3. Results

The inertial focusing microfluidic device is schematically shown in figure 1(a) (see the detailed fabrication procedure in figure S1). The microfluidic channel has a rectangular cross-section, hosting particles of average sizes of diameter $a_0$ = 30 μm. Figure1(b) depicts the migration and focusing of soft particles with distinct deformability in microfluidic flow. Lateral migration of microparticles is driven by the total inertial forces exerted on the particles, and the equilibrium focusing position is reached as the total lift force ( $F_L$ ) becomes 0 (Jannesari Ghomsheh, Jafari & Funfschilling Reference Jannesari Ghomsheh, Jafari and Funfschilling2023). These forces stem from two main sources (Loth & Dorgan Reference Loth and Dorgan2009): first, the transverse background shear within the fluid (known as shear-induced lift, $F_S$ ); second, the interaction between a particle and the channel wall (referred to as wall-induced lift, $F_W$ ). The final equilibrium focusing position is reached when $F_S=F_W$ (figure 1 b). Different particle deformability is expected to alter the focusing positions (Doddi & Bagchi Reference Doddi and Bagchi2008).

Hydrogel microparticles with diameter of around 30 μm were generated by microfluidic emulsions (figure S2) and diluted to 0.1 % (w/v) before being introduced to the microchannels to minimise interparticle interactions. Water solution containing 0.01 % Span 80 surfactant was used to mitigate particle aggregation. A high-speed camera was used to observe and measure the particle positions $x_{eq}$ under different flow Reynolds number ( $Re$ ) (see detailed experimental set-up in figure S4). Here, $Re$ is defined as $Re=\rho _f U_m D_W/\mu$ , where $\rho_{f}$ is the fluid density, $\mu$ is the dynamic viscosity of the fluid, $U_m$ is the maximum flow velocity and $D_W=2WH/(W+H)$ is the hydraulic diameter with $W$ and $H$ being the width and height of the channel, respectively. We define the dimensionless equilibrium position $X_{eq}=2x_{eq}/W$ .

For a pilot study, microparticles with the same average diameter of around $a_0$ = 30 μm, made from poly(methyl methacrylate) (PMMA) and 2 % alginate (ALG) hydrogel, were used (figure S5). The PMMA and ALG hydrogel particles have substantially different stiffness, with Young’s modulus of $E$ = 3 GPa (Weihs, Lee & Kramer Reference Weihs, Lee and Kramer1994) and $E$ = 40 kPa (Lee & Mooney Reference Lee and Mooney2012), respectively. To clearly observe their equilibrium positions, the PMMA beads were pre-stained with TX-red dye and ALG particles were stained with FITC fluorescent dye (figure 1 c), which were observed using a fluorescent microscope. From figure 1(d–f), the focusing and separation processes of the beads can be clearly observed. Based on the work of Di Carlo (Reference Di Carlo2009), the channel used in this work was designed to be sufficiently long to allow the equilibrium position to be reached even under the highest flow rates. Figure 1(g) illustrates the variation in focusing and separation locations under different flow rates. The results show that, at $Re$ = 100, the beads start focusing. When $Re$ reaches 120 and above, the two kinds of beads can be separated at a distance of around 42 μm, with the channel width being 300 μm. This proves the current inertial chip has the ability to separate the particles with similar sizes based on deformability. To investigate the underlying mechanisms and optimise the separation process, we employed a set of PEG hydrogel beads as a model system and conducted a series of experimental tests.

To systematically investigate how the particle stiffness and flow condition affect the deformation and focusing of soft particles in microfluidic flow, we fabricate PEG hydrogel microparticles with similar sizes (around 30 μm) and controlled concentration adjusted from 8 % to 16 % (w/v), leading to elastic shear modulus ranging from 14.5 to 58.7 kPa (figure S3). These PEG beads serve as model particles, facilitating the investigation of the correlation of particle deformation, orientation and equilibrium focusing position with the Reynolds number and particle stiffness. To obtain clearer shape outlines after the deformation of PEG particles, the optical lens was replaced with a high-speed camera. The deformation of the PEG microparticles under different experimental conditions was recorded with Reynolds numbers ranging from 80 to 200 in increments of 20.

Figure 2.

Experimental observation of microparticle deformation and tilt in microfluidic flow. Hydrogel microparticles are made from PEG with concentration varying from 8 % to 16 % (w/v). The Reynolds numbers of 80, 120, 160, 200 are shown for illustration. The images are taken by a high-speed camera. The PMMA beads are included for comparison. Scale bar: 10 μm.

Typical high-speed camera images of deformed hydrogel microparticles are shown in figure 2 (see figure S8 for more details). The deformation occurring within the channel as the PEG microparticles migrate to their equilibrium positions, with respect to varying particle elastic modulus and flow speed, is clearly observable. The particles exhibit an elliptical shape when viewed from the top, and as the velocity increases, the softer beads (those with lower PEG concentration) deform to a greater extent and have higher aspect ratios. Additionally, the orientation of those deformed particles (tilt angle) is also observed especially for softer ones at relatively higher Reynolds number. Moreover, the spinning of deformed particles is also observed, and this rotation occurs along the major axis (figure S7). This allows the particles to move in the direction of the flow at a constant angle, avoiding the wobbling motion reported in the literature (Alghalib et al. Reference Alghalib, Rosti and Brandt2019; Lauricella et al. Reference Lauricella, Zhou, Luan, Papautsky and Peng2022). Analysis of particle rotation across multiple frames revealed conservation of the short-axis length, indicating that particles undergo deformation to adopt a spindle morphology rather than a dish-like configuration under the imposed channel flow conditions. This consistent preservation of the short-axis dimension during rotation provides compelling evidence for the spindle-shaped deformation hypothesis.

Under deformation, the initially spherical particles transformed into a spindle-like geometry, as previously discussed. Assuming conservation of particle volume during deformation yields

(3.1) \begin{equation} D_T=\big(a-\sqrt {{a_0}^3/a}\big)/\big(a+\sqrt {{a_0}^3/a}\big) ,\end{equation}

where $a_0$ is the initial diameter and $a$ is the final length of major axis measured from experiment. Figure S6 f presents the experimental measurements of $D_T$ and $a$ , which shows agreement with the theoretical predictions derived from our spindle geometry assumption. This correspondence provides strong validation for the spindle-shaped deformation model.

Figure 3.

Experimental dependence of microparticle deformation and tilt on the Reynolds and capillary numbers. (a) Variation of the Taylor deformation parameter ( $D_T$ ) as a function of the Reynolds number ( $Re$ ). Inset shows the outline of the deformed microparticles fitted by the general equation of an ellipse with principal semi-axes denoted by a and b, respectively (figure S6). (b) Variation of the Taylor deformation parameter ( $D_T$ ) versus the capillary number ( $Ca_e$ ). Symbols: experimental data; black solid line: linear fitting by $D_T= 1.25Ca_e$ . (c) Variation of the tilt angle ( $\theta$ ) versus $Re$ . Inset illustrates the definition of $\theta$ . (d) Variation of the tilt angle ( $\theta$ ) versus $Ca_e$ . Symbols: experimental data; black solid line: linear fitting by $\theta = 45 - 57.325Ca_e$ . Error bars in (a–d) indicate the variation of experimental data for at least three tests.

The edge profile of the deformed particles is fitted by an elliptical function (figure S6) and measured by the Taylor deformation parameter, $D_T=(a-b)/(a+b)$ , where a and b are the major and minor axes of the ellipse (see inset in figure 3 a). The resulting values of $D_T$ as a function of the Reynolds number for hydrogel particles with different deformability are shown in figure 3(a). Parameter $D_T$ is observed to increase almost linearly with $Re$ in the range 50–200, and the slope increases with the deformability of those particles (i.e. lower PEG concentrations). To evaluate the effect of deformability of the particles under shear flow, the capillary number, $Ca_e=\eta \dot {\gamma }/G$ , is introduced. The experimental images reveal a small $Ca_e$ and $D_T$ demonstrating that equation (2.2) can be applied. The plot of $D_T$ as a function of $Ca_e$ collapses all the experiment data in a straight line, revealing a linear dependence of $D_T$ on $Ca_e$ (figure 3 b). The slope of 1.25 is consistent with the linear theoretical prediction.

The orientation of deformed microparticles under different Reynolds number is also examined. The tilt angle, $\theta$ , defined by the angle between the major axis of the deformed particle and the flow direction, is plotted as a function of $Re$ (figure 3 c). The decreasing trend of $\theta$ versus $Re$ indicates the deformed particles tend to ‘lay down’ as the Reynolds number increases, resulting in a smaller tilt angle. Similar to $D_T$ , the tilt angle changes are more pronounced for softer microparticles (with lower PEG concentration). The plot of all the experimental data of $\theta$ versus $Ca_e$ converges to a master straight line with a starting angle of 45 $^\circ$ (figure 3 d). The linear correlation between $\theta$ and $Ca_e$ also agrees with theoretical prediction by Roscoe (Reference Roscoe1967). However, the relationship between $\theta$ and $Ca_e$ exhibits slightly distinct slopes across varying PEG concentrations, with the master curve demonstrating subtle deviations from reference data under equivalent flow conditions (Villone & Maffettone Reference Villone and Maffettone2019). These variations can be attributed to modifications in the coefficient $k_2$ , which arise from alterations in material parameters, particularly the elastic moduli. The concentration-dependent response suggests a systematic influence of PEG on the material properties governing particle dynamics. Through the experimental validation, we have demonstrated the accuracy of Roscoe’s linear approximation model in systems with low capillary number and monodisperse particles. The close agreement between theoretical predictions and experimental measurements demonstrates the reliability of our experimental apparatus.

To reveal how the flow condition and particle stiffness affect the inertial focusing of soft particles, typical high-speed camera images of different microparticles at their equilibrium positions under varied Reynolds number are shown in figure 4 (see more in figure S9). It is evident that the equilibrium position of soft particles shows clear dependence on the Reynolds number as well as deformability. Hydrogel microparticles with the same deformability migrate further towards the wall under higher flow rates, and, under the same flow rate, softer microparticles (with lower PEG concentration) tend to move further towards the centre.

Figure 4.

Experimental observation of microparticle equilibrium positions in microchannels. (a) Comparison of the equilibrium positions of PEG microparticles in microfluidic flow with different Reynolds number ( $Re$ ). (b) Comparison of the equilibrium positions of PEG microparticles with different concentration (‘Conc.’). Scale bars in (a,b) are 30 μm. The black line in each image indicates the position of the microfluidic wall.

We conduct inertial focusing measurements of hard particles to validate our set-up, and the experimental results are plotted in figure S10. These results were compared with works done by Segré & Silberberg (Reference Segré and Silberberg1961), Matas et al. (2004, 2009) and Asmolov (Reference Asmolov1999) that experimentally and numerically modelled the particle focusing in the differently shaped tubes. The general agreement with the numerical results in the literature validates the measurement accuracy of the experimental set-up. The discrepancy is attributed to the different tube geometries.

Figure 5.

Parametric dependence of microparticle equilibrium position in microfluidic flow. (a) Variation of the equilibrium position ( $X_{eq}$ ) of microparticles made from PEG hydrogel of different concentrations versus the Reynolds number ( $Re$ ). (b) Variation of the equilibrium position ( $X_{eq}$ ) versus the shear modulus ( $G$ ) of the hydrogel at different Reynolds number. (c) Variation of the equilibrium position ( $X_{eq}$ ) versus the capillary number ( $Ca_e$ ) for different PEG hydrogel microparticles. (d) Plot of $(1-X_{eq})/Re$ versus normalised major axis ( $a/a_0$ ) of deformed microparticles made from different PEG hydrogel.

The variation of the equilibrium position of soft microparticles with the Reynolds number is examined in figure 5(a). The decreasing trend of the dimensionless equilibrium position $X_{eq}$ (defined as $2x_{eq}/W$ ) versus $Re$ indicates the migration of microparticles towards the wall as the Reynolds number increases (see figure 1 b). The decrease of $X_{eq}$ versus $G$ for all Reynolds numbers confirms the observation of deformation-induced shift of microparticles towards the centre. Furthermore, $X_{eq}$ as a function of $Ca_e$ is plotted in figure 5(c), demonstrating a linear relationship with $Ca_e$ but with different slopes corresponding to the variation in the particle stiffness.

As discussed in § 2, the major axis of deformed particles ( $a/a_0$ ) serves as a key indicator of the particle size under confined flow. The method for calculating $a/a_0$ and $X_{eq}$ from captured images is described in figure S6. The modulus ( $G$ ) of different PEG microparticles and their corresponding capillary number ( $Ca_e$ ) shown in figure 5(b) can be determined based on their mechanical properties and the experimental conditions, as detailed in figure S3. We observe a roughly linear dependence of the equilibrium position on the Reynolds number (figure 5 a), with larger particles tending to migrate towards the centre of the channel ( $X_{eq} \rightarrow 1$ ). Figure 5(d) shows a plot of $(1-X_{eq})/Re$ as a function of $a/a_0$ , where all experimental data of PEG microparticles collapse onto a single master curve. This suggests a linear correlation between the equilibrium position and the major axis of the deformed microparticles in the microfluidic flow. Experimental data further support this linear correlation, as shown in figure S11 a, where the equilibrium position exhibits a clear relationship with the dimensionless major axis $a/a_0$ . These results confirm that $X_{eq} \propto a/a_0$ , with similar slopes for the plots of $1-X_{eq}$ versus $a/a_0$ , differing only in their yintercepts that correspond to varying flow speeds.

The capillary number ( $Ca_e$ ), which quantifies the particle deformability under confined flow, also plays a crucial role in determining the equilibrium position. We observe a complex dependence of the focusing position on both the flow rate and particle deformability. Specifically, different slopes in the relationship between $X_{eq}$ and $Ca_e$ suggest that particle deformability, characterised by $Ca_e$ , influences the focusing behaviour of the particles. Figure 5(d) illustrates the linear relationship between $X_{eq}$ and $a/a_0$ , and the plot of $(1-X_{eq})/Re$ versus $Ca_e$ shows a linear dependence, confirming that $X_{eq}$ is influenced by $Ca_e$ as well. Furthermore, the Taylor deformation parameter, which is correlated with $Ca_e$ as $D_T=1.25Ca_e$ (figure 3 b), can also be used to describe the relationship between the major axis and the capillary number. Figure S6 f reveals a nearly linear relationship between $D_T$ and $a/a_0$ , suggesting that $Ca_e$ and $a/a_0$ are highly correlated for small deformations. This relationship allows the linear function in figure 5(d) to be expressed in terms of $Ca_e$ . The linear dependence of the equilibrium position on both Reynolds number and capillary number provides a framework for designing microchip separation systems based on sample modulus, flow rate and particle size. As shown in figure S11 b, for a rectangular channel with a width of 300 μm and a height of 100 μm, the general equation for the equilibrium position is $X_{eq} = 1 - 0.0054Re(1 - 2.6Ca_e)$ .

To test specific applications, such as cell separation, human prostate cancer cells (PC-3, with a shear modulus of G = 287 $\pm$ 52 Pa (Molter et al. (Reference Molter, Muszynski, Tao and Trivedi2022)); and LNCaP, with G = 540 $\pm$ 28 Pa (Faria et al. Reference Faria, Ma, Gazi, Gardner, Brown, Clarke and Snook2008) were injected into the microchannel. These cells were diluted to 5000 cells ml⁻¹ and pre-stained with TX-red and FITC dyes separately for better observation. A 1:1 mixture of cell solution was then injected into the channel with the same velocities. Cell separation was tested at $Re$ = 100, 120 and 140, as shown in figure 6. The separation of the two cell lines happens at $Re$ = 120. The similar separation behaviour observed between microparticles and cancer cells (figures 1 g, 4 and 6 b–d) indicates a significant potential of this chip in the field of cell separation and other biological applications, and highlights the potential of using deformable hydrogel microparticles to mimic the deformation and dynamics of rare target cells in microfluidic flows. The capillary number ( $Ca_e$ ) of the cells was subsequently calculated using their shear moduli and the average shear rate. The inertia focusing results were compared with our PEG model with including the data for ALG beads (figure S11 c). Those ALG beads indeed follow the linear master line predicted according to the PEG model. Although a consistent trend is observed for the cells, the clear deviation from linear behaviour indicates that the PEG model is only suitable for low capillary numbers (i.e. $Ca_e\lt 0.2$ ). This highlights the need for a more comprehensive theoretical framework capable of incorporating nonlinearities and cell-specific mechanical properties.

Figure 6.

Separation of cancer cell line samples. (a) Optical images of LNCaP (upper) and PC-3 (lower) human prostate cancer cells with similar sizes. (b–d) Experimental observation of different focusing positions of the two cell lines, PC-3 (red) and LNCaP (green), under Reynolds numbers of $Re$ = 100, 120 and 140, respectively. Scale bars: 50 μm.