Research report: Diffusion in the liquid Al–In alloy when approaching the miscibility gap

Aluminum and indium are a classic example. When combined in the liquid state at intermediate compositions, they separate into two distinct liquids, a behavior known as liquid-liquid phase separation. The team, led by Toni Schiller, Benedikt Reiplinger, Asbjørn Torsvik Krüger, Elke Sondermann and Andreas Meyer, measured interdiffusion in liquid aluminum-indium alloys at aluminum-rich compositions and at temperatures close to the miscibility gap.

Interdiffusion, sometimes called chemical diffusion, describes the transport of mass driven by a gradient in chemical potential. In a fully miscible alloy, this atomic traffic smooths out concentration differences. The conventional expectation, rooted in the Darken equation, is that interdiffusion is the product of a kinetic contribution, built from the self-diffusion coefficients of the two species, and a thermodynamic factor derived from the curvature of the Gibbs energy.

For an ideal mixture the thermodynamic factor equals one, but as a system approaches its miscibility gap, that factor plunges toward zero. On paper, interdiffusion should therefore collapse as the alloy nears demixing. The experiments tell a different story.

To obtain reliable numbers, the researchers used an in situ shear cell furnace combined with X-ray radiography, a technique that has been refined over many years. The shear cell holds four adjacent capillaries, each 26 millimeters long and 1.3 millimeters in diameter, filled with alloys of different indium content. Once the target temperature is reached and the samples are fully liquid, the lower segment of the cell is shifted, bringing pairs of capillaries into contact and forming three diffusion couples per experiment, each initially characterized by a sharp step in concentration.

A microfocus X-ray source operating at 150 kilovolts and a cadmium-telluride detector with an effective pixel size of about 50 micrometers then record the blurring of that step in real time at one frame per second. Because the process is observed directly in the liquid, the method sidesteps the artifacts introduced by melting and solidification that plague conventional post-mortem diffusion measurements. Converting the X-ray photon counts into concentration profiles using reference alloys of known composition, the team obtained the concentration as a function of position and time, c(x,t).

For a one-dimensional step profile in an effectively infinite sample, Fick's second law has an exact solution in terms of the error function, and the width of the concentration profile grows linearly with the product of the diffusion coefficient and time. Fitting this solution to the measured profiles yields the interdiffusion coefficient for the concentration range spanned by each couple. A critical quality check is the stability of the fitted center of the profile: if the diffusion coefficient varied strongly with concentration, the profile would become asymmetric and the center would drift over time.

The team also analyzed the left, center and right portions of each vertical capillary separately to estimate the influence of residual convection, the dominant error source in such measurements, and excluded any data with relative errors above fifteen percent. Across compositions from about 1.5 to 10.5 atomic percent indium, the interdiffusion coefficient shows only a weak concentration dependence. At 1033 kelvin, it drops by roughly twenty percent over that range, while the thermodynamic factor calculated from the thermodynamic description of Kaban and colleagues falls by about fifty percent.

In other words, the steep slowdown expected from thermodynamics is largely absent. Compared with the self-diffusion of pure liquid aluminum, measured at 980 kelvin via quasi-elastic neutron scattering as 7.2 times ten to the minus nine square meters per second, interdiffusion in the alloy is about twenty percent slower even at very low indium contents, a drop consistent with earlier observations of solutes such as nickel, copper, silver and gold in liquid aluminum. On the indium-rich side, previous ex situ shear cell measurements found interdiffusion coefficients of about 8.4 to 9.3 times ten to the minus nine square meters per second, of similar magnitude to the aluminum-rich values reported here.

The temperature dependence, by contrast, behaves conventionally. All the data, averaged over a composition of roughly six atomic percent indium, fit a single Arrhenius function with an activation energy of 271 plus or minus 27 millielectronvolts and a pre-exponential factor of 12.8 times ten to the minus eight square meters per second. That activation energy sits almost exactly at the concentration-weighted linear combination of the activation energies of the pure elements, about 280 millielectronvolts for liquid aluminum and 108 for liquid indium, which yields 270 millielectronvolts.

Notably, measurements in the lithium-sodium system, which also has a miscibility gap, showed deviations from Arrhenius behavior near the critical temperature, so the clean Arrhenius behavior of aluminum-indium is itself informative. Two data points taken right at the boundary of the miscibility gap, where the diffusion couple was partially inside the demixing regime, showed apparent mass transport, but the authors caution these should not be interpreted as pure interdiffusion coefficients, since fine dispersed droplets of the minority phase may have contributed without ever being resolved in the X-ray images. To explain why the kinetic side of the equation seems to compensate for the collapsing thermodynamic factor, the researchers turned to density.

Denser, more tightly packed liquids generally exhibit slower diffusion, because atoms need free volume to move. Using electromagnetic levitation combined with optical dilatometry, the team measured the density of aluminum-rich aluminum-indium melts up to twenty atomic percent indium and at temperatures between 850 and 1400 kelvin. In levitation, a water-cooled copper coil carrying roughly 200 amperes at 250 kilohertz induces eddy currents that both heat and suspend the spherical sample, eliminating container contact and the contamination it brings.

A laser shadowgraph projected onto a CCD camera and analyzed with an edge-detection algorithm yields the sample volume, and hence the density, with a carefully quantified uncertainty budget accounting for mass loss, calibration, temperature and sample shape distortions. First, the temperature dependence of density, and therefore the thermal expansion, is identical inside and outside the miscibility gap. At 950 kelvin, a composition of twenty atomic percent indium lies well within the two-liquid region, yet the thermal expansion above and below the segregation temperature of about 1100 kelvin is the same.

Second, and more dramatically, the concentration dependence deviates strongly from ideality. Between five and fifteen atomic percent indium, the measured densities fall below the ideal-solution values, corresponding to a negative excess volume with a relative magnitude of up to fourteen percent. Earlier work on demixing systems such as lithium-sodium and bismuth-gallium reported deviations of less than 0.1 percent, and a previous study of aluminum-indium itself found no more than 0.4 percent.

A deviation of this size has never been reported for any other demixing liquid alloy, and molecular dynamics simulations suggest such excess volumes arise from an interplay between repulsive and attractive interactions between the unlike atoms. From the measured densities, the team calculated the volume packing fraction of the melt, treating the atoms as hard spheres with their covalent radii of 121 picometers for aluminum and 142 picometers for indium. The result, between 0.33 and 0.38 at 1200 kelvin, is remarkably low compared with other liquid alloys such as germanium-nickel at 0.43 or zirconium-nickel at 0.55 at the same temperature.

A loose packing means atoms can move with relative ease, which implies fast self-diffusion. That is precisely the mechanism the authors propose to explain their diffusion results: the kinetic contribution, dominated by the self-diffusion coefficients and possibly enhanced by the velocity cross-correlation factor S in the generalized Darken equation, counteracts the falling thermodynamic factor, leaving interdiffusion nearly flat as the miscibility gap approaches. The picture remains incomplete, because no self-diffusion data exist yet for aluminum-rich aluminum-indium melts, and future measurements are expected to settle the question.

Subject of Research: Interdiffusion and density of liquid aluminum-indium alloys approaching the liquid-liquid miscibility gap

Article Title: Diffusion in the liquid Al–In alloy when approaching the miscibility gap

Article References: Schiller, T., Reiplinger, B., Krüger, A. T., Sondermann, E., & Meyer, A. (2026). Diffusion in the liquid Al–In alloy when approaching the miscibility gap. Journal of Materials Science, 61(43), 34015-34026. https://doi.org/10.1007/s10853-026-13727-3

Image Credits: AI Generated

DOI: 10.1007/s10853-026-13727-3

Keywords: interdiffusion, miscibility gap, monotectic alloys, liquid metals, aluminum-indium, shear cell, X-ray radiography, electromagnetic levitation, density, excess volume, self-diffusion, Darken equation