Ultrasound-induced acoustophoretic motion of microparticles in three dimensions

We derive analytical expressions for the three-dimensional (3D) acoustophoretic motion of spherical microparticles in rectangular microchannels. The motion is generated by the acoustic radiation force and the acoustic streaming-induced drag force. In contrast to the classical theory of Rayleigh streaming in shallow, infinite, parallel-plate channels, our theory does include the effect of the microchannel side walls. The resulting predictions agree well with numerics and experimental measurements of the acoustophoretic motion of polystyrene spheres with nominal diameters of 0.537 and 5.33 $\mu \text{m}$. The 3D particle motion was recorded using astigmatism particle tracking velocimetry under controlled thermal and acoustic conditions in a long, straight, rectangular microchannel actuated in one of its transverse standing ultrasound-wave resonance modes with one or two half-wavelengths. The acoustic energy density is calibrated in situ based on measurements of the radiation dominated motion of large 5-$\mu \text{m}$-diameter particles, allowing for quantitative comparison between theoretical predictions and measurements of the streaming-induced motion of small 0.5-$\mu \text{m}$-diameter particles.

Figure 1

A cross-sectional sketch in the $yz$ plane of the classical Rayleigh-Schlichting streaming pattern in the liquid-filled gap of height $h$ between two infinite, parallel rigid walls (black) in the $xy$ plane. The bulk liquid (light shade) supports a horizontal standing sinusoidal pressure half-wave $p_1$ (dashed lines) of wavelength $\lambda$ in the horizontal direction parallel to the walls. In the viscous boundary layers (dark shade) of submicrometer thickness $\delta$, large shear stresses appear, which generate the boundary-layer (Schlichting) streaming rolls (light thin lines). These result in an effective boundary condition $\langle v_{2y}^{{\mathrm{bnd}}_{}}\rangle$ (thick light arrows) with periodicity $\lambda /2$ driving the bulk (Rayleigh) streaming rolls (black thin lines). Only the top and bottom walls are subject to this effective slip boundary condition.

Figure 2

Analytical results for the streaming velocity $\langle {\mathit{v}}_2\rangle$ in parallel-plate channels. (a) Plot of the analytical expressions (19) and (20) for $\langle {\mathit{v}}_2\rangle$ (arrows) and its magnitude [color plot from 0 (black) to $v_{\mathrm{str}}$ (white)] in the vertical $yz$ cross section of a parallel-plate channel (Fig. 1) with $\lambda /2=w$ $(n=1)$ and aspect ratio $\alpha =1.2$. (b) The same as (a), but for $\alpha =0.2$. (c) The same as (b) but for a standing full wave $\lambda =w$ $(n=2)$. (d) Line plot of the amplitude $\langle v_{2y}(\tilde{y},0)\rangle$ of the streaming velocity, in units of $v_{\mathrm{str}}$, along the first half of the center axis [white dashed lines in (a) and (b)] with $\lambda /2=w$ for aspect ratios $\alpha =0.2$, 0.5, 0.8, and 1.2. (e) Line plot of the maximum ${\langle v_{2y}(\tilde{y},0)\rangle }_{\max }$ of the center-axis streaming velocity, in units of $v_{\mathrm{str}}$, as function of aspect ratio for the resonances $n\lambda /2=w$, with $n=1$, 2, and 3, respectively.

Figure 3

Analytical results comparing the streaming velocity field $\langle {\mathit{v}}_2\rangle$ in the parallel plate and the rectangular channel. (a) Color plot from 0 (black) to $v_{\mathrm{str}}$ (white) of the analytical expression for $\langle v_2\rangle$ [Eqs. (19) and (20)] in the classical parallel-plate geometry with a half-wave resonance $\lambda /2=w$ $(n=1)$. Due to symmetry, only the left half ($-1<\tilde{y}<0$) of the vertical channel cross section is shown. (b) As in (a) but for $\langle v_2\rangle$ in the rectangular channel [Eqs. (25) and (31)], including the first 20 terms of the Fourier series. (c) Line plots of $\langle v_{2y}(\tilde{y},0)\rangle$ in units of $v_{\mathrm{str}}$ along the left half of the center line for the parallel-plate channel (dashed lines) and the rectangular channel (full lines) for aspect ratios $\alpha =0.1$, 0.4, and 0.8 and the half-wave resonance $\lambda /2=w$. (d) As in (c) but for the full-wave resonance $\lambda =w$ $(n=2)$.

Figure 4

Measured particle trajectories (thin black lines) obtained using the 3D-APTV technique in the microchannel (gray walls) actuated at the 1.94-MHz horizontal half-wave resonance. For selected trajectories, the particle positions are represented by dots. (a) 5-$\mu \text{m}$-diameter particles moving (thick arrows) to the vertical center plane $y=0$, and (b) 0.5-$\mu \text{m}$-diameter particles exhibiting circular motion as in Fig. 2b.

Figure 5

Comparison between experimental, analytical, and numerical studies of the acoustophoretic particle velocities ${\mathit{u}}^{\mathrm{p}}$ of 0.5-$\mu \text{m}$-diameter polystyrene particles in water. The particle velocities ${\mathit{u}}^{\mathrm{p}}$ (vectors) and their magnitude [color plot ranging from 0 $\mu$m/s (black) to 63 $\mu$m/s (white) in all three plots] are shown in the vertical cross section of the microchannel, divided into a pixel array consisting of $37\times 15$ square bins of side length 10 µm. The axes of the plot coincide with the position of the channel walls. (a) The APTV measurements of the 0.5-$\mu$m-diameter particles, shown in Fig. 4b, projected onto the vertical cross section. The maximum velocity is 63 $\mu$m/s. Close to the side walls, experimental data could not be obtained, which is represented by hatched bins. (b) Analytical prediction of ${\mathit{u}}^{\mathrm{p}}$ based on Eq. (32), taking both the radiation force and the streaming-induced drag force into account. The first 20 terms of the Fourier series for $\langle {\mathit{v}}_2\rangle$ [Eq. (25)] have been included in the calculation. The maximum velocity is 59 $\mu$m/s. There are no free parameters in this prediction as the acoustic energy density was calibrated in situ based on measurements of large 5-$\mu$m-diameter particles, shown in Fig. 4a. (c) Numerical validation of the analytical result for ${\mathit{u}}^{\mathrm{p}}$ using the method described in Muller et al. [22]. The numerical solution has been scaled by the thermoviscous prefactor to the streaming amplitude  (13). The maximum velocity is 59 $\mu$m/s.

Figure 6

(a) Color plot of the number of times the velocity has been measured in each square bin. (b) Color plot of standard error of the mean (SEM) particle velocity in each square bin.

Figure 7

(a) Color plot of the difference between the experimental and analytical acoustophoretic particle speeds $\Delta u^{\mathrm{p}}$ [Eq. (38)]. (b) Line plots of $\Delta u^{\mathrm{p}}$ along the dashed lines in (a), marked A, B, C, D, E, and F, with error bars indicating the $1\sigma$ error of $\Delta u^{\mathrm{p}}$. The lines are positioned at $y=0\mu$m, $y=\pm 91.7\mu$m, $z=0\mu$m, and $z=\pm 52.3\mu$m. The off-center lines go through the rotation centers of the flow rolls, and consequently ${\mathit{u}}^{\mathrm{p}}\approx u_y^{\mathrm{p}}{\mathit{e}}_y$ in B, D, and F, while ${\mathit{u}}^{\mathrm{p}}\approx u_z^{\mathrm{p}}{\mathit{e}}_z$ in A, C, and E.

Figure 8

Experimental data from Ref. [45] compared with the theoretical predictions of Eqs. (36) and (37). $\mu$PIV measurements, in the center plane $z=0$, of the $y$ component of the acoustophoretic velocity ${\langle u_y^{\mathrm{p}}(y,0)\rangle }_x$ (open and closed dots) for 0.6-$\mu$m-diameter polystyrene particles in water, small enough that streaming dominates and ${\mathit{u}}^{\mathrm{p}}\approx \langle {\mathit{v}}_2\rangle$. The observed motion (thick arrows) in (a) and (b) resembles the analytical results shown in Figs. 2b and 2c, respectively. For each value of $y$, the measured velocity $u_y^{\mathrm{p}}$ is averaged along the $x$ coordinate, with resulting SEM smaller than the size of the dots. The sinusoidal parallel-plate prediction (thin line) [Eq. (36)] is fitted to the data points far from the side walls (open dots), while the rectangular-channel prediction (thick line) [Eq. (37)] is fitted to all data points (open and closed dots). In both fits the acoustic energy $E_{\mathrm{ac}}$ is treated as a free parameter. (a) The half-wave resonance $\lambda /2=w$ $(n=1)$ with $f=1.940\text{MHz}$ and $U_{\mathrm{pp}}=1$ V. (b) The full-wave resonance $\lambda =w$ $(n=2)$ with $f=3.900\text{MHz}$ and $U_{\mathrm{pp}}=1$ V.