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

The research, published in the International Journal of Aeronautical and Space Sciences by Zijian Jiang, Huisong Wu, Zhenjun Zhao, Busen Li, and Ligeng Yu, describes an integrated vertical beam–electromagnetic support system designed specifically for full-mode flutter wind tunnel tests. Flutter is one of the most dangerous phenomena in aeronautics, a self-reinforcing oscillation in which aerodynamic forces feed energy into a structure’s natural vibration modes until the wing or tail can tear apart in seconds. Testing for flutter requires the model to move as freely as possible, yet traditional supports, whether rigid stings piercing the fuselage or cable-suspension rigs stretching across the test section, all add stiffness, damping, and aerodynamic interference that contaminate the measurement of the aircraft’s true dynamic behavior.

The team’s concept centers on a vertical beam running through the test section, along which a slider carrying the aircraft model can move. The critical innovation is electromagnetic: instead of relying on mechanical contact to hold the slider against gravity, the system uses a pair of coils and a permanent magnet arrangement to generate an upward electromagnetic force that counteracts the slider’s weight. The motivation, the authors explain, is the strict mass requirement of the balance slider, the instrumented component that measures the aerodynamic forces acting on the model. By offloading the weight to magnetic forces, the design frees the slider to be lighter and more responsive, which matters enormously when the model must be allowed to pitch, heave, and flutter naturally under the influence of the airflow.

Making a magnet hold something up in mid-air is deceptively hard. As the authors note in their references, the nineteenth-century physicist Samuel Earnshaw proved that static magnetic fields cannot stably levitate a ferromagnetic object in free space; any small displacement causes the object to fall toward or fly away from the magnet rather than return to equilibrium. Passive levitation is therefore impossible in this configuration, and active feedback control becomes essential. The system must continuously sense the slider’s position and adjust the coil currents thousands of times per second to keep the model hovering at its intended station, all while the aerodynamic forces of the tunnel flow push and pull on the aircraft model attached below.

To understand and tame this coupled electromechanical machine, the researchers built a comprehensive mathematical model of the entire system, capturing the vertical beam, the slider, the aircraft model, and the electromagnetic circuit together. From this model they derived the Lagrange–Maxwell equations, a classical framework that unites the mechanics of moving masses with the electrodynamics of currents and magnetic fields. These equations describe how the slider’s motion changes the air gap in the magnetic circuit, how that change alters the magnetic flux linkage and the resulting force, and how the electromagnetic force in turn accelerates the slider, a feedback loop that exists in the physics itself, before any controller is added. The authors also obtained an analytical expression for the electromagnetic force that counteracts the slider’s weight, which allowed them to pin down the critical design parameters: the coil geometry, the number of turns, the properties of the permanent magnet, and the baseline voltages and currents needed to establish equilibrium.

With the baseline condition established, in which the aerodynamic forces on the model and the model’s weight are in balance and the electromagnetic support holds the slider steady, the team turned to the question of what happens when something disturbs this delicate balance. A gust of air, a vibration from the tunnel, or the onset of flutter itself will push the model away from equilibrium, and the support system must respond. The researchers used a first-order Taylor series expansion of the electromagnetic force and the magnetic flux linkage to construct a linearized model valid for small perturbations around the operating point. Linearization is the standard workhorse of control engineering: it replaces the nonlinear magnetic force law with a linear approximation that is accurate near equilibrium and amenable to the powerful tools of linear systems theory.

The first result of this analysis was sobering. A complex modal analysis of the open-loop system, meaning the support system left to its own devices without feedback, revealed that the system is unstable. Left uncontrolled, any small disturbance would grow rather than decay, with the slider drifting away from its equilibrium position as the electromechanical coupling between the beam dynamics and the magnetic circuit amplified the motion. This instability is the direct fingerprint of Earnshaw’s theorem playing out in a real machine: the same magnetic forces that hold the slider up also push it sideways or vertically away from balance whenever it strays even slightly from the sweet spot.

The remedy was a position-velocity-acceleration feedback control strategy. Instead of measuring only where the slider is, the controller also monitors how fast it is moving and how quickly that motion is changing, feeding all three signals back into the coil drive voltages with carefully chosen gains. This triple feedback gives the controller a much richer picture of the system’s state than position alone, allowing it to damp oscillations aggressively while still holding the mean position steady. The authors then repeated their complex modal stability analysis on the closed-loop system, the combination of plant and controller, and confirmed that with the feedback gains in place, all system modes converge: disturbances decay, and the levitated slider returns to equilibrium. Time-domain simulations of the closed-loop dynamic response, initiated with small initial disturbance velocities, demonstrated the efficacy of the proposed control approach, showing the system settling smoothly back to its operating point after being perturbed.

The broader significance of this work lies in the long-running quest for what aerodynamicists call free-flight testing in ground facilities. The references in the paper trace decades of effort: NASA Langley’s pioneering free-flight mounts and the cable mount system in its Transonic Dynamics Tunnel, more recent three-cable and four-cable suspension systems analyzed by Chinese groups for full-model flutter tests, and suspension devices developed for gust response experiments in large low-speed tunnels. Each approach trades off intrusion against stability and payload capacity. Cable systems allow large motions but their own dynamics, analyzed in detail in prior multibody studies, couple with the model’s aeroelastic behavior. Rigid mounts are stiff and intrusive. An electromagnetic support promises something closer to the ideal: the model hangs in the airstream with only a slender vertical beam as its mechanical backbone, and the magnetic field does the invisible work of weight support without adding aerodynamic drag or mechanical stiffness in the directions that matter for flutter.

Electromagnetic actuation is already proving its worth elsewhere in aerospace engineering, from fractionated spacecraft payloads controlled entirely by magnetic forces to semi-active vibration suppression in solar panels using adjustable-stiffness magnetic joints, both cited by the authors as evidence of the technology’s maturity. What this new study adds is a rigorous treatment of the specific coupling problem that arises when a levitation system must also accommodate the flexible, aeroelastic dynamics of a full aircraft model in a wind tunnel. By deriving the governing Lagrange–Maxwell equations, extracting the design parameters analytically, proving open-loop instability, and demonstrating closed-loop convergence through complex modal theory and time-domain simulation, the team has laid the theoretical groundwork for a support system that could make flutter testing of full-scale aircraft models cleaner, more accurate, and ultimately safer for the flying public. The next step, as with any simulation-backed design, will be building the hardware and letting a real model hover in a real airstream, where the messy realities of turbulence and sensor noise will test whether the mathematics holds up in the wind.

Subject of Research: Electromagnetic levitation support system dynamics and control for full-model flutter wind tunnel testing

Article Title: Electromechanical Coupling Dynamics and Control of Vertical Beam-Electromagnetic Support System for Wind-Tunnel Tests

Article References: Jiang, Z., Wu, H., Zhao, Z., Li, B., & Yu, L. (2026). Electromechanical Coupling Dynamics and Control of Vertical Beam-Electromagnetic Support System for Wind-Tunnel Tests. International Journal of Aeronautical and Space Sciences. https://doi.org/10.1007/s42405-026-01254-z

Image Credits: AI Generated

DOI: 10.1007/s42405-026-01254-z

Keywords: wind tunnel testing, flutter, electromagnetic levitation, Lagrange-Maxwell equations, feedback control, aeroelasticity, stability analysis, complex modal analysis, vertical beam support, mechatronics, aircraft models, magnetic suspension