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Unformatted text preview: 60 Point Defect Thermodymanics Equilibrium Concentration of Solute Atoms We consider a dilute substitutional solution of B atoms in an A matrix. We let N be the number of A atoms in the system (which is fixed) and xN be the number of B atoms dissolved in the lattice. Thus x is the fraction of atomic sites where the wrong kind of atom is located. We follow the regular solution/quasi- chemical approach in which the formation energy of the point defect is dominated by the energies of the chemical bonds associated with the impurity defect (quasi-chemical) and where the mixing entropy is that for an ideal solution (regular solution). The system under consideration is one with an infinite supply of B atoms. Each time a B atom is dissolved in the lattice the A atom it replaces takes up a site at the A/B interface and extends the A lattice by one atomic volume. Unit process of dissolving one B atom into an A lattice We wish to describe the equilibrium state of the system. Will any B atoms dissolve in A at equilibrium? Adding B atomic defects to the A lattice increases the internal energy of the system and that discourages the introduction of foreign, defect atoms. But the entropy of the system also increases when B atoms are introduced and that causes some B atoms to be present at equilibrium, in spite of the increased internal energy of the system. We know this result in another context. Everyone is familiar with the fact that solutes usually dissolve spontaneously in pure components and the equilibrium concentration increases with temperature. This is shown as the solvus line on a phase diagram. At a particular temperature, the equilibrium concentration of solute is given by the intersection of the isotherm with the solvus line. Solvus line in a simple phase diagram 61 If the pressure and temperature are held constant, then the equilibrium state of the system is determined by the lowest possible Gibbs free energy. So we consider the change in Gibbs free energy that occurs when N B = xN B atoms are introduced into the A lattice. This may be written as ¡ G = ¡ H ¢ T ¡ S , where ¡ H and ¡ S are the change in enthalpy and entropy, respectively, when the defects are introduced. We first consider the factors that make up ¡ H . Enthalpy The change in enthalpy can be expressed as ¡ H = ¡ E + p ext ¡ V , where ¡ E and ¡ V are the change in internal energy and volume associated with creation of the defects and p ext is the external hydrostatic pressure. The internal energy change can be expressed as ¡ E = xN ¡ e f , where ¡ e f is the formation (internal) energy of each defect, usually just called the formation energy of the defect. It, in turn, can be composed of two parts, ¡ e f = ¡ e f bond + ¡ e f strain , where ¡ e f bond is the internal energy change associated with making the wrong kinds of chemical bonds at the defect and ¡ e f strain is the elastic strain energy associated with creating the defect. We may estimate ¡ e f bond if the energies of the...
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This note was uploaded on 12/01/2011 for the course MS&E 206 taught by Professor Nix during the Spring '08 term at Stanford.
- Spring '08