Effect of particle-particle interaction on dielectrophoretic single particle trap in a sudden contraction flow Open Access

Dielectrophoresis (DEP) has been an effective way to manipulate tiny objects by varying the voltage, frequency or phase of the electrical signal applied to the electrodes in micro-devices.^1–8 Micro-trap for single particle has potential applications in biology and analytical chemistry.^9–13 Among the constructed micro-traps, some were based on the DEP platform.^14–21 According to the principle proposed by Hughes,⁶ these DEP traps can be classified into two subcategories. Methods in the first subcategory try to reduce the size of the electrode to the same or smaller order of the trapped particles.^14–16 Applied DEP force acts mainly on one particle because its strength decays rapidly with the increase of the distance to electrode. Methods in another subcategory isolate a single particle in the first step, followed by the operation of preventing other particles from entering into this trap.^17–21 The former ones are suitable for high concentration solution and the latter ones for dilute solution.

Designing a general and robust micro-trap is always desired. Different from previous works, we put forward a novel strategy about single particle trapping to retain only one particle after two particles are held in the trap already. With this method, the way to weaken particle-particle interaction force becomes very important. In this way, a pillar array trap has been developed in our previous work by introducing the discontinuous or time-varying DEP force.^22,23 Compared with continuous DEP, the temporally disappearing of DEP force during “off” period seems to be responsible for the improvement of trapping behavior. Meanwhile, the highly non-uniform flow field near the stagnation points of pillars was produced to improve the selectivity and stability of the trap. However, it is difficult to carry out a thorough mechanism analysis because these trapped particles not only contact the top surface but also lean against the pillar array. It is inevitable to introduce extra interaction forces from these solid boundaries. The effect of interaction force in non-uniform electric field on DEP manipulation has been noticed by Aubry and the co-workers,^24–26 and attention has been paid to quantifying this effect in the experiment.²⁷ Some new interaction types of solid particle with air/liquid interface were observed as well.²⁸ Therefore, it is necessary to conduct a specific study to clarify the effect of particle-particle interaction in this complicated issue.

In this paper, we put forward a microdevice combining a sudden contraction flow field and pulsed nDEP to study particle-particle interaction as shown in Fig. 1. The sudden contraction flow field refers to the rapid change of cross-section area near the outlet. Considering that an open channel without lid is adopted, the particle will not contact the solid boundary and, as a result, only the interaction force among them will be focused. This platform will help to clarify the effect of particle-particle interaction on single particle trap.

Based on the point dipole model,^29,30 the DEP force on a spherical particle (with radius r, relative permittivity ε_p and conductivity σ_p) suspended in a medium (with relative permittivity ε_f and conductivity σ_f) is given by

where E_rms is the root mean square of electric field strength, ε₀ is the permittivity in vacuum, and ω is the angular frequency of applied electric field. The direction of DEP force depends on the sign of Clausius-Mossotti (CM) factor β, and it takes on values from −0.5 to 1. The DEP force is classified as positive DEP force (pDEP) when β > 0 and negative DEP force (nDEP) when β < 0. When a particle experiences nDEP, it is repelled from the region of high electric field. In our situation, the particles are much less polarizable than the medium, and thus the CM factor takes on a value of −0.47. In this work, a commercial finite element software (COMSOL, Inc.) was adopted to analyze the steady state electric field. Due to the nature of long strip electrodes, 3-D distribution of electric potential V, is simplified to the 2-D distribution on x-z symmetrical plane. DEP force can be expressed as

Here, the subscripts, x and z, represent the differentiation in the corresponding direction and the number of the subscripts represents the order of differentiation.

On the other hand, the electric interaction force between two or more particles always appears along with the applied electric field. With the similar point dipole approximation, the interaction force F_D, on the ith particle due to the jth particle in the non-uniform electric field has been given by^24,30

where s is the distance between the centres of the two particles, and s_ij is the unit vector in the direction from the centre of jth particle to ith particle. For the simulation of the particle-particle interaction force, we only considered the situation that two particles are located on the z-plane. The alignment of the two particles can be parallel (∥) or vertical (⊥) to x-direction. In the parallel (∥) configuration, the trailing particle is chosen as the reference particle and the resultant s_ij is [−1, 0, 0]. In the vertical (⊥) configuration, the bottom particle is chosen as the reference particle and the resultant s_ij is [0, −1, 0]. Introducing 2-D simplification and the value of s_ij into Eq. (3), the expression of interaction force is obtained,

Here, the subscripts, i and j, represent ith and jth particles, and the meaning of x or y is the same as that in Eq. (2). According to Eq. (4), the interaction force for the two parallel alignment particles has no y-component. According to Eq. (5), the interaction force for the two vertical alignment particles has no x- and z-components. In the following parts, only the distribution of x-component of F_D∥ is studied because it is the only component that could interact with F_DEP.

We applied the above model to the present design and solved the distributions on the z = 15 μm plane. The property parameters used are listed in Table 1. In the simulation, the unit peak-to-peak voltage is applied to one electrode and another electrode is grounded. DEP force or interaction force with different voltages can be calculated according to the relationships, |F_DEP| ∼ V² or |F_D| ∼ V². From the results shown in Fig. 2, several points need to be noted. For the distribution of F_DEP (Fig. 2(a)), the whole region can be divided into two sub-regions by the center of two electrodes (x = 70 μm), where F_DEP is zero. On the left side of this point, F_DEP is less than zero and the minimum of F_DEP appears at the downstream edge of the left electrode (x = 60 μm). Signs of F_DEP determine the direction in which the F_DEP pushes the particle, forward for positive or backward for negative. From the distribution of F_D (Fig. 2(b)), it can be seen that F_D∥ is larger than zero in most of the region, which indicates that the leading particle applies an attractive force to the trailing one. In Fig. 2(c), the ratio of F_D∥/F_DEP gives the relative importance of F_D∥ and F_DEP. Typically, at the location (x = 60 μm), F_D∥ is about 1.4 times larger than F_DEP, and, at another location (x = 70 μm), the ratio is infinite due to a zero DEP force. Basically, the magnitude of F_D∥ is in the same order as F_DEP and the effect of F_D∥ cannot be neglected.

In the micro-fluidic device, fluid applies a drag force on the particle. Because of the very small Reynold (Re) number, the drag force acting on a sphere can be described by Stokes equation,

where μ is the viscosity of fluid, U_f is the local flow velocity, and U_p is the velocity of the particle. As the density of the particles (1.05 g/cm³) is close to that of water, the buoyancy force is neglected in the present analysis. Given the viscous nature of our experimental system, the particle attains its terminal velocity immediately and the acceleration can be neglected. The total force is expressed as the sum of fluid drag and DEP force,

This expression can be used to determine where the particle is trapped in the micro-well. Provided that the strength of DEP is strong enough, for example, with 10V_pp voltage, the resultant distribution is shown in Fig. 3(a). Two points with the total force of zero show the two locations where the sphere could come to rest. According to stable condition, the location satisfied the condition, dF·dx < 0, gives the stable equilibrium point (ep). The left-most location is the ep, where a particle from upstream is stopped and cannot move forward. The right-most location is an unstable equilibrium point, which is denoted as the separation point (sp). When the particle passes this point, it will be pushed away to the right side.

The relative strength of DEP force and Stokes force plays an important role in determining the location of the trapped particle. As shown in Fig. 3(b), with the decrease of applied voltages, the curve of total force is lifted accordingly, and ep and sp tend to merge together. It is noted that, with about 6V_pp voltage, they turn into the same point. At this moment, the minimum nDEP force is just matched with Stokes force and there is only one intersection point on the total force curve. The location of intersection point is defined as releasing point (rp), where a particle will release if the strength of nDEP force becomes smaller. This location is very close to the point with the minimum nDEP force. Finally, as the applied voltage is further reduced, the particle cannot be held by nDEP force any more. An example with 4V_pp voltage is shown in Fig. 3(b). There is no intersection point of zero total force in this curve.

Different from single particle’s situation, to determine a unique status of two trapped particles requires at least two parameters, namely, the location of the centre of the two particles and the alignment direction. Force moment dominates the latter. Generally, the total force moment of the two particles under a stable status must be zero. However, it does not mean that one status with zero total force moment is the terminal position of the two particles. It is because the status could be unstable. Similar to the situation of force stability, the arrangement satisfied the condition, dM·dθ < 0, providing the stable position. In the x-z symmetrical plane of micro-well, both the arrangements of the two particles parallel and vertical to x-axis can satisfy the condition of zero force moment. Through introducing a tiny artificial perturbation, dθ (>0), into the system, we can find out which arrangement is stable. Consider the situation in which the two particles are presented parallel to x-axis (left case in Fig. 4(a)) firstly, the force moment generated by Stokes force is given by

Here, u and v are the x- and y-components of fluid velocity respectively and u_x is the gradient of u with respect to x-direction. The force moment from nDEP force is given by:

Here, DEP_x is the gradient of nDEP with respect to x-direction. Applying Eqs. (8) and (9) and dθ → 0, the stable condition is rewritten as:

According to the above equation, the simulation result is depicted in Fig. 4. From this figure, it can be seen that the intersection point of two curves is just located at the downstream edge of left electrode, where the first order differential of nDEP force is zero. It needs to be noted that the force moment from nDEP force is far larger than that from Stokes force and dominates the direction of alignment of the two particles. Therefore, the sign of the gradient of nDEP force will decide the form of the two particles. In other words, the two particles align in the direction vertical to flow if they are presented on the left side (DEP_x(x) < 0) and align in the direction parallel to the flow if they are presented on the right side of this point (DEP_x(x) > 0). Similar result can also be obtained from another arrangement where the two particles are placed parallel to the flow originally (right case in Fig. 4(a)),

From the analysis above, we can obtain an outlined description on how to trap the particles in micro-well by the continuous nDEP force. When a particle comes from upstream, it experiences nDEP force and drag force. The relative strength of the two forces will determine whether the particle can be held or not. Practically, we keep the flow rate constant and then decrease the voltage from a pre-set higher value. Thus, the particle comes to rest at ep firstly because the total force is zero at this point (Fig. 3(a)). Because the strength of DEP is higher than the required value to hold the single particle, more particles might be trapped around this location. Considering that two particles are trapped, they should be in the direction vertical to x-axis because DEP_x(x) < 0 at ep (Fig. 4). The interaction between them is a repellent force according to Eq. (5). To achieve a single particle trap, it is straightforward to reduce the applied voltage. With the decrease of voltage, the particles move forward gradually and arrive at rp finally (Fig. 3(b)). As stated above, rp also represents the transition location of the alignment of the two particles (Fig. 4). Hence, the two particles change into the direction parallel to x-axis and the interaction force between them becomes the attractive force (Eq. (4) and Fig. 2(b)). This ratio of F_D/F_DEP is about 1.4 (Fig. 2(c)), meaning that the interaction force is larger than the strength to trap single particle. Once the leading particle is flushed out of micro-well by fluid, it is inevitable to pull the trailing one out of micro-well by the attractive interaction force. Therefore, it is impossible to obtain single particle trap by simply decreasing the strength of continuous DEP force in micro-well.

Next, pulsed nDEP method is put forward to replace the continuous nDEP force. Through switching DEP force between “on” and “off” during a given period, the particle-particle interaction force can be eliminated for a while. The mechanism of how the trap works can be explained using Fig. 5. During the “off” period, the alignment of the two particles should be in the direction parallel to flow because no nDEP force moment is induced. Since the velocity field shaped by micro-well is highly non-uniform, the released particles move at different speeds based on their locations. After a short period of time, one of the particles would move over the sp, while the other one may not. When the nDEP force is turned on again, the trailing particle will be pushed back, while the leading one will continue to flow downstream. The upper bound and lower bound of the applied pulsed frequency are decided by two typical positions of particles as shown in the two panels of Fig. 5. The corresponding gap between two particles near sp is dependent upon the ratio of velocities. The micro-well used in the present work has a sudden contraction geometry, which is capable of inducing the big gap. From the simulation result listed in Table 2, the gap range is 5–6.6 μm, which almost doubles its original distance. Based on this principle, the two trapped particles can be separated by choosing the suitable pulsing frequency.

To verify the proposed method, we carried out a proof-of-concept experiment. ITO coated glass is used because of its high transparency and good electrical conductivity. Standard photolithography is applied to pattern the interdigital electrodes. The specimen is then etched in a solution of ferric chloride (35 g), hydrochloric acid (250 mL) and deionized water (250 mL). The etching time in our experiment was about 2 min, then the specimen was immersed in a 5% sodium carbonate solution to remove the surplus etching solution. The electrical connection was realized through affixing a copper wire to a reserved ITO pad with silver conductive glue. SU-8 micro-well was positioned accurately on the ITO substrate. The micro-device is thus obtained.

A sinusoidal voltage generated by signal generator (Agilent 33220A) is applied to the electrodes. Built-in amplitude modulation function with ∼1 Hz square wave is adopted to generate pulsed nDEP. Because the strength of nDEP will decrease dramatically with the increase of the distance to electrode, the evaporation-driven method is applied to an open channel to form a steady low-velocity flow. Because of the hydrophobic nature of SU-8 channel wall, a droplet of aqueous sample is pulled from outlet to inlet along the channel and assures that the whole micro-channel is immersed thoroughly. After a few minutes, a steady flow is constructed, whose possible driven force are capillary force, evaporation³¹ or the gravity of droplet. Different heights ranging from 40 μm to 5 μm were tested and the optimum about 15 μm was found. The flow formed via this method is stable and can last for more than half an hour, long enough to finish the testing. The shortcoming of this method is the unadjustable flow rate. Only the electric parameter is adjustable in the entire process. The typical velocity at the outlet of micro-well was estimated about 150 ± 50 μm/s. PS fluorescent microspheres with nominal diameter of about 5 μm were purchased from DUKE Science Corp. The microspheres were mixed with deionized water, and the conductivity of the solution was monitored using a Thermo Electron Russel portable conductivity meter. Finally, the chip was mounted on the stage of an upright microscope (DMLM, Leica) which was supplied with an updating 100 W Hg-lamp. The motion of the fluorescent microspheres was recorded by an image acquiring system. The experiment was conducted at room temperature, 18–20 °C, and the property parameters were regarded as constant.

A 10 MHz sine wave is used to generate continuous nDEP force. PS beads are introduced into micro-well from the inlet. After a number of beads enter the micro-well, nDEP gate is activated by turning on the power of electrode. If the applied voltage is high enough, all beads inside the micro-well will be blocked. Sometimes, if the quantity of trapped beads is too big, some of them could be flushed out. For these trapped particles, it is obvious that they are concentrated at a fixed position, namely ep. This is because there is no DEP force to balance the component of hydrodynamic force parallel to the strip electrodes. Hereafter, through decreasing the applied voltage, we attempted to flush the extra trapped PS beads and have only one bead trapped inside the trap. The operation was done very carefully with 0.01Vpp step by adjusting the knob on the panel. Once the applied voltage is less than a critical value (about 6V_pp–7V_pp), it was found that all trapped beads (from two trapped beads up to tens of trapped beads) were released simultaneously. It is impossible to retain just one particle even with such a careful adjustment. Snapshots extracted from the experimental video are listed in Fig. 6(a) to show the pattern of beads at the moment just before they were released.

Pulsed nDEP force was generated from the built-in amplitude modulation function. The applied voltage is fixed at 10V_pp. This signal includes two frequencies: one is 10 MHz sine wave for nDEP force, and the other one is ∼1 Hz square wave for modulation signal. The low-frequency modulation signal is used to switch nDEP force between “on” and “off” conditions. From the experimental observation, one particle presented in the micro-well experiences two possible motions, a stable oscillation inside the micro-well or an unstable releasing from the micro-well. If both the strength of DEP and the pulsing frequency are high enough, the particle moves forward during “on” period and moves backward during “off” period, which is the state of stable oscillation. The frequency of oscillation is the same as that of the applied modulation signal, and the amplitude of oscillation relies on the applied voltage. If either the strength of DEP or the pulsing frequency is not high enough, this particle will be flushed out of the micro-well.

More attention was paid to the situation when the two particles were presented in the micro-well. Different from what happens for single particle, there is still the third situation for the two trapped beads. That is, one particle experiences the stable oscillation and the second one is flushed away. From the experimental observation, the particles move at different speeds based on their locations. After a short period of time, the leading particle will spread over a larger region than the trailing particle does. When the nDEP force is turned on again, the trailing one will be pushed back, while the leading one will be flushed out. Generally, a higher amplitude of the AC voltage is needed (compared to the continuous DEP) to trap the equivalent number of particles due to the effect of time-averaged strength. The frequency of pulsed signal to separate the two particles is about 1.9 ± 0.25 Hz. With more particles coming from upstream, the isolating process will be repeated many times, and only the last particle is left in this trap, eventually. The pulsing frequency is decreased further and it is found that the last particle will be released at about 1.2 Hz. The difference between trapping and releasing single particle is ∼0.7 Hz, which is a considerable quantity compared with the resolution of the signal generator.

In this section, the direction of the two trapped particles is considered. It is found that, under the continuous nDEP, the alignment of the two particles changed with the applied voltages when the background flow field is fixed. With the higher voltage, the two particles line up in the direction parallel to the strip electrodes. With the decrease of the applied voltage, the alignment of the two particles becomes unstable and finally tends to align in the direction vertical to the strip electrode (parallel to flow direction). The applied voltages at this moment are approaching the critical value to release them (6V_pp–7V_pp). Similarly, under the pulsed nDEP mode, the two trapped PS beads experience not only the periodic oscillation transversely but also the rotation-like motion in terms of the applied pulsing frequency. The alignments of the two particles are switched between the direction parallel to the strip electrode during “on” period and the direction vertical to the strip electrode during “off” period. To our best knowledge, this rotation-like motion that is induced intentionally by DEP force has not been noticed in previous works yet.

The proof-of-concept to isolate single particle from two or more same-sized particles is proved. These particles are trapped in non-uniform flow field and do not contact any solid boundary. By comparing two kinds of signal sources, this experiment is deemed as evidence that the interaction between the two particles cannot be ignored. Although the experiment is in agreement with simulation results, it does not indicate that the present model is good enough to predict the behaviour of particle under pulsed DEP force. The uncertainty of measurement of height is still rather high. Several possible factors, including air/liquid interface and the force balance in the vertical direction, were not involved in the simulation and they could have an influence on particle trapping. Meanwhile, the fluidic interaction between the two particles is also neglected because it is believed that it is not so important for the flow with the very low Re number.³² A more accurate model is needed to verify the assumptions.

In this paper, a non-uniform flow field formed by micro-well and the nDEP force generated by a pair of strip electrodes were used to realize a single particle trap. The simplified 2-D simulation was carried out to study the distribution of several possible factors, including flow field, DEP force, particle-particle interaction force and force moment. From the distribution curves, three characteristic locations, namely, the equilibrium point, separation point and releasing point, are denoted. The results also show that for two releasing particles, the attractive interaction force is about 1.4 times of the required nDEP. Then, pulsed nDEP is put forward to replace the continuous one to trap single particle. Since the interaction force F_D is inversely proportional to s⁴, the increase of the distance reduces the strength of F_D significantly. It is about 1/2⁴ in the present design. In comparison, pulsed nDEP is a better excitation power source than the continuous one. In experimental aspect, the ITO micro-electrode and the micro-well made of SU-8 photoresistence were fabricated. This proof-of-concept design was verified by 5 μm polystyrene particle. A single particle has been isolated successfully from a group of particles using pulsed nDEP with about 1.9 ± 0.25 Hz pulsing frequency.

The authors are grateful for the financial support from the National Natural Science Foundation of China (11602187) and the Shaanxi Provincial Natural Science Foundation (2018JM1029).