Nonlinear electrophoresis of dielectric particles in Newtonian fluids

In classical electrokinetics, the electrophoretic velocity of a dielectric particle is a linear function of the applied electric field. Theoretical studies have predicted the onset of nonlinear electrophoresis at high electric fields because of the nonuniform surface conduction over the curved particle. However, experimental studies have been left behind and are insufficient for a fundamental understanding of the parametric effects on nonlinear electrophoresis. We present in this work a systematic experimental study of the effects of buffer concentration, particle size, and particle zeta potential on the electrophoretic velocity of polystyrene particles in a straight rectangular microchannel for electric fields of up to 3 kV/cm. The measured nonlinear electrophoretic particle velocity is found to exhibit a 2(±0.5)‐order dependence on the applied electric field, which appears to be within the theoretically predicted 3‐ and 3/2‐order dependences for low and high electric fields, respectively. Moreover, the obtained nonlinear electrophoretic particle mobility increases with decreasing buffer concentration (for the same particle) and particle size (for particles with similar zeta potentials) or increasing particle zeta potential (for particles with similar sizes). These observations are all consistent with the theoretical predictions for high electric fields.


INTRODUCTION
Electrophoresis is an electrokinetic phenomenon widely adopted for particle transport and manipulation in microand nano-fluidic devices [1][2][3]. It is the movement of an linear dependence on the electric field, , and particle zeta potential, , via the Smoluchowski equation under the thin EDL limit [10,11] where is the fluid permittivity, and is the fluid viscosity. However, recent studies indicate that the linearity for is valid only in the limit of a weak electric field, = ∕ ≤ 1, and a small particle zeta potential, ∕ < 1, where is the particle radius, and is the thermal voltage. Under these conditions, the ions within the EDL of the particle can maintain the equilibrium state yielding a homogeneous electrostatic potential and ionic concentration [12][13][14].
Increasing the electric field and/or particle zeta potential distorts the EDL and induces ionic fluxes across the EDL because of the surface conduction effect [15][16][17][18], leading to a nonlinear dependence of on both and [19][20][21][22], where (1) is the linear electrophoretic particle mobility, (3) is the nonlinear electrophoretic particle mobility, is the Dukhin number characterizing the surface conduction effect, is the inverse of the Debye length, and − is the dimensionless drag coefficient for counterions. The formula for (3) in Equation (3) was obtained by Schnitzer and Yariv [23] for ≪ 1 at small Peclet numbers, = ∕ ≪ 1, where is the effective diffusion coefficients of ions. A similar formula to Equation (3) was also reported by Mishchuk and Dukhin [24], whereas a slightly different formula was later obtained by Shilov et al. [25] for arbitrary values of . Schnitzer et al. [21] also obtained an expression for (3) in the weak-field limit, ≤ 1, for arbitrary values of , which, however, shows inconsistencies with that from Shilov et al. [25] because of the ignored ion advection and other salt-related effects in the latter. In all these formulae except that from Schnitzer et al. [21], (3) increases with increasing that may be a consequence of the increasing Debye length, 1∕ , via the decrease of buffer concentration or the increasing particle zeta potential. It also increases with the particle radius, , even though itself actually gets smaller for larger particles. At large Peclet numbers, ≫ 1, or equiva-lently strong electric fields, ≫ 1, Schnitzer and Yariv [23] predicted an 3∕2 -dependent nonlinear electrophoretic particle velocity, where ( ) is a function of . Therefore, (3∕2) increases with the increase of particle zeta potential or the decrease of buffer concentration and particle size. Mishchuk and Dukhin [24] reported a different formula for (3∕2) , which decreases with the increase of particle zeta potential. Other theoretical and numerical studies on nonlinear particle electrophoresis can be referred to a recent review article from Khair [26].
There have also been a few experimental studies on nonlinear electrophoresis of dielectric particles. The earliest experiment seems to be reported by Kontush et al. [27] in a Russian colloidal journal that is unfortunately not accessible to the authors of this work. However, Mishchuk and Dukhin [24] noted that the prediction of (3∕2) in Equation (6) agrees closely with the experimental result of Kontush et al. [27] for spherical latex particles. Shilov et al. [25] measured the lateral drift of sedimenting polystyrene particles of 30 µm diameter in water and KCl solution under electric pulses. Their observed cubic electrophoresis for electric fields stronger than 0.1 kV/cm agrees with the theoretical prediction of (3) in Equation (3). Later, Barany [28] reported the measurement of polymer-coated polystyrene particles using the same experimental setup as in Shilov et al. [25], where the cubic electrophoresis is found as theoretically predicted to increase with the particle diameter. Mishchuk and Barninova [29] also observed a greater nonlinear electrophoretic velocity for larger latex particles for electric fields of up to 0.2 kV/cm, in-line with the prediction of (3) in Equation (3). In contrast, the nonlinear electrophoretic velocity of larger latex particles was found smaller for larger electric fields of up to 0.8 kV/cm, corresponding to the theoretical prediction of (3∕2) in Equation (6), though the Peclet number was reported to remain on the order of 1 in both experiments. In another study, Youssefi and Diez [30] measured the electrophoretic velocity of carboxyl-treated 0.2 µm diameter polystyrene particles for electric fields over the range of 0.1-250 kV/cm. They observed a 3/2-order dependence of their electrophoresis measurements on electric fields of up to 40 kV/cm, in agreement with the prediction of Equation (6). For even higher electric fields, their measured electrophoretic velocity still increases with the electric field but slower than the 3/2-order dependence.
Tottori et al. [31] studied the electrophoretic motion of highly charged polystyrene and poly(methyl methacrylate) (PMMA) particles of 0.5 µm diameter for electric fields of up to several kV/cm. Their measured nonlinear electrophoretic velocity exhibits a 3-order dependence on the imposed electric field, in good agreement with the theoretical prediction of Equation (3). In a more recent study, Cardenas-Benitez et al. [32] reported a reversed electrokinetic motion for carboxylated polystyrene particles of 1.0, 1.9, and 5.1 µm diameters in dilute KCl solutions when the imposed electric field is beyond a threshold magnitude (smaller than 1 kV/cm for all cases). The authors termed this state the electrokinetic equilibrium condition (EEC) and explained it using the nonlinear electrophoretic particle velocity in Equation (2) that increases more quickly with the electric field than the opposing linear electroosmotic fluid velocity. They later used the EEC to obtain the nonlinear electrophoretic mobilities of other types of particles [33,34] and achieve the separation of almost identical particles [35] as well as sub-100 V particle trapping [36].
However, the current experimental studies are still insufficient for a systematic understanding of the parametric effects of fluid and particle properties on nonlinear electrophoresis. We carry out a set of experiments in this work to investigate the respective effects of buffer concentration, particle size, and particle zeta potential on the nonlinear electrophoretic velocity of dielectric particles in aqueous electrolyte solutions through a straight rectangular microchannel. Specifically, we will study if and how the nonlinear electrophoretic particle mobility, ( ) , and nonlinear index, ≠ 1, vary with each of these fluid and particle properties.

Microchannel and chemicals
The microchannel was fabricated from polydimethylsiloxane with the standard soft lithography technique [37]. The channel is straight and 1 cm long with a uniform width and depth of 50 µm each. The experiment studies the effects of three individual parameters on nonlinear particle electrophoresis. The first parameter is buffer concentration, for which 5 µm diameter plain polystyrene particles (Sigma-Aldrich) were resuspended in phosphate buffer solutions with concentrations ranging from 0.01 to 0.05, 0.075, and 0.1 mM. These solutions were all prepared by diluting the original 50 mM buffer solution (pH = 7) with DI water. The second parameter is particle size, for which 3, 5, and 10 µm diameter plain polystyrene particles (Sigma-Aldrich) were each resuspended in 0.075 mM phosphate buffer. The third parameter is particle zeta potential, for which three types of (nearly) 5 µm diameter polystyrene particles, including 5 µm plain particles from Sigma-Aldrich, 4.95 µm fluorescent carboxyl particles from Bangs Laboratories, and 4.8 µm fluorescent carboxylatemodified particles from Thermo Scientific, were each resuspended in 0.075 mM phosphate buffer. These particles were noticed to travel at different speeds in the same solution under the same electric field, indicating that they have dissimilar zeta potentials probably because of their intrinsic surface groups.

Experimental technique
The prepared particle suspensions were each driven through the microchannel by a high-voltage DC power supply (Glassman High Voltage) via platinum electrodes inserted into the end-channel reservoirs. The voltages varying from 0.1 to 3 kV were imposed upon the 1 cm long channel, yielding the average electric fields of 0.1-3.0 kV/cm. The corresponding dimensionless electric field, = ∕ , for = 2.5 µm particles was calculated to range from 1 to 30. For each applied voltage, the direction of electric field was reversed once via a two-way electric switch to repeat the test for the purpose of canceling the potential influence of backflow. Moreover, each run of test was kept no more than 30 s (i.e., 15 s for each direction) to minimize both the backflow [38] and Joule heating effects [39]. In addition, the reservoirs were intentionally made large to minimize the impact of pH change due to electrolysis at high electric fields, which also facilitates reducing the backflow. The motion of particles was observed to remain along the direction of the applied electric field, indicating stronger fluid electroosmosis (which is along the electric field direction) than particle electrophoresis (which is against the electric field direction) in all tested cases. It was recorded using an inverted microscope imaging system (Nikon Eclipse TE2000U, Nikon Instruments). The CCD camera (Nikon DS-Qi1Mc) was run in a binning mode for increasing the frame rate to around 50 fps at a reduced concentration. The captured images were processed using the Nikon imaging software (NIS-Elements AR 2.30).
The velocity of particles was measured using the particle tracking velocimetry, where three to five particles traveling along the channel centerline (only) were tracked to obtain an average value. To quantify the effect of the potential pressure-driven backflow at high electric fields, we seeded 1 µm diameter polystyrene particles (Bangs Laboratories) into the reference solution, that is, 0.075 mM buffer, for a real-time recording of the fluid velocity immediately after the electric field was turned off. The measured velocity of the tracer particles along the channel centerline was found no more than 5% of that of our test particles under the highest electric field. We also monitored the temporal variation of electric current in the highest concentration 0.1 mM buffer for estimating the Joule heating effects and the accompanying electrothermal flow [39]. The electric current rise was found to remain less than 10% of the initial value within the 15 s application of the highest 3 kV/cm electric field, indicating a fewer than 5 • C increase in the average fluid temperature for an assumed 2% temperature coefficient of the electric conductivity [40]. This small temperature elevation was assumed to have an insignificant impact on the fluid properties and, hence, the particle motion. In addition, we estimated that under pure DC electric fields, the induced charge electroosmotic flow at the reservoir-microchannel junction [39] is weak with no significant influence on the particle motion inside the microchannel.

Experimental data analysis
The measured particle velocity, , in the straight microchannel is the sum of the electroosmotic fluid velocity, , and electrophoretic particle velocity, We split into the linear component, (1) , and the nonlinear component, ( ) , where the nonlinear index, > 1. Thus, the measured particle velocity can be rewritten as = + ( ) where is the electrokinetic particle velocity that has been long accepted to scale linearly with the applied electric field in classical electrokinetics [9,12], is the (linear) electrokinetic particle mobility [11,14], and ( ) is the nonlinear electrophoretic particle mobility. The primary objective of this work is to study if and how ( ) and vary with the fluid and particle properties. To do so, we utilize the same method as in Tottori et al. [31] to extract ( ) from the experimental data. Briefly, the linear electrokinetic particle velocity, , was determined through a linear fit (the slope denotes the electrokinetic particle mobility, ) of the measured particle velocity, , at the three smallest electric fields, that is, 0.1, 0.2, and 0.25 kV/cm. This analysis was based on the assumption that ( ) ≪ and, hence, ≅ at small electric fields. The nonlinear electrophoretic particle velocity, ( ) , was then calculated by subtracting the obtained from the measured . The log-log transformation was then used to determine the nonlinear electrophoretic particle mobility, ( ) , and nonlinear index, , via the intercept and slope of the linear fit for ( ) as a function of . Figure 1 plots the experimentally measured velocities of 5 µm-diameter Sigma-Aldrich particles in buffer solutions with concentration varying from 0.01 to 0.05, 0.075, and 0.1 mM under different electric fields. The error bars (note some of them are within the symbol size and become invisible) highlight the maximum variations of the measured of three to five particle velocities with respect to their average for each electric field. The measured particle velocity, , in each buffer solution is observed to increasingly deviate from the linear electrokinetic particle velocity, (reflected by the linear trend lines in Figure 1), at higher electric fields. This upward trend goes against that reported by Cardenas-Benitez et al. [32], the reason behind which is currently unclear. One possible explanation could be that the electrophoretic particle velocity, , in our experiment decreases nonlinearly with the increase of electric field because of, for example, the predicted retardation effect of surface conduction [23] and/or dielectric-solid polarization at strong fields [41]. We will work on revising the experimental technique to obtain directly. The discrepancy between and , that is, the nonlinear electrophoretic particle velocity, ( ) , exhibits an apparent dependence on the buffer concentration in Figure 1. The Peclet number in this experiment was estimated to vary from around 2 to 60 using = ∕ based on the effective diffusion coefficient, = 0.5 × 10 −9 m 2 /s [42], and the average of 0.4 and 12 mm/s for the lowest and highest electric fields of 0.1 and 3 kV/cm, respectively. This range of covers both 1 < < 10 and ≫ 1, implying that ( ) may be inclined toward (3∕2) in Equation (6).

Effect of buffer concentration
A summary of ( ) in the four buffer solutions is shown in Figure 2A as a function of the electric field. There is a clear trend that the nonlinear particle electrophoresis gets enhanced in lower concentration buffers, which should be attributed to the thicker EDL therein (characterized by the Debye length, 1∕ ) and, hence, the stronger surface conduction effect. This trend is consistent with the predictions of both (3) in Equation (3) and (3∕2) in Equation (6) in terms of the increased Dukhin number, . The experimentally obtained data for ( ) in each buffer solution are found to be best fitted with a positive power trend line as illustrated in Figure 2A. We used the log-log transformation (see Figure S1 for the log-log plot) as noted above to determine the nonlinear electrophoretic particle mobility, ( ) , and nonlinear index, , from the linear fit of ( ) against . Figure 2B presents the extracted ( ) and that each exhibit a linear decreasing trend with the increase of buffer concentration. Specifically, the value of decreases from approximately 2.4 in 0.01 mM buffer to 1.6 in 0.1 mM buffer, both of which appear to be within the theoretically predicted = 3 and 3∕2 for small and large electric fields [23,24], respectively. The value of ( ) , the unit of which is noted to vary roughly around = 2, decreases from approximately 0.18 to 0.08 mm/(s (kV/cm) 2 ) when the buffer concentration increases from 0.01 to 0.1 mM.
As the change of buffer concentration often modifies the particle zeta potential [43], the observed trend for ( ) may be associated with both factors. The sole  Figure 3 shows the experimentally measured velocities for 3 and 10 µm-diameter Sigma-Aldrich particles in 0.075 mM buffer solution. The electrokinetic particle velocity, (see the linear trend lines), is found insensitive to the particle size (including 5 µm), so is the electrokinetic particle mobility, = 4.3 × 10 −8 m 2 /V s, and particle zeta potential. However, the deviation of the measured particle velocity, , from is clearly greater for the smaller 3 µm particles. The estimated Peclet number spans from around 1.2 to 36 for 3 µm particles and from 4 to 120 for 10 µm ones over the range of electric fields tested. Both  Figure 4A, which shows a generally increasing trend with the decrease of particle diameter over the range of electric fields. They are again each best fitted with a positive power trend line, the intercept and slope of which in the log-log space (see Figure S2) gives the nonlinear electrophoretic particle mobility, ( ) , and nonlinear index, , respectively. The extracted values of ( ) and both decrease with the increase of particle diameter, which are each best fitted with a negative power trend line as viewed in Figure 4B. It is noted from Equation (4) that the Dukhin number, , gets larger for smaller particles, which should yield stronger surface conduction effects. However, the theoretically predicted (3) in Equation (3) for small Peclet numbers turns out to be a positive function of the particle diameter. Such a trend goes against our observation in Figure 4B, where the value of ( ) decreases from approximately 0.18 to 0.08 mm/(s (kV/cm) 2 ) for assumed = 2 when the particle diameter increases from 3 to 10 µm. It, however, appears consistent with the prediction of (3∕2) ∼ −1∕2 in Equation (6) for large Peclet numbers because our estimated are indeed more inclined toward the high regime as noted above. Moreover, the extracted range of ( ) for particles of different sizes in Figure 4B is found to match that in Figure 2B for 5 µm particles in buffers of varying concentrations. In addition, our extracted value of decreases from approximately 2.5 for 3 µm particles to 1.6 for 10 µm particles. This range is also consistent with the experimentally obtained variation of in Figure  2B and is again within that of the theoretically predicted = 3 and 3∕2 for small and large electric fields [23,24], respectively.

3.3
Effect of particle zeta potential shows a greater deviation from than that of Bangs particles, indicating stronger nonlinear electrophoresis. The estimated Peclet number is 1≤ ≤ 43 for Bangs particles and 0.6 ≤ ≤ 50 for Thermo particles over the range of electric fields tested. Therefore, the nonlinear electrophoretic velocities, ( ) , should be in theory inclined toward (3∕2) for both types of particles. Figure 6A compares the experimentally determined, ( ) , for the three types of particles, which are each best fitted with a positive power trend line. It is apparent that ( ) grows larger with the decrease of over the range of electric fields, where the linear electrokinetic particle velocity, , as traditionally defined in Equation (9), depends on the particle zeta potential via the following (linear) electrokinetic mobility, , under the thin EDL limit [9,11,14], The wall zeta potential, , for 0.075 mM buffer was found to be around −123 mV from the experimentally measured electroosmotic fluid velocity via the electric current monitoring method [44]. The particle zeta potential, , was then calculated from Equation (11) using the experimentally determined . Figure 6B shows the extracted values of ( ) and as a function of | | from the power trend lines in Figure 6A (see Figure S3 for the log-log plot). The nonlinear index increases slightly from = 1.9 for Sigma particles at  (4)] increases from 0.067 for Sigma particles to 0.14 and 0.16 for Bangs and Thermo particles, respectively, with the increase of | |. This trend is noted to also agree with the prediction of (3) in Equation (3) for low Peclet numbers. It, however, goes against that reported by Vaghef-Koodehi et al. [35], the reason behind which is currently unclear. Overall, our observed buffer concentration, particle size, and particle zeta potential effects on nonlinear electrophoresis are all in good agreement with the prediction of (3∕2) in Equation (6). This phenomenon seems to align with our estimated values of Peclet number that are more inclined toward the high regime.

CONCLUDING REMARKS
We have experimentally studied the effects of buffer concentration, particle size, and particle zeta potential on the nonlinear electrophoresis of polystyrene particles in a straight rectangular microchannel. The measured data for the nonlinear electrophoretic particle velocity as a function of the applied electric field are best fitted with a positive power trend line for each case. The nonlinear electrophoretic particle mobility, ( ) , and nonlinear index, , extracted from the trend lines are both found to increase with the decrease of buffer concentration and particle size or the increase of particle zeta potential. However, the nonlinear index, , stays at the value of 2 with a deviation of no more than ±0.5 in all the tested cases, which appears to be within the 3-and 3/2-order dependences for low and high electric fields, respectively. Moreover, the obtained trends for ( ) as a function of the tested fluid and particle properties are all consistent with the theoretical prediction of (3∕2) in terms of the Dukhin number. This observation turns out to be in-line with our estimated values of Peclet number that are inclined toward the high regime in all cases. For future work, we will study if biological cells experience nonlinear electrophoresis [45] that may be utilized for enhanced detection throughput [46]. We will also look into the influences of other factors on nonlinear particle electrophoresis, such as dielectric polarization and hydrophobicity [41,[47][48][49][50].

A C K N O W L E D G M E N T S
This work was supported in part by NSF under grant numbers CBET-2100772 and CBET-2127825 and by Clemson University through the Creative Inquiry Program.

C O N F L I C T O F I N T E R E S T S TAT E M E N T
The authors have declared no conflict of interest.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.