Analysis and Simulation of Blood Cells Separation in a Polymeric Serpentine Microchannel under Dielectrophoresis Effect

Abstract

The current work presents a novel microfluidic approach, allowing a full separation of blood cells. The approach relies on using a polydimethylsiloxane serpentine microchannel equipped with a series of electrodes, providing two separation zones. The proposed design exploits the unique configuration of the channel along with the inherent difference in dielectric properties of the three kinds of blood cells to achieve a size-based sorting. The platelets (PLTs) are subjected to a larger dielectrophoretic force than red blood cells (RBCs) and white blood cells (WBCs), forcing them to be separated in the first zone. This leaves RBCs and WBCs to be separated in the second zone. The model developed in this work has been used intensively to examine the feasibility of the proposed approach. The model results showed a full separation of blood content can be achieved over a range of phase flow rates and AC frequencies.

1. Introduction

Using fluidic devices at a microscale level in various biological and chemical applications has grown rapidly in the last couple of decades [1,2]. The concept of using a complete lab on a chip (commonly called a lab-on-a-chip device) was developed for the first time in the 1990s [3]. This miniaturized lab is based on lithographing different configurations of microfluidic channels on a single chip of only a few square centimeters in area, allowing full control of various cell screening functions (culture, separation, sorting, and analysis). Based on the mechanism of cell separation, microfluidic separation techniques can be categorized into three main types: a passive, active, and a hybrid system, which is based on both [4,5,6,7,8]. The cell separation in passive techniques merely relies on the design of microchannels that manipulates fluid dynamics inside them and, hence, the drag and lift hydrodynamic forces acting on the cells, driving them for separation. Examples of passive separation techniques are hydrodynamic filtration, deterministic lateral displacement method, micro-hydrocyclone, micro-vortex, Dean flow fractionation, and pinched flow fractionation [9,10,11,12,13,14,15,16]. The active separation techniques, on the other hand, are radically different. They depend, mainly, on applying an external force (magnetic, optical, electric, or acoustic), and manipulating a certain size of particle from a fluid sample to achieve a separation. Perhaps, one of the most well-known methods of an active separation technique is that based on using an electric force field (dielectrophoresis). Compared to other methods, the dielectrophoretic method (DEP) offers many advantages such as ease of operation, a high degree of separation, purity, and low cost [17].

The application of the dielectrophoretic phenomenon emerged in the twentieth century. In 1923, Hatschek and Thorne [18] introduced a similar concept of dielectrophoresis. A year later, Hatfield [19] filed a patent for using dielectric separation in chemical processing. In 1951, Pohl [20] observed that the polarization forces of solid particles produced by a non-uniform electric field can separate them from a polymer solution. In 1966, the idea of using DEP in biological applications was introduced for the first time by Pohl and Hawk [21]. The authors used DEP to separate dead and live yeast cells. Following this work, many papers were published particularly from 1990 to the present day about the manipulation of biological cells in microfluidic devices using the dielectrophoretic effect. Rosenthal and Voldman [22], in 2005, introduced a novel micro-dielectrophoretic device that can trap single particles against flow rates. The size-selection trap was explored experimentally using different sizes of polystyrene particles over different flow rates. Kim et al. [23] designed a dielectrophoretic separation cell equipped with two electrodes at different angles. In the work, different cell types are labeled with specific dielectrophoretic tags. This alters the permittivity of the cells, allowing them to be separated with high purity. In 2011, Piacentini et al. [24] used a series of liquid electrodes to generate a non-uniform electric field. The generated field separates blood cells (red blood cells and platelets) according to their size in a single-stage microchannel. The authors reported that approximately 99% purity had been achieved. In 2013, Li et al. [25] introduced a novel s-shaped microchannel to separate 10 and 15 μm polystyrene particles suspended in a water solution. The authors used a non-uniform electric field generated by micro-DC power. The results showed that the proposed method is efficient and a high degree of separation can be achieved. Lewpiriyawong and Yang [26], in 2014, used five insulating blocks of polydimethylsiloxane (PDMS) to generate a non-uniform electric field to separate fluorescent and nonfluorescent particles in a microchannel, and a high degree of purity was also reported (~99%). Two years later, Tao et al. [27] presented a chip based on the DEP method to separate different sizes of polystyrene microparticles (3, 10, and 25 μm). The author inserted pair of needle electrodes in the proposed microchannel to work as obstacles, resulting in purity of more than 90%. Kale et al. [28], in 2018, investigated the effect of applied voltage on the separation efficiency of 5 μm latex beads in a DEP device. Heidari et al. [29], in 2019, examined the location of non-uniform electrodes and electric intensity and other design factors on the emulsion process of dielectric liquids and the instability of the interface between them. In 2019, Aghaamoo et al. [30] designed a system based on deterministic lateral displacement (DLD) and dielectrophoresis approaches to separate circulating tumor cells from patient blood samples. The authors claimed that the proposed system is a promising approach in the future in terms of both throughput and efficiency. Similar to that work, Wang et al. [31] demonstrated, in 2021, a two-stage separation device of circulating tumor cells based on the DLD and DEP methods. The experimental results showed that tumor cells can be separated with high separation efficiency (91%) and purity (80.7%) without affecting cell activity. In 2022, Zaman et al. [32] designed a novel microfluidic device that is based on an array of microelectrodes. The device generates a dielectrophoretic force such that can drag a single cell across a straight two-dimensional corridor. The transport of single polystyrene beads of diameters 20 and 10 μm was examined experimentally. The results showed that 20 μm polystyrene beads can transport linearly for a distance of 840 μm while those with a diameter of 10 μm traveled 1100 μm.

Different than these studies, the work presented here focuses on developing a dielectrophoretic-serpentine microchannel, allowing a complete separation of whole blood cells (red, white, and platelets cells). Two effects are considered in the proposed design: flow fractionation (passive method) and dielectrophoretic effect (active method). The first effect is achieved using a serpentine microchannel along with supplying the buffer and blood samples at a relatively low Reynolds number to avoid any mixing. The second effect is applied using a series of liquid electrodes, generating a non-uniform electric field. A model was developed to understand the separation mechanism. The details of the design, theory, and the developed model are given in the subsequent section.

2. Channel Design and Operation Principles

The design of the proposed microchannel is shown in Figure 1a. The channel consists of two main separation zones along with two inlets (for blood sample and the buffer) and three different outlets distributed along the channel. Outlet No.1 (10 μm) is located at the end of the first separation zone (No.1) while the others (28 μm) are at the end of separation zone No.2. Further, the channel is equipped with a series of liquid electrodes (40 × 40 μm). The electrodes were placed in chambers such that they point vertically in the direction of the main flow (e.g., Piacentini et al. [24]). The distribution of electrodes shown in Figure 1 (i.e., ± V) ensures generating a non-uniform electrical field over the total channel length without disrupting the main flow. In practice, this channel can be made simply on a block of polydimethylsiloxane (PDMS) using a soft lithography process.

In terms of operation, as depicted in Figure 1b, the blood sample is injected at Inlet No.1 (22 μm) while the buffer is supplied from Inlet No.2 (18 μm). Since the flow is laminar in the main channel, any mixing process between the inlet streams is not expected to occur. Dielectrophoretic signals can be applied on liquid electrodes. The generated dielectrophoretic force along with the effect of hydrodynamic forces starts sorting the blood cells (RBCs, WBCs, and PLTs) according to their sizes. The dielectrophoretic force repels the bigger cells (WBCs and RBCs) while attracting the smaller ones (PLTs). Accordingly, it should be expected that the platelets would be close to the electrodes and hence exit from Outlet No.1, leaving only a buffer enriched by WBCs and RBCs to flow through separation zone No.2 (Figure 1b). In this zone, the electrodes repel WBCs to migrate to the upper end of the channel and attract RBCs to be closer to the lower end (bottom of the channel). This situation forces WBCs and RBCs to be separated and, accordingly, RBCs and WBCs are collected from Outlet No.2 and 3, respectively, and a successful separation can be achieved.

Table 1. Channel Dimensions where channel depth is fixed along the channel to be 20 μm.

Parameter	Inlets Width (μm)	Outlets Width (μm)	Main Channel (μm)	Electrode (μm)	Separation Zones (μm)
Dimensions	No.1 = 22 No.2 = 18	No.1 = 10 No.2 = 28 No.3 = 28	Length ≈ 2840 Width = 40	Length = 40 Width = 40	Length = 2 × 1120 Distance 1 = 60 Distance 2 = 260

lists the details of the channel dimensions.

3. Theory

For a particle–fluid system, there are five principal forces acting on the suspended particles: (1) Magnus force, (2) Saffman force, (3) viscous drag force, (4) wall-lift force, and (5) shear gradient lift force. Magnus and Saffman’s forces have a negligible effect compared to the other forces [33]. Moreover, since the channel is not straight, it is expected that there is a drag effect in both a longitudinal and transverse direction on the particles. The transversal drag effect is mainly due to the effect of circulating motions, which is a direct result of channel curvature. The remaining forces are only the lift forces due to the shear and wall effect, and these are responsible for the particles’ lateral migration. The former (shear-gradient force) pushes the suspended particles toward the channel walls, while the latter (wall-lift force) pushes the particles toward the channel center line. So, clearly, the two forces act against each other and, hence, it should be expected that the position of suspended particles inside the channel is determined by the net balance of these two forces (F_L). Segré and Silberberg [34], in 1961, investigated the effect of F_L and they observed that this force causes the particles to reside above the channel center (i.e., in the middle distance between the center and channel wall). For Poiseuille flow, the F_L value can be predicted using the equation below, assuming that the particles are spherical in shape and have a small size [35]:

$$F_L=f_L{\rho }_f{U}^2{a}^4/H^2$$

(1)

where $f_L$ is the coefficient of lift effect and it is correlated with the Reynolds number of the particle and its lateral position, H is the hydraulic mean diameter of the channel, ${\rho }_f$ is fluid density, $U$ is the velocity of the fluid, and $a$ is the diameter of the particle.

In addition to the effect of hydrodynamic forces discussed above, the effect of the dielectrophoretic force must be considered. According to the dipole moment theory, the time average effect of this force on spherical particles can be expressed in terms of the permittivity of the dielectric medium (${\epsilon }_m$) and particle radius ($a$) as [30]:

$$〈F_{DEP}(t)〉=2\pi {\epsilon }_m{a}^3\mathrm{Re}[K_{CM}(\omega )]\nabla {\left|E_{rms}\right|}^2$$

(2)

The term $\mathrm{Re}[K_{CM}(\omega )]$ represents the real part of the Clausius–Mossotti factor. This term can be defined as:

$$K_{CM}(\omega )=\frac{\overline{{\epsilon }_{cell}}-\overline{{\epsilon }_m}}{\overline{{\epsilon }_{cell}}+2\overline{{\epsilon }_m}}$$

(3)

The terms in Equation (3) are complex functions of permittivity and conductivity of cell and medium and the angular frequency ($\omega$). So, they can be written as:

For Cell:	$$\overline{{\epsilon }_{cell}}={\epsilon }_{cell}-j\frac{{\sigma }_{cell}}{\omega }$$	(4)
For Medium:	$$\overline{{\epsilon }_m}={\epsilon }_m-j\frac{{\sigma }_m}{\omega }$$

where j is the imaginary part of the complex number. Looking at Equation (3), one can conclude that there are three possible scenarios. First, when values of $K_{CM}(\omega )$ > 0. In this case, the cells are more polarizable than the carrier medium and will be affected by positive $F_{DEP}$ and, hence, the cells migrate toward the high electric field areas. Second, when values of $K_{CM}(\omega )$ < 0. Here, the cells will be affected by negative $F_{DEP}$ and this force will repel them from the high electric field areas. The third scenario occurs when a specific value of frequency is applied, resulting in zeroing $F_{DEP}$ so that there is no dielectrophoretic force applied to the cells. This value of frequency is called crossover frequency and must be avoided if a successful dielectrophoretic separation is required.

3.1. Modelling of Blood Cells Separation

3.1.1. Flow Equations

The blood sample is treated in the computation as a Newtonian and incompressible fluid. Further, since the flow inside the channel is at low Reynolds number values, it is taken to be laminar. Moreover, the flow is not affected by other processes such as the transfer of species or heat, and hence, a complete developed flow can be achieved inside the channel. Considering these conditions and taking the fluid density and viscosity to be ${\rho }_{f\hspace{0.33em}}$ and ${\mu }_{f\hspace{0.33em}}$, the continuity and two-dimensional flow equations can be expressed as follows [36]:

Continuity:

$$\frac{\partial u_f}{\partial x}+\frac{\partial v_f}{\partial y}=0$$

(5)

Flow equation in x-direction:

$${\rho }_{f\hspace{0.33em}}\left(u_f\frac{\partial u_f}{\partial x}+v_f\frac{\partial u_f}{\partial y}\right)=-\frac{\partial p}{\partial x}+{\mu }_f\left(\frac{{\partial }^2{u}_f}{\partial x^2}+\frac{{\partial }^2{u}_f}{\partial y^2}\right)$$

(6)

Flow equation in y-direction:

$${\rho }_f\left(u_f\frac{\partial v_f}{\partial x}+v_f\frac{\partial v_f}{\partial y}\right)=-\frac{\partial p}{\partial y}+{\mu }_f\left(\frac{{\partial }^2{v}_f}{\partial x^2}+\frac{{\partial }^2{v}_f}{\partial y^2}\right)+{\rho }_f{g}_y$$

(7)

It is important to emphasize here that the convection terms in Equations (6) and (7) are expected to have a small effect since the flow examined in the work is at low Reynolds. However, it is considered for the sake of completeness.

3.1.2. Electric Potential

At a steady state, the current conservation equation is written as:

$$\nabla .J=0$$

(8)

where $\mathrm{J}$ is the current density and based on Ohm’s law can be expressed as:

$$\begin{gathered} J=\sigma E \\ E=-\nabla \mathrm{V} \end{gathered}$$

(9)

where E is the electric field, V is the voltage, and $\sigma$ is the conductivity.

3.1.3. Force Balance Equation

The balance of forces acting on particles can move within a fluid can be expressed as [37,38]:

$$\frac{d(m_p{V}_P)}{dt}={\overrightarrow{F}}_{Drag}+{\overrightarrow{F}}_{lift}+{\overrightarrow{F}}_{DEP}$$

(10)

Using the definition of each force term in Equation (10), one can obtain:

$$\frac{d(m_p{V}_P)}{dt}=3\pi {\mu }_{f}a(V_f-V_p)+{\rho }_f\frac{a^2}{4D^2}\beta (\beta G_1(L)+\gamma G_2(L))n+2\pi {\epsilon }_f{a}^3\mathrm{Re}[K_{CM}(\omega )]\nabla {\left|E_{rms}\right|}^2$$

(11)

where

$n$ is a unit vector normal to the walls of the channel.

$m_p$ is the mass of the particle.

$V_f$ is a vector representing the velocity of the fluid ($V_f=u_{f}i+v_{f}j+w_{f}k$).

$V_p$ is the particle velocity vector ($V_p=u_{p}i+v_{p}j+w_{p}k$).

$a$ is the particle diameter.

$D$ is the gap separating the walls of the channel.

$\beta$ and $\gamma$ are correlations. Ho and Leal [37] define $\beta$ and $\gamma$ correlation in terms of $D$ and $V_p$ as $\left|D(n.\nabla )V_p\right|$ and $\left|\frac{D^2}{2}{(n.\nabla )}^2{V}_p\right|$, respectively.

L is the distance from a particle to one of the walls normalized by the gap separating the walls of the channel ($D$).

$G_1(L)$ and $G_2(L)$ are lengthy correlations. Ho and Leal [37] defined these correlations as a function of the L ratio.

Up to this point, the governing equations of flow, electric field, and particle motion are presented. Clearly, Equation (10) or (11), is a function of fluid velocities and the location of particles inside the channel, so one can infer that the numerical solution of Equations (5)–(10) gives the details of the flow and electric field distribution and determines the particle position at any position in the channel.

3.1.4. Numerical Solution

The solution of Equations (5)–(10) is performed numerically using Comsol Multiphysics software (version 5.5). Comsol uses the finite element technique, which is based on the discretization of the domain into tiny areas to find the solution in each node formed between them. The numerical solution is divided into two steps: (1) solving (Equations (5)–(9)), and this gives the flow and electric field at a steady-state condition; (2) using the velocity field computed in step one to solve Equation (10), but at unsteady-state conditions to determine the particle locations along the channel with time. To achieve steps one and two in Comsol Multiphysics software, two modules were used: the general PDE and electric current module to solve Equations (5)–(9), and the particle tracing for fluid flow module to solve Equation (10). The boundary conditions used throughout the computation are summarized in

Table 2. Boundary Conditions.

Flow Boundary Conditions
Condition Type	Location
no-slip conditions	At the channel walls
Pressure = 0	At the channel outlets
no possibility of backflow	-
Electric conditions
Condition Type	Location
Electric Insulation	At the channel walls except for the electrode locations
Electric Potential	At electrode locations (±5 V alternatively)
Particle tracing conditions
bounce condition	At the channel walls
time step = 0.005 s	-

. Furthermore, there is a list of input properties and parameters that are needed to solve the governing equations along with the proposed boundary conditions. First, the physical properties of the fluid and those of the particles moving within the fluid are needed and these are given in

Table 3. Fluid–particle Physical Properties.

Fluid Properties
Density (kg/m3)	Viscosity (Pa.s)			Fluid Medium Conductivity (S/m)			Fluid Relative Permittivity	Ref.
1000	0.001			0.055			80	[35]
Particle Properties
Type	Diameter (µm)	Density (kg/m3)	Particle Relative Permittivity	Particle Electric Conductivity (S/m)	Shell Thickness (nm)	Shell Relative Permittivity	Shell Electrical Conductivity (S/m)	Ref.
WBCs	10	1050	150.9	0.6	7	6	1.4 × 10−7	[39]
RBCs	6	1050	59	0.31	9	4.44	1.0 × 10−6	[39]
PLTs	1.8	1050	50	0.25	8	6	1.0 × 10−6	[39]

. It is important to consider, here, that the fluid used in the proposed channel is a diluted blood sample so, as shown in Table 3, the properties of water at standard conditions are used in the numerical solution. Second, the channel geometry and fluid velocity are also needed. In terms of the geometry, the dimensions shown in Figure 1 and Table 1 are used. The fluid velocity at Inlet No.1 (Figure 1) is changed in the current study to cover a flow rate range of blood samples (Q₁) from 0.1 to 10 µm³/h with a fixed buffer flow rate (Q₂) of 5 µm³/h.

3.1.5. Mesh Dependence Study

The mesh dependence was explored here to ensure that a reliable solution was obtained. This was done by reducing the elements’ size (mixed triangles and rectangles) and observing the changes in the maximum velocity (u_{f max}) at the middle of the channel in separation zone 1 at Q₁ of 0.1, 5, and 10 µm³/h. The other operating conditions are V = 5 V, f = 100 kHz, and Q₂ = 5 µm³/h. The statistics of the mesh are shown in

Table 4. The statistics of the grid used in the mesh study.

No. of Elements (N)	Max. Element Size (µm)	Min. Element Size (µm)
8459	204	0.942
16,223	104	0.471
37,837	58.1	0.196
65,652	15.7	0.031
102,560	5.2	0.0132

. The values of u_{f max} against the number of elements (N) are plotted in Figure 2. It is clear from the figure that the values of u_{f max} do not change considerably when N > 60,000 elements for the three selected flow rates. Therefore, using N of about 65,000 and the corresponding parameters shown in Table 4 gives a reliable numerical solution.

3.1.6. Model Verification

The verification of the proposed model is important to ensure that reliable computational results were obtained. This was done here by predicting the experimental results reported by Piacentini et al. [24] in 2011. In that work, as mentioned, the authors used a series of liquid electrodes to separate RBCs and PLTs according to their size in a single-stage microchannel. The velocity ratio of a blood sample to the buffer is 134/853, each in µm/s. The applied voltage is 10 V at 100 kHz. The model results showed that the general trend of cell trajectories was captured very well, achieving a complete separation that is greater than that measured experimentally by 1%. The difference is negligible, in general, and could be attributed to either a computational or experimental error. While the channel design in the work of Piacentini et al. [24] is quite different from that used here, both works are based on the same principles and, hence, using the model predictions to understand the separation mechanism in the proposed channel is not expected to give rise to any differences. The results of the validation study are shown in

Table 5. The predicted values by the proposed model against the corresponding values reported in Piacentini et al. [24].

Parameter	Values Reported in Ref. [24]	Model Values
Velocity Ratio	134/853 (µm/s)	133.98/853.1
Separation Efficiency	98.8%	100%

below.

4. Results

The results of the Clausius–Mossotti factor and the computational results of blood cell separation in the proposed channel are introduced here. The Clausius–Mossotti factor was predicted using a single-shell model [40]. The computational results, on the other hand, were obtained using the model developed here, following the solution approach given in Section 3.1.4. As mentioned, the boundary conditions and the required properties used throughout the current study are listed in Table 3 and Table 4. In the study, the buffer flow rate (Q₂) and the electrode voltage are fixed at 5 µm³/h and ±5 V, respectively, while the effect of blood sample flow rate (Q₁) and AC frequency (f) is changed. The studied range of Q₁ is from 0.1 to 10 µm³/h and that for f is from 0 to 500 kHz. The number of cells in the blood sample is taken to be 61 cells for each type, making altogether 183 cells moving in the channel.

4.1. Clausius–Mossotti Factor

It is helpful, first, to look at the real part of the Clausius–Mossotti factor ($K_{CM}$). Figure 3 shows the plot of this factor against a wide range of conceivable values of frequency for the three types of blood cells. The calculations are made using MyDEP software [41], which is based on a single-shell model. The model approximates the cell to a sphere and has a cytoplasm enclosed by only a single thin shell.

Looking at Figure 3, it is clear that the cells have different Re[K_CM] at f values greater than 100 kHz. This is mainly due to the inherent difference in dielectric properties of those cells shown in Table 3, which becomes dominant at high-frequency ranges. However, the dielectrophoretic behavior becomes highly independent of these properties at low-frequency values. Moreover, the results of Figure 3 indicate that the cells will be affected by negative $F_{DEP}$ over the studied range of frequency. So, it should be expected that the cells are less polarizable than the carrier medium and migrate toward the low electric field areas inside the channel. Over the studied range of frequency, $F_{DEP}$ equals zero only for WBCs at f = 244 kHz. Accordingly, there is no electrophoretic force applied to WBCs at this particular value of frequency, and this was avoided in the numerical study.

4.2. Flow, Electric Field, and Cell Motion Characteristics

To explore the flow, electric field, and cell motion characteristics, the computation at Q₁ = 1 µm³/h and f = 100 kHz is taken as a reference case. Figure 4 shows the results (velocity and voltage distribution and cell trajectories along the channel) under these certain conditions. The colors in Figure 4a,b are shaded from blue to red corresponding to the largest values of streamwise velocity and voltage to the smallest values of them. In the cell trajectory plot (Figure 4c), the black, grey, and white colors are used to label the platelets, RBCs, and WBCs, respectively.

In terms of flow (Figure 4a), it is clear that the velocity of the sample is the lowest (blue) at the walls and the highest (red) in the middle, forming a typical parabola profile. Similarly, for the electric field shown in Figure 4b, the results show that a non-uniform electric field is generated where the maximum voltage values (red) and the minimum blue areas) are at the electrodes distributed over the total channel length. These observations were examined closely by looking at the velocity and voltage profile in the middle of the channel (Section A-A in Figure 4a), considering the two separation zones. The profiles are shown in Figure 5a,b.

As can be seen in Figure 5a, there are two obvious observations. First, the velocity profile in separation zone 1 is larger than that in the second zone. Since the channel has a constant section in the two zones, this could be explained in terms of mass conservation. In separation zone 1, the flow rate is constant until reaching Outlet 1 where the platelets are collected. Here, the flow rate reduces and only some of the sample counties flow into the second zone. So, a smaller amount of sample flows in Section 2 at a lower velocity, causing the observed difference in the velocity profile of the two separation zones. Second, the velocity profile of the two separation zones is close to a standard parabola profile of Poiseuille flow, except for the distortion observed at the end of the two profiles. This asymmetry is, mainly, due to the presence of electrodes. However, the distortion is small and does not have a pronounced effect on the general profile. Further, such profiles could be disturbed by the secondary motion that is generated due to the curvature of the channel. This effect is not revealed in the results since the studied range of the flow gives low Reynolds numbers (<1) and, hence, low Dean numbers (i.e., the ratio of the viscous force to the centrifugal force). It should be considered here that operating at a higher Reynolds number could stimulate the drag effect of secondary motion in the perpendicular direction of the main flow and, hence, accelerate the particles to reach their equilibrium positions faster in the flow section, but that is out of the scope of the current work.

Figure 5b shows the voltage profile in the two separation zones. As expected, there is a clear distribution of voltages in the two zones, and it is identical, with a maximum voltage of 5 V supplied by the electrodes. As mentioned, this non-uniform electric field is essential to achieve a successful dielectrophoretic separation.

In terms of cell trajectories, solving Equation (10) allows the tracing of RBCs, WBCs, and PLTs along the channel, and the results are shown in Figure 4c. As can be seen, the blood cells (RBCs, WBCs, and PLTs) start to take an equilibrium position along the channel depending on their sizes. This is mainly due to the effect of the generated dielectrophoretic force along with the effect of hydrodynamic forces. This was tested by switching off the effect of the dielectrophoretic force, and the results showed that the hydrodynamic forces are not enough to achieve a successful separation. The presence of a dielectrophoretic force separates the blood cells by attracting the small particles (PLTs) and repelling the bigger ones (WBCs and RBCs). Accordingly, the platelets are close to the electrodes and hence exist from Outlet No.1, leaving only a buffer enriched by WBCs and RBCs to flow through separation zone No.2 (as shown in Figure 4c). In this zone, the electrodes repel WBCs to move away toward the upper end of the channel while attracting RBCs to be closer to the lower end (bottom of the channel). This situation forces WBCs and RBCs to be separated and, accordingly, RBCs and WBCs are collected from Outlet No.2 and 3, respectively, and a successful separation can be achieved (as can be seen in Figure 4c).

4.3. Separation Performance

The separation performance of the proposed design was examined in the current work. This was made by looking at the separation degree over a range of blood volume flow rates (0.1 to 10 µm³/h) and AC frequencies (0–500 kHz). The separation degree is defined here as the number of a certain cell divided by the total number of that cell in each outlet. It is useful to consider that Outlet No.1 (Figure 1) is dedicated to collecting the platelets (61 cells) and Outlet No.2 and 3 for RBCs (61 cells) and WBCs (61 cells), respectively. So, as an example, collecting 6 platelets, 30 RBCs, and 61 WBCs in Outlet No.1, 2, and 3, respectively, under certain conditions means that the separation degree is 98% for platelets, 49% for RBCs, and 100% for WBCs.

The computed separation degree of each cell type is shown in Figure 6 over the studied range of volume flow rates and f = 100 kHz, where the bars with black, red, and grey colors are used for the platelets, RBCs, and WBCs, respectively. The corresponding values of the volume flow rate ratio (Q₂/Q₁) are also shown for a reference on the top x-axis of the figures.

Figure 6 shows clearly that it is possible to achieve a complete separation of the three main blood cells in a serpentine channel under the effect of a dielectrophoretic force. For instance, a full separation is attained at Q₁ from 0.5 to 5 µm³/h (corresponding to a flow rate ratio from 10 to 1). At Q₁ = 0.1, 0.2, and 0.3 µm³/h, however, the degree of separation of RBCs is not fully achieved, decreasing the purity of WBCs. Moreover, at Q₁ = 10, the best separation degree of platelets is less than 10%, decreasing the purity of the RBCs, since they are collected at Outlet No.2.

The effect of AC frequency on the separation degree of blood cells is shown in Figure 7 at constant blood and buffer flow rates (Q₁ = 0.1 and Q₂ = 5 µm³/h). As can be seen, the best separation is achieved at f value between 100 and 300 kHz. Exceeding the lower or the upper limit of this range affects directly the separation of RBCs, decreasing the purity of other cells, as can be seen in Figure 7.

5. Discussion and Conclusions

An efficient approach to separate blood cells has been presented. The approach is based on using a serpentine channel, containing a series of electrodes to generate a dielectrophoretic force. The approach was examined by studying the effect of both volume flow rates (0.1–10 µm³/hr) and different current frequencies (0–500 kHz) on the separation of blood cells (platelets, RBCs, and WBCs). The two-dimensional model developed here gives a prediction of flow, electric field, and cell separation under these operating conditions. It is argued, based on the proposed design, that a full separation of blood content could be achieved in a single channel with two separation zones. This has been demonstrated here. The results showed that the platelets, RBCs, and WBCs can be sorted completely at Q₁ from 0.5 to 5 µm³/h (corresponding to a flow rate ratio from 10 to 1), f = 100 kHz, and V = 5V. It was shown that the presence of a dielectrophoretic force separates the blood cells. This force attracts the small particles (platelets), repelling the bigger ones (WBCs and RBCs), and causing the platelets to be separated in zone 1. This leaves WBCs and RBCs to flow through zone 2 and to be separated there under the effect of the same force. The results also indicate that the best separation is achieved at f value between 100 and 300 kHz at constant blood and buffer flow rates (Q₁ = 0.1 and Q₂ = 5 µm³/h).

Although it appears that the suggested separation technique is efficient, one might argue that the throughput is small, affecting the downstream post-analysis. This can be avoided by using multiple channels connected in parallel or scaling up the channel size taking full advantage of the inherent secondary motion to improve the separation. In such cases, active and passive methods are utilized and, hence, better performance and wider operation conditions are expected. Further, it is believed that this system will be a useful tool to sort complex biological systems such as a blood sample containing circulating tumor cells.

Finally, the work lays down the theory of blood cell separation in a dielectrophoretic serpentine. However, the theory work presented here can be seen as a first step toward understanding this technique and how it works, and definitely, further work is necessary. It is believed that carrying out a series of experiments with targeted computations could advance the current work.