Measurement of Equilibrium Vacancy Concentrations in Dilute Aluminum-Magnesium Alloys

Abstract

Precise measurements were made of the differential length expansion ($\frac{\ensuremath{\Delta}{L}^{\ensuremath{'}}}{{L}^{\ensuremath{'}}}\ensuremath{-}\frac{\ensuremath{\Delta}{L}^{0}}{{L}^{0}}$) and differential x-ray lattice-parameter expansions ($\frac{\ensuremath{\Delta}{a}^{\ensuremath{'}}}{{a}^{\ensuremath{'}}}\ensuremath{-}\frac{\ensuremath{\Delta}{a}^{0}}{{a}^{0}}$), between specimens of pure aluminum and aluminum-base alloys containing 0.56 and 1.11 at.% magnesium, during slow reversible heating and cooling between the solidus and 185\ifmmode^\circ\else\textdegree\fi{}C. The differential-expansion data are used to determine absolute differences between the equilibrium vacancy concentrations ${C}_{v}$ in the dilute alloys and the pure metal from the relation $\ensuremath{\Delta}{C}_{v}={{C}_{v}}^{\ensuremath{'}}\ensuremath{-}{{C}_{v}}^{0}=3(\frac{\ensuremath{\Delta}{L}^{\ensuremath{'}}}{{L}^{\ensuremath{'}}}\ensuremath{-}\frac{\ensuremath{\Delta}{L}^{0}}{{L}^{0}})\ensuremath{-}3(\frac{\ensuremath{\Delta}{a}^{\ensuremath{'}}}{{a}^{\ensuremath{'}}}\ensuremath{-}\frac{\ensuremath{\Delta}{a}^{0}}{{a}^{0}}).$ Here $\frac{\ensuremath{\Delta}L}{L}$ and $\frac{\ensuremath{\Delta}a}{a}$ are length and lattice-parameter expansions, and the prime and zero superscripts refer to the alloy and pure aluminum, respectively. Since ${{C}_{v}}^{0}$ is known from previous measurements, the vacancy concentrations in each alloy ${{C}_{v}}^{\ensuremath{'}}$ are then determined. The differential length and lattice-parameter measurements were carried out using the same technique previously employed in the determination of equilibrium vacancy concentrations in dilute aluminum-silver alloys. The technique allows the measurement of concentration differences as small as \ifmmode\pm\else\textpm\fi{}4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$. The addition of magnesium caused only exceedingly small changes in the equilibrium vacancy concentrations. The changes did not exceed a few percent of the concentrations in pure aluminum. The results are compared at 650\ifmmode^\circ\else\textdegree\fi{}C to the simple first-order vacancy-solute-atom-binding model of Lidiard and others. It is concluded that the vacancy-magnesium-atom binding free energy was ${{G}_{v,\mathrm{Mg}}}^{b}(650\ifmmode^\circ\else\textdegree\fi{}\mathrm{C})⋜0.01$ eV. Possible values of binding energies and entropies consistent with these results are discussed. The results are compared to the previous measurements for dilute aluminum-silver alloys where larger (but still small) solute-induced vacancy increments were found, and where ${{G}_{v,Ag}}^{b}(650\ifmmode^\circ\else\textdegree\fi{}\mathrm{C})\ensuremath{\approx}0.08$ eV.