Measurement of Equilibrium Vacancy Concentrations in Dilute Aluminum-Magnesium Alloys

Precise measurements were made of the differential length expansion ($\frac{\Delta L^{\prime }}{L^{\prime }}-\frac{\Delta L^0}{L^0}$) and differential x-ray lattice-parameter expansions ($\frac{\Delta a^{\prime }}{a^{\prime }}-\frac{\Delta a^0}{a^0}$), 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°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 $\Delta C_v={C_v}^{\prime }-{C_v}^0=3(\frac{\Delta L^{\prime }}{L^{\prime }}-\frac{\Delta L^0}{L^0})-3(\frac{\Delta a^{\prime }}{a^{\prime }}-\frac{\Delta a^0}{a^0}).\mathrm{}$ Here $\frac{\Delta L}{L}$ and $\frac{\Delta a}{a}$ 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 ${C_v}^{\prime }$ 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 ±4×${10}^{-5\mathrm{}}$. 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°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 ${G_{v,\mathrm{Mg}}}^b\left(650\mathrm{}°\mathrm{C}\right)⋜0.01$ 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 ${G_{v,Ag}}^b\left(650\mathrm{}°\mathrm{C}\right)\approx 0.08$ eV.