Nonlinear electrophoresis of nonspherical particles in a rectangular microchannel

Nonlinear electrophoresis offers advantageous prospects in microfluidic manipulation of particles over linear electrophoresis. Existing theories established for this phenomenon are entirely based on spherical particle models, some of which have been experimentally verified. However, there is no knowledge on if and how the particle shape may affect the nonlinear electrophoretic behavior. This work presents an experimental study of the nonlinear electrophoretic velocities of rigid peanut‐ and pear‐shaped particles in a rectangular microchannel, which are compared with rigid spherical particles of similar diameter and surface charge in terms of the particle slenderness. We observe a decrease in the nonlinear electrophoretic mobility, whereas an increase in the nonlinear index of electric field when the particle slenderness increases from the peanut‐ to pear‐shaped and spherical particles. The values of the nonlinear index for the nonspherical particles are, however, still within the theoretically predicted range for spherical particles. We also observe an enhanced nonlinear electrophoretic behavior in a lower concentration buffer solution regardless of the particle shape.

length, 1∕) because of the surface conduction effect [3][4][5][6].The consequence is the onset of nonlinear electrophoresis whose velocity is predicted based upon a spherical particle model to exhibit a 3-to 3/2-order dependence on the electric field strength [7][8][9][10].This phenomenon has been experimentally investigated with spherical dielectric particles by several research groups [11][12][13][14][15].It has also been utilized to enhance the trapping and separation of spherical particles [16][17][18][19][20].A brief overview of these earlier studies was provided in our previous work in early 2023 [21] and is therefore skipped here.Readers interested in this topic are also suggested to refer to the review paper from Khair [22] for a more complete discussion of those theoretical and experimental works published before 2022.We present below a summary of only those papers published since our previous work [21].
Lapizco-Encinas et al. published four papers pertaining to nonlinear electrophoresis during this time period.Two of these papers are dedicated to the fundamental understanding of the significant factors in nonlinear electrophoresis.Ernst et al. [23] studied the particle size and charge dependencies of nonlinear electrophoretic velocity for a total of nine distinct types of spherical polystyrene particles.They assessed the experimental data under both the 3-and 3/2-order electric field scaling and obtained the corresponding nonlinear electrophoretic mobilities for each type of particles.They reported that the mobilities in both regimes increase with increasing particle size and decrease with increasing particle charge.Later, Lomeli-Martin et al. [24] divided the commercially available spherical polystyrene particles into three categories based on the difference in their nonlinear electrophoretic behaviors: "type 1" particles travel along with the electroosmotic fluid flow but reverse once the imposed electric field goes beyond a threshold; "type 2" particles travel against the fluid flow and have very small values of nonlinear electrophoretic mobility; "type 3" particles travel along with the fluid flow exhibiting a linear electrophoretic velocity even at extremely high electric fields (∼6 kV/cm).The authors concluded from the common features among these particles that size, surface functionalization, and electrical charge can all be determining factors in electrophoresis.
The other two papers from Lapizco-Encinas et al. are focused upon the application of nonlinear electrophoresis in size-or charge-based separation of particles and cells.Vaghef-Koodehi et al. [25] presented a continuous separation of particles and cells of similar characteristics through the combined linear and nonlinear DC electrokinetic phenomena in an insulator-based electrokinetic system.The authors developed a spherical particle model in COMSOL to predict the retention times of particles and cells in four distinct separations of binary mixtures at increasing difficulty, from spherical polystyrene particles of different sizes to Escherichia coli versus Saccharomyces cerevisiae, Bacillus cereus versus S. cerevisiae, and B. cereus versus Bacillus subtilis.Their predictions were reported to agree with the experimentally measured particle/cell retention times with acceptable deviations and variations.In a later work, Ahamed et al. [26] demonstrated the use of DC-biased low-frequency AC voltage to achieve in a similar insulatorbased electrokinetic system the separation of same-sized spherical polystyrene particles with ∼14 mV zeta potential difference.They again used the spherical particle model in COMSOL, which considers both linear and nonlinear electrophoresis, to examine the effect of fine-tuning AC voltage frequency, amplitude, and DC bias, respectively.
The numerically optimized value for each of these parameters was used in the experiment, which was found to improve the separation resolution by more than fivefolds.
In a very recent theoretical paper, Cobos and Khair [27] developed a spectral element algorithm to compute the electrophoretic velocity of a spherical dielectric particle with arbitrary EDL thickness over a wide range of DC electric fields.They reported that the nonlinear contribution to the electrophoretic velocity of moderately charged particles (∕ ∼ 1) grows as the electric field increases, whose onset is a function of the dimensionless particle radius, .It, however, vanishes at high electric fields (∕ ≫ 1) with the electrophoretic velocity approaching the Hückel limit [27].The authors further reported that their computed values for the electrophoretic velocity of highly charged particles (∕ ≫ 1) under the thin EDL limit ( ≫ 1) match the asymptotic result from Schnitzer and Yariv [10] and as well the experimental result from Tottori et al. [15].Our previous work [21] presented a systematic experimental study of the effects of buffer concentration, particle size, and surface charge on the electrophoretic velocity of spherical polystyrene particles in a straight rectangular microchannel.We demonstrated that the measured nonlinear electrophoretic particle velocity exhibits a 2(±0.5)-orderdependence on the applied electric field of up to 3 kV/cm, within the theoretically predicted 3-and 3/2-order dependences [7][8][9][10].We also found that the nonlinear electrophoretic mobility and index both decrease with increasing buffer concentration and particle size but increase with increasing particle charge, consistent with the theoretical predictions for high electric fields (∕ ≫ 1).
We will also study the nonlinear electrophoretic velocity of nonspherical particles in buffers of varying concentrations and compare the results with those obtained for spherical particles in our previous work [21].

Microchannel and chemicals
A straight rectangular microchannel was used in the experiment.It was fabricated from polydimethylsiloxane via the standard soft-lithography technique [43].The channel is 1 cm long with a uniform width and depth of approximately 50 µm each.Our experiment studied the nonlinear electrophoretic motion of various shaped rigid polystyrene particles, including 5.0 µm-diameter spherical particle (Sigma-Aldrich), 3.5 µm-diameter/6.0 µm-length peanut-shaped particle (Magsphere Inc.), and 3.8 µmdiameter/5.1 µm-length pear-shaped particle (Magsphere Inc.).The equivalent spherical diameters of the two nonspherical particles, which were obtained from their calculated total volumes in COMSOL, are approximately identical and are only about 15% smaller than that of the spherical particle (Table 1).These particles were each resuspended in 0.025 mM phosphate buffer solution for an investigation of the particle shape effect on nonlinear electrophoresis.The particle concentration was kept low in each suspension (around 10 5 particles per mL) to minimize the particle-particle interactions.The Debye length in this solution was estimated to be about 1∕ = 63 nm, such that the dimensionless particle radius is  = 40 ≫ 1 for the 5.0 µm-diameter spherical particle, satisfying the thin EDL condition.It is noted that the threshold value of  for this assumption may vary among different studies (e.g., 40).To quantify the analysis, we define a dimensionless particle slenderness, : where  is the maximum radius of the particle perpendicular to its long axis (or the half-length of the particle's short-axis), and  is the half-length of the particle along its long axis (or the half-length of the particle's long-axis).We also studied the effect of buffer concentration on the nonlinear electrophoresis of peanut-shaped particles.Table 1 summarizes the dimensions and slenderness values of the three types of particles used in the experiment.

Experimental techniques
The electrokinetic motion of particles through the microchannel was driven by a high-voltage DC power supply (Glassman High Voltage).The electric field was varied from 0.1 to 5 kV/cm in each test, corresponding to 1 ≤  ≤ 50 for 5.0 µm diameter particles.The run of each test was kept no more than 15 s for each direction of electric field to minimize the influences of both Joule heating and backflow as detailed in our previous work [21].Briefly, the effect of Joule heating was estimated to be insignificant because the temporal variation of electric current was observed to be no more than 10% even in the highest concentration buffer under the highest electric field [44].Moreover, the liquid levels in the end-channel reservoirs were balanced prior to every test to avoid the pressure-driven particle motion.The spherical and nonspherical particles were observed to move in the direction of the imposed DC electric field in all cases tested.This phenomenon indicates that the electroosmotic fluid flow is stronger than the electrophoretic particle motion, the latter of which is against the direction of electric field because of the naturally negative charge of particles [45,46].The particle motion was visualized using an inverted microscope imaging system (Nikon Eclipse TE2000U, Nikon Instruments) and recorded through a CCD camera (Nikon DS-Qi1Mc) in a binning mode.The captured images were processed using the Nikon imaging software (NIS-Elements AR 2.30).The particle velocity was measured using the particle tracking velocimetry, where (at least) five particles traveling along the centerline of the microchannel were tracked to obtain an average for each electric field.

Experimental data analysis
We used the approach detailed in our previous work [21] to process the experimentally measured data of particle velocity,   =   +   , which is a result of the summation of the electroosmotic fluid velocity,   , and electrophoretic particle velocity,   .Briefly, we break down   into the linear component, , and nonlinear component,  where   =   + (1)  =    is the traditionally defined (linear) electrokinetic particle velocity with   being the (linear) electrokinetic mobility, and  ()  is the nonlinear electrophoretic mobility with the nonlinear index of electric field  > 1.Under the assumption that  ()  ≪   and hence   ≅   at small electric fields [15], we determined   through a linear regression of   for  ≤ 500 V/cm.The nonlinear electrophoretic velocity,  ()  , was then obtained by subtracting    from the measured   values at higher electric fields.The intercept and slope of the plot of  ()  versus  in the log-log space give the nonlinear electrophoretic mobility,  ()  , and nonlinear index, , respectively.

Orientation of nonspherical particles in electrophoresis
Figure 1A shows an image of the peanut-and pear-shaped particles, which were mixed with the spherical particles in 0.025 mM buffer for easy visualization, under the application of 0.2 kV/cm DC electric field.Both types of nonspherical particles were observed to quickly align their long-axes with the electric field direction and travel along with the spherical particle (nearly) at the center plane of the microchannel.These observations are consistent with the phenomena reported in previous studies, which arise from the combined action of the Maxwell and hydrodynamic stresses in the presence of the insulating channel walls [47][48][49].We also noticed that the pear-shaped particles may travel with their heads or tails (highlighted in Figure 1A) leading, the percentage of which is approximately 50% each.We measured the velocity of pear-shaped particles,   , at either orientation for electric field ranging from 0.1 to 0.4 kV/cm (note the identification of particle orientation gets more difficult at higher electric fields).As viewed from Figure 1B,   scales linearly with the electric field strength as nonlinear electrophoresis is negligible at small electric fields such that   ≅   =   .Moreover, it exhibits an insignificant dependence (less than 5% difference between the slopes of the two linear trendlines, i.e.,   ) on the particle orientation at every electric field.Therefore, we did not attempt to identify the orientation of pear-shaped particles at higher electric fields for a convenient study of nonlinear electrophoresis.We F I G U R E 1 Electrophoresis of nonspherical particles in 0.025 mM buffer in a rectangular microchannel under electric field ranging from 0.1 to 0.4 kV/cm: (A) microscopic images of the peanut-and pear-shaped particles along with a spherical particle, whose long-axes are aligned with the imposed DC electric field of 0.2 kV/cm.The lengths of the short-and long-axes of a nonspherical particle are highlighted; (B) plot of the measured particle velocity (symbols),   , for the pear-shaped particles with heads and tails (highlighted on the image in A) leading the motion, respectively.The dotted lines are the linear trendlines to the experimental data for these two orientation cases with the corresponding equations and R-squared values being both displayed.admit this treatment may cause certain errors, for example, the influence of particle orientation on nonlinear electrophoresis may no longer be negligible at high electric fields.

Effect of particle shape on nonlinear electrophoresis
Figure 2A shows the experimentally measured   for the three types of particles in 0.025 mM buffer under electric field ranging from 0.1 to 5 kV/cm.There is an insignificant gap among the three linear trendlines (i.e.,   ) to the data points for 0.5 kV/cm and below, indicating approximately identical values of electrokinetic mobility (with 5% F I G U R E 2 Electrophoresis of spherical, pear and peanut-shaped particles in 0.025 mM buffer under electric field ranging from 0.1 to 5 kV/cm: (A) experimentally measured velocity (symbols with error bars; note some of the error bars are within the symbol size and become invisible),   , where the linear trendlines are the best fits for the experimental data points at 0.5 kV/cm and below (assumed to represent the linear electrokinetic particle velocity,   ); (B) experimentally obtained (symbols with error bars) nonlinear electrophoretic velocity,  ()  =   −   , versus electric field, where the curves are the positive power trendlines best fitted for the experimental data points.variation),   = 2.23(±0.11)× 10 −8 m 2 /V s, for the spherical and nonspherical particles.Therefore, the particle zeta potential can be viewed to remain similar among these particles under the thin EDL limit [39], so that any different nonlinear behaviors witnessed in Figure 2A can be viewed more closely associated with the particle shape.For electric fields above 1 kV/cm, the data of   start increasingly deviating from the linear trendline for each type of particles in Figure 2A.Moreover, this deviation exhibits a visible dependence on the particle shape, which is evidenced from the dissimilar power trendlines to the data of nonlinear electrophoretic velocity,   To further compare the nonlinear electrophoretic behaviors of spherical and nonspherical particles, we replot the data of  ()  versus electric field in the log-log space.As seen from Figure 3A, the power trendline in Figure 2B for each type of particles now turns into a linear trendline, whose y-intercept and slope yield the nonlinear electrophoretic mobility,  ()  , and nonlinear index of electric field, , respectively.Interestingly, the three linear trendlines in Figure 3A are roughly parallel indicating marginal differences in  among the three types of particles.However, the peanut-shaped particle is apparently greater  ()  than the spherical and pear-shaped ones.Figure 3B compares the obtained values of  ()  and  among the three types of particles in terms of the particle slenderness,  = ∕, in Equation (1).One can see a decrease of  ()  , whereas an increase of  with the increase of  from the peanut to pear and spherical particles.However, the value of  still stays at around 2, which is consistent with our recent experiment [21] and within the range of theoretical predictions [7][8][9][10] for spherical particles at high electric fields.Referring to the findings in our study that  ()  and  both become greater for smaller spherical particles [21], we speculate that the decreasing trend of  ()  with the increase of  may be a result of the increasing particle radius, , perpendicular to the particle moving direction (i.e., the direction of the imposed DC electric field, see Figure 1A and Table 1), which plays an important role in the drag force [50].In contrast, the increasing trend of  with the increase of  may arise from the decreasing particle length along the electric field direction, leading to a larger curvature of the particle surface and hence a stronger surface conduction effect within the EDL [22,27].

Effect of buffer concentration on nonlinear electrophoresis of nonspherical particles
Our previous work demonstrates that spherical particles exhibit stronger nonlinear electrophoresis in lower concentration buffer solutions [21] because of the thicker EDL and hence stronger surface conduction effects therein [7][8][9][10].This trend should remain valid for nonspherical particles as the impact of buffer concentration on the ionic fluxes within and across the EDL is intuitively independent of particle shape.Figure 4A displays the experimentally obtained data of  ()  versus electric field for the peanut-shaped particles in 0.01, 0.025, and 0.05 mM buffers along with the corresponding power trendlines.Like the spherical particles in our previous study [21], nonspherical particles overall also have larger values of  ()  in the lower concentration buffers at each imposed electric field.Moreover, the differences in  ()  among the three buffer concentrations get increasingly large under higher electric fields.Figure 4B shows the extracted nonlinear electrophoretic components  ()  and  as a function of the buffer concentration.As expected, both  ()  and  exhibit a decreasing trend with the increase of buffer concentration for the peanut-shaped particles.Moreover, the values of  are still within the range of 3/2 and 2, consistent with the theoretical prediction of nonlinear electrophoresis for spherical particles at high electric fields [7][8][9][10].Similar results are also obtained for the pear-shaped particles in buffers of varying concentrations (see Figure S1)

CONCLUDING REMARKS
We have built upon our previous work [21] to experimentally study the effect of particle shape on the nonlinear electrophoresis of rigid particles in a rectangular microchannel under high electric fields.Both peanut- and pear-shaped particles have been tested along with spherical particles with approximately similar diameter and surface charge.A dimensionless parameter, that is, particle slenderness , is defined to quantify the particle shape, which increases from the peanut-to pear-shaped and spherical particles.We find that the nonlinear electrophoretic mobility  ()  decreases with the increasing particle slenderness, whereas the opposite goes to the nonlinear index  of electric field.It is speculated that these two trends may be associated with the particle dimension along and perpendicular to the electric field direction, respectively.We also find that the nonlinear index  for each type of nonspherical particles is still within the theoretically predicted range for spherical particles at high electric fields.Moreover, both  ()  and  are found to increase in a lower concentration buffer solution regardless of the particle shape.It is important to note that our experiments in both this and the earlier work [21] have been restricted to dilute particle suspensions.We will study in future work if and how the particle-particle interaction

TA B L E 1
Dimensions and slenderness values of the three types of particles.
()  =   −   , in Figure 2B.The peanut-shaped particles appear to have the largest  ()  .The spherical and pear-shaped particles display weaker, whereas overall similar nonlinear behaviors in  ()  over the range of electric fields under test.

F U R E 3
Nonlinear electrophoresis of spherical, pear and peanut-shaped particles in 0.025 mM buffer: (A) experimentally obtained (symbols with error bars) nonlinear electrophoretic velocity,  ()  , versus electric field in the log-log space, where the linear trendlines are the best fits to the data points; (B) comparison of the nonlinear electrophoretic mobility,  ()  , and nonlinear index of electric field, , as a function of the particle slenderness.The lines are used to guide the eyes only.
Nonlinear electrophoresis of peanut-shaped particles in buffer solutions with varying concentrations: (A) experimentally obtained (symbols with error bars) nonlinear electrophoretic velocity,  ()  , versus electric field, where the curves are the positive power trendlines best fitted for the data points; (B) comparison of the nonlinear electrophoretic mobility,  ()  , and nonlinear index of electric field, , with respect to the buffer concentration.The lines are used to guide the eyes only.