Segregation (materials science)

This is the earliest theory specifically for grain boundaries, in which McLean[3] uses a model of P solute atoms distributed at random amongst N lattice sites and p solute atoms distributed at random amongst n independent grain boundary sites. The total free energy due to the solute atoms is then:

${\displaystyle G=pe+PE-kT[\ln(n!N!)-\ln(n-p)!p!(N-P)!P!]}$

where E and e are energies of the solute atom in the lattice and in the grain boundary, respectively and the kln term represents the configurational entropy of the arrangement of the solute atoms in the bulk and grain boundary. McLean used basic statistical mechanics to find the fractional monolayer of segregant, ${\displaystyle X_{b}}$, at which the system energy was minimized (at the equilibrium state), differentiating G with respect to p, noting that the sum of p and P is constant. Here the grain boundary analogue of Langmuir adsorption at free surfaces becomes:

${\displaystyle {\frac {X_{b}}{X_{b}^{0}-X_{b}}}={\frac {X_{c}}{1-X_{c}}}\exp \left({\frac {-\Delta G}{RT}}\right)}$

Here, ${\displaystyle X_{b}^{0}}$ is the fraction of the grain boundary monolayer available for segregated atoms at saturation, ${\displaystyle X_{b}}$ is the actual fraction covered with segregant, ${\displaystyle X_{c}}$ is the bulk solute molar fraction, and ${\displaystyle \Delta G}$ is the free energy of segregation per mole of solute.

Values of ${\displaystyle \Delta G}$ were estimated by McLean using the elastic strain energy, ${\displaystyle E_{\text{el}}}$, released by the segregation of solute atoms. The solute atom is represented by an elastic sphere fitted into a spherical hole in an elastic matrix continuum. The elastic energy associated with the solute atom is given by:

${\displaystyle E_{\text{el}}={\frac {24\pi \mathrm {K} \mu _{0}r_{0}(r_{1}-r_{0})^{2}}{3\mathrm {K} +4\mu _{0}}}}$

where ${\displaystyle \mathrm {K} }$ is the solute bulk modulus, ${\displaystyle \mu _{0},}$ is the matrix shear modulus, and ${\displaystyle r_{0},}$ and ${\displaystyle r_{1},}$ are the atomic radii of the matrix and impurity atoms, respectively. This method gives values correct to within a factor of two (as compared with experimental data for grain boundary segregation), but a greater accuracy is obtained using the method of Seah and Hondros,[10] described in the following section.

Using truncated BET theory (the gas adsorption theory developed by Brunauer, Emmett, and Teller), Seah and Hondros[10] write the solid-state analogue as:

${\displaystyle {\frac {X_{b}}{X_{b}^{0}-X_{b}}}={\frac {X_{c}}{X_{c}^{0}}}\exp }$ ${\displaystyle \left({\frac {-\Delta G'}{RT}}\right)}$

where ${\displaystyle \Delta G=\Delta G'+\Delta G_{\text{sol}}}$

${\displaystyle X_{c}^{0}}$ is the solid solubility, which is known for many elements (and can be found in metallurgical handbooks). In the dilute limit, a slightly soluble substance has ${\displaystyle X_{c}^{0}=\exp \left({\frac {\Delta G_{\text{sol}}}{RT}}\right)}$, so the above equation reduces to that found with the Langmuir-McLean theory. This equation is only valid for ${\displaystyle X_{c}\leq X_{c}^{0}}$. If there is an excess of solute such that a second phase appears, the solute content is limited to ${\displaystyle X_{c}^{0}}$ and the equation becomes

${\displaystyle {\frac {X_{b}}{X_{b}^{0}-X_{b}}}=\exp \left({\frac {-\Delta G'}{RT}}\right)}$

This theory for grain boundary segregation, derived from truncated BET theory, provides excellent agreement with experimental data obtained by Auger electron spectroscopy and other techniques.[12]

Other models exist to model more complex binary systems.[12] The above theories operate on the assumption that the segregated atoms are non-interacting. If, in a binary system, adjacent adsorbate atoms are allowed an interaction energy ${\displaystyle \omega \,}$, such that they can attract (when ${\displaystyle \omega \,}$ is negative) or repel (when ${\displaystyle \omega \,}$ is positive) each other, the solid-state analogue of the Fowler adsorption theory is developed as

${\displaystyle {\frac {X_{b}}{X_{b}^{0}-X_{b}}}={\frac {X_{c}}{1-X_{c}}}\exp \left[{\frac {-\Delta G-Z_{1}\omega \,{\frac {X_{b}}{X_{b}^{0}}}}{RT}}\right].}$

When ${\displaystyle \omega \,}$ is zero, this theory reduces to that of Langmuir and McLean. However, as ${\displaystyle \omega \,}$ becomes more negative, the segregation shows progressively sharper rises as the temperature falls until eventually the rise in segregation is discontinuous at a certain temperature, as shown in the following figure.

Guttman, in 1975, extended the Fowler theory to allow for interactions between two co-segregating species in multicomponent systems. This modification is vital to explaining the segregation behavior that results in the intergranular failures of engineering materials. More complex theories are detailed in the work by Guttmann[13] and McLean and Guttmann.[14]

The Langmuir–McLean equation for segregation, when using the regular solution model for a binary system, is valid for surface segregation (although sometimes the equation will be written replacing ${\displaystyle X_{b}}$ with ${\displaystyle X_{s}}$).[15] The free energy of surface segregation is ${\displaystyle \Delta G_{s}=\Delta H_{s}-T\,\Delta S}$. The enthalpy is given by

${\displaystyle -\Delta H_{s}=\gamma _{0}^{s}-\gamma _{1}^{s}-{\frac {2H_{m}}{ZX_{c}(1-X_{c})}}\left[Z_{1}(X_{c}-X_{s})+Z_{v}\left(X_{c}-{\frac {1}{2}}\right)\right]+{\frac {24\pi \mathrm {K} \mu _{0}r_{0}(r_{1}-r_{0})^{2}}{3\mathrm {K} +4\mu _{0}}}}$

where ${\displaystyle \gamma _{0}}$ and ${\displaystyle \gamma _{1}}$ are matrix surface energies without and with solute, ${\displaystyle H_{1}}$ is their heat of mixing, Z and ${\displaystyle Z_{1}}$ are the coordination numbers in the matrix and at the surface, and ${\displaystyle Z_{v}}$ is the coordination number for surface atoms to the layer below. The last term in this equation is the elastic strain energy ${\displaystyle E_{\text{el}}}$, given above, and is governed by the mismatch between the solute and the matrix atoms. For solid metals, the surface energies scale with the melting points. The surface segregation enrichment ratio increases when the solute atom size is larger than the matrix atom size and when the melting point of the solute is lower than that of the matrix.[12]

A chemisorbed gaseous species on the surface can also have an effect on the surface composition of a binary alloy. In the presence of a coverage of a chemisorbed species theta, it is proposed that the Langmuir-McLean model is valid with the free energy of surface segregation given by ${\displaystyle \Delta G_{\text{chem}}}$,[16] where

${\displaystyle \Delta G_{\text{chem}}=\Delta G_{s}+(E_{B}-E_{A})\Theta \,}$

${\displaystyle E_{A}}$ and ${\displaystyle E_{B}}$ are the chemisorption energies of the gas on solute A and matrix B and ${\displaystyle \Theta }$ is the fractional coverage. At high temperatures, evaporation from the surface can take place, causing a deviation from the McLean equation. At lower temperatures, both grain boundary and surface segregation can be limited by the diffusion of atoms from the bulk to the surface or interface.