compositions of the reacting phases. Eutectoid reactions, analogous but occurring entirely in the solid state, transform a single solid solution into two different solid phases, such as the decomposition of (γ) into ferrite (α) and (Fe₃C) in the iron-carbon system at 727°C, resulting in the lamellar microstructure known as . These invariant points (where F=0 per the ) indicate no , fixing both temperature and overall composition for the reaction to proceed. Tie-lines, or isothermals, connect the equilibrium compositions of coexisting phases at a given within two-phase s of the . The Gibbs , F = C - P + 1 (for condensed systems at constant pressure), governs the interpretation: in a single-phase solid solution (P=1, C=2), F=2, allowing independent variation of and composition; in a two-phase (P=2), F=1, where alone determines phase compositions along the tie-line, with the quantifying relative phase fractions. Lens-shaped miscibility gaps commonly appear in phase diagrams for systems with limited solid solubility, bounded by solvus lines that converge at a critical temperature (consolute point) above which complete mixing occurs. Below this point, the gap indicates thermodynamic instability of the homogeneous solution, driving into two immiscible phases whose compositions follow the gap boundaries.

Formation and Kinetics

Mechanisms of Formation

Solid solutions form through solid-state , a process that homogenizes atomic distributions within a crystalline lattice by enabling solute atoms to migrate to substitutional sites. This migration predominantly occurs via vacancy-mediated jumps, where thermal activation allows solute atoms to exchange positions with adjacent vacancies, progressively reducing compositional gradients. Annealing processes accelerate this by elevating temperatures to levels that increase vacancy concentration and mobility, thereby achieving equilibrium solid solutions over extended periods. For instance, in Al-clad iron systems, annealing at 450–640°C for 2–72 hours promotes phase formation through homogenization. In the melting and solidification route, components dissolve into a common liquid phase before co-crystallizing into a during cooling, with the final composition governed by the solidus boundary. Equilibrium solidification yields compositions up to the maximum , but rapid techniques, such as , kinetically trap extended solubilities by limiting atomic rearrangement, resulting in metastable supersaturated solutions. An example is the Ni-Mo system, where extends Mo solubility in Ni from an equilibrium maximum of 28 at.% to 37.5 at.% by suppressing . Mechanical alloying provides a room-temperature pathway to solid solution formation by subjecting mixtures to high-energy ball milling, which induces repeated deformation, , and cycles that refine particle sizes and enhance interatomic mixing. This severe deformation generates excess defects, such as dislocations and vacancies, that lower barriers and drive solute dissolution into the host lattice, even in immiscible systems. In Cu-Co alloys, for example, mechanical alloying dissolves Co particles into the Cu matrix to form a supersaturated face-centered cubic solid solution, stabilized by the stored energy from processing. Nonequilibrium processing methods like and vapor deposition create thin-film solid solutions by directly incorporating solute atoms under conditions far from , bypassing limits. accelerates ions into the substrate surface, creating a rapid quench that forms supersaturated surface alloys through ballistic mixing and defect-enhanced , independent of phase stability. Vapor deposition techniques, including and , sequentially deposit atomic layers onto substrates, enabling controlled formation of supersaturated solutions in otherwise immiscible binaries; in Cu-Cr thin films, these methods yield metastable solid solutions with tunable Cr content up to several atomic percent.

Diffusion Processes

Diffusion processes in solid solutions govern the atomic mobility required for homogenization and phase equilibration, quantifying how solute atoms redistribute within the host lattice over time. These processes are fundamentally described by Fick's laws, which model diffusion as a response to concentration gradients. Fick's relates the diffusive flux JJ to the concentration gradient ∇C\nabla C: J=−D∇CJ = -D \nabla C where DD is the diffusion coefficient, representing the material's propensity for atomic transport. This law holds for one-dimensional cases as J=−D∂C∂xJ = -D \frac{\partial C}{\partial x} and extends to higher dimensions in isotropic media. Fick's , combining the first law with conservation, yields the : ∂C∂t=D∇2C\frac{\partial C}{\partial t} = D \nabla^2 C or in one dimension, ∂C∂t=D∂2C∂x2\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}, predicting concentration evolution under non-steady-state conditions. These equations apply directly to solid solutions, where CC denotes solute concentration and boundary conditions reflect experimental setups like diffusion couples. In solid solutions, diffusion manifests as self-diffusion or interdiffusion, each characterized by distinct coefficients. Self-diffusion involves identical atoms exchanging positions in a pure crystal or solvent-rich matrix, measured via tracer isotopes to yield the self-diffusion coefficient D∗D^*, which reflects intrinsic lattice mobility. Interdiffusion, occurring between dissimilar atoms in alloys, produces a chemical diffusion coefficient D~\tilde{D} that accounts for coupled fluxes and thermodynamic factors. A key demonstration of their difference is the Kirkendall effect, observed in marker experiments where inert markers at the diffusion couple interface shift toward the slower-diffusing component due to unequal atomic fluxes and resultant vacancy imbalances. This was first evidenced in Cu-Zn alloys, with Zn diffusing faster than Cu at 780°C, leading to interface velocity proportional to the flux difference. Atomic jumps underlying these processes occur via vacancy or interstitial mechanisms, each dominating based on solute size and lattice type. Vacancy diffusion, prevalent in substitutional solid solutions like FCC metals, requires thermal generation of lattice vacancies; an atom adjacent to a vacancy exchanges positions with jump frequency Γ=νexp⁡(−ΔGm/kT)\Gamma = \nu \exp(-\Delta G_m / kT), where ν\nu is attempt frequency, ΔGm\Delta G_m is migration , kk is Boltzmann's constant, and TT is . The overall diffusion coefficient is D=16a2cvfΓD = \frac{1}{6} a^2 c_v f \Gamma, with cvc_v as vacancy fraction, aa as jump distance, and ff as correlation factor (~0.78 for FCC random walks) correcting for directional preferences in successive jumps. Interstitial diffusion, common for small solutes in open structures like BCC iron, involves atoms hopping between interstitial sites without vacancies, yielding higher DD due to lower barriers; the coefficient follows D=16a2Γ′D = \frac{1}{6} a^2 \Gamma', where Γ′\Gamma' is the interstitial jump rate, often 10^4-10^6 times faster than vacancy-mediated at homologous temperatures. Diffusion coefficients in solid solutions depend strongly on and composition, typically plotted as Arrhenius relations to reveal parameters. The form D=D0exp⁡(−Q/RT)D = D_0 \exp(-Q/RT) captures thermally activated behavior, with QQ encompassing formation and migration energies; linear Arrhenius plots of ln⁡D\ln D vs. 1/T1/T allow extrapolation across temperatures. Composition dependence arises from lattice distortions and site availability, often modeled via D~(x)=(xBDA∗+xADB∗)ϕ\tilde{D}(x) = (x_B D_A^* + x_A D_B^*) \phi, where xx are mole fractions, D∗D^* are tracer coefficients, and ϕ\phi is the thermodynamic factor. In the Al-Cu system, interdiffusion in the α-phase shows parameters with Arrhenius plots confirming exponential increase from ~10^{-14} cm²/s at 400°C to ~10^{-10} cm²/s at 600°C, highlighting sensitivity to Cu content that accelerates homogenization in age-hardenable alloys.

Applications

In Metallurgy and Alloys

Solid solution strengthening in metallic alloys primarily involves the incorporation of substitutional solute atoms into the host lattice, distorting the crystal structure and impeding dislocation glide to elevate yield strength. This lattice strain arises from atomic size mismatches between solute and solvent elements, creating elastic interactions that increase the stress required for plastic deformation. In austenitic stainless steels, such as those based on Fe-Cr-Ni compositions, chromium and nickel solutes provide significant solid solution strengthening, contributing up to several hundred MPa to the overall strength while maintaining ductility for structural applications. The homogeneous atomic distribution in solid solutions also enhances resistance by eliminating phase boundaries that could act as galvanic cells, promoting uniform passivation. For instance, in alpha brasses (Cu-Zn alloys with up to 35% Zn), the single-phase solid solution structure resists dezincification and general in atmospheric and marine environments better than multiphase alternatives. Historical alloys exemplify these benefits, such as (92.5% Ag-7.5% Cu), where copper atoms in solid solution increase hardness from ~25 HV for pure silver to ~80-100 HV, enabling durable jewelry and utensils without compromising luster. Similarly, (Al-4% Cu-0.5% Mg), developed in the early 20th century, relies on a supersaturated solid solution of Cu and Mg in aluminum to achieve high strength-to-weight ratios for frames after appropriate processing. In modern applications, nickel-based superalloys like utilize from Cr, Mo, and Nb solutes to deliver creep resistance and high tensile strength, with ultimate tensile strengths of approximately 760 MPa at 650°C, critical for gas turbine blades and exhaust components. To form and stabilize these solid solutions, metallurgical processing employs solutionizing heat treatments, where alloys are heated to 800-1100°C (depending on composition) to fully dissolve solutes, followed by rapid to retain the supersaturated state and prevent premature precipitation. Subsequent aging at lower temperatures (100-200°C) refines the microstructure, further optimizing strength through controlled solute distribution without inducing full decomposition. This sequence is essential for alloys like austenitic steels and superalloys, ensuring reproducible enhancement of mechanical and environmental performance.

In Ceramics and Semiconductors

In semiconductors, solid solutions are primarily formed through substitutional doping, where dopant atoms replace host lattice atoms to modify electrical properties. For n-type doping, phosphorus (P) atoms substitute silicon (Si) atoms in the lattice, introducing an extra valence electron that enhances electron conductivity. Similarly, boron (B) atoms create p-type doping by substituting Si, resulting in acceptor sites that generate holes as majority charge carriers. These substitutional solid solutions enable precise control over carrier concentration, forming the basis for p-n junctions in devices like transistors and solar cells. Small dopants may also occupy interstitial sites briefly, but substitution dominates for stable conductivity. Band gap tuning in solid solutions further expands their utility by adjusting optical and electronic responses through compositional variation. In systems, such as Ga₂O₃-Al₂O₃, the can be varied continuously from 4.8 eV to 6.6 eV by altering the Al content, enabling tailored absorption and emission properties. This tunability arises from the of band edges in the solid solution, as seen in halides where Sr²⁺ substitution for Pb²⁺ in CsPbBr₃ widens the for visible-light applications. Ceramic solid solutions leverage similar principles to enhance dielectric and ionic functionalities. In perovskites like (Ba,Sr)TiO₃, the substitution of Sr for Ba forms a complete solid solution that optimizes permittivity for multilayer capacitors, achieving high density and temperature stability. For ionic conductivity, (ZrO₂-Y₂O₃), particularly at 8 mol% Y₂O₃, stabilizes the cubic phase and boosts oxygen mobility, making it a key solid in solid oxide cells with conductivities exceeding 0.1 S/cm at 1000°C. Optoelectronic devices benefit from solid solutions in III-V semiconductors, such as GaAs_{1-x}P_x alloys used in light-emitting diodes (LEDs). The phosphorus content x controls the band gap from ~1.4 eV (GaAs) to ~2.3 eV (GaP), enabling tuning from (~655 nm) to green light for efficient emission. Challenges in these materials include phase segregation at high temperatures, which disrupts homogeneity in solid solutions like ceria-zirconia, leading to reduced performance in electrolytes or dielectrics. Sol-gel synthesis addresses this by enabling low-temperature processing to form uniform solid solutions in ceramics and semiconductors, promoting nanoscale mixing and avoiding segregation during gelation and calcination.

Exsolution

Exsolution refers to the process in which a homogeneous solid solution decomposes into two distinct phases, typically triggered by cooling below the solvus line where the of mixing (ΔG_mix) becomes positive for intermediate compositions, rendering the single-phase state thermodynamically unstable. This positive ΔG_mix arises from the dominance of enthalpic interactions over entropic contributions at lower temperatures, driving the system toward minimization of free energy through unmixing. The primary mechanisms of exsolution are and and growth, distinguished by the presence or absence of a compositional barrier to . is a continuous process occurring within the spinodal region of the , where small composition fluctuations amplify spontaneously via without , leading to interconnected domains and initially coherent interfaces that maintain lattice continuity between phases. In contrast, and growth is a discontinuous mechanism outside the spinodal, requiring an energy barrier for forming discrete nuclei that expand by solute attachment, often resulting in isolated precipitates with coherent interfaces at small sizes that transition to incoherent as misfit strains accumulate and dislocations form. A classic example of exsolution occurs in alkali feldspars, where cooling of a high-temperature Na-K solid solution produces perthite textures featuring lamellar or rod-like intergrowths of Na-rich (albite) and K-rich (orthoclase) phases, with lamellae oriented along directions like (601) to minimize elastic strain energy. These microstructures, such as fine-scale cryptoperthites or coarser vein perthites, arise from coupled diffusion of Na and K, and they play a key role in geological minerals by recording thermal histories in igneous rocks. The development of exsolution textures can be controlled through cooling rates, as slow cooling enhances kinetics, promoting coarser lamellar or rod-like precipitates that improve texture and in applications like ceramics.

Ordering and Precipitation

In solid solutions, ordering transitions occur when atoms rearrange from a random distribution to a more structured configuration below a critical , enhancing such as strength and stability. A classic example is the Cu₃Au , which undergoes a second-order at approximately 663 from a disordered face-centered cubic (fcc) solid solution to an ordered L1₂ structure, where copper and atoms occupy distinct sublattices. This order-disorder transformation is driven by thermodynamic favorability at lower temperatures, with the critical marking the point where contributions from disorder balance the enthalpic gains of ordering. Precipitation hardening in solid solutions involves the controlled formation of fine secondary phases from a supersaturated matrix, leading to significant strengthening through mechanisms like coherency strains and interactions. In aluminum-copper alloys, the process follows a well-defined sequence: starting from a supersaturated solid solution obtained by from high temperature, Guinier-Preston (GP) zones—coherent clusters of atoms—form first during low-temperature aging, providing initial hardening. These evolve into metastable θ″ precipitates (coherent Al₃Cu discs), followed by semi-coherent θ′ plates, and finally the stable incoherent θ phase (CuAl₂), with peak hardness typically occurring at the θ′ stage due to optimal size and distribution of obstacles to motion. This sequence relies on diffusion-controlled and growth, tailored by aging time and temperature to balance strength and . Bainite formation in steels exemplifies the interplay between diffusional and shear transformations within solid solutions, where partial decomposition of the parent austenite phase produces a microstructure of ferrite plates with dispersed carbides. Unlike fully diffusional transformations like pearlite, bainite involves a displacive (shear) mechanism for the initial ferrite formation, coupled with carbon diffusion to adjacent austenite, leading to carbide precipitation and partial solution decomposition without complete solute partitioning. This hybrid nature—debated between "diffusion school" (emphasizing carbon enrichment) and "shear school" (focusing on invariant plane strain)—results in bainite's fine-scale structure, offering improved toughness over martensite while avoiding pearlite's coarseness. In beta-titanium alloys, ordered phases such as the B2 structure contribute to high-temperature performance in applications, where atomic ordering in the body-centered cubic matrix enhances creep resistance and stability under load. For instance, B2-ordered Ti–Mo–Al alloys exhibit yield strengths such as 818 MPa at 1073 K for Ti50Mo35Al15, with the ordered phase forming through that promotes sublattice occupation by aluminum and , enabling use in components and airframes. These ordered structures in beta alloys, stabilized by alloying elements like , provide a balance of and strength superior to disordered solid solutions, critical for lightweight designs.

References

  1. http://dspace.mit.edu/bitstream/handle/1721.1/35863/99/3-091Fall-2002/NR/rdonlyres/Materials-Science-and-Engineering/3-091Introduction-to-Solid-State-ChemistryFall2002/DEFC3DC2-4486-4B97-9F81-F1D12F898417/0/Notes_10.pdf
  2. https://www2.tulane.edu/~sanelson/eens211/mineral_chemistry.htm
  3. https://dl.asminternational.org/technical-books/monograph/141/chapter/2418290/Structure-of-Metals-and-Alloys
  4. https://www.researchgate.net/publication/235762217_Solid_solutions_Background_history_and_scientific_perspective
  5. https://www.phase-trans.msm.cam.ac.uk/2004/titanium/hume.rothery.html
  6. https://www.sciencedirect.com/topics/engineering/primary-solid-solution
  7. https://www.sciencedirect.com/topics/materials-science/solid-solution-hardening
  8. https://pubs.acs.org/doi/10.1021/accountsmr.3c00126
  9. https://royalsocietypublishing.org/doi/10.1098/rsta.1934.0014
  10. https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Map%253A_Inorganic_Chemistry_%28Housecroft%29/06%253A_Structures_and_Energetics_of_Metallic_and_Ionic_solids/6.07%253A_Alloys_and_Intermetallic_Compounds/6.7A%253A_Substitutional_Alloys
  11. https://www.researchgate.net/figure/Equilibrium-Cu-Zn-phase-diagram-5_fig2_316598481
  12. https://www.sciencedirect.com/topics/materials-science/solid-solution
  13. https://www.roymech.co.uk/Useful_Tables/Matter/Phase_diagram.html
  14. https://www.engineeringenotes.com/engineering/materials-engineering/types-of-solid-solutions-material-engineering/34508
  15. https://www.doitpoms.ac.uk/tlplib/solid-solutions/thermodynamics.php
  16. https://www.geo.arizona.edu/~ganguly/SolidSoln.pdf
  17. https://www.phase-trans.msm.cam.ac.uk/mphil/MP4-3.pdf
  18. https://www.sciencedirect.com/science/article/abs/pii/S1359645409004558
  19. http://www.diva-portal.org/smash/get/diva2:558929/FULLTEXT01.pdf
  20. https://ui.adsabs.harvard.edu/abs/2009PhRvB..80m4112A/abstract
  21. https://pubs.geoscienceworld.org/msa/rimg/article/88/1/689/614940/Synthesis-of-Diamonds-and-Their-Identification
  22. https://pubs.aip.org/aip/jap/article/111/5/053506/345644/Pressure-effects-on-phase-equilibria-and-solid
  23. https://www.researchgate.net/publication/338849672_Ternary_Interactions_and_Implications_for_Third_Element_Alloying_Potency_in_Al-Ce-Based_Alloys
  24. http://www.icaa-conference.net/ICAA12/pdf/P088.pdf
  25. https://www.jstage.jst.go.jp/article/matertrans/61/11/61_MT-MBW2019010/_html/-char/ja
  26. https://deringerney.com/hume-rothery-rules-and-the-solid-solubility-of-binary-systems/
  27. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/13%3A_The_Phase_Rule_and_Phase_Diagrams/13.02%3A__Phase_Diagrams-_Binary_Systems
  28. https://ocw.mit.edu/courses/3-091sc-introduction-to-solid-state-chemistry-fall-2010/pages/solid-solutions/35-binary-phase-diagrams-limited-solubility/
  29. https://www2.tulane.edu/~sanelson/eens211/2compphasdiag.html
  30. https://www.princeton.edu/~maelabs/mae324/glos324/solvus.htm
  31. https://www.chem.tamu.edu/rgroup/hughbanks/courses/462/handouts/Phase_Diagram_supplement.pdf
  32. https://www3.nd.edu/~amoukasi/cbe30361/Lecture_Phase_Diagram_2014.pdf
  33. https://ocw.mit.edu/courses/3-091sc-introduction-to-solid-state-chemistry-fall-2010/8561973350a2e44d7ee4e8304272c56f_MIT3_091SCF09_aln10.pdf
  34. https://web.eng.fiu.edu/wangc/EGN3365-9.pdf
  35. https://wikis.mit.edu/confluence/pages/viewpage.action?pageId=94020136
  36. https://www.andrew.cmu.edu/user/dl0p/laughlin/pdf/460.pdf
  37. https://www.researchgate.net/publication/296838174_SOLID-STATE_DIFFUSION_REACTION_AND_FORMATION_OF_INTERMETALLIC_PHASES_IN_THE_Fe-Al_SYSTEM
  38. https://www.sciencedirect.com/science/article/pii/B9780125309851500059
  39. https://ntrs.nasa.gov/api/citations/19860012188/downloads/19860012188.pdf
  40. https://www.tandfonline.com/doi/full/10.1080/21663831.2022.2075243
  41. https://link.aps.org/doi/10.1103/PhysRevB.48.13244
  42. https://apps.dtic.mil/sti/tr/pdf/ADA179880.pdf
  43. https://www.mpie.de/3023328/Cu-Cr_thin_film_structures
  44. https://pubs.acs.org/doi/10.1021/ed041p397
  45. http://li.mit.edu/Stuff/PM/Reading/Nakajima97JOM.pdf
  46. https://www.doitpoms.ac.uk/tlplib/diffusion/diffusion_mechanism.php
  47. https://srd.nist.gov/JPCRD/jpcrd61.pdf
  48. https://www.sciencedirect.com/science/article/abs/pii/S0921509305010683
  49. https://www.sciencedirect.com/science/article/abs/pii/S104458031832816X
  50. http://www.diva-portal.org/smash/get/diva2:621120/FULLTEXT01.pdf
  51. https://www.nature.com/articles/s41529-018-0027-4
  52. https://www.azom.com/article.aspx?ArticleID=6280
  53. https://escholarship.org/content/qt4bx8355x/qt4bx8355x.pdf
  54. https://nvlpubs.nist.gov/nistpubs/ScientificPapers/nbsscientificpaper347vol15p271_A2b.pdf
  55. https://asm.matweb.com/search/SpecificMaterial.asp?bassnum=NINC33
  56. https://www.bodycote.com/services/heat-treatment/solution-and-age/solution-and-age-aluminium-alloys/
  57. https://www.bodycote.com/services/heat-treatment/solution-and-age/
  58. https://www.jrfurnace.net/what-is-solution-heat-treatment-and-why-is-it-important/
  59. https://ocw.tudelft.nl/wp-content/uploads/solar_energy_section_6B.pdf
  60. https://chem.libretexts.org/Courses/Barry_University/CHE360%253A_Inorganic_Chemistry/08%253A_The_Crystalline_Solid_State/8.05%253A_Semiconductors-_Band_Gaps_Colors_Conductivity_and_Doping
  61. https://pmc.ncbi.nlm.nih.gov/articles/PMC8307042/
  62. https://www.sciencedirect.com/science/article/abs/pii/S0272884222017424
  63. https://chemrxiv.org/engage/chemrxiv/article-details/6213ab944e899e670e55a634
  64. https://ir.library.oregonstate.edu/downloads/ff365c492
  65. https://iopscience.iop.org/article/10.1149/2.0641707jes
  66. https://marktechopto.com/the-history-of-leds-and-led-technology/
  67. https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2010.0512
  68. https://www.sigmaaldrich.com/US/en/technical-documents/technical-article/materials-science-and-engineering/solid-state-synthesis/sol-gel-science-for
  69. https://www.doitpoms.ac.uk/tlplib/solid-solutions/free-energy.php/exsolution.php
  70. https://link.springer.com/chapter/10.1007/978-1-4613-9019-0_10
  71. https://ajsonline.org/api/v1/articles/61786-exsolution-by-spinodal-decomposition-ii-perthite-formation-during-slow-cooling-of-anatexites-from-ngoronghoro-tanzania.pdf
  72. https://pubs.geoscienceworld.org/msa/ammin/article/100/5-6/1277/40546/Routine-characterization-and-interpretation-of
  73. https://link.aps.org/pdf/10.1103/PhysRevResearch.3.023239
  74. https://www.colorado.edu/faculty/zunger-matter-by-design/sites/default/files/attached-files/531.pdf
  75. https://www.sciencedirect.com/science/article/abs/pii/S1005030216302468
  76. https://www.mdpi.com/2076-3417/11/11/4906
  77. https://www.sciencedirect.com/science/article/abs/pii/S1359028606000490
  78. https://www.twi-global.com/technical-knowledge/faqs/faq-what-are-the-microstructural-constituents-austenite-martensite-bainite-pearlite-and-ferrite
  79. https://www.sciencedirect.com/science/article/abs/pii/S0966979519308593
  80. https://www.sciencedirect.com/science/article/pii/S104458031830295X
Add your contribution
Related Hubs
Contribute something
User Avatar
No comments yet.