FR33 · Acoustofluidics

Acoustic Radiation Force: Trapping Particles in a Standing Wave

How does a standing sound wave push particles into lines, and when does the stirred liquid win instead? A standing sound wave pushes small particles toward the planes where its time-averaged energy is lowest for them. Polystyrene beads and most cells are denser and stiffer than water, so they gather at pressure nodes; softer, lighter particles gather at antinodes. The sound also stirs the liquid: at 2 MHz, polystyrene beads below about 1.4 micrometres follow those flow rolls instead of lining up.

A 2 MHz standing wave across a 377 µm water channel herds microbeads into the pressure node. Drag a bead out and watch it return.

Acoustic contrast factor $0.165$

Return time to the node $0.466\,\mathrm{s}$

Radiation drift over streaming drift $12.8$

Diameter where streaming takes over $1.4\,\mu\mathrm{m}$

Choose a particle and a sound field

Particle material

Particle diameter

5

2a\;(\mu\mathrm{m})

Acoustic energy density

20

E_{\mathrm{ac}}\;(\mathrm{J/m^3})

Half-wavelengths across the channel

1

n

Include acoustic streaming

Drag the orange probe bead anywhere in the channel and let go.

Cloud seed

Seed of the scattered cloud

3301

Model, units and sources

$E_{\mathrm{ac}}=\frac{\kappa_0p_a^2}{4},\quad u_0=\frac{4E_{\mathrm{ac}}}{\rho_0c_0},\quad f=\frac{nc_0}{2w}$

Particles move overdamped at radiation drift plus streaming velocity. Streaming uses Rayleigh’s parallel-plate solution, valid outside the boundary layer when the wavelength is much larger than the height; side walls and heating are left out. The contrast factor includes the viscous dipole term; its change from the inviscid value is listed among the extra numbers.

Barnkob et al. (2012), Eqs. 1–14, Table II · Settnes & Bruus (2012), Eqs. 48b–e, Table I

Derive the radiation force step by step

Model assumptions

Two-dimensional cross-section of a water-filled channel 377 µm wide and 157 µm high, driven at a half-wave resonance across its width. Beads feel the Gor’kov radiation force of an ideal standing wave plus Stokes drag; boundary-driven Rayleigh streaming between parallel plates can be switched on. Particles do not interact. Wall corrections to drag, heating, side-wall streaming, gravity, Brownian motion and the full three-dimensional resonance are omitted. Water and polystyrene follow Barnkob et al. (2012); pyrex and the cell follow Settnes and Bruus (2012); the soft particle is hypothetical. All motion is a model simulation, not measured data.

Radiation force and potential across the channel

Probe return after release

Radiation versus streaming by particle size

More numbers and how they are checked

Drive frequency: $0$
Pressure amplitude: $0$
Streaming velocity scale u₀: $0$
Largest radiation drift speed: $0$
Centre-plane streaming speed: $0$
Monopole coefficient f₁: $0$
Dipole coefficient f₂: $0$
Viscous change of the contrast factor: $0$
Viscous boundary layer δ: $0$
Trap stiffness at the stable plane: $0$
Particle inertial time: $0$
Probe release position: $0$
Stable plane the probe returns to: $0$
Largest gap between simulated and exact return: $0$

White dots are minus the slope of the Gor’kov potential, taken by central differences from the field samples; the line is the closed-form force. The exact return solves the drift equation in closed form and does not use the integrator. On the centre plane, streaming adds a drift with the same sinusoidal shape, so an exact curve exists there too.

Physics tutorial

A standing wave is a landscape for particles

BackgroundA piezo transducer under a microchannel drives the water at about 2 MHz. At this frequency half a wavelength of sound just fits across the 377 µm channel, so a standing wave forms with a pressure node in the middle.

Why it mattersThe sound oscillates two million times a second, yet beads drift steadily over fractions of a second. Something averaged over many cycles must push them, and something else must stir the liquid around them.

Start with the essentials

Focus question: How does a standing sound wave push particles into lines, and when does the stirred liquid win instead?
One-sentence intuition: The time-averaged pressure and velocity of the wave define a potential for each particle. Its slope is the radiation force, and its sign depends on whether the particle is denser and stiffer than water. The force grows with the particle volume, while the streaming drag on a particle does not depend on its size, so small particles follow the streaming rolls instead.

Core mathematical model

Gor’kov potential

U=\frac{4\pi a^3}{3}\left[f_1\frac{\kappa_0}{2}\langle p^2\rangle-f_2\frac{3\rho_0}{4}\langle v^2\rangle\right]

For a sphere much smaller than the wavelength. The brackets are averages over one acoustic cycle of the squared pressure and velocity.

Radiation force in a half-wave field

F=-\frac{\partial U}{\partial y}=4\pi\Phi k a^3E_{\mathrm{ac}}\sin(2ky)

Here $y$ is measured from the left wall, where the pressure is largest, and $k$ is the wavenumber.

Acoustic contrast factor

\Phi=\frac{f_1}{3}+\frac{f_2}{2},\qquad f_1=1-\frac{\kappa_p}{\kappa_0},\qquad f_2=\frac{2(\rho_p-\rho_0)}{2\rho_p+\rho_0}

A positive contrast factor sends particles to pressure nodes; a negative one sends them to pressure antinodes. Polystyrene in water gives about 0.17. The dipole term is shown without viscosity; the Lab adds the thin viscous layer around each particle, which matters for small dense beads.

Drift speed and return time

u_{\mathrm{rad}}=\frac{F}{6\pi\eta a}=\frac{2\Phi k a^2E_{\mathrm{ac}}}{3\eta}\sin(2ky),\qquad \tau=\frac{3\eta}{4\Phi k^2a^2E_{\mathrm{ac}}}

Inertia relaxes in under a microsecond, so the bead moves at the speed where Stokes drag balances the radiation force. Near the node the return is exponential with time constant τ.

Where streaming takes over

2a_c=2\delta\sqrt{\frac{3\psi}{\Phi}},\qquad \delta=\sqrt{\frac{2\eta}{\rho_0\omega}},\qquad \psi=\frac{3}{16}

Here $\delta$ is the viscous boundary-layer thickness and $\psi$ the centre-plane streaming coefficient between parallel plates. For polystyrene at 2 MHz the critical diameter is about 1.4 µm.

Common difficulties

Pushed toward the quietest point

Typical misconceptionSound pushes particles away from where it is loudest, so every particle ends at a pressure node.

Better mental modelWhere a particle ends depends on how it differs from the liquid. A particle softer and lighter than water has a negative contrast factor and gathers at pressure antinodes, the walls of this channel.

Run the experiment

01
Watch the beads line up
Start with Beads line up at the node.
What to observe: Within about a second of model time the scattered beads gather into a vertical line at the centre, where the pressure colour stays faint.
02
Pull one bead out
Drag the orange probe toward a side wall and let go.
What to observe: The probe returns along the exact curve in the return chart: slow near the wall and the node, fastest a quarter of the way across.
03
Shrink the beads
Choose Tiny beads ride the rolls, or slide the diameter below 1.4 µm.
What to observe: The beads circle in four rolls instead of lining up, because streaming now moves them faster than the radiation force.