Measurement of Equilibrium Vacancy Concentrations in Dilute Aluminum-Silver Alloys

Precision measurements were made of the differential length expansions ($\frac{\Delta L^{\prime }}{L^{\prime }}-\frac{\Delta L^0}{L^0}$) 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 two dilute aluminum base alloys Containing 0.52 and 0.94 at.% silver during slow reversible heating and cooling between the solidus and the solubility limit temperature. Absolute differences between the equilibrium vacancy concentrations in each alloy and the pure metal were then obtained 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, $C_v$ is the equilibrium vacancy concentration and $\frac{\Delta L}{L}$ and $\frac{\Delta a}{a}$ are length and lattice parameter expansions. The prime and zero superscripts refer to the alloy and pure aluminum, respectively. Since ${C_v}^0$ is known from previous measurements, these differential data serve to determine ${C_v}^{\prime }$. The differential length and lattice parameter measurements were carried out using the same general technique previously employed in the determination of equilibrium vacancy concentrations in a series of pure face-centered cubic metals, and yielded a relatively high precision in the determination of the extremely small differences involved. The addition of silver caused only small increases in the vacancy concentration. Values of $\Delta C_v$ equal to (13±5)×${10}^{-5\mathrm{}}$ and (12±5)×${10}^{-5\mathrm{}}$ were found for the 0.52 and 0.94 at.% silver alloys, respectively, at their solidus temperatures. These increments correspond to ⋜ 23% of the concentration in pure aluminum. The results for both alloys could be fitted, within the estimated uncertainty of the data, to a simple first-order vacancy-solute atom binding model, where $\Delta C_v=12\mathrm{}{C}_s{C_v}^0\left[\exp \right(-\frac{{S_{\mathrm{vs}}}^b}{k}\left)\exp \right(\frac{{E_{\mathrm{vs}}}^b}{\mathrm{kT}})-1].\mathrm{}$ Here, $C_s$ is the solute concentration, and the best value of the binding energy, ${E_{\mathrm{vs}}}^b$, was found to be 0.08 eV for an assumed binding entropy ${S_{\mathrm{vs}}}^b=0\mathrm{}$. This value was of the order generally expected from previous experiments. The significance of the apparent agreement between the data and the model were discussed.