A 3D modeling framework for accurate trajectory-based prediction of critical diameter in deterministic lateral displacement microfluidics

Abstract

Deterministic Lateral Displacement (DLD) is a high-precision microfluidic technique for particle separation based on size differences. However, the lack of an accurate predictive model for the critical diameter (Dc) limits both the design flexibility and understanding of DLD behavior. In this study, we propose a novel Dc prediction framework based on a 3D physical model, achieving high accuracy and computational efficiency. Experimental validation shows excellent agreement between predicted and actual particle trajectories. Remarkably, we discover that Dc exhibits a U-shaped variation along the vertical direction of the DLD channel, revealing a transition zone. Numerical simulations show that particles within this zone undergo vertical oscillations, causing trajectory switching between zigzag and bump modes, resulting in an altered zigzag trajectory. This framework reveals the mechanism behind altered zigzag formation from a 3D perspective and provides a powerful tool for the rapid, accurate, and customizable design of DLD microfluidic separation devices.

Introduction

Microfluidic separation technology is an advanced method based on microscale fluid manipulation, offering advantages such as miniaturization, low cost, and minimal reagent consumption^1,2. It enables the isolation of target biological particles or molecules from complex mixed samples, providing a powerful tool for cutting-edge research in areas such as rare cell isolation^4,5, exosome purification⁷, and pathogen detection⁸. This has greatly advanced applied research in fields like precision medicine, liquid biopsy, and point-of-care diagnostics.

Among the many microfluidic separation techniques, deterministic lateral displacement (DLD) has received widespread attention since its introduction by Huang³, due to its high precision, reliability, and structural simplicity⁹. DLD has achieved significant applications, including particle separation^10,11, droplet separation¹², blood cell separation^13–17, and the isolation of circulating tumor cells from whole blood^18–20.

The separation performance of DLD depends on the geometric configuration of its internal periodic micropillar array^21,22. As shown in Fig. 1a, neighboring 2 × 2 micropillars form a parallelogram. The longitudinal periodic distance between micropillars is λx, the lateral periodic distance is λy, the downstream shift per row is Δλ, Gx is the longitudinal gap, and Gy is the lateral gap. The DLD structural period is defined as Np = λy / Δλ.

Fig. 1. Schematic diagram of typical DLD working principle.

a Geometric features of a DLD device. b Flow field characteristics inside the DLD channel and two typical particle trajectories: bump and zigzag

Taking the micropillar array with Np = 5 in Fig. 1b as an example, the DLD separation process can be analyzed. The flow direction in the device is horizontal, and the stream occupying width r at the top side of a micropillar flows into the gap below the next downstream micropillar. This implies that after passing through five columns, the fluid reconverges into the same gap. Thus, the flow between micropillars is equally divided into five streams, and the higher the velocity, the narrower the stream width. Based on this ideal equal-flow partitioning model, particles are treated as point mass. If the particle diameter dp > 2r, it follows a bump trajectory (red curve in Fig. 1b); if dp < 2r, it follows a zigzag trajectory (blue curve in Fig. 1b). This enables size-based separation. The particle diameter at the transition between these two trajectories is defined as the critical diameter, Dc.

Accurately predicting Dc is crucial for DLD device design, saving time and cost. However, the equal-flow partitioning model²³, which treats particle as point mass, ignores particle size effects on motion and thus cannot accurately predict Dc.

To improve Dc prediction accuracy, Davis, John Alan et al. proposed an empirical formula:

$$Dc=1.4G{\epsilon }^{0.48}$$ 1

where G = Gx = Gy, ε = Δλ/λ, and λ = λx = λy. While this formula is relatively accurate for circular micropillar arrays with equal longitudinal and lateral periodic distance²⁴, it fails for non-circular or unevenly spaced micropillars.

To address the effects of non-standard micropillar shapes and irregular spacing on flow fields, researchers have turned to finite element simulation tools. These tools allow precise geometric modeling of micropillar arrays and simulation of flow behavior under complex boundary conditions using fluid dynamics governing equations. For instance, J. C. Wang et al. proposed the MOPSA algorithm²⁵, which uses COMSOL to obtain fluid velocity fields and MATLAB to predict particle trajectories. While this method intuitively displays trajectories, it introduces a non-physical fitting coefficient β, which varies with geometry, limiting its ability to predict Dc for new structures. Furthermore, they attempted to mitigate the shortcomings of MOPSA by introducing an artificial intelligence-based approach to predict the β value of the device^26,27. However, due to the limited training parameters, it still fails to accurately predict the Dc of arbitrary DLD devices.

To better capture particle–fluid interactions, C. Wang et al. proposed a unified-field monolithic fictitious domain–finite element method for fluid–structure–contact interactions⁶. This method dynamically updates flow field data to obtain more accurate flow information around particles, which in theory allows for more precise prediction of Dc. However, this approach (1) involves complex formulations and procedures, making implementation difficult; (2) requires continuous remeshing and recalculation of the fluid domain, leading to high computational costs; and (3) its accuracy remains to be thoroughly evaluated. More importantly, all these methods simplify the 3D DLD device into a 2D model, which assumes an infinitely tall channel and thus only captures the mid-height flow field. In real DLD devices with finite height, wall effects in the vertical (Z) direction significantly impact velocity and pressure distributions. Therefore, 2D simulations cannot capture the true 3D flow field or accurately predict particle trajectories.

To improve flow field accuracy, K. M. Lai et al. introduced the 3D-MOPSA algorithm²⁸, which builds a 3D COMSOL model and uses MATLAB to predict Dc after solving the flow field. However, this method inherits MOPSA’s shortcomings and still fails to provide accurate Dc predictions.

In addition, DLD inevitably encounters the altered zigzag trajectory, where the lateral displacement distance lies between those of the zigzag and bump modes²⁹. This broadens Dc into a range, causing particles with sizes within the altered zigzag region to experience insufficient lateral displacement and be misclassified as background particles, thereby degrading the separation performance. To elucidate the underlying mechanism, several studies have reported the occurrence of altered zigzag trajectories^30,31. Among them, Vandersman’s group proposed that anisotropic permeability within the micropillar array induces this behavior. Specifically, finite-size micropillars disturb the ideal periodic flow field established by point-sized pillars, leading to altered zigzag motion³². However, such explanations consider only 2D flow disturbances and neglect the coupling between particle size and the real flow field.

In summary, existing trajectory prediction methods for DLD devices mostly rely on 2D models, which cannot reflect realistic flow conditions and thus suffer from limited accuracy and applicability. Although 3D physical modeling approaches have emerged, they have yet to systematically investigate the effects of vertical flow distribution on Dc. Consequently, a method that accurately predicts Dc based on the underlying physical interaction between particles and the flow field remains lacking.

This work proposes a method for accurately predicting the critical diameter Dc of DLD devices. The method requires only a single stationary flow field solution of 3D model in finite element software. A 2 × 2 data source, which is a collection of physical values within the minimal rectangle enclosing four adjacent micropillars arranged in a parallelogram distribution, is extracted from the 3D model to compute the trajectories of particles of different sizes. The Dc value is then determined based on the particle sizes corresponding to the bump and zigzag trajectories.

In this method, the particle surface is divided into small curved areas of equal size. At each area, the local flow velocity is decomposed into radial and tangential components, with the radial component contributing to particle displacement. By calculating the drag force (including viscous and pressure components) and lift force acting on the particle, and applying displacement equations along with collision handling between the particle and the micropillar, the particle trajectory is successfully predicted. The predicted results agree well with both literature data and our experimental results.

We further analyzed how the choice of 2 × 2 data source location in the 3D model affects the trajectory prediction and proposed a strategy for selecting the effective data source. Additionally, Numerical results revealed that Dc exhibits a U-shaped distribution along the height direction, with the smallest Dc at mid-height. The U-shaped region corresponds to a transition zone, where lateral displacements of the particle are between those of zigzag and bump trajectories. Based on the simulation results, we explain the formation of altered zigzag trajectories from a 3D perspective: particles within the transition zone undergo periodic vertical motion, resulting in trajectories that combine features of both zigzag and bump patterns. Finally, we discuss the computational efficiency, fabrication tolerances, strategies for minimizing the transition zone, and directions for future automated design efforts.

Results

Influence of the 2×2 data source

In an ideal world, the DLD device has an infinite micropillar array and a stable flow field. A stable flow field refers to a region where each 2 × 2 micropillar array has a consistent velocity field and pressure gradient. Based on this, our method calculates the particle trajectory by performing repeated mapping using a selected 2 × 2 data source. The detailed computation approach is described in 'Trajectory prediction workflow' section. If the size of the 3D simulation model matches the actual device, it is easy to determine that the central region is within a stable flow field. However, due to limitations in computational resources and time, the size of the simulated 3D model cannot match that of the full-scale DLD channel. However, in smaller 3D models, the stable flow domain is limited or may even be absent. Therefore, selecting the correct data source is crucial to prediction accuracy.

To investigate the effect of the 2 × 2 data source within the model, the example described in³ is adopted. The corresponding structural parameters are listed in the first row of Table 1, with a structural period of Np = 10. A 3D simulation model consisting of 16 rows and 31 columns was built, as shown in Fig. 2a, and Fig. 2b shows the XY-plane projection of the 3D model. As reported in this work, this device can distinguish between 0.7 μm and 0.9 μm particles. Therefore, a 0.5 μm particle must follow a zigzag trajectory in this actual device. We defined Npz as the number of micropillars within one zigzag period, and then initialized 0.5 μm particles at the mid-height of the channel to investigate Npz corresponding to the 2 × 2 data sources labeled 1–13 in Fig. 2b.

Table 1.

Information of predicted results and experimental results

Device	Micropillar		λx (μm)	λy (μm)	Δλ (μm)	Channel height (μm)	Tested separation performance	Our predicted separation performance
	Cross section	Feature (μm)
3	Circle	Diameter: 6.4	8	8	0.8	5	0.7 μm and 0.9 μm particles	0.7 μm and 0.9 particles
6	Circle	Diameter: 33	50	45	1.5	30	4 μm and 5 μm particles	4 μm and 5 μm particles
Ours	IETRC	Altitude: 28 Fillet: 3.5	60	60	4	30	12 μm and 13 μm particles	12 μm and 13 μm particles

Fig. 2. Relationship between the 2×2 data source position and zigzag trajectories in the 3D model.

a The 3D physical fields obtained in COMSOL, including the solved pressure and velocity fields, as well as the distributions of pressure and velocity within the 2 × 2 micropillar region. b Projection of the 3D model onto the XY plane, where the small rectangles represent the positions of 2 × 2 data sources, and each is labeled with its corresponding number. c Zigzag trajectories of 0.5 μm particles at positions numbered 1, 3, 5, 12, and 13. d Npz for data sources numbered 1−8 along the central horizontal flow direction. e Npz for data sources numbered 9–13 in the lateral direction

Figure 2c displays the zigzag trajectories of the 0.5 μm particles originating from data sources labeled 1, 3, 5, 12, and 13, visually demonstrating that Npz depends on the location of the 2 × 2 data source. Figure 2d shows Npz for data sources labeled 1–8. It is evident that along the central horizontal flow direction, Npz follows a concave distribution: lower in the middle and higher at both ends. Specifically, the number is 10 in the central region and >10 near the inlet and outlet. Some experiments have demonstrated that in the stable flow domain of a DLD array, Npz equals the geometric period Np of the micropillar array^3,33. In this case, Np = 10, so data sources that produce zigzag cycles containing 10 micropillars are more representative of the actual stable flow region. The reason why >10 micropillars appear per zigzag cycle near the inlet and outlet is due to the boundary condition setup in the simulation, where the inlet is set to average velocity and the outlet to atmospheric pressure. These conditions prevent the flow near the boundaries from reaching stable state.

Figure 2e plots Npz for data sources labeled 9–13 along the lateral direction, exhibiting a convex distribution. The central region of the channel yields the expected number of 10 micropillars per zigzag cycle, while the lateral regions result in fewer than 10. This is attributed to the lateral boundary effects of the DLD device³⁴, where the lateral velocity near the channel sidewalls is higher than in the stable flow region, reducing Npz value.

If a 2 × 2 data source with Npz more than Np is used to predict particle trajectories, the lateral force in the flow field will be underestimated, leading to a small Dc. Conversely, if Npz less than Np, the lateral force will be overestimated, causing a big Dc. Therefore, only a 2 × 2 data source in which Nps equals Np is consistent with the actual stable flow domain and yields more accurate predictions.

Based on this, we summarize a method for determining the appropriate 2 × 2 data source. First, an initial particle diameter is set that will certainly follow the zigzag mode in the DLD channel. This is easy to achieve, for example, by choosing half the empirical Dc (1). Then, various 2 × 2 data sources at different positions are tested outward from the model center, until one is found where the particle’s zigzag trajectory includes exactly Np micropillars. This data source is then selected for subsequent performance analysis of the DLD device.

Comparison between prediction and experiment

To validate the accuracy of the proposed method, we compared the predictions with experimental data from both published studies and our experiments. Table 1 lists the device geometry parameters and the corresponding comparison results. In these simulations, the initial vertical position of particle was set to the mid-height of the channel.

Comparison with published studies

The first row of Table 1 corresponds to the structural data from³, which demonstrates the separation of 0.7 μm and 0.9 μm particles. We constructed a simulation model based on the parameters provided in the reference (assuming a height of 5 μm) and obtained the predicted trajectories for 0.7 μm and 0.9 μm particles, as shown in Fig. 3a. The simulation reveals that the 0.7 μm particle follows a zigzag trajectory, while the 0.9 μm particle follows a bump trajectory, indicating that the device can effectively distinguish between these two particle sizes. This prediction aligns with the results reported in the literature.

Fig. 3. Comparison between predicted and published trajectories.

a Top view of predicted trajectories for a device from [3]. The red circles represent the predicted trajectory of 0.9 μm particles, and the blue circles correspond to 0.7 μm particles. b–f present an analysis of the trajectories from Reference [6]. b Top view of predicted trajectories for device parameters. The red and bule respectively represent predicted trajectories for 5 μm and 4 μm particles. c Overlay of experimental and predicted trajectories: Region I shows the 4 μm particle trajectory from the literature, and Region II shows the 4.2 μm particle trajectory predicted by our method. Green outlines represent the micropillar contours in the simulation model and blue lines indicate the predicted particle trajectories. d Euclidean distance between the particle center trajectory obtained from the experiment in (c) and the predicted particle center trajectory. The ribbon area indicates the range of mean ± standard deviation (SD). The mean Euclidean distance is 0.246 μm and SD is 0.182 μm. e Overlay of the trajectory of 5 μm particles reported in the literature and the trajectory predicted (in red) by our method. f Euclidean distance in (e) between the particle center trajectory from the literature and the predicted particle center trajectory. The mean Euclidean distance is 0.317 μm and SD is 0.209 μm

The second row of Table 1 refers to the structural parameters from⁶, which allow for the separation of 4 μm and 5 μm particles. We built the simulation model according to the data in the paper, and the predicted trajectories of the 4 μm and 5 μm particles are shown in Fig. 3b. The predictions are consistent with the experimental results reported in the literature.

To further analyze the agreement between the predicted and experimental trajectories, we compared them in Fig. 3c. Part I shows the trajectory of 4 μm particles from the publication, and Part II shows the predicted trajectory for 4.2 μm particles using our method. The green circles indicate the simulated micropillar array. The predicted particle trajectory almost completely overlaps with the experimental trajectory when the micropillar arrays in the simulation and experiment are aligned. Considering the coefficient of variation in particle size, the presence of 4.2 μm particles among 4 μm particles is reasonable. Furthermore, we manually traced the particle centerline from Fig. 3c-I using multi-points tool in ImageJ software, as shown in Fig. S1A. We aligned the extracted center coordinates with those of the simulated trajectory, resulting in Fig. S1B. The Euclidean distance between the two was calculated and shown in Fig. 3d, which mainly falls within the range of 0.246 ± 0.182 μm and the maximum value is 0.822 μm. Using the same approach, as shown in Fig. S1C and Fig. S1D, we obtained the overlaid trajectories of 5 μm particles from the experimental result in Fig. 3e–I and the predicted result in Fig. 3e-II. It can be visually observed that the two trajectories almost completely overlap. The average Euclidean distance between them is 0.317 ± 0.209 μm, and the maximum Euclidean distance is 0.983 μm, as shown in Fig. 3f. These results demonstrate a high degree of consistency between the predicted and experimental trajectories, indicating the effectiveness of the proposed framework.

Comparison with our experimental tests

We designed and fabricated a DLD device based on SU-8 mold and Polydimethylsiloxane (PDMS) replication techniques. The test device is shown in Fig. 4a, featuring three inlets and two outlets. The cross-section of the micropillars is an inverted equilateral triangle with rounded corners (IETRC), and the geometric period of the structure is Np = 15. A microscopic image of the micropillar array is shown in Fig. S2, and the extracted parameters are listed in the bottom row of Table 1. We predicted the device could distinguish between 12 μm and 13 μm particles, as shown in Fig. 4b.

Fig. 4. Comparison between predicted and our experimental trajectories.

a The test DLD device. b Predicted trajectories for 12 μm and 13 μm particles using our method. C Overlay of experimental and predicted center trajectories (blue curve) for the 12 μm particles, obtained by aligning the micropillar arrays (green contours) in experimental image I and predicted image II. d Euclidean distance between the experimental particle center trajectory and the predicted particle center trajectory in (c). The ribbon area indicates the range of mean ± standard deviation (SD). The mean Euclidean distance is 0.512 μm and SD of that is 0.320 μm. e Overlay of experimental and predicted trajectories (red curve) for the 13 μm particles, based on alignment of the micropillar arrays (green contours) in images I and II. f Euclidean distance between the experimental and predicted particle center trajectories in (e). The mean Euclidean distance is 0.449 μm and SD of that is 0.362 μm

We built a test system as illustrated in Fig. S3. High-speed cameras were used to capture the instantaneous positions of particles, and ImageJ was used to overlay frames using the minimum intensity projection to obtain particle trajectories, as shown in Figs. 4c–I and e–I. Subsequently, the simulated micropillar array (outlined in green) was aligned with the experimental array to facilitate a comparison between the predicted and observed particle trajectories. Figure 4c shows the comparison for the 12 μm particle. The simulated particle center trajectory (blue curve) almost overlaps with the experimental trajectory. The experimentally obtained particle center trajectory and the alignment between the experimental and predicted center trajectories are shown in Fig. S4. The Euclidean distances between the two aligned trajectories are mainly distributed around 0.512 ± 0.320 μm, with a maximum value of 1.335 μm, as shown in Fig. 4d. Similarly, the simulated 13 μm particle center trajectory (red curve) closely matches the experimental 13 μm trajectory in Fig. 4e. The aligned trajectories exhibit Euclidean distances mostly around 0.449 ± 0.362 μm, with a maximum of 1.796 μm, as illustrated in Fig. 4f. These experimental results confirm that the Dc of the DLD device lies between 12 μm and 13 μm, consistent with the simulation predictions and confirming the accuracy of the predicted trajectories.

Effect of particle vertical position

Relationship between particle vertical position within channel and Dc value

To the best of our knowledge, no studies have yet investigated the effect of particle vertical position within the channel on the Dc in DLD devices. This is primarily because most analyses simplify the problem using 2D models, which ignores the vertical effect. However, our proposed method utilizes a 3D flow field model, allowing us to analyze how particle trajectories vary with vertical position in the channel.

We created simulation cases for DLD devices in which the micropillar cross-sections were circular and IETRC, while keeping all geometric parameters and simulation conditions identical except for micropillar shape. The specific parameters are listed in Table 2. To obtain the Dc values at different vertical positions within the channel, we fixed the particle’s vertical position during trajectory prediction. The resulting relationship between Dc value and vertical position is shown in Fig. 5a, where the x-axis represents the normalized vertical position within the channel, and the y-axis shows the Dc value that normalized by the Dc value at mid-height. Solid endpoints on the curves indicate particle contact with the top or bottom wall.

Table 2.

Simulation parameters for studying the effect of particle vertical position

Device	Micropillar		λx (μm)	λy (μm)	Δλ (μm)	Channel height (μm)	Simulation conditions	Transition zone # (μm)
	Cross section	Feature (μm)
1	Circle	Diameter: 28	60	60	4	30	Mean velocity: 5 mm/s Fluid density: 1.05 g/cm3 Particle density: 1.05 g/cm3	11.7 - 12.6
2	IETRC	Altitude: 28 Filled: 3.5						12.1 - 12.5

# During the prediction of the particle trajectory, the particle remained at the same height within the channel as its initial value

Fig. 5. Effects of particle size and vertical position on trajectory.

a Normalized Dc values for particles located at different normalized vertical position within the channel. b–f are all based on the data from Device 2 in Table 2. b The predicted number of micropillars within one period of the altered zigzag trajectories corresponding to particles of different sizes, and their fitting curves. c Altered zigzag trajectory of a 12.3 μm particle located within the transition region; the blue represents the particle trajectory projected onto the XY plane, while the magenta curve shows the trajectory of the particle center in the vertical (Z) direction. d Slice of the Z-component of velocity within the 2 × 2 micropillar unit. e Correspondence between the Z-direction lift force and the XY-plane projected trajectory of a 12.3 μm particle. f Vertical motion characteristics of particles of different sizes within the altered zigzag mode. The black curve indicates the maximum displacement during the Non-collision phase, and the orange curve represents the average velocity per micropillar during the bump phase

From Fig. 5a, both DLD structures exhibit the same pattern: the Dc value forms a U-shaped distribution with the minimum at middle height of the channel. The Dc value of the DLD device increases as it approaches the top or bottom wall. This leads to several conclusions: (1) When the particle size is smaller than the Dc value at mid-height, it will follow a zigzag mode regardless of its vertical position. (2) When the particle size is larger than the Dc value at the endpoints of the U-curve, it will always follow a bump mode. (3) When the particle size lies within the U-curve range of Dc, the trajectory depends on particle vertical position. This interval is called the transition zone. Due to lift and drag forces, particle vertical position fluctuates dynamically. As a result, particles in this transition zone exhibit trajectories combining both zigzag and bump behaviors, referred to as altered zigzag trajectories.

Hypothesis on altered zigzag trajectories

We propose a new hypothesis: The Dc value varies with vertical position within the DLD device, as shown in Fig. 5a, and particles in the transition zone oscillate vertically within the channel. This periodic vertical movement causes the particle trajectory to switch between zigzag and bump modes, forming an altered zigzag trajectory.

To verify this hypothesis, we used the parameters from Device 2 in Table 2, which has a structural period of 15 micropillars. Simulation results show that its transition zone lies between 12.1–12.5 μm. We plotted the particle trajectories from 12.1 μm to 12.5 μm in 0.1 μm intervals. The resulting trajectories are shown in Fig. S5. From these, we extracted the period lengths and plotted them in Fig. 5b. We observed that the altered zigzag period length increases with particle size in a nonlinear, approximately exponential manner.

To analyze the motion process of the altered zigzag mode, we predicted the trajectory of a 12.3 μm particle, as shown in Fig. 5c. The blue curve represents the projection of the particle trajectory onto the XY plane, with a period of 24 micropillars, which is longer than the structural period of 15, indicating an altered zigzag trajectory. The magenta curve shows the vertical (Z-direction) position of the particle over time. Combining these plots, we see that the particle exhibits periodic vertical oscillation during its altered zigzag motion.

The mechanism of altered zigzag formation is as follows: The DLD structure can be regarded as a channel with alternating expansions and contractions, which induces secondary flows. Ignoring gravitational effects, slight pressure differences exist along the height of the channel, influencing the vertical motion of particles. Additionally, the fluid viscosity also affects particle motion in the vertical direction. The drag force acting on the particle includes both viscous and pressure-induced resistance, whose general expression can be written as:

$${\mathit{F}}_{\mathit{D}}=\frac{1}{2}{\rho }_{\mathrm{sol}}{v}^2{C}_{D}A$$ 2

where, F_D represents the drag force, ρ_sol is the solution density, v denotes the relative velocity between the particle and the fluid, C_D is the drag coefficient, and A is the projected area in the direction of motion. When considering only drag, the particle moves along the flow direction.

From COMSOL, we extracted the Z-direction flow field within the 2 × 2 micropillar region. The field was sliced at equal intervals along the flow direction, generating images I–VI, as shown in Fig. 5d, in which the positive values represent flow in the positive Z direction, while negative values indicate flow in the negative Z direction. In Fig. 5c, the particle resides in the lower half of the channel, between 0 and H/2, where H is the channel height. For clarity, each slice was divided into three regions (1, 2, and 3), corresponding to three stages experienced by the particle within the 2 × 2 micropillar array.

The fluid viscosity induces a nonuniform velocity distribution, which generates a lift force on the particle. For the results shown in Fig. 5c, Fig. 5e presents the lift force in the Z direction along with the particle trajectory projected onto the XY plane during an altered zigzag period. Since the particle is close to the channel wall, with a distance to the bottom wall of less than 3.5 μm, the wall-induced lift force becomes the dominant component, and the resulting lift force is directed along the positive Z direction.

During the altered zigzag motion, after a particle leaves a zigzag turning point (red arrow in Fig. 5c, with the spatial overlap of the particle trajectory and streamlines shown in Fig. S6), its motion can be divided into three stages, corresponding to the three regions labeled 1, 2, and 3 in red text in Fig. 5c, e:

Stage 1: Within each 2 × 2 micropillar unit, the particle sequentially passes through regions I-1, II-1, III-1, IV-1, V-1, and VI-1 as shown in Fig. 5d. During this stage, the particle’s velocity increases with the XY-plane flow velocity, resulting in the particle spending more time passing through II-1 than VI-1. This produces a net positive displacement due to drag. As shown in Fig. 5e, lift force increases—the higher surrounding XY-plane flow velocity increases the relative velocity between the particle and fluid, thereby enhancing the lift. The combined effects of drag and lift drive rapid particle motion in the positive Z direction.

Stage 2: Within each 2 × 2 micropillar, the particle passes through regions I-2, II-2, III-2, IV-2, V-2, and VI-2. Here, the Z-direction flow velocity is nearly zero, so the net displacement due to drag is negligible, and vertical displacement is primarily driven by lift. Figure 5e shows that lift gradually decreases during this stage—the surrounding XY-plane flow slows down, reducing the relative velocity and thus the lift, which slows the upward (positive Z) displacement.

Stage 3: Within each 2×2 micropillar, the particle passes through regions I-3, II-3, III-3, IV-3, V-3, and VI-3. In this stage, the particle is in contact with, or very close to, the micropillars. Consequently, the time to pass III-3 is shorter than that for V-3, resulting in a negative net Z-displacement contribution from drag. Collisions reduce the particle’s speed, increasing the relative velocity with the surrounding fluid and slightly enhancing lift, as shown in Fig. 5e, which slows the downward (negative Z) displacement but cannot prevent it entirely.

After multiple collisions, when the particle diameter is smaller than the local Dc at the given height, it reaches the next turning point, completing one altered zigzag trajectory cycle.

We further analyzed the relationship between particle size and vertical displacement based on Fig. S5. During the non-collision phase (the stage 1 and 2 in Fig. 5c), larger particles experience greater vertical displacement toward the channel center, as shown black curve in Fig. 5f. In this case, the particle size varies from 12.1 μm to 12.5 μm, while the corresponding vertical displacement ranges from 1.1 μm to 3.0 μm. However, during the bump phase (the stage 3 in Fig. 5c), larger particles move toward the wall with smaller average velocity, as shown orange curve in Fig. 5f. Hence, larger particles require more micropillar collisions to return to the zigzag mode, resulting in longer bump paths. Since all altered zigzag trajectories share the same non-collision length, a longer bump phase leads to a longer altered zigzag period and larger lateral displacement. This aligns with the well-known trend: larger particles follow trajectories closer to the bump mode.

Simulation time consumption

We evaluated the time required for the entire simulation process. In both models, the meshing strategy set the maximum and minimum mesh sizes in the XY plane to 10 μm and 5 μm, respectively, including boundary layers. Along the vertical direction, a symmetric exponential distribution was used, with a maximum spacing of <3 μm. The computational resources used were: AMD Ryzen 9 5900HS processor and 32 GB DDR4 3200 MHz memory. The time required for each key step in the trajectory prediction of both parameter models is listed in Table 3.

Table 3.

Simulation time consumption of key steps

DLD simulation parameter	COMSOL (12 rows × 25 columns)		MATLAB (Passing through15 micropillars)
	Mesh generation	Stationary solution	mode	Zigzaga	Zigzagb	Bumpb	Bumpa
Device 1 in Table 2	43 s	28 min 36 s	size	9.4 μm	11.6 μm	11.7 μm	14 μm
			time	82 s	106 s	202 s	150 s
Device 2 in Table 2	55 s	52 min 22 s	size	9.6 μm	12.0 μm	12.1 μm	14.5 μm
			time	86 s	116 s	242 s	151 s

a Indicates a larger difference between the particle size and the Dc value

b Indicates a smaller difference between the particle size and the Dc value

From Table 3, it can be observed that in the COMSOL simulation stage, the majority of the time was spent on stationary solution. The model with IETRC micropillar cross-sections required significantly more time, mainly due to its more complex contour, which generated a higher mesh count and therefore increased computation time. In MATLAB, for the same model and the same number of micropillars passed, particles of different sizes consumed different amounts of time. Particles with sizes closer to the Dc value required more time, as they entered the critical transition region between bump and zigzag modes. In this region, particle velocity tends to be zero, causing the particle to dwell longer. This is consistent with the experimentally observed fluorescence images in Fig. S7, where brightness increases significantly in the transition zone. This implies more iterations are needed, resulting in a notable increase in overall computation time.

We evaluated the time required to determine the Dc value for a single model, from initial setup to final determination. The total estimated time for the current procedure is ~100 min. The evaluation process is as follows:

(1) Using the established parameterized 3D modeling framework in COMSOL, modeling can be completed by importing a 2D micropillar contour model and modifying structural parameters. This step typically takes no more than 10 min.

(2) According to Table 3, mesh generation and stationary solution in COMSOL take about 55 min.

(3) Following the method described in ‘Influence of the 2× 2 data source' section, the 2 × 2 data source position for simulation can usually be determined through fewer than six trial runs, which can be done within 10 min.

(4) Using the bisection method, the Dc value with an accuracy of 0.1 μm can be determined within 10 trajectory simulations, taking up to 25 min.

Discussion

Summary of work

This study proposes a high-precision particle trajectory prediction framework for DLD devices, based on a co-simulation platform combining COMSOL and MATLAB. The stationary fluid flow is computed in COMSOL, while the iterative calculation of particle trajectories is conducted in MATLAB. The framework improves prediction accuracy by using a 2 × 2 data source determined by the principle that the number of micropillars in a zigzag period (Npz) equals to the number of micropillars defined by the structural period (Np). Based on the physical laws governing particle–fluid interactions, the method incorporates surface velocity decomposition, force-displacement equations, and handling particle–wall overlap to predict trajectories. To verify the accuracy of the proposed method, we modeled both published and self-fabricated devices. The predicted trajectories show high consistency with the experimental results. Subsequent analysis based on the simulation model revealed that the Dc in DLD devices depends on the particle position within the channel height, exhibiting an overall U-shaped distribution: Dc is minimal at intermediate channel heights and increases toward the channel walls, with the maximum and minimum values defining a transitional region. This finding further explains that particles within the transitional region experience vertical periodic motion due to the combined effects of lift and drag forces, resulting in the altered zigzag trajectory. However, during the altered zigzag phase, the vertical displacement of the particle within the channel was limited to 1–3 μm, a very small distance. Owing to obstruction by the micropillars in the DLD device, these vertical variations cannot be observed from the side. Validation of this phenomenon will require more advanced instrumentation in future studies.

We compared our method with existing approaches for predicting the performance of DLD devices, as shown in Table S1. Since the comparative studies did not provide quantitative data for their respective models, Table S1 presents a qualitative comparison of algorithm performance. As shown in Table S1, benefiting from the 3D physical field and more comprehensive force analysis, our framework achieves better trajectory overlap and more accurate prediction of Dc.

Enhancing DLD device resolution

Due to the 3D model, we observed that the Dc varies with the particle vertical position within the channel, exhibiting a U-shaped distribution. We predicted the transition region range of a given model from a simulation perspective.

When using a DLD device to separate two samples with relatively small size differences, improving device resolution is crucial. Higher resolution corresponds to a narrower transition region. As shown in Fig. 5a, the normalized curve of micropillars with circular cross-sections lies significantly above that of micropillars with IETRC cross-sections, indicating that the IETRC design leads to a smaller transition region and thus higher resolution. This insight suggests that optimizing micropillar shapes can enhance DLD device performance.

Although our method provides a predicted transition range, the actual transition region in physical devices is wider. This discrepancy is attributed to unavoidable inconsistencies in micropillar structures during fabrication, leading to slight flow field variations. Consequently, A particle with a size slightly larger than the transition zone should have exhibited bump trajectory, but it shows an altered zigzag trajectory, enlarging the practical transition region. Additionally, unstable flow near the device inlet and outlet may also result in similar altered zigzag trajectories.

Optimization of computation speed

Our evaluation shows that the total time required from model construction to determining Dc can be kept under 100 min, which is relatively long. Therefore, further optimization is possible. In terms of physical field computation, introducing lateral periodic boundary conditions where appropriate could also speed up physical field solving. For trajectory calculations, we observed that CPU usage by MATLAB remains at ~12%. Future improvements could include enhancing parallel processing capabilities to increase CPU utilization and reduce simulation time. Moreover, a trajectory recognition function could be integrated into the iterative process to intelligently skip redundant calculations, significantly reducing the time spent in MATLAB. In addition, the entire workflow could also be deployed on a more powerful computing platform to further reduce the overall processing time.

Impact of fabrication and materials on prediction accuracy

The commonly used fabrication method combining SU-8 molds with PDMS casting offers high precision, mature processing, and scalability, making it widely used in DLD device manufacturing. However, it also has limitations. First, the SU-8 process cannot produce ideal sharp corners and is not suitable for fabricating micropillar arrays with aspect ratios over 5. Therefore, the geometric design of DLD devices should account for fabrication-induced errors, which can be estimated based on prior manufacturing experience. Second, due to the elasticity of PDMS, high flow rates can cause deformation in channels and micropillars, altering the device’s Dc. Thus, if higher throughput is required in the application, the cross-sectional area perpendicular to the flow direction in the DLD device can be increased. This approach reduces the internal pressure by lowering the average flow velocity, thereby minimizing the impact of deformation on device performance.

Effect of particle concentration

In this study, it is assumed that at any given time, no more than one particle exists within a 2 × 2 pillar unit domain. However, if the particle concentration is too high, two or more particles may simultaneously appear within the same 2 × 2 micropillar unit, leading to mutual interference such as collisions and flow disturbances, which can affect the particle trajectories. To ensure stable performance of the DLD device, the particle concentration should therefore be constrained.

Taking the second parameter in Table 2 as an example, the liquid volume within a 2 × 2 micropillar unit is ~9.156 × 10⁻⁵ μL. Ideally, this volume contains only one particle, corresponding to a solution concentration of 1.09 × 10⁷ particles/mL. In practical, non-ideal conditions, to ensure that each 2 × 2 micropillar unit contains at most one particle, the ideal concentration can be diluted by a factor of 10–100, resulting in a particle concentration >1.09 × 10⁵ particles/mL. The above calculation represents the particle concentration inside the DLD array. For DLD systems with sheath flow, the actual sample concentration can be obtained by multiplying by the ratio of the total flow rate to the sample flow rate, which is greater than one.

In our experiments, a lower concentration of 1 × 10³ particles/mL was used to ensure that only one particle was captured within the microscope field of view, allowing for better trajectory superposition. Nevertheless, the device also maintained consistent performance at higher concentrations, consistent with the theoretical predictions.

Future improvements

Although our method accurately predicts the Dc of DLD devices, it lacks automation. There are two main areas for improvement. First, in the current workflow, determining the Dc of a given DLD model relies on manual identification of trajectory types and manual adjustment of particle sizes. In the future, trajectory types could be automatically recognized based on trajectory features and device structure parameters. The particle size for the next simulation can then be automatically set based on prior trajectory–size data until the size difference between bump and zigzag trajectories falls within a defined threshold.

Second, a fully automated DLD device design tool could be developed. A feasible approach would involve using MATLAB as the control interface, leveraging its application programming interface (API) with COMSOL to operate the simulation model. This tool would automatically adjust structural parameters based on predicted Dc, solve the physical fields, and evaluate the Dc of update device, repeating the process until the predicted Dc meets the target range and all structural constraints are satisfied. This would enable a fully automated DLD device design process.

Significance of this study

Compared to existing numerical methods for predicting Dc, the proposed method achieves high precision by utilizing the physical laws of particle–fluid interaction. This framework has the potential to be extended to predict particle trajectories in other microfluidic devices. For instance, in spiral³⁵ or S-shaped microchannels³⁶, the Dean force essentially arises from secondary vortices induced by centrifugal effects. The vortices exert drag on the particles, causing lateral migration across the channel cross-section, where the balance between drag and lift determines stable particle positions^37,38. The proposed framework employs finite element simulations to obtain the steady-state flow field and comprehensively accounts for drag and lift effects, making it capable of predicting particle motion in microchannels with arbitrary geometries. In the future, when combined with automation, this framework could evolve into a powerful design tool for microfluidic systems, lowering the barriers and cost of device development and promoting the application of particle-sorting technologies in precision and personalized medicine.

Materials and Methods

Trajectory prediction workflow

Obtaining the 3D fluid field

A 3D fluid domain model of the DLD device is first constructed in COMSOL Multiphysics. Material properties are then defined, including the fluid density and viscosity. Boundary conditions are configured next, with the channel walls specified as no-slip, the average fluid velocity at the inlet, and standard atmospheric pressure at the outlet. Finally, the mesh is generated, the physic is solved, and the project file is saved. For meshing, a swept mesh is used instead of a free tetrahedral mesh. This approach offers two main advantages: (1) it speeds up the meshing process; (2) it preserves consistent micropillar contours along the height direction. In contrast, a free tetrahedral mesh results in irregular micropillar contours along the vertical axis, as shown in Fig. S8. Therefore, the micropillar cross-section at any vertical position can be approximated using the surface profile of the model, which reduces the complexity of the trajectory prediction algorithm.

In MATLAB, the COMSOL-provided API is used to load the simulation data locally. The mpheval function is used to extract flow velocity components (u, v, w) and pressure p, while the mphinterp function is used to retrieve the fluid density and viscosity.

Particle position remapping

When the DLD micropillar array is sufficiently large, the fluid field within any 2 × 2 sub-array exhibits the same velocity distribution and pressure gradient. In this method, the pressure acting on a particle originates from the pressure difference across the particle surface, rather than the absolute pressure magnitude. Based on this, the flow field within a 2 × 2 micropillar region can be periodically extended to simulate the flow field across the entire DLD device.

In the proposed algorithm, the smallest rectangle containing a 2 × 2 micropillar region is extracted from the 3D model as the data source, as indicated by the blue box in Fig. 6a, and Fig. 6b showing a 3D view of this region. In Fig. 6c, the orange parallelogram inside the box, whose four vertices are located at the centers of the four micropillars, defines the area in which the particle center is allowed to move. When the particle center exits this orange region, it is remapped back into the parallelogram.

Fig. 6. Schematic diagram illustrating the steps of the algorithm.

a Top view of a local DLD array, where the blue dashed box marks a 2 × 2 micropillar array region selected as the data source. b 3D perspective of the data point distribution extracted from the 2 × 2 micropillar array region. c During trajectory calculation, the particle center is constrained within the orange parallelogram. Upon exiting, the position is remapped back into the designated region. d Division of the particle surface: equally divided into M layers along the Z-axis and into N segments by equal angles in the XY-plane. Point P is the center of a small curved surface. e Relationship between point P and tetrahedron HIJK, used to determine the tetrahedral mesh containing point P. f Difference in fluid velocity at point P on the particle surface is decomposed into radial and tangential components. g A collision-sliding method is used to prevent overlap between the particle domain and the micropillar domain

This remapping is implemented using the parallelogram rule of vectors, as illustrated in Fig. 6c. Let the coordinates of the four points be A, B, C, and D, and the center position of the particle be point P. Define the vectors CB = B − C, CD = D − C, and CP = P − C. According to the parallelogram rule, the position of P can be expressed as:

$$\mathit{CP}=\alpha \cdot CD+\eta \cdot CB$$ 3

This can be solved using matrix operations:

$$\left[\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right]={\left[\begin{array}{c}\mathit{C}\mathit{D}\hfill \\ \mathit{C}\mathit{B}\hfill \end{array}\right]}^{-T}\times {\left[\mathit{C}\mathit{P}\right]}^T$$ 4

The particle coordinates are then remapped based on the values of α and η. If 0 ≤ α ≤ 1, the particle lies within the left and right boundaries of the parallelogram; if α < 0, the particle is to the left of the parallelogram; if α > 1, it is to the right. The remapping formula in the CD direction is:

$$\left\{\begin{array}{c}P=P0\le \alpha \le 1\hfill \\ P=P+(D-C)\alpha <0\hfill \\ P=P-(D-C)\alpha >1\hfill \end{array}\right)$$ 5

Similarly, for the CB direction: If 0 ≤ η ≤ 1, the particle lies within the top and bottom boundaries of the parallelogram; if η < 0, it is below; if η > 1, it is above. The remapping formula is:

$$\{\begin{array}{c}P=P0\le \eta \le 1\hfill \\ P=P+(B-C)\eta <0\hfill \\ P=P-(B-C)\eta >1\hfill \end{array}$$ 6

The selected region (blue box) is larger than the actual fluid field region used for computation. This approach ensures the continuity of fluid field values even when the particle has partially exited the orange parallelogram region.

Surface discretization of the particle

Due to the viscosity of the solution, the physical field inside the channel is non-uniform, which results in varying physical quantities acting on different positions of the particle surface. To more accurately obtain the net physical quantities acting on the particle, the surface of the sphere is discretized using a finite element approach, as illustrated in Fig. 6d. The discretization method is as follows: First, set the current center of the particle as the origin of the coordinate system. Then, sliced into M equally thick layers along the Z-axis. Next, divide each layer into N equal-angle areas in the XY plane. This process results in a total of M × N small curved surface elements. Based on geometric relationships, the center coordinates (x, y, z) of each small surface element are calculated as:

$$\begin{array}{c}\mathit{x}=\sqrt{R^2-z^2}\cdot cos\frac{\left(2\cdot n-1\right)\cdot \pi }{N}\hfill \\ y=\sqrt{R^2-z^2}\cdot sin\frac{\left(2\cdot n-1\right)\cdot \pi }{N}\hfill \\ z=\frac{M+1-2\cdot m}{M}\cdot R\hfill \end{array}$$ 7

where, m is the index of the spherical slice, n is the index of the angular sector, and R is the radius of the sphere. According to the spherical cap area formula³⁹, the area of each small surface element is:

$$S_{scf}=\frac{4\pi R^2}{M\cdot N}$$ 8

where, the subscript scf stands for small curved face, indicating that each surface element is assigned an equal weight. This uniform weighting facilitates subsequent calculations of the particle velocity and pressure.

Obtaining velocity and pressure on the particle surface

The data returned by the mpheval function is based on tetrahedral mesh elements. In this case, physical quantities are only available at the mesh nodes, and values at non-node points must be estimated from existing data. Since the tetrahedral elements are small, the physical quantities within a single tetrahedron can be approximated as varying linearly.

The barycentric coordinate method is first used to identify the tetrahedral element that contains a given point P. As shown in Fig. 6e, let the coordinates of the four vertices of a tetrahedron be H, I, J, and K, and the coordinate of the center point of a small surface element be P, all in the form of (x, y, z). The calculation method is:

$$P={\gamma }_{H}H+{\gamma }_{I}I+{\gamma }_{J}J+{\gamma }_{K}K$$ 9

$$\left[\begin{array}{c}{\gamma }_H\hfill \\ {\gamma }_I\hfill \\ {\gamma }_J\hfill \\ {\gamma }_K\hfill \end{array}\right]={\left[\begin{array}{c}H,1\hfill \\ I,1\hfill \\ J,1\hfill \\ K,1\hfill \end{array}\right]}^{-T}\times {\left[P,1\right]}^T$$ 10

Here, γ_H, γ_I, γ_J, and γ_K are the weights corresponding to the vertices H, I, J, and K, respectively. A necessary and sufficient condition for point P to lie within the tetrahedron is that all four γ values are greater than or equal to zero. Next, the physical quantity at point P is estimated using the values and weights at the four vertices. Let V_H, V_I, V_J, and V_K be the values of the physical quantity (e.g., velocity or pressure) at points H, I, J, and K, respectively. Then the physical quantity at point P is calculated by:

$$V_P={\gamma }_H{V}_H+{\gamma }_I{V}_I+{\gamma }_J{V}_J+{\gamma }_K{V}_K$$ 11

In this way, the velocity components (u_scf, v_scf, w_scf) and the pressure (p_scf) at the center of each small surface element are obtained. This process is repeated for all surface elements.

Viscous force acting on the particle surface

Due to the viscosity of the fluid, the surface of a particle immersed in it is subjected to a viscous force τ:

$$\mathit{\tau }=\mu \nabla \mathit{V}\cdot S$$ 12

where ∇V denotes the velocity gradient, and S is the area. In this method, it is assumed that the physical quantities on the particle surface are equal to those of the flow field at the corresponding positions. However, since the velocity gradient inside the DLD device varies spatially, the viscous forces acting on different points of the particle surface differ accordingly. Given that the particle is a rigid body, these non-uniform viscous forces lead to both rotational and translational movement, see Video S1 and Fig. S9.

As shown in Fig. 6f, let point O be the center of the particle and point P be the center of a small surface element, with the radial vector defined as OP. Assume the velocity V_P on the surface of the sphere and the fluid velocity V_O at the center are obtained by (11). According to geometric relations, the projection of the difference between V_P and V_avg onto OP is:

$${\mathit{V}}_{\mathit{rad}}=\frac{\left({\mathit{V}}_{\mathit{P}}-{\mathit{V}}_{\mathit{O}}\right)\cdot \mathit{OP}}{{\Vert \mathit{OP}\Vert }^2}\cdot \mathit{OP}$$ 13

Following vector decomposition rules, the tangential velocity component is:

$${\mathit{V}}_{\tan }=\left({\mathit{V}}_{\mathit{P}}-{\mathit{V}}_{\mathit{O}}\right)-{\mathit{V}}_{\mathit{rad}}$$ 14

As described in ‘Surface discretization of the particle’ section, the area of each small surface element is the same, meaning the weights are equal. Therefore, the net radial velocity V_rad acting externally on the particle and the net tangential velocity V_tan on the particle surface are computed as averages:

$$V_{rad\_net}=\frac{1}{M\times N}\sum _{m=1}^M\sum _{n=1}^N{V}_{rad\_mn}+V_{\mathit{O}}$$ 15

$$V_{tan\_net}=\frac{1}{M\times N}\sum _{m=1}^M\sum _{n=1}^N{V}_{tan\_mn}$$ 16

where the subscript mn indicates the index of each small surface element. It is worth noting that V_{tan_net} induces particle rotation, leading to a Magnus force⁴⁰. However, the force generated by the Magnus effect is much smaller than the drag force and can thus be neglected in terms of its impact on displacement. On the other hand, V_{rad_net} contributes to particle translation.

Since Stokes’ formula only considers the drag force on an object caused by fluid viscosity under low Reynolds number conditions (Re ≪ 1)⁴¹, the viscous force F_vis acting on the particle can be approximated using the Stokes equation:

$$F_{vis}=F_{stoke}=6\pi \mu R·\times (V_{rad\_net}-V_{ps})$$ 17

where the V_ps is the velocity of the particle

Net pressure acting on the particle surface

The mechanism of buoyancy arises from the pressure difference acting on the particle surface in the gravitational (Z) direction, which results in vertical movement of the particle. Similarly, pressure differences in the X and Y directions can also cause particle motion along those axes.

In ‘Obtaining velocity and pressure on the particle surface’ section, the pressure p_scf at the center of each small surface element was obtained. This pressure value can be regarded as the average pressure over that small surface. According to the pressure formula, the pressure force F_scf acting on the small surface is:

$$F_{scf}=p_{scf}\times S_{scf}$$ 18

the pressure on each small surface element is projected through its center point to the center of the particle, and decomposed into pressure components F_x, F_y, and F_z.

$$\begin{array}{c}{F}_x=-\sum _{m=1}^M\sum _{n=1}^N{F}_{sc{f}_{mn}}\cdot sin\left({\theta }_m\right)\cdot cos\left({\varphi }_n\right)\hfill \\ F_y=-\sum _{m=1}^M\sum _{n=1}^N{F}_{sc{f}_{mn}}\cdot sin\left({\theta }_m\right)\cdot sin\left({\varphi }_n\right)\hfill \\ F_z=-\sum _{m=1}^M\sum _{n=1}^N{F}_{sc{f}_{mn}}\cdot cos\left({\theta }_m\right)\hfill \end{array}$$ 19

where m represents the index of the layer in the Z direction, n represents the index in the angular direction, θ denotes the angle between the line connecting the center of the small surface and the particle center with respect to the Z-axis. ϕ denotes the angle between the projection of the pressure vector in the XOY-plane and the positive X-axis. As a result, the net pressure force F_press acting on the suspended particle can be expressed in vector form as:

$${\mathit{F}}_{\mathit{press}}=\left(F_x,F_y,F_z\right)$$ 20

Lift force calculation

When a particle is located in a flow field with a significant velocity gradient, it inevitably experiences a fluid-induced lift force. This lift force can affect the particle trajectory³⁶. Two types of lift forces are considered: one is the wall-induced lift caused by wall effects, and the other is the shear-induced lift. For the wall-induced lift, we adopt the simplified expression proposed by the team of Ho, B. P⁴²., with the calculation formula given as:

$$\begin{array}{c}{\mathit{F}}_{\mathit{wL}}={\rho }_{sol}\frac{r_p^4}{D^2}{\left|D\left(\mathit{N}\cdot \nabla \right){\mathit{u}}_p\right|}^2{G}_1\left(s\right)\mathit{N}\hfill \\ {\mathit{u}}_p=\left(\mathit{I}-\left(\mathit{N}\otimes \mathit{N}\right)\right){\mathit{V}}_{\mathit{pf}}\hfill \end{array}$$ 21

where ρ_sol is the density of the solution, r_p is the radius of the particle, V_pf is the fluid velocity, I is the identity matrix, D is the distance between the channel walls, N is the normal vector of the reference wall, s is the ratio of the particle’s distance from the reference wall to D, and G₁ is functions of s. The shear-induced lift is calculated using the classical Saffman lift formula:

$$\begin{array}{c}{\mathit{F}}_{s\mathit{L}}=6.46r_p^2{\mathit{L}}_v\sqrt{\mu {\rho }_{\mathit{sol}}\frac{\left|{\mathit{V}}_{\mathit{pf}}-{\mathit{V}}_{\mathit{ps}}\right|}{\left|{\mathit{L}}_v\right|}}\hfill \\ {\mathit{L}}_v=\left({\mathit{V}}_{\mathit{pf}}-{\mathit{V}}_{\mathit{ps}}\right)\times \left[\nabla \times \left({\mathit{V}}_{\mathit{pf}}-{\mathit{V}}_{\mathit{ps}}\right)\right]\hfill \end{array}$$ 22

Both the wall-induced lift (F_wL) and the shear-induced lift (F_sL) involve the gradient operator (∇) in their respective formulations. To evaluate them, the first-order shape functions of the tetrahedral mesh are first determined, followed by the computation of their partial derivatives. The total lift force acting on the particle is then given by:

$$F_L=F_{wL}+F_{\mathit{sL}}$$ 23

Prediction of particle position and velocity

The core idea of this method for generating particle trajectories is to predict the particle state at the next time step based on its current state and the forces acting on its surface. After the viscous force, net pressure, and lift force acting on the particle are obtained, respectively, according to Newton’s second law:

$$F_{vis}+F_{press}+F_L=m_p\times a$$ 24

where a is the acceleration of the particle and m_p is the mass of the particle. The change in particle velocity during Δt is:

$$\mathrm{\Delta }{V}_{ps}=a\times Dt$$ 25

Since Δt is very small, the particle displacement during this time is minimal. Therefore, it can be assumed that the particle surroundings remain unchanged over Δt, meaning that F_press, F_L, and V_{rad_net} are constant during this interval. Combining the above expressions, the particle velocity after Δt is:

$$\begin{array}{c}{\mathit{V}}_{ps}\left(t\right)=\left(\frac{{\mathit{F}}_{fix}}{{\zeta }_{stoke}}+{\mathit{V}}_{rad\_net}\right)\left(1-e^{-\frac{{\zeta }_{stoke}}{m_p}\times t}\right)+{\mathit{V}}_{ps}\left(0\right)\times e^{-\frac{{\zeta }_{stoke}}{m_p}\times t}\hfill \\ {\zeta }_{stoke}=6\pi \mu R\hfill \\ {\mathit{F}}_{fix}={\mathit{F}}_{press}+{\mathit{F}}_L\hfill \end{array}$$ 26

where V_ps(0) is the particle velocity at the beginning of the time step, and V_ps(Δt) is the particle velocity after Δt. Integrating with respect to t, the displacement of the particle over this time interval can be expressed as:

$$\mathit{s}\left(t\right)=\left(\frac{{\mathit{F}}_{fix}}{{\zeta }_{stoke}}+{\mathit{V}}_{rad\_net}\right)t+\left[\left(\frac{{\mathit{F}}_{fix}}{{\zeta }_{stoke}}+{\mathit{V}}_{rad\_net}-{\mathit{V}}_{ps}\left(0\right)\right)\frac{m}{{\zeta }_{stoke}}\right]\left(e^{-\frac{{\zeta }_{stoke}}{m}\times t}-1\right)$$ 27

Assuming the particle’s center coordinate at the start of Δt is S_cur, the coordinate at the next time step, S_next, is:

$$S_{next}=S_{cur}+s\left(\mathrm{\Delta }t\right)$$ 28

Handling particle-wall overlap

Although (28) provides the predicted particle position at the next time step, it does not account for wall constraints, which may result in the particle overlapping with the wall. In this method, when the particle domain overlaps with the wall domain, a collision-sliding method is applied to estimate the particle position at the end of Δt, as illustrated in Fig. 6g. The process is described as follows:

(1) Obtain the particle displacement vector S_disp:

$${\mathit{S}}_{\mathit{disp}}=S_{next}-S_{cur}$$ 29

(2) Calculate the maximum distance D_max along the displacement direction within the overlap region. The endpoint Q₁ of D_max on the wall contour is the initial collision point. Then compute the unit normal vector of the wall at the collision point, n_wall. Furthermore, the other endpoint Q₂ of D_max on the particle contour is also obtained.

(3) Assume that point G is the projection of point Q₂ onto the collision wall surface. The displacement of the particle from point Q₂ to point G can be expressed as:

$$\mathrm{\Delta }\mathit{s}=G-Q_2=[(Q_1-Q_2)\cdot {\mathit{n}}_{\mathit{wall}}]{\mathit{n}}_{\mathit{wall}}$$ 30

(4) Adjust the predicted particle center coordinate:

$$S_{next}=S_{next}+\mathrm{\Delta }s$$ 31

The effect of the collision-sliding method is that the particle first collides with the wall at point Q₁, and subsequently slides along the wall surface to point G.

If there is no overlap between the particle domain and the wall domain, then Δs = 0, and no correction to the predicted position is needed.

Plotting the particle trajectory in the DLD device

This method extracts only a 2×2 micropillar region from the 3D model to predict the particle trajectory. To better illustrate the relationship between the particle trajectory and the micropillar array, a custom function is used to extract the geometric parameters of the micropillar array from the 3D model. Then, by selecting a single micropillar and arraying it according to the extracted parameters, it is possible to render a micropillar array of arbitrary scale. Finally, the particle position data at each time point is plotted onto the reconstructed array to visualize the particle trajectory.

Iterative algorithm flow

The individual functions are connected in logical computational order to form the algorithm flowchart, as shown in Fig. S10. Given the file path of the COMSOL model, the program is executed after the initial particle states and the stopping conditions for the iterative calculations are specified. Upon completion, the program directly outputs the particle trajectories.

Steps to determine the critical diameter Dc

(1) First, set an appropriate particle diameter Dp, such as half the gap width (e.g., Dp = Gy/2), and run the method to obtain the particle trajectory. At this point, the trajectory will exhibit either a zigzag or bump pattern.

(2) If the particle exhibits a bump trajectory, record the current particle diameter as Dp₊, then reduce the particle diameter parameter and rerun the method. Similarly, if the particle shows a zigzag trajectory, record the diameter as Dp₋, then increase the particle diameter and rerun the method.

(3) Repeat step 2 until the difference between Dp₊ and Dp₋ meets the desired tolerance. The critical diameter Dc of the device lies between Dp₊ and Dp₋. The smaller the difference between Dp₊ and Dp₋, the higher the precision of the determined Dc.

Sample preparation

The base solution is a glycerol-water mixture composed of glycerol, Tween 20, and phosphate buffer solution (PBS) solution in an approximate mass ratio of 21.5: 3.5: 75.0, resulting in a density of ~1.05 g/cm³. Tween 20 is added to prevent particle aggregation. During testing, this base solution is also used as the sheath fluid.

The 12 μm and 13 μm particles are individually diluted using the base solution to a final particle concentration of ~1×10³ particles/mL. Before transferring the particle samples into syringes, the centrifuge tubes containing the sample solutions are placed in an ultrasonic cleaner and treated for 5 min to ensure complete dispersion of the particles.

Trajectory overlap evaluation

After extracting the recorded particle center positions from the experimental images, the simulation trajectories are mapped onto the coordinate system of the test images under the prerequisite that the micropillar array contours in both datasets are aligned. Using the particle center (x_T,y_T) in the test image as a reference point, we extract the corresponding coordinate (x_T,y_P) from the simulated trajectory. Since the simulation data points usually do not contain the exact value of x_T, the two adjacent points enclosing x_T on the predicted trajectory are used to estimate y_P through linear interpolation. Finally, the Euclidean distance (D_Euclidean) can be simplified as:

$$D_{Euclidean}=\left|y_P-y_T\right|$$ 32

Device fabrication and testing platform

Refer to the supplementary information.

Author contributions

Conceptualization: J.C., X.H., Methodology: J.C. Software: J.C., Validation: J.C., X.H., W.X., Formal analysis: J.C., Investigation: J.C., X.H., Resources: X.H., W.X., L.S., Data curation: J.C., Writing–Original Draft: J.C., Writing–Review and Editing: J.C., X.H., W.X., L.S., Visualization: J.C., X.H., Supervision: X.H., L.S., Project administration: X.H., L.S., Funding acquisition: X.H.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41378-025-01139-3.