Acoustophoretic Control: Fundamentals & Applications

• ACPs are processes that use tailored acoustic fields to steer particles, cells, and droplets without direct contact.
• They integrate multimodal acoustic actuation with control theory to achieve precise positioning, levitation, and separation under various constraints.
• Advanced modeling, feedback control, and distributed actuation are leveraged in ACPs to optimize performance in complex, dynamic environments.

Acoustophoretic Control Problems (ACPs) are manipulation tasks in which an acoustic field is designed so that particles, cells, droplets, or larger levitated objects move toward prescribed states without mechanical contact. In a formal multimodal formulation, an ACP is written as $$(\mathcal{S}_P,\mathbf{F}_M,\mathbf{q}_s,\mathbf{q}_r,\theta,T)$$, where the state space, available acoustic modes, initial state, target state, tolerance, and time limit define feasibility; in broader experimental practice, the same notion covers microchannel focusing, droplet patterning, volumetric levitation, rotational actuation through squeeze films, and distributed phased-array manipulation (Perticarari et al., 7 Oct 2025, Muller et al., 2013, Yu et al., 2012, Gabai et al., 2016, Kemsaram et al., 12 May 2025). This suggests a unifying view of ACPs as acoustic field synthesis under dynamical, geometric, and material constraints.

1. Formal scope and canonical objectives

ACPs are defined by the requirement that acoustic actuation produce specific translational or rotational outcomes while respecting the fact that one field typically acts on all objects simultaneously. In the multimodal control setting with $P$ particles and $M$ acoustic modes, the global state is the concatenation of particle positions, and the controlled dynamics take the form

$$\dot{\mathbf{q}} = \mathbf{F}_M(\mathbf{q})\,\mathbf{w},$$

with non-negative modal weights $\mathbf{w}$ satisfying $\sum_i w_i \le 1$; feasibility is then expressed by the requirement that each particle reach its target within tolerance $\theta$ by time $T$ (Perticarari et al., 7 Oct 2025). In microfluidic ACPs, the same problem is often cast in terms of lateral focusing, separation, or outlet recovery rather than exact point-to-point steering, whereas in levitation systems it is cast as trapping, transport, or orientation control.

Across the literature, the recurrent control objectives are stable levitation, in-plane or cross-stream translation, simultaneous manipulation of multiple objects, and robustness to disturbances. A contactless air-ultrasound demonstration explicitly states the ability to “stably levitate and move along a plane multiple objects in air,” including metal sodium chunks and water droplets, before, during, and after a violent exothermal reaction that introduces hydrogen gas; the same demonstration emphasizes applicability to hazardous, chemical, or radioactive samples (Foresti et al., 2013). In microchannels, the objectives are commonly finite-time focusing to pressure nodes, separation of populations with different acoustophoretic responses, or maintenance of a prescribed pressure field close to a reference during excitation (Harshbarger et al., 2023, Bucci et al., 2018).

A second canonical distinction is between single-particle and population-level ACPs. Single-particle formulations dominate force, torque, and levitation studies, whereas free-flow separation and clinical acoustophoretic devices treat populations with continuously distributed radii or material properties. In that setting, the controlled outcome is not only a trajectory but also a probability distribution over positions or outlets, so ACPs naturally include overlap minimization, side-stream recovery, and robustness to uncertainty in particle parameters (Garofalo, 2018, Garofalo, 2019).

2. Governing models and state representations

The basic ACP plant model is a linear acoustic field coupled to overdamped particle motion. In frequency-domain descriptions, the acoustic pressure often satisfies a Helmholtz-type equation, while the particle experiences a time-averaged acoustic radiation force derived from a Gor’kov potential. For small spherical particles in a standing wave, the contrast-factor formulation gives

$$\Phi(\tilde{\kappa},\tilde{\rho})=\frac{1}{3} f_1(\tilde{\kappa})+\frac{1}{2} f_2(\tilde{\rho}),$$

with

$$f_{1}(\tilde{\kappa}) = 1 - \tilde{\kappa}, \qquad f_{2}(\tilde{\rho}) = 2 \,\frac{\tilde{\rho} - 1}{2 \tilde{\rho} + 1},$$

and the radiation force is

$P$0

For a 1D standing wave,

$P$1

so the acoustophoretic response is governed by the product of field strength and acoustic contrast factor (Harshbarger et al., 2023).

In microchannels, inertia is usually negligible and the dynamics reduce to a first-order drift law. For cell migration in a 1D standing wave,

$P$2

and analytic trajectory formulas can be fitted to measured paths to identify $P$3 (Harshbarger et al., 2023). In rectangular channels where streaming cannot be neglected, the particle velocity is the sum of the streaming field and the radiation-driven drift,

$P$4

which provides a 3D overdamped state equation directly suitable for control and optimization (Muller et al., 2013).

At the device scale, ACP plant models may include full electromechanical coupling. Silicon-glass devices are modeled with coupled elastic, acoustic, and piezoelectric fields derived in a variational framework, with frequency-dependent Hamiltonian and Lagrangian densities used to define both mechanical and electrical indicators (Garofalo et al., 2016). Polymer devices require a whole-system model including PZT transducer, PMMA chip, glycerol coupling layer, and fluid-filled channel; in that setting the relevant resonance is a whole-system ultrasound resonance rather than the conventional half-wave channel resonance (Lickert et al., 2021). At the PDE-control end of the spectrum, the acoustic pressure itself is taken as the state of a third-order-in-time Moore–Gibson–Thompson system with boundary actuation,

$P$5

and the control objective is to keep $P$6 close to a reference pressure in $P$7 (Bucci et al., 2018).

3. Controllability, uncertainty, and performance metrics

A central ACP question is whether the available acoustic actuation can generate sufficient local directional freedom. In the multimodal formulation, a state is locally controllable if and only if the origin lies in the interior of the convex hull of the mode vector fields $P$8 at that state. The ratio of locally controllable volume to state-space volume was shown to correlate strongly with the probability of success $P$9 of random manipulation tasks, with a reported Pearson correlation of $M$0 (Perticarari et al., 7 Oct 2025). In noise-free 1D systems the same work reports

$M$1

while in noisy 1D and 2D systems $M$2 is accurately predicted by Wendel’s Theorem, leading to the practical rule that fixed $M$3 yields approximately fixed success probability and that $M$4 corresponds to roughly 50% local controllability (Perticarari et al., 7 Oct 2025).

In separation ACPs, controllability is not usually expressed as reachability of exact states but as evolution of means, variances, and overlap between populations. The mean-and-covariance dynamics approach propagates the mean state $M$5 and covariance $M$6 via

$M$7

providing a reduced-order surrogate for distributed particle populations (Garofalo, 2019). For Gaussian-mixture population models, the same structure underlies acoustophoretic separation metrics such as side-stream recovery, whose component form is

$M$8

with total population recovery obtained by weighted summation over Gaussian components (Garofalo, 2018).

Performance metrics in ACPs are correspondingly heterogeneous. Geometry-based and population-based measures coexist with field-based indicators. For microparticle separations, the literature discussed here includes bandwidth, resolution index, separation resolution, ideal separation efficiency, and separation efficiency (Garofalo, 2019). For acoustofluidic resonators, performance is quantified by the energy localized in the channel, the acoustofluidic yield

$M$9

and the acoustophoretic mean orientation

$$\dot{\mathbf{q}} = \mathbf{F}_M(\mathbf{q})\,\mathbf{w},$$0

which distinguishes horizontal focusing fields from vertically oriented modes (Garofalo et al., 2016). In particle focusing experiments, the polymer-device literature adds focusability as the fraction of particles that reach a prescribed central band within a finite focusing time (Lickert et al., 2021).

A recurrent source of uncertainty is the material response lumped into the acoustic contrast factor. For biological cells, the measured contrast factor is determined by dynamic density and bulk modulus at the actuation frequency, not by static stiffness alone. Static $$\dot{\mathbf{q}} = \mathbf{F}_M(\mathbf{q})\,\mathbf{w},$$1-modulus changes may correlate with changes in $$\dot{\mathbf{q}} = \mathbf{F}_M(\mathbf{q})\,\mathbf{w},$$2, but static $$\dot{\mathbf{q}} = \mathbf{F}_M(\mathbf{q})\,\mathbf{w},$$3-modulus alone is not a reliable predictor of the dynamic ACF; the same study shows that many combinations of density and bulk modulus yield the same $$\dot{\mathbf{q}} = \mathbf{F}_M(\mathbf{q})\,\mathbf{w},$$4, and that varying static $$\dot{\mathbf{q}} = \mathbf{F}_M(\mathbf{q})\,\mathbf{w},$$5-modulus while keeping $$\dot{\mathbf{q}} = \mathbf{F}_M(\mathbf{q})\,\mathbf{w},$$6 and $$\dot{\mathbf{q}} = \mathbf{F}_M(\mathbf{q})\,\mathbf{w},$$7 fixed does not affect $$\dot{\mathbf{q}} = \mathbf{F}_M(\mathbf{q})\,\mathbf{w},$$8 (Harshbarger et al., 2023). This directly motivates robust and stochastic ACP formulations.

4. Device architectures and actuation modalities

ACPs are instantiated in markedly different physical architectures, but the control variables remain recognizable: frequency, amplitude, phase, geometry, and in some cases actuator pose. In droplet acoustophoresis, the droplet itself is the resonator, with free surfaces acting as reflectors and shape controlled by edge pinning or hydrophilic/hydrophobic interface pinning. Frequency selects cavity eigenmodes, amplitude governs migration speed, and droplet geometry determines the topology of nodal lines; experimentally, circular, annular, and rectangular droplets produce mode-dependent particle patterns, and the same platform was used to manipulate Caenorhabditis elegans (Yu et al., 2012). This suggests an ACP formulation in which geometry and frequency are co-designed to realize a target nodal pattern.

In microfluidic silicon-glass devices, a coupled elastic–acoustic–piezoelectric model supports frequency-domain optimization. Mechanical indicators such as average pressure-gradient direction and channel-localized acoustic energy are correlated with electrical indicators such as impedance, power, and resonance $$\dot{\mathbf{q}} = \mathbf{F}_M(\mathbf{q})\,\mathbf{w},$$9-value, and acoustophoretic operating frequencies are selected by maximizing channel energy within frequency bands where the pressure-gradient orientation is favorable (Garofalo et al., 2016). In polymer devices, a 3D finite-element model and focusing experiments show that the relevant operating point is a whole-system ultrasound resonance; under anti-symmetric 30-V peak-to-peak excitation between 0.5 and 2.5 MHz, acoustic energy densities of $\mathbf{w}$0 and particle focusing times of $\mathbf{w}$1 were reported in a PMMA chip, with acoustophoretic action comparable in quality and strength to conventional silicon-glass or pure glass devices (Lickert et al., 2021).

Macro-scale levitation and rotation introduce a different ACP architecture: a vibrating annulus and thin squeeze film, driven by three equally spaced actuators. Virtual-work analysis maps the three actuator inputs to cosine and sine doublet modes, and an algebraic transformation reduces the over-actuated setup to a single scalar control parameter $\mathbf{w}$2 that tunes the standing/traveling-wave ratio. Along the nominal control path,

$\mathbf{w}$3

so the acoustic torque becomes a function of one bounded parameter while levitation is preserved; a closed-loop PID implementation then achieves rapid angular positioning (Gabai et al., 2016).

Distributed acoustophoretic systems extend the actuator set itself into the control space. AcoustoBots mount a phased array of transducers on mobile robots, each mini-PAT being a one-sided $\mathbf{w}$4 array of 64 elements, operating at $\mathbf{w}$5, with hinge-controlled orientations of $\mathbf{w}$6, $\mathbf{w}$7, and $\mathbf{w}$8 for haptics, audio, and levitation respectively (Kemsaram et al., 12 May 2025). The acoustic planner computes per-element phases using GS-PAT, while a central UDP server coordinates robot motion, orientation, and array holograms. This transforms ACPs into coupled field-synthesis and multi-robot coordination problems.

5. Shape, material, and multiphase complexity

A common simplification in acoustophoretic modeling is to replace real objects by spheres or other symmetric surrogates. The literature surveyed here repeatedly shows the limits of that approximation. For arbitrary small scatterers, a polarizability-tensor formalism augments the classical monopole and dipole responses with Willis coupling terms, yielding generalized force and torque expressions in which asymmetry contributes weakly to force but strongly to torque. In standing waves, the additional terms shift equilibrium positions and angles; in traveling waves, Willis contributions remain significant for torque even when their direct effect on force is small (Sepehrirahnama et al., 2021). The same work concludes that it is essential, in general, to account for shape for objects undergoing acoustophoretic manipulation, with particular implications for biological cells.

Axisymmetric irregular particles require even more geometry-sensitive treatment. A semi-analytical framework based on conformal mapping computes the acoustic radiation force and torque for user-customized transducer arrays by solving the scattering problem in a computation coordinate system whose axis coincides with the particle symmetry axis, while particle rotation is converted into an opposite rotation of the transducer array in that coordinate system. The reported computational efficiency is more than 100 times higher than that of the full numerical method, and dynamic trajectories were found to differ completely across geometric features, indicating that geometric features can be a potential degree of freedom to tune acoustophoretic process (Tang et al., 2022).

Material complexity enters ACPs through the acoustic contrast factor and its frequency dependence. In biological-cell studies at approximately $\mathbf{w}$9, measured $\sum_i w_i \le 1$0 values include $\sum_i w_i \le 1$1 for untreated SaOs-2, $\sum_i w_i \le 1$2 after 4% formaldehyde fixation, $\sum_i w_i \le 1$3 after cytochalasin D treatment, and $\sum_i w_i \le 1$4 for LM5 despite a lower static $\sum_i w_i \le 1$5-modulus than SaOs-2 (Harshbarger et al., 2023). The result directly contradicts the common assumption that static stiffness differences alone are sufficient for acoustophoretic separation. A plausible implication is that ACPs based on static mechanical proxies are intrinsically miscalibrated unless dynamic density and bulk modulus are measured or robustly inferred.

Multiphase and strongly disturbed environments add another layer of difficulty. Contactless mixing of levitated sodium and water demonstrates that acoustic traps can remain effective before, during, and after a violent exothermal reaction that produces a hydrogen gas phase (Foresti et al., 2013). In microchannels, long-time experiments across particle diameters from $\sum_i w_i \le 1$6 down to $\sum_i w_i \le 1$7 reveal a crossover from radiation-dominated to streaming-dominated motion, late-time depletion of particles below the crossover size, and bulk focusing of submicrometer particles in inhomogeneous fluids where acoustic body forces suppress streaming (Qiu et al., 2020). ACPs in such regimes must therefore accommodate gas generation, evolving particle shape or mass, boundary-layer streaming, and Brownian motion rather than treating the force field as static.

6. Optimization, feedback, and research directions

Optimization in ACPs ranges from heuristic frequency selection to full feedback synthesis. Frequency-domain device studies select operating points by maximizing channel energy or focusability within admissible mode-orientation bands, using easy-to-measure electrical quantities such as impedance and power as proxies for hard-to-measure mechanical indicators (Garofalo et al., 2016). Population models use Gaussian mixtures and mean-and-covariance dynamics to evaluate outlet statistics and side-stream recovery at approximately $\sum_i w_i \le 1$8 times lower computational cost than direct ensemble simulation, which makes them suitable for design-space exploration and model-based optimization (Garofalo, 2018).

Feedback control has been developed at the acoustic-field level as well. For the linearized Moore–Gibson–Thompson equation with boundary actuation, the optimal finite-horizon control problem

$\sum_i w_i \le 1$9

leads, despite the singular nature of the problem, to a non-standard operator Riccati equation and a time-varying feedback law for the boundary input (Bucci et al., 2018). This is directly relevant to ACPs that seek to keep an acoustic pressure field close to a desired spatiotemporal reference rather than merely excite a nominal resonance.

Several objective misconceptions are corrected by recent work. One is that half-wave channel resonance suffices to determine acoustophoretic operation in compliant devices; polymer chips instead exhibit whole-system ultrasound resonances that control the actual focusing behavior (Lickert et al., 2021). Another is that global resonance always implies optimal acoustophoretic performance; silicon-glass studies show that channel energy can peak even when the device as a whole does not present the clearest electrical resonance signature (Garofalo et al., 2016). A third is that more modes simply increase control authority without a quantitative law; multimodal controllability results indicate that success probability is governed, to a first approximation, by the ratio $\theta$0 and by the geometry of the attainable velocity cone (Perticarari et al., 7 Oct 2025).

Current research directions suggest several converging trajectories. One is reduced-order but geometry-aware modeling: the conformal-mapping framework for irregular particles and the polarizability-tensor framework with Willis coupling both point toward ACP formulations in which geometry is an explicit control-relevant parameter rather than a perturbation (Tang et al., 2022, Sepehrirahnama et al., 2021). Another is distributed actuation: AcoustoBots already combine phased-array holography, robot motion, and hinge orientation in a centralized acoustic control framework with two mobile robots, explicitly positioning this as groundwork for larger swarms (Kemsaram et al., 12 May 2025). A third is robust biophysical identification: cell-separation ACPs increasingly require direct characterization of dynamic $\theta$1 rather than reliance on static mechanical measurements (Harshbarger et al., 2023).

Taken together, these developments establish ACPs as a general control-theoretic and design-theoretic framework for contactless acoustic manipulation. The common structure is the same across microchannels, droplets, levitated reactions, squeeze-film rotors, and mobile phased arrays: acoustic fields are the actuators, pressure or radiation-force landscapes are the intermediate state, and the ultimate objectives are positional, orientational, or distributional control under uncertainty, geometry dependence, and frequency-selective resonance structure.