Solid solution strengthening

Solid solution strengthening is a type of alloying that can be used to improve the strength of a pure metal. Atoms of one element are added to a crystalline lattice comprised of atoms of another. The alloying element will diffuse into the matrix, forming a "solid solution". In most binary systems, when alloyed above a certain concentration, a second phase will form and the material will enjoy the benefits of precipitation strengthening.

Types

Depending on the size of the alloying element, a substitutional solid solution or an interstitial solid solution can form.

In a substitutional solid solution, solute atoms replace solvent atoms in their lattice positions. Based on the Hume-Rothery Rule, solvent and solute atoms must differ in atomic size by less than 5% in order to form this type of solution. Because both elements exist in the same crystalline lattice, both elements in their pure form must be of the same crystal structure. Examples of substitutional solid solutions include the Cu-Ni and the Ag-Au binary systems.

When the solute atom is much smaller than the solvent atoms, an interstitial solid solution forms. This typically occurs when the solute atoms are less than half as small as the solvent atoms. The diameter of the atoms of elements must not differ by more than 15% too, and that the electronegativiy difference should not be too big. Elements commonly used to form interstitial solid solutions include H, N, C, and O.

Strengthening mechanisms

The strength of a material is a measurement of 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 of the material. This gain is a result of both lattice distortion and the modulus effect.

When solute and solvent atoms differ in size, local stress fields are created (if solute atom size is larger than solvent atom size, this field is compressive, and similarly, when solute atoms are smaller than solvent atoms, this field is tensile). Depending on their relative locations, solute atoms will either attract or repel dislocations in their vicinity, requiring a higher force to overcome the obstacle. This is known as the size effect. This allows the solute atoms to relieve either tensile or compressive strain in the lattice, which in turn puts 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 both edge, screw, and mixed dislocations.

The energy density of a dislocation is dependent on its Burgers vector as well as the modulus of the local atoms. When the modulus of solute atoms differs from that of the host element, the local energy around the dislocation is changed, increasing the amount of force necessary to move past this energy well. This is known as the modulus effect. Meanwhile, in the specific case of a lattice distortion, the difference in lattice parameter leads to a high stress field around that solute atom that impedes dislocation movement.

Governing equations

Solid solution strengthening increases yield strength of the material by increasing the 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, b is the magnitude of the Burger's vector, and $\epsilon$ is the lattice strain due to the solute. This is composed of two terms, one due to lattice distortion and one due to local modulus change.

$\epsilon =\epsilon _{a}\epsilon _{G}$ Here, $\epsilon _{a}$ is the lattice distortion term, $\β$ a constant dependent on the solute atoms and $\epsilon _{G}$ the term that captures the local modulus change.

The lattice strain due to the solute can be described as:

$\epsilon ={\dfrac {\Delta a}{a\Delta c}}$, where a is the lattice parameter of the material.

As the concentration of the solute atoms increase, so does the stress, and therefore the yield strength of the material also increases. --Zaklouta (talk) 03:06, 24 November 2007 (UTC)