# Efficient paradigm to enhance particle separation in deterministic lateral displacement arrays

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## Abstract

Microfluidics based particle sorting and separation methods are gaining momentum to be applied for various applications. Deterministic lateral displacement (DLD) methods are prominent for high resolution in separation and there has been extensive studies to develop more efficient devices based on the DLD. However, it is still challenging to fully eliminate negative effects of the boundaries that degrade particle separation efficiency by perturbing the fluid flow in the channel. In this article, we present two equations to optimize channels’ geometry near the boundaries. Implementing the equations, the fluid behavior is improved around the pillars and thereby, separation efficiency is increased. The Boundary Correction Paradigm (BCP) enhances the microchannel’s functionality as much as 2–3 times and can be highly beneficial in microchannels. Also, an equation is proposed in order to recalibrate the BCP in microchannels with desired pillar diameters. The calibration equation assures high accuracy and resolution of the DLD devices corrected with the BCP.

## Keywords

Microfluidics Deterministic lateral displacement Particle separation efficiency Boundary correction paradigm## 1 Introduction

One of the mostly employed capabilities of microfluidics towards accomplishment of Lab-on-a-chip technology is separation of specific bio particles in a whole sample. This application paves the way for diagnosing different diseases and developing efficient therapeutics. Separation methods and particle analysis are two significant steps need to be swift, low-cost, efficient, and continuous [1, 2, 3, 4, 5, 6]. Microfluidics is widely utilized for this purpose and enables separation of target particles with low concentration in biofluids like blood [7]. The separation of bioparticles on microfluidic platforms mostly relies on particle physical properties like mass, size, charge, etc. and mainly fall into two categories of active and passive [8, 9, 10]. Active sorting is implemented on devices that use external fields to manipulate and act on the particles and the flow. Some of the active separation methods include dielectrophoresis, magnetophoresis, and acoustophoresis. Dielectrophoresis [11, 12, 13] sorts particles based on their size and capacitance by imposing variant electric field in the channel. Acoustophoresis transmits bulk [14, 15] or surface [16] ultrasound waves so that the particles in the channel would be separated based on differences in size, density, and compressibility [17]. Magnetophoresis offers particle sorting based on the magnetic susceptibility of the particles which could be intrinsic [18] or superparamagnetic [19, 20]. On the contrary, passive methods mainly rely on channel geometry and hydrodynamic forces for separation which their simplicity and throughput is comparable to active methods [21]. Passive methods include, but are not limited to, hydrodynamic or hydrophoretic filtration, pinched-flow fractionation (PFF), and deteministic lateral displacement (DLD) [8, 9, 22, 23, 24]. The DLD, first proposed by Huang et al. [25], sorts particles based on their diameters. This method is used for variety of purposes such as separating blood components [26, 27, 28, 29], measuring platelet activity [30], separating circulating tumor cells (CTCs) [31, 32, 33], spores [34], bacteria [35], parasites [36] and even nanoparticles [37, 38]. Compared to other methods, the DLD is cost-efficient, easy to use, accurate, and leaves morphology and properties of the particles intact. In addition, it has highest resolution among separation methods [25].

In the DLD method, the channel’s geometry is of high importance. The design and arrangement of pillar arrays play the main role in separation. The angle of pillar arrays relative to the flow direction is called migration angle (\(\theta\)). Also, the gap (G) between each two pillars are equal. The gap (G) and migration angle (\(\theta\)) define a critical size (\(\hbox {D}_{\mathrm{c}}\)) for separation. Therefore, particles will follow either zigzag or bumping streamlines [25, 39]. Particles with smaller diameters than the \(\hbox {D}_{\mathrm{c}}\) will go through zigzag mode with approximately no lateral displacement. On the contrary, particles with larger diameters than the \(\hbox {D}_{\mathrm{c}}\) will have bumping trajectory and migrate upward [25, 39]. In this way, the particles with diameters less and over the critical diameter (\(\hbox {D}_{\mathrm{c}}\)) will be separated.

One of the challenges in the DLD method, however, is particle clogging near the boundary interfaces. According to the DLD theory [25], it is assumed that there are no boundaries to disturb the Poiseuille flow and particle separation. Due to limitations in the fabrication and inevitable presence of boundaries, the Poiseuille flow pattern is disturbed in experimental setups. Therefore, anticipated streamlines and critical diameters (\(\hbox {D}_{\mathrm{c}_{\mathrm{th}}}\)), according to the DLD theory, won’t be realized in the experiment. Each pillar in the array would have a different \(\hbox {D}_{\mathrm{c}}\) and the pillars close to the boundaries would malfunction, thereby, zigzag and bump streamlines will be merged and particles of all sizes flow near walls.

In this paper a more efficient wall correction method based on the original theory is proposed to further bridge the gap between theory and practice. Similar to the Inglis et al.’s method, the Boundary Correction Paradigm (BCP) also includes two equations for the microchannel boundary correction. Derivation of the BCP’s equations are first explained. Afterward, on a completely similar microchannel, the BCP and Inglis et al.’s method are used to correct the boundaries and then the fluid streamlines and resulted critical diameters by both methods are compared. The BCP showed to enhance homogeneity of streamlines and critical diameters in the microchannel. Also, the critical diameters converged to the \(\hbox {D}_{\mathrm{c}_{\mathrm{th}}}\).

## 2 Materials and methods

### 2.1 Theory of DLD

### 2.2 Improving effect of using obround pillars before the DLD array

### 2.3 Derivation of boundary correction equations

Although it is not practical to realize unlimited boundaries for the channel, the wall correction methods attempt to accomplish this boundary condition as much as possible. To obtain an efficient wall correction method, it is required to comprehend flow behavior, especially near channel walls and boundaries. Therefore, two different equations must be developed for the lower and upper walls. These equations are developed by curve fitting of flow flux profile data passing through modified gap quantities, calculated in the last section. Among various curve fitting possibilities, the least error was obtained by power equation with two variables. Having these assumptions and the following comprehensions, the equations are developed.

#### 2.3.1 Lower wall equation

#### 2.3.2 Upper wall equation

*n*thgap, equaling G, that has 2Q passing flow. Therefore, similar to lower wall equations, we have,

## 3 Results and discussion

In this section, derivation of the BCP and it’s two equations and variables are elaborated. Also, a microchannel’s boundaries are corrected by BCP and Inglis et al.’s method and resulting flow patterns and critical diameters (\(\hbox {D}_{\mathrm{c}}\)) are compared.

### 3.1 Determination of BCP equations’ variables

As the equations are developed, derivation of variables to find the optimized wall correction method are essential. To do so, the obtained data from simulation of modified gaps and their relative flux are used. The curve fitting of the results with power equation pattern (\({{y}}=ax^{b}\)) with two variables resulted in the least error. However, this question was raised that how changing basic boundary conditions like \(\varDelta {P}\) affects the output. Therefore, three different \(\varDelta {P}\) values (0.1, 1, 10 Pa) were utilized to repeat the simulation and compare the outcome. Aftermath, it was observed that only quantity of “a” is influenced in power fitting by different amounts of pressure which does not affect the equation. Figure 3b illustrates how flux is increased by higher pressure differences (\(\varDelta {P}\)).

### 3.2 Calibration formula for the BCP method

### 3.3 Numerical analysis

## 4 Conclusion

Particle separation is a significant step toward personalized medicine and lab-on-a-chip technology. Among particle separation methods, the DLD has the highest resolution and throughput compared to other methods. However, the ideal DLD boundary conditions (no boundaries to disturb the flow) cannot be implemented in the fabrication and experiment. There has been attempts to further extend the original theory and maximizing the resolution and efficiency of DLD devices by boundary correction methods. Using wall correction methods becomes crucial when the DLD arrays are utilized in nanochannels and for sensitive applications. In this paper, the Boundary Correction Paradigm (BCP) is presented to eliminate disharmonies caused by boundary interfaces which can severely affect the process of particle or cell separation when the DLD method is employed. The presented method introduces two equations to correct the boundaries and further extends and optimizes the original theory. Also, a calibration equation is proposed to assure high accuracy and resolution of the BCP in microchannels with desired pillar diameters. The standard deviation of the flow regime from the theory was enhanced 31–56% in the considered microchannel compared to previous methods.

## Notes

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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