Acoustic radiation force on a heated spherical particle in a fluid including scattering and microstreaming from a standing ultrasound wave

TL;DR

This work extends the classical theory of acoustic radiation forces by incorporating a heated spherical particle in a viscous fluid, deriving analytical expressions for the time-averaged force $\bm{F}^\mathrm{rad}$ that include particle vibrations, scattering, and thermoviscous microstreaming. A quasi-steady background temperature field $T_0$ is computed from diffusion, and its gradients couple to the acoustic response, yielding a heating-dependent correction $D_n^{\Delta T_0}$ to the force coefficients. The key finding is that bulk heating alters the acoustic contrast factor $\Phi_{\mathrm{ac}}$ and can even reverse the direction of $\bm{F}^\mathrm{rad}$ in certain fluid–solid combinations, with microstreaming patterns shifting toward unidirectional dipolar structures as heating grows. This introduces a new control parameter for acoustofluidic manipulation, enabling tunable particle focusing, trapping, and sorting via controlled heating, and suggests feasible optical heating strategies to realize such control in microscale devices. The results are illustrated for standing plane waves across multiple particle sizes and fluids, highlighting both the magnitude and sign changes of the forcing as a function of time and temperature rise.

Abstract

Analytical expressions are derived for the time-averaged, quasi-steady, acoustic radiation force on a heated, spherical, elastic, solid microparticle suspended in a fluid and located in an axisymmetric incident acoustic wave. The heating is assumed to be spherically symmetric, and the effects of particle vibrations, sound scattering, and acoustic microstreaming are included in the calculations of the acoustic radiation force. It is found that changes in the speed of sound of the fluid due to temperature gradients can significantly change the force on the particle, particularly through perturbations to the microstreaming pattern surrounding the particle. For some fluid-solid combinations, the effects of particle heating even reverse the direction of the force on the particle for a temperature increase at the particle surface as small as 1 K.

Figures (3)

• Figure 1: (a) The standard constant-temperature case $\Delta T_0 = 0$Doinikov1994a of a spherical particle in an incident acoustic wave $\phi_\mathrm{1c}^\mathrm{in}$ (green arrow), which gives rise to the acoustic radiation force $\bm{F}^\mathrm{rad}$ (red arrow) on the particle through scattered waves (such as $\phi_\mathrm{1c}^\mathrm{sc,0}$, dashed lines) and microstreaming $\langle \bm{v}_2^\mathrm{sc,0} \rangle$ (quadrupolar-like blue arrows). (b) Emphasizing further details of the standards case: the viscous scattering $\bm{\psi}^\mathrm{sc}_1$ in the boundary layer (light blue) of width $\delta_s$outside and the transmitted waves $\phi_\mathrm{1c}'$ and $\bm{\psi}^\mathrm{\prime}_1$inside the particle of radius $a$. (c) Heating of the particle gives rise to the temperature deviation $\Delta T_0$ (light red) in the surrounding fluid. The scattered acoustic wave $\phi_\mathrm{1c}^\mathrm{sc}$ (warped dashed lines) for $\Delta T_0 > 0$ is significantly changed compared to $\phi_\mathrm{1c}^\mathrm{sc,0}$ in (a) and (b). (d) Acoustic scattering on the heated sphere with thermal contributions to both scattering and microstreaming $\langle \bm{v}_2^\mathrm{sc} \rangle$ (blue), now with a significant directional component that leads to a modified acoustic radiation force $\bm{F}^\mathrm{rad}$ (red arrow).
• Figure 2: The acoustic contrast factor $\Phi_\mathrm{ac}$ at frequency $f=1~\textrm{MHz}$ plotted as function of time $t$ for a polystyrene particle with radius $a=1$, $2$, or $5\,\textrm{\textmu{}m}$ heated to $\Delta T_0^\mathrm{surf}=1~\textrm{K}$ in different liquids: (a) water with $t^\mathrm{diff}_\lambda = 2.6~\textrm{s}$, (b) ethanol with $t^\mathrm{diff}_\lambda = 2.5~\textrm{s}$, and (c) oil with $t^\mathrm{diff}_\lambda = 4.0~\textrm{s}$, where $t^\mathrm{diff}_\lambda =\lambda^2/(6 D^\mathrm{th}_0)$ is the heat diffusion time across one wavelength $\lambda$.
• Figure 3: Results for a polystyrene particle of radius $a=2~\textrm{\textmu{}m}$ in water in a standing plane wave at 1 MHz (wavelength $\lambda = 1.5~\textrm{mm}$) and at $T^\infty_0 = 300~\textrm{K}$. (a) The time evolution of the radial temperature deviation $\Delta T_0(\hat{r},t)$ in the fluid for $1<\hat{r} < 5000$ computed analytically (orange line) from Eq. (\ref{['eq:delT0_limit']}) and numerically (red dashed line). (b) A unit-vector plot of the direction of the streaming velocity $\langle \bm{v}_2 \rangle$ and a color plot of its amplitude $|\langle \bm{v}_2 \rangle|$ for $\Delta T_0^\mathrm{surf}=0~\textrm{K}$ (left half) and $\Delta T_0^\mathrm{surf}=1~\textrm{K}$ (right half) computed numerically in Comsol Multiphysics.