The research, carried out by Jeyapradhap Thirisangu and Karthick Subramani of the Indian Institute of Information Technology, Design and Manufacturing, Kancheepuram in Chennai, builds on a growing body of work showing that droplets beyond the Rayleigh limit behave in strikingly non-intuitive ways under acoustic fields. Within the Rayleigh limit, where the droplet diameter is much smaller than the acoustic wavelength, the minimum acoustic energy density needed to suspend a droplet against gravity is essentially independent of size. Beyond that limit, the two authors demonstrate theoretically that this critical energy density becomes highly size-dependent and, remarkably, non-monotonic: it rises to a peak, falls, and then alternates in a pattern of ever-larger peaks and troughs as droplet size increases.

The origin of this strange behavior lies in the internal structure of the standing wave as seen by a large droplet. When a droplet spans multiple pressure nodes and antinodes, different regions of its volume sit in zones of opposing acoustic force. The portion located a quarter-wavelength below a pressure node experiences an upward force, while the portion a quarter-wavelength above the node experiences a downward force. These opposing contributions partially cancel each other, reducing the net acoustic force available to fight gravity. As the droplet grows, the volume contributing to the positive, gravity-defying force shrinks relative to the whole, so the energy needed for suspension climbs. Past a certain size, however, the balance shifts again: more of the droplet volume begins to contribute to the positive net force, and the critical energy density falls. The result is an alternating pattern in which, counterintuitively, some larger droplets require less acoustic energy to levitate than smaller ones in the very same wavefield.

Thirisangu and Subramani exploit this non-monotonic physics to build what they call acoustic sorting beyond the Rayleigh limit. In their simulations, performed with COMSOL Multiphysics 6.0 on a two-dimensional channel 12 millimeters wide and 6 millimeters tall, water droplets are suspended in mineral oil and subjected to a plane standing acoustic wave applied along the height of the channel. The computational domain, discretized with more than 800,000 grid elements and a maximum grid size of 15 microns, solves the incompressible mass continuity and momentum equations together with an advection-diffusion equation for the phase field that tracks the droplet interface, with the acoustic force incorporated as a body force and gravity and interfacial tension included in the framework.

The sorting demonstrations come in three flavors. In the first case, at a frequency of a few kilohertz corresponding to a wavelength of several millimeters, the authors apply an acoustic energy density at which the larger droplet, of a size that sits in the suspending regime, remains levitated, while the smaller droplet, whose critical energy density is higher, settles under gravity. In the second case, simply by changing the frequency and adjusting the energy density, the situation reverses: the bigger droplet settles while the smaller one suspends. The third case is the most intriguing of all. At the same applied conditions under which both the smallest and largest droplets settle, a droplet of intermediate size can be suspended, because its critical energy density happens to fall below the applied value while the critical values of its smaller and larger neighbors do not.

This intermediate-size selectivity is impossible within the Rayleigh limit. There, because the suspension threshold is size-independent, a given acoustic energy density either levitates droplets of all sizes or lets them all settle. Beyond the Rayleigh limit, the non-monotonic curve of critical energy density versus size creates windows in which droplets of a chosen size band can be held aloft while everything else falls. The authors show that the settling and suspending regimes are separated by a simple criterion: if the applied acoustic energy density is below the critical value for a given droplet size, the droplet settles; if it equals or exceeds it, the droplet suspends. A non-dimensional version of this curve allows the same analysis to be transferred to any wavelength, so the sorting conditions can be computed for whatever frequency the experimenter chooses.

The authors go further and lay out a three-step recipe for extracting arbitrary intermediate-sized droplets from a broad size range. First, the wavelength is chosen so that the ratio of droplet size to wavelength places the target size at the first local minimum of the critical energy density curve; applying the corresponding energy density then settles all droplets larger than the target while leaving the target and everything smaller suspended. Second, the frequency is retuned so that the target size sits at the first local maximum of the curve, and an energy density slightly below the target’s critical value is applied; now the smaller droplets remain suspended while droplets of the target size begin to settle. Third, while the target droplets are settling, the energy density is raised slightly above their critical value, halting their descent and leaving them suspended in the fluid between the smaller droplets, which stay at the top, and the larger droplets, which settle at the bottom. The net effect is a clean isolation of any desired intermediate size.

The method even extends to droplets that are identical in size but differ in interfacial tension. In the simulations discussed so far, the maximum Bond number, which measures the ratio of acoustic to interfacial forces, is only 0.063, far below one, so surface tension keeps the droplets intact. When the interfacial force weakens, for example by adding surfactants to reduce the interfacial tension, the opposing acoustic forces within a large droplet can compete strongly enough to split it into daughter droplets. The authors demonstrate a two-step sorting scheme for two droplets of the same size: applying a high acoustic energy density splits the droplet with the weak interfacial tension into two daughters while the droplet with the strong interfacial tension remains whole, and then reducing the energy density to a value between the two critical levels lets the daughter droplets settle while the intact droplet suspends. Sorting by interfacial tension, through size-preserving splitting, thus becomes possible with the same acoustic hardware.

The theoretical findings hold promise for advanced droplet-sorting devices, but the authors are careful to spell out the practical hurdles. The acoustic energy densities and pressure amplitudes required are high enough that any experimental implementation would need pressurized systems to prevent cavitation, the formation of vapor bubbles that would otherwise disrupt the wavefield. The study also rests on simplifying assumptions: a predetermined acoustic energy density is employed without solving the first-order acoustic fields, and the acoustic body force is derived using a Boussinesq approximation in which density and sound speed variations are small. Solving the bidirectionally coupled equations of the fast-time acoustic fields and the slow-time hydrodynamic motion, with the divergence of the Reynolds stress tensor as the body force, would yield more accurate predictions of droplet dynamics. Even so, the work establishes a clear theoretical roadmap: by tuning frequency and acoustic energy density, researchers can now in principle hold droplets of any selected size, or any selected interfacial tension, motionless against gravity while everything around them falls, a capability that could reshape how emulsions, biological droplets, and encapsulated materials are sorted in the laboratory and beyond.

Subject of Research: A theoretical study of acoustic sorting of droplets beyond the Rayleigh limit under combined acoustic and gravity forces

Article Title: Sorting droplets beyond the Rayleigh limit under acoustic-gravity forces: a theoretical approach

Article References: Thirisangu, J., & Subramani, K. (2025). Sorting droplets beyond the Rayleigh limit under acoustic-gravity forces: a theoretical approach. Frontiers in Acoustics, 3, Article 1544464. https://doi.org/10.3389/facou.2025.1544464

Image Credits: AI Generated

DOI: 10.3389/facou.2025.1544464

Keywords: acoustofluidics, droplet sorting, Rayleigh limit, acoustic radiation force, gravity, interfacial tension, droplet splitting, microfluidics, Sorting, droplets, beyond, Rayleigh