|Home | About | Journals | Submit | Contact Us | Français|
We present here a study on overlooked aspects of alternating current (AC) electrokinetics-AC electrophoretic (ACEP) phenomena. The dynamics of a particle with both polarizability and net charges in a non-uniform AC electric trapping field is investigated. It is found that either electrophoretic (EP) or dielectrophoretic (DEP) effects can dominate the trapping dynamics, depending on experimental conditions. A dimensionless parameter γ is developed to predict the relative strength of EP and DEP effect in a quadrupole AC field. An ACEP trap is feasible for charged particles in ‘salt-free’ or low salt concentration solutions. In contrast to DEP traps, an ACEP trap favors the down scaling of the particle size.
Dielectrophoretic (DEP) traps have been successfully implemented for many applications [1, 2, 3]. Such traps operate through the interaction of induced polarization charges with non-uniform electric fields. According to the classic DEP theory , the sign associated with the real part of Clausius-Mossotti (CM) factor (K(ω), figure 1(a)) dictates the behavior of the particle. For Re[K(ω)]>0, particles will be directed toward a local electric field maxima (positive DEP, pDEP), while for Re[K(ω)]<0, particle is attracted to a local electric field minima (negative DEP, nDEP). Negative DEP traps offer distinct advantages for many applications since the particles are trapped away from electrodes, mostly with a quadrupole geometry [5, 6, 7, 8, 9]. The frequency that produces a change from pDEP to nDEP is referred to as the crossover frequency fco and is given by (1/2π)[(σp − σm)(σp + 2σm)/(εm − εp)(εp + 2εm)]1/2 (in Hz, ), where σ and ε are the conductivity and permittivity, respectively, with p and m denoting the particle and medium. This prediction is in good agreement with experimental data in salt solutions for nanoscale to microscale latex beads . However, we observe in our experiment that the charged polystyrene beads suspended in ‘salt-free’ deionized (DI) water can be trapped in the center of a quadrupole electric field for frequencies significantly below the predicted crossover frequency (figure 1 and experimental details in Appendix A). This ‘anomalous’ center trapping behavior in the pDEP region (shaded area in figure 1) motivates us to investigate other contributions in the trapping dynamics. The first possible mechanism for this ‘anomalous’ trapping behavior is AC electroosmosis (ACEO), which would play a dominant role in the case of low frequencies and low conductivities . Indeed, we observe ACEO in our experiments, but only at frequencies around 1 kHz, far below the frequency regime of consideration here (a lower limit around 20 kHz). As a result, ACEO is not responsible for the anomalous center trapping behavior at frequencies higher than 20 kHz.
It is well known that most polarizable particles and molecules suspended in aqueous solutions will develop surface charges by either dissociation of surface chemical groups or adsorption of ions from the solution . Therefore, when placed in a liquid with a spatially non-uniform electric field, colloidal particles experience not only a dielectrophoretic (DEP) force but also an electrophoretic (EP) force [12, 13, 14]. It is a widely-held notion that the EP effect in aqueous solution will ‘vanish’ upon high frequency AC fields due to the linearity of EP with electric field [13, 14, 15]. As a result, EP contributions are not taken into account in most of the high frequency electrokinetic experiments [6, 7, 8, 9]. However, the assumption that EP effects will vanish is only true for a spatially homogeneous AC field. A charged particle exposed to an oscillating inhomogeneous electric field can experience a cycle-averaged force. This so-called ponderomotive force  plays a significant role in a variety of physical systems such as Paul traps  and laser-based particle acceleration .
Here we show both experimentally and theoretically the non-vanishing ponderomotive EP effect in high frequency electric field and its application for trapping charged particles in aqueous solutions. In contrast to DEP traps [5, 6, 7, 8, 9], an AC electrophoretic (ACEP) would favor the down scaling of the particle size.
We start our analysis with the one-dimensional EP motion of a homogeneous spherical particle with mass m, charge Q and radius a in a high frequency AC electric field. The damping force due to the viscosity of the liquid is of the form −ξ, where the Stokes drag coefficient ξ can be approximated by ξ = 6πηa, and η is the dynamic viscosity. Here we assume that the hydrodynamic memory effect  can be neglected for the dragging force (i.e. the friction force is only dependent on the current velocity). Without loss of generality, we assume the particle moves in an electric potential consisting of two parts: a static part U (x) and a harmonically oscillating part V (x) cosωt, where ω is the angular frequency. The magnitude of V (x) is not assumed small in comparison with U (x). Note that both potentials have a spatial dependence. This potential gives a static force and an oscillating force f(x, t) = QE0(x) cosωt, where .
Under the conditions considered in this paper, the dynamics of the particle can be described by a secular motion S(t) on a time scale typically longer than one oscillating cycle τ = 2π/ω, which is decoupled from the rapidly oscillating micromotion R(S, t) (Appendix B). Therefore, the particle motion can be written in form, X(t) = S(t)+R(S, t), with the constraint R(S, t) = 0 (bracket denotes time averaging). If the amplitude of the rapidly oscillating motion is much smaller than the characteristic length of the non-uniform electric field , it is reasonable to assume |S| |R| (where the dependence on t and S are henceforth dropped for brevity). Therefore R can be considered a small perturbation to S, and thus the equation of motion, mẌ = −ξẊ + F(X) + f(X, t) can be expanded to the first order in R,
The rapidly oscillating terms on each side of equation (1) must be approximately equal, m ≈ −ξ + f(S,t). The oscillating term term is neglected by assuming , which is reasonable for highly damped environments such as water. By integration, we obtain the rapid micromotion component as,
where f0(x) = QE0(x). The rapid micromotion R(S, t) is thus an oscillation at the same frequency as f(x, t). The oscillating amplitude depends on the position of the secular motion through f0(S), the driving frequency, the damping coefficient, and the particle mass.
The secular motion S(t) can therefore be found by averaging equation (1) over one period of the rapid micromotion, and by replacing f0(x) with QE0(x),
It should be noted that the above analysis is valid for any viscosity of the medium when (Appendix B). This inequality is valid for most experiments [6, 7, 8, 9] performed in an aqueous environment around MHz range.
Let us consider three related cases based on equation (3).
Under this circumstance, the averaged secular motion is described by m = −ξ. By integration, we obtain S = A1e−ξt/m + A2, where A1 and A2 are constants depending on initial conditions. This is simply a transient response and will not contribute to a long time scale drift motion. Since the amplitude of the superimposed rapid micromotion R(S, t) is usually negligible at high frequency (~1/ω2, equation (2)), high frequency ACEP effects vanish in this case. This is consistent with O'Brien's results with a parallel plate geometry . Note that O'Brien assumed a uniform electric field. It is thus invalid to apply O'Brien's result directly for the cases of non-uniform electric fields [21, 22].
The third term on the right hand side of equation (3) provides a ponderomotive EP force for the secular motion due to the non-uniformity of the electric field. The particle is directed towards a point with weaker electric field. Moreover, due to the squared dependence on charge Q in equation (3), the repelling of the particle from regions of high electric field intensity holds true for both positive and negative charges. We need to emphasize that even though the time average over one period of both f(S, t) and R(S, t) is zero at a fixed point, averaging over the micromotion in the non-uniform electric field is the essential mechanism that causes the movement of charged particles.
In this case, it is convenient to express F(x) and E0(x) in equation (3) by potential energy derivatives and the motion equation is obtained as, , where Φsp(x) defines an AC pseudopotential, which is given by,
As a consequence, the particles will move towards a point where and oscillate there, which is described by equation (2). In other words, under this situation the charged particle will oscillate at the bottom of the effective pseudopotential. Since f0(S) is a complex function of time, the motion toward and around the bottom of the effective pseudopotential depends on the detailed form of E0(x), and other parameters of the system (frequency ω, viscosity ξ, etc.).
To this end, we have investigated in detail the EP behavior of a charged particle in a high frequency AC electric field. A translational motion is unarguably possible for charged particles in an non-uniform AC field. As a result, simply ignoring the EP effect in AC trapping field may not be correct.
Taking the EP effect into consideration for the quadrupole trapping field shown in the inset figure 1(b), the total instantaneous force on the particle will be m − ξ + Fep + Fdep = 0, where Fep and Fdep are the instantaneous EP force and instantaneous DEP force, respectively. Although analytical solutions for a general case can not be obtained, information about the relative contribution of EP and DEP to the trapping dynamics can be obtained from a ponderomotive force point of view, which could give a more intuitive insight. The ponderomotive force for DEP effect is well studied [4, 11] and has the form in the point dipole approximation. We note that this point dipole approximation gives an upper bound to the DEP force . The ponderomotive force for the EP effect can be expressed as for AC-only case. As a result, spatial non-uniformity of the electric field (aforementioned case II and III) is critical for both DEP and EP ponderomotive forces.
The ratio of the EP versus the DEP ponderomotive force will determine the dominant mechanism. For arbitrarily complex trap geometries or potentials, this requires a case by case numerical calculation. We here choose a symmetric geometry for illustrative purposes, which is a good approximation to many experimentally realized traps [5, 6, 7, 8, 9]. Assuming a quadrupole AC electric potential (note that we have considered the one-dimensional motion without loss of generality. Namely, in the case of a quadrupole trap the motions of a particle are independent in each dimension), the ponderomotive force for both DEP and EP can be written as Fdep = −kdep and, Fep = −kep, where kdep and kep (defined as the trap stiffness for EP and DEP effect) are of the forms,
In order to hold the particle in the center of the trap, the ponderomotive force for both EP and DEP should be a restoring force (i.e. kep and kdep should be positive). For DEP trapping, this means a negative value of Re[K(ω)] is needed (nDEP).
From the above analysis we see that both EP and DEP effects are able to trap the particle in the center of the device. Since the trap stiffness can be experimentally estimated through the equipartition theorem as k = κBT/δ2, where κB is Boltzmann's constant, T the absolute temperature, and δ the thermal fluctuations due to Brownian noise, experiments on extracting the position fluctuations of a trapped particle in ‘salt-free’ water are performed to examine the dominant mechanism. figure 2 shows the trap stiffness in both x and y direction as a function of V2. According to equation (5) and (6), both kep and kdep have a linear dependence on V2. We attempted a linear fit with the experimental data for both the EP and DEP cases. Since the real part of the CM factor is bounded within (-0.5, 1) , the maximum slope for DEP falls very short of the experimental data (dashed blue line). This indicates that the trapping in our ‘salt-free’ situation cannot be due to the DEP mechanism. In contrast, a fitting by EP effect (red line) is achievable and estimates the net charge Q as 8.4 × 104 e. We note that a significant amount of charge is critical for EP trapping to dominate over DEP trapping, as will be shown below.
To compare the relative strengths of the EP and DEP in trapping behavior, a dimensionless parameter γ is defined as,
In order for EP effect to dominate in trapping dynamics, it requires γ 1. This will happen at a large net charge Q, a small particle size a, or a low working frequency ω. figure 3 (a) plots the γ value as a function of Q and a at a fixed frequency (1 MHz), while figure 3 (b) shows the γ dependence on Q and ω at a fixed particle size (0.5 μm). It should be emphasized that even though the γ parameter derived here is based on a quadrupole electric field, the general conclusion holds true for other geometries: the EP effect can dominate the trapping dynamics in the case of a sufficiently high charge in a non-uniform electric field.
Based on this γ parameter, we are then able to make a consistent explanation for our experiment and other DEP trapping experiments with a quadrupole electric field [6, 7, 8, 9]. The key parameter involved is the amount of net charge. Notice that the charge we used in the derivations above is the effective net charge rather than the bare charge . A charged surface in contact with a highly conductive liquid creates an induced electric double layer (EDL). A significant fraction of the particle's charge is neutralized by the strongly bounded counterions in the Stern layer. The charged particle plus the thin Stern layer is further screened by diffusive counterions within a characteristic Debye length λD. To determine the effective charge Qeff, we look at the motion of charged particles, which is induced by electrostatic forces, friction, and electrophoretic retardation forces. Among them, the electrophoretic retardation force originates from the delayed response of the surrounding ionic atmosphere to the motion of a charged particle. The electrophoretic mobility including this retardation effect can be described by Henry's formula ,
where α = a/λD is the ratio of particle radius to the Debye length of the electrolyte solution, ε is the dielectric constant of the electrolyte, ζ is the zeta-potential, and η is viscosity of the solution. Ohshima et al.  showed that f (α) is a monotonic increasing function that varies from 1 to 3/2. Since ζ-potential can be expressed in Debye-Hückel form ,
where Qbare is the bare charge of particle. The electrophoretic mobility μE can be rewritten as,
The effective charge of particle can thus be estimated as,
At a high ionic concentration c, the Debye length (λD ~ c−1/2) becomes very small, as a result, a/λD 1 and f (a/λD) → 3/2. Therefore, . The effective charge is greatly reduced in salt solutions. This reduced effective charge in high salt concentrations pushes the γ parameter in figure 3 into the DEP dominating region (γ 1). For most nDEP trapping experiments reported, high salt concentrations were intentionally added to adjust the conductivity of the suspension medium [1, 6, 7, 8, 9], therefore EP effects in trapping dynamics could be safely neglected for those experiments.
In contrast, it is easy to see from equation (11) that Qeff → Qbare when a → 0 and/or λD → ∞. This happens for ultrasmall particles or very low ionic concentrations. For our experiment solutions, a = 0.5 μm, λD ~ 1 μm, Qeff ~ Qbare. This unscreened large amount of effective charge will direct the γ parameter in figure 3 towards the EP dominating region (γ 1). An estimation for our 0.5 μm radius beads which showed the ‘anomalous’ DEP trapping behavior at 20 kHz gives the γ value ~10. Under this circumstance, the EP effect is the dominant mechanism for the center trapping behavior and therefore the predicted pDEP/nDEP boundary becomes invalid (figure 1).
Finally, we briefly comment on the scaling performance of both DEP and EP trap stiffness in the quadrupole trapping device. As shown in equation (5) and (6), reducing the device size r0 helps to enhance the trapping strength for both EP and DEP in a same fashion . Secondly, increasing the applied voltage has the same impact on the trap stiffness (~ V2) and the maximum voltage that can apply is limited by the breakdown field and other electrokinetic effects (e.g. electro-thermal flow ). Most importantly, the DEP trap strength decreases with the volume of the particle. Conversely, EP traps prefer smaller particles since the trap stiffness increases with decreasing the mass of the charged particle, which makes the ACEP effect very attractive for single molecule trapping.
In summary, we have elucidated the importance of EP effects in a non-uniform AC electric field. The relative contributions of both DEP and EP effects in a quadrupole trapping field are studied and an important dimensionless parameter γ is obtained, which presents a consistent explanation for both the anomalous trapping behavior in ‘salt-free’ deionized water and most other DEP trapping experiments with salt solution. EP traps prefer smaller particles, as long as the particles are sufficiently charged. Therefore, it might be feasible to trap single molecules by the EP effect.
This research is supported by the U.S. National Human Genome Research Institute of the National Institutes of Health under grant No. 1R21HG004764-01.
Our planar quadrupole trapping device was fabricated on a SiO2/Si wafer. The insulating SiO2 layer has a thickness of 3 μm. Four Au/Cr (~ 400/50 nm) electrodes were formed on top of this insulating substrate by UV-lithography and a double layer liftoff process. The tip to tip distance (2r0) for each electrode pair ranges from 2 to 8 μm. The microfluidic channel was formed by poly(dimethylsiloxane) (PDMS) using SU-8 as a molding master . Oxygen plasma treatment was used to permanently bond the PDMS to the device surface and form an anti-evaporation microfluidic channel. An inlet and an outlet were punched through before assembling. Once the device was assembled, it could be repeatedly used for a long time.
The particles used in the experiments are polystyrene beads (Polysciences, Warrington, PA) of two diameters (0.481±0.004 μm and 0.982±0.013 μm). The surface of these particles is functionalized with carboxylate group (-COOH). The parking area for each group is around 320 Å2/group. Table A1 summarizes the properties for these two kinds of beads. These COOH surface groups are the origin of the negative charges (COO−). Scanning electron microscopy (SEM) revealed that all the particles had a pronounced spherical shape.
The solution used in the experiment was prepared by the following steps: (i)The beads were firstly diluted to a density of ~ 106 particle/mL by deionized (DI) water (milli-Q grade, resistivity 18 MΩ · cm), in order to eliminate the particle-particle interactions during the experiment. (ii)In order to thoroughly remove the residual ions from the stock solution, the beads prepared in step 1 were washed five times in DI water by centrifuging the beads in a 10 mL tube at 13500 G for 10 min, re-suspending in DI water each time. Salt solutions with 0.1X, 1X and 10X phosphate buffered saline (PBS) (Sigma, St. Louis, MO) were also prepared, following the same steps (i) and (ii). The final conductivity of ‘salt-free’ solution was measured as 0.1 μS/cm (EC 215 Multirange Conductivity Meter, Hanna Instruments) and this slowly goes up to maximum 2.0 μS/cm during the course of an experiment (mostly due to the gas absorption).
The particle conductivity is estimated as σp = σpbulk + 2Ks/a, where σpbulk = 0 S, and Ks = 1 nS is the surface conductance, which was confirmed for polystyrene beads from various techniques [30, 31].
|Diameters (μm)||Parking area (Å2/group)||Number of -COOH groupsa||Net charges Q (e)b|
|0.481±0.004||158.2||4.59 × 105||8.4 × 104|
|0.982±0.013||7.9||3.83 × 107||1.8 × 105|
The crossover frequency for salt solutions (PBS solutions) obeys the classic DEP theory very well. In contrast, the crossover frequency for ‘salt-free’ solution is abnormally low (down to around 20 kHz). This abnormal trapping behavior for ‘salt-free’ solution was repeated for at least five separate experiments with more than one single particle trapping observations per experiment.
The device was wire-bonded and mounted onto a printed circuit board (PCB). Potentials in the form of U – V cosωt, produced by a function generator (Tektronix AFG3252) together with a voltage amplifier (Tabor Electronics, Model 9250), were delivered to the device through 50 Ω BNC cables and monitored by an oscilloscope (Tektronix DPO 4104).
The motion of the charged particle was monitored by an optical microscope (Olympus BX51) and the video was taken by a high-sensitivity digital CCD camera (Olympus DP70) with the highest shutter speed as fast as 1/44000 s. We use a particle tracking algorithm which has been described in detail elsewhere  to extract the motion fluctuations. The videos were decomposed into frame sequences using the software VirtualDub (http://www.virtualdub.org). The particle tracking was then carried out using the NIH ImageJ platform (http://rsbweb.nih.gov/ij/) with a particle tracking algorithm (https://weeman.inf.ethz.ch/ParticleTracker).
It should be noted that the video based position extraction method does not measure the instantaneous particle position and has a problem of ‘motion blur’, which results from time-averaging a signal over a finite integration time (shuttle time or acquisition time) . This will lead to the underestimation of the real variance and overestimation of the trap stiffness for each data point in figure 2 (main text). However, for a fixed optical setup, the relative relations between each measured variance (the slope) remain unchanged.
Considering a particle of charge Q and mass m and in a quadrupole linear (2D) trap of characteristic radius r0 and electric potential, , which by definition provides a spatially non-uniform electric field, with DC and AC components in x-dimension
and the trap dimensionless parameters are defined as
The equation of motion in the x direction is
Introducing a dimensionless viscosity b and a dimensionless time τ
then equation B.5 takes the form
Without loss of generality, the solution of equation B.8 can be written in the form of the Floquet expansion
where Pn(τ) and σ(τ) are slowly varying functions on the time scale such that the external electric field is switched ‘on’ adiabatically. By equating the same Floquet components and neglecting time derivatives of Pn and σ, the infinite, homogeneous system of equations follows,
where the dependence on τ is henceforth dropped for brevity, and n takes all integers. To secure a nontrivial solution, one equates the infinite tridiagonal determinant of the system with zero. This yields the equation for the infinite number of Floquet-Lyapunov (FL) exponents σ, which define an infinite number of solutions for the system. We will seek the solution for the case of particular experimental interest, q 1, under arbitrary dragging parameter b and DC parameter a. Looking for the non-oscillating (P0) and oscillating terms to the lowest non-vanishing powers in q (P±1), equation B.11 simplifies into
reducing the infinite homogenepus system into the finite one of order 3. Thus
which, when replaced in equation B.12, gives the FL exponents
Note that the above analysis is valid for an arbitrary drag parameter b and arbitrary DC parameter a. The equation B.13 yields
Notice that it is easy to show that P±2 are proportional to q2, P±3 to q3, therefore P±1 are the leading coefficients in expansion of small q of the rapid oscillating part of the Floquet expansion in equation B.10, oscillating with the driving frequency ω. Since in P0(t) ~ exp(± ∫ σ0dτ), σ0 can be either real or imaginary, the latter case producing secular oscillations of the frequency ωσ0 which is in the limit b = 0, a = 0 equal to , obviously much smaller than ω when q 1. However, when b > 1 (typical for aqueous environments), ωσ0 is real, and when combined with the exponent in equation B.7 yields − (b/4 σ0) < 0 when q 1. This means that x(t) is proportional to an exponentially decreasing function of time, exp [− (b/4 σ0)ωt]. It is interesting to note that the two exponent factors are quite different in size. Thus, , while . This means only the solution corresponding to σ1 will define long time motion in water, the other being a short-time transient.
By neglecting high order terms in q, we can now write P(τ) in equation B.10 as,
where i=1, 2 correspond to two eigen solutions in equation B.15,
The full solution for x(t) is then
where A1 and A2 are the integration constants, depending on the initial conditions. If one assumes zero initial velocity at an initial position x0, as well as adiabatic switching on of the external potential, then for b q, the solution takes the form
When only an AC field is applied (a = 0), it follows
However, when also b=0 (vacuum case)
defining the secular angular frequency .
To make a connection of these results with the more general case of non-uniform electric field in the main text, it can be seen from equation B.3 and equation B.4 that , where in the case of quadrupole field, . By the same token, f0(S)/(mω2) in equation 2 in the main text takes the form qS/2 in case of the quadrupole field. equation 3 in the main text takes the form
where the last term at the right-hand side of equation B.25 defines the ponderomotive potential for the case of a quadrupole AC potential, yielding the ‘simplest’ spatially inhomogeneous potential, linear in x.
The harmonic EP pseudopotential of equation 4 in the main text can be written as
Finally, the EP ‘trap stiffness’ for a linear quadrupole trap has the form