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In metallurgy, solid solution strengthening is a type of
alloy An alloy is a mixture of chemical elements of which at least one is a metal. Unlike chemical compounds with metallic bases, an alloy will retain all the properties of a metal in the resulting material, such as electrical conductivity, ductility, ...
ing that can be used to improve the strength of a pure
metal A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typica ...
. The technique works by adding atoms of one element (the alloying element) to the
crystalline lattice A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macrosc ...
of another element (the base metal), forming a solid solution. The local nonuniformity in the lattice due to the alloying element makes plastic deformation more difficult by impeding dislocation motion through
stress field A stress field is the distribution of internal forces in a body that balance a given set of external forces. Stress fields are widely used in fluid dynamics and materials science. Consider that one can picture the stress fields as the stress cr ...
s. In contrast, alloying beyond the solubility limit can form a second phase, leading to strengthening via other mechanisms (e.g. the precipitation of intermetallic compounds).


Types

Depending on the size of the alloying element, a substitutional solid solution or an interstitial solid solution can form. In both cases, atoms are vizualised as rigid spheres where the overall crystal structure is essentially unchanged. The rationale of crystal geometry to atom solubility prediction is summarized in the
Hume-Rothery rules Hume-Rothery rules, named after William Hume-Rothery, are a set of basic rules that describe the conditions under which an element could dissolve in a metal, forming a solid solution. There are two sets of rules; one refers to substitutional solid ...
and
Pauling's rules Pauling's rules are five rules published by Linus Pauling in 1929 for predicting and rationalizing the crystal structures of ionic compounds. First rule: the radius ratio rule For typical ionic solids, the cations are smaller than the anion ...
. Substitutional solid solution strengthening occurs when the solute atom is large enough that it can replace solvent atoms in their lattice positions. Some alloying elements are only soluble in small amounts, whereas some solvent and solute pairs form a solution over the whole range of binary compositions. Generally, higher solubility is seen when solvent and solute atoms are similar in
atomic size The atomic radius of a chemical element is a measure of the size of its atom, usually the mean or typical distance from the center of the nucleus to the outermost isolated electron. Since the boundary is not a well-defined physical entity, there ...
(15% according to the
Hume-Rothery rules Hume-Rothery rules, named after William Hume-Rothery, are a set of basic rules that describe the conditions under which an element could dissolve in a metal, forming a solid solution. There are two sets of rules; one refers to substitutional solid ...
) and adopt the same crystal structure in their pure form. Examples of completely miscible binary systems are Cu-Ni and the Ag-Au face-centered cubic (FCC) binary systems, and the Mo-W
body-centered cubic In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties ...
(BCC) binary system. Interstitial solid solutions form when the solute atom is small enough (radii up to 57% the radii of the parent atoms) to fit at interstitial sites between the solvent atoms. The atoms crowd into the interstitial sites, causing the bonds of the solvent atoms to compress and thus deform (this rationale can be explained with
Pauling's rules Pauling's rules are five rules published by Linus Pauling in 1929 for predicting and rationalizing the crystal structures of ionic compounds. First rule: the radius ratio rule For typical ionic solids, the cations are smaller than the anion ...
). Elements commonly used to form interstitial solid solutions include H, Li, Na, N, C, and O. Carbon in iron (steel) is one example of interstitial solid solution.


Mechanism

The strength of a material is dependent on how easily dislocations in its crystal lattice can be propagated. These dislocations create stress fields within the material depending on their character. When solute atoms are introduced, local stress fields are formed that interact with those of the dislocations, impeding their motion and causing an increase in the
yield stress In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and ...
of the material, which means an increase in strength of the material. This gain is a result of both lattice distortion and the
modulus effect Modulus is the diminutive from the Latin word ''modus'' meaning measure or manner. It, or its plural moduli, may refer to the following: Physics, engineering and computing * Moduli (physics), scalar fields for which the potential energy function ...
. When solute and solvent atoms differ in size, local stress fields are created that can attract or repel dislocations in their vicinity. This is known as the size effect. By relieving tensile or compressive strain in the lattice, the solute size mismatch can put the dislocation in a lower energy state. In substitutional solid solutions, these stress fields are spherically symmetric, meaning they have no shear stress component. As such, substitutional solute atoms do not interact with the shear stress fields characteristic of screw dislocations. Conversely, in interstitial solid solutions, solute atoms cause a tetragonal distortion, generating a shear field that can interact with edge, screw, and mixed dislocations. The attraction or repulsion of the dislocation to the solute atom depends on whether the atom sits above or below the slip plane. For example, consider an
edge dislocation In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to s ...
encountering a smaller solute atom above its slip plane. In this case, the interaction energy is negative, resulting in attraction of the dislocation to the solute. This is due to the reduced dislocation energy by the compressed volume lying above the dislocation core. If the solute atom were positioned below the slip plane, the dislocation would be repelled by the solute. However, the overall interaction energy between an edge dislocation and a smaller solute is negative because the dislocation spends more time at sites with attractive energy. This is also true for solute atom with size greater than the solvent atom. Thus, the interaction energy dictated by the size effect is generally negative. The elastic modulus of the solute atom can also determine the extent of strengthening. For a “soft” solute with elastic modulus lower than that of the solvent, the interaction energy due to modulus mismatch (''U''modulus) is negative, which reinforce the size interaction energy (''U''size). In contrast, ''U''modulus is positive for a “hard” solute, which results in lower total interaction energy than a soft atom. Even though the interaction force is negative (attractive) in both cases when the dislocation is approaching the solute. The maximum force (''F''max) necessary to tear dislocation away from the lowest energy state (i.e. the solute atom) is greater for the soft solute than the hard one. As a result, a soft solute will strengthen a crystal more than a hard solute due to the synergistic strengthening by combining both size and modulus effects. The elastic interaction effects (i.e. size and modulus effects) dominate solid-solution strengthening for most crystalline materials. However, other effects, including charge and stacking fault effects, may also play a role. For ionic solids where electrostatic interaction dictates bond strength, charge effect is also important. For example, addition of divalent ion to a monovalent material may strengthen the electrostatic interaction between the solute and the charged matrix atoms that comprise a dislocation. However, this strengthening is to a less extent than the elastic strengthening effects. For materials containing a higher density of stacking faults, solute atoms may interact with the stacking faults either attractively or repulsively. This lowers the stacking fault energy, leading to repulsion of the
partial dislocations In materials science, a partial dislocation is a decomposed form of dislocation that occurs within a crystalline material. An ''extended dislocation'' is a dislocation that has dissociated into a pair of partial dislocations. The vector sum of t ...
, which thus makes the material stronger. Surface carburizing, or
case hardening Case-hardening or surface hardening is the process of hardening the surface of a metal object while allowing the metal deeper underneath to remain soft, thus forming a thin layer of harder metal at the surface. For iron or steel with low carbon ...
, is one example of solid solution strengthening in which the density of solute carbon atoms is increased close to the surface of the steel, resulting in a gradient of carbon atoms throughout the material. This provides superior mechanical properties to the surface of the steel without having to use a higher-cost material for the component.


Governing equations

Solid solution strengthening increases yield strength of the material by increasing the shear stress, \tau, to move dislocations: \Delta \tau =Gb \epsilon^\tfrac 3 2 \sqrt c where ''c'' is the concentration of the solute atoms, ''G'' is the
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: :G \ \stackr ...
, ''b'' is the magnitude of the
Burger's vector In materials science, the Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted as , that represents the magnitude and direction of the lattice distortion resulting from a dislocation in a crystal lattice. The vecto ...
, and \epsilon is the lattice strain due to the solute. This is composed of two terms, one describing lattice distortion and the other local modulus change. \epsilon = , \epsilon_G - \beta \epsilon_a , Here, \epsilon_G the term that captures the local modulus change, \beta a constant dependent on the solute atoms and \epsilon_a is the lattice distortion term. The lattice distortion term can be described as: \epsilon_a = \dfrac , where ''a'' is the lattice parameter of the material. Meanwhile, the local modulus change is captured in the following expression: \epsilon_G = \dfrac , where ''G'' is shear modulus of the solute material.


Implications

In order to achieve noticeable material strengthening via solution strengthening, one should alloy with solutes of higher shear modulus, hence increasing the local shear modulus in the material. In addition, one should alloy with elements of different equilibrium lattice constants. The greater the difference in lattice parameter, the higher the local stress fields introduced by alloying. Alloying with elements of higher shear modulus or of very different lattice parameters will increase the stiffness and introduce local stress fields respectively. In either case, the dislocation propagation will be hindered at these sites, impeding plasticity and increasing yield strength proportionally with solute concentration. Solid solution strengthening depends on: * Concentration of solute atoms * Shear modulus of solute atoms * Size of solute atoms * Valency of solute atoms (for ionic materials) For many common alloys, rough experimental fits can be found for the addition in strengthening provided in the form of: \Delta\sigma_s = k_s\sqrt{c} where k_s is a solid solution strengthening coefficient and c is the concentration of solute in atomic fractions. Nevertheless, one should not add so much solute as to precipitate a new phase. This occurs if the concentration of the solute reaches a certain critical point given by the binary system phase diagram. This critical concentration therefore puts a limit to the amount of solid solution strengthening that can be achieved with a given material.


See also

* Strength of materials *
Strengthening mechanisms of materials Methods have been devised to modify the yield strength, ductility, and toughness of both crystalline and amorphous materials. These strengthening mechanisms give engineers the ability to tailor the mechanical properties of materials to suit a vari ...


References


External links


The Strengthening of Iron and Steel
Metallurgy Strengthening mechanisms of materials de:Mischkristallverfestigung ru:Диффузионное насыщение металлами