Forces acting on a small particle in an acoustical field in a thermoviscous fluid

We present a theoretical analysis of the acoustic radiation force on a single small spherical particle, either a thermoviscous fluid droplet or a thermoelastic solid particle, suspended in a viscous and heat-conducting fluid medium. Within the perturbation assumptions, our analysis places no restrictions on the length scales of the viscous and thermal boundary-layer thicknesses ${\delta }_{\mathrm{s}}$ and ${\delta }_{\mathrm{t}}$ relative to the particle radius $a$, but it assumes the particle to be small in comparison to the acoustic wavelength $\lambda$. This is the limit relevant to scattering of ultrasound waves from nanometer- and micrometer-sized particles. For particles of size comparable to or smaller than the boundary layers, the thermoviscous theory leads to profound consequences for the acoustic radiation force. Not only do we predict forces orders of magnitude larger than expected from ideal-fluid theory, but for certain relevant choices of materials, we also find a sign change in the acoustic radiation force on different-sized but otherwise identical particles. These findings lead to the concept of a particle-size-dependent acoustophoretic contrast factor, highly relevant to acoustic separation of microparticles in gases, as well as to handling of nanoparticles in lab-on-a-chip systems.

Figure 1

Sketches of the physical mechanisms responsible for various multipole components in the scattering of an incident acoustic wave on a particle. (a) Compressibility contrast: the incident periodic pressure field compresses the particle relative to the fluid, which leads to monopole radiation. (b) Thermal contrast: the incident periodic temperature field leads to thermal expansion of the particle relative to the fluid, which also gives rise to monopole radiation and the development of a diffusive thermal boundary layer (pink). (c) Density contrast: a difference in inertia between particle and fluid causes the particle to oscillate relative to the fluid, which gives rise to dipole radiation and the development of a viscous boundary layer (blue). (d) Particle resonances: acoustic wavelengths comparable to the particle size lead to complex shape changes, which give rise to multipole radiation and a complex thermoviscous boundary layer (pink and blue).

Figure 2

A compressional wave ${\varphi }_{\mathrm{i}}$ propagating in a thermoviscous fluid medium with parameters ${\rho }_0,{\eta }_0,{\kappa }_s,{\alpha }_p,c_p,\gamma ,$ and $k_{\mathrm{th}}$, is incident on a thermoviscous fluid droplet with parameters ${\rho }_0^{\prime },{\eta }_0^{\prime },{\kappa }_s^{\prime },{\alpha }_p^{\prime },c_p^{\prime },{\gamma }^{\prime }$, and $k_{\mathrm{th}}^{\prime }$, which results in a compressional scattered wave and highly damped thermal and shear waves both outside in the fluid medium (${\varphi }_{\mathrm{r}},{\varphi }_{\mathrm{t}},{\psi }_{\mathrm{s}}$) and inside in the fluid droplet (${\varphi }_{\mathrm{c}}^{\prime },{\varphi }_{\mathrm{t}}^{\prime },{\psi }_{\mathrm{s}}^{\prime }$). Viscous and thermal boundary layers are described by the highly damped waves both outside and inside the fluid droplet. In the long-wavelength limit the droplet radius $a$ and the boundary-layer thicknesses ${\delta }_{\mathrm{s}},{\delta }_{\mathrm{t}},{\delta }_{\mathrm{s}}^{\prime },{\delta }_{\mathrm{t}}^{\prime }$ are mutually unrestricted, but all much smaller than the acoustic wavelength $\lambda$. For a thermoelastic particle, the shear mode ${\psi }_s^{\prime }$ describes a propagating transverse wave instead of an internal viscous boundary layer.

Figure 3

Acoustic contrast factor ${\mathrm{\Phi }}_{\mathrm{ac}}$ plotted as a function of ${\delta }_{\mathrm{s}}/a$, the viscous boundary-layer thickness in the medium normalized by particle radius. The curves are calculated using ideal theory (green), viscous theory (thermoviscous theory with $D_{\mathrm{th}}=0$, blue), and thermoviscous theory (red) for (a) an oil droplet in water (wa-oil) and (b) a water droplet in oil (oil-wa), both at $20{}^{\circ }\mathrm{C}$. Thermoviscous theory leads to corrections to the acoustic radiation force around $100\%$. The vertical dashed lines indicate examples of particle sizes corresponding to the given value of ${\delta }_{\mathrm{s}}/a$ at $f=1\mathrm{MHz}$. Note that the acoustic contrast factor changes sign at a critical particle radius for the case of water droplets in oil considered in (b).

Figure 4

Acoustic contrast factor ${\mathrm{\Phi }}_{\mathrm{ac}}$ for particles in air plotted as a function of ${\delta }_{\mathrm{s}}/a$, the viscous boundary-layer thickness in the medium normalized by particle radius. The curves are calculated using ideal theory (green), viscous theory (thermoviscous theory with $D_{\mathrm{th}}=0$, blue), and thermoviscous theory (red) for (a) a polystyrene particle in air (air-ps) and (b) a water droplet in air (air-wa), both at 300 K. Ideal theory predicts a constant value of ${\mathrm{\Phi }}_{\mathrm{ac}}=5/6$ independent of particle size. For particles much smaller than the boundary-layer thickness, however, thermoviscous theory predicts huge deviations from ideal theory leading to acoustic contrast factors two orders of magnitude larger than expected from ideal-fluid theory. The vertical dashed lines indicate examples of particle sizes corresponding to the given value of ${\delta }_{\mathrm{s}}/a$ at $f=1\text{kHz}$.