In this work, an analytic solution for the hydrodynamic dispersion of silver colloidal nanoparticles released into an oscillatory electroosmotic flow between microelectrodes of axially variable shape is presented. The longtime colloid concentration response is derived using the homogenization method together with multiplescale analysis. The results indicate that the deposition of nanoparticles onto the surface of the microelectrodes depends on the rate constant β of solute reaction at the wall, on the angular frequency ω, and mainly on the induced pressure gradient that arises due to the variable geometrical shape of the walls. For suitable values of the previous parameters, we show that colloidal nanoparticle concentration can be enhanced as well as choosing the location where it will happen.
I. INTRODUCTION
Having its origins in microanalytical methods, microfluidics enables the controlled manipulation and processing of fluids down to attoliter volume range, with high accuracy.^{1} In the context of sensing, microfluidics has advanced the sensitivity and resolution of detection systems by employing minute volumes of samples and reagents that would require high volumes (several orders of magnitude in comparison) when using established macroscopic techniques. Even with the remarkable advances made in the field of microfluidics and their demonstrated impact in the analytical chemistry field, detailed analysis of the transport of analyte to the active sensing surface of micro and nanostructured sensors is yet to be performed. As a sensing mechanism, surfaceenhanced Raman scattering (SERS) is an emerging technology that provides a specific spectral fingerprint of an analyte via inelastic light scattering from a sample,^{2} amplifying singlemolecule detection, thus enabling its identification.^{3} A major obstacle of SERS is its low signal intensity. The use of metallic (e.g., Ag, Au, or Al) nanostructures has been proposed for increasing the low SERS sensing signal,^{4–6} where the enhancement of analyte detection (illicit drugs, food contaminants, pesticides, etc.) efficiency has been achieved.^{7,8} In addition, it has been shown that the SERS signal intensity of target analytes was enhanced by means of concentration amplification when alternating current (AC) electric field effects were used for the collection of the analyte onto the detection site. These electric field effects were made possible with the incorporation of microelectrodes into the SERS active substrate and were shown to be dependent on the microelectrode shape and configuration (e.g., round,^{7} triangular,^{9} and rectangular^{10} shapes). Thus, a combination of electrokinetic phenomena, such as electroosmosis and electrophoresis, with a correct microelectrode design can further improve the SERS signal.
Electroosmosis consists of the movement of an electrolyte solution relative to a stationary charged surface induced by an external electric potential.^{11,12} If an oscillatory electroosmotic flow (OEF) is used, as a consequence of using AC, the solute being carried by the fluid will be subject to mass transfer beyond molecular diffusion, known as Taylor dispersion.^{13} This dispersion effect has been shown to be dependent on the fluid speed, crosssectional geometry, and the AC angular frequency,^{14} where the latter has already enabled maximum electrodeposition in a parallel flat porous plate by operating near resonant frequency.^{15} In this sense, a lot of analytical, experimental, and numerical studies in oscillatory flows have been developed to enhance the dispersion of a passive solute in a variety of circumstances.^{16–23}
Under an externally applied AC field, an induced dipole dependent on the electric frequency will develop on the polarized particles, creating an interaction with the AC electric field known as the dielectrophoretic effect (DEP)^{24} and a particle motion called electrophoresis.^{25} This particle motion is caused by the action of a force, known as dielectrophoretic force, which can be either attractive or repulsive according to the orientation on the dipole. On this subject, Hölzel et al.^{26} conclude that even when electrohydrodynamic effects (i.e., electroosmotic flow) interfere with dielectrophoresis,^{27} a large DEP force will be felt in the nanoparticles as a consequence of charge movements both in the Stern layer and in the diffuse double layer, which surrounds the particles in an aqueous solution. In a similar manner, Dies et al.^{28} considered the DEP force as the primary force of nanoparticle assembly. In their analysis, the electrical double layer (EDL) at the microelectrodes is AC frequency dependent, where a threshold between fully developed and not formed is shown, obtaining an intermediate frequency where the AC electroosmotic flow is maximized. From another standpoint, Green et al.^{29} stated that the force exerted by the fluid on the particles was more than sufficient to override the DEP force at low frequencies. Furthermore, the fluid flow shows a great dependence on the AC frequency, increasing and falling in magnitude as the frequency is decreased.^{30}
In this work, we determine the potential of oscillatory electroosmotic flow (OEF), at low frequencies with irregular shape microelectrodes, for improving colloidal transport. While the effect of crosssectional shape on hydrodynamic dispersion has been previously studied,^{29,31,32} there has not been an analytical solution that fully explains the main parameters that drive the colloids between microelectrodes. In this context, by modeling a previous experiment,^{7} our results lead to the conclusion that the microelectrode's shape is responsible for inducing a pressure gradient, which follows an inverse cubic power relationship with the former, and the enhancement of colloidal transport as its consequence. Therefore, the present analysis aims to broaden the understanding of the importance of OEF in colloidal transport, where the significance of hydrodynamic forces in enhancing mass transport cannot be overstated.
II. HYDRODYNAMIC FORMULATION
In Fig. 1(a), a quadrupolar electrode array connected to an AC is shown. Uncharged colloidal silver nanoparticles (AgNPs) are transported through an axially variable isothermal channel mainly due to an OEF of a Newtonian fluid. This flow will appear from the electroosmotic forces caused by an external imposed oscillatory electric field E and the induced electric field in the electrical double layer (EDL), where AgNPs do not contribute to the latter. Figure 1(b) shows a quarter of the quadrupolar array where NPs enter the left side at a constant concentration $ C \u2217 $. Since observing the electrical deposition between microelectrodes is of interest and considering the symmetry of the quadrupolar electrode array regarding the electrokinetic phenomena, this closeup will be our main domain for the analytical solution. Its length is defined as $ L = 2 R e h 0 $, where R_{e} is the radius of the microelectrodes, and h_{0} is the minimum gap halfheight. The variable shape of the microelectrodes walls is defined with the aid of the function $ h ( x \u0303 ) $, e.g., for a round shape $ h ( x \u0303 ) = \xb1 h 0 ( 1 + x \u0303 2 2 R e h 0 ) $, where $ \xb7 \u0303 $ indicates that the variable has dimensional units. For convenience, the origin of the coordinate system $ x \u0303 , y \u0303 $ is placed at the center between the gap of microelectrodes, as shown in Fig. 1(b), where $ x \u0303 $ and $ y \u0303 $ are longitudinal and transverse coordinates, respectively.
A. Governing equations
B. Nondimensional mathematical model
C. Homogenization method
As mentioned, the concentration of external NPs is governed by Eq. (9), and for the analysis, we focus on the transport at a scale 2 L greater than the gap between microelectrodes $ 2 h 0 $. To obtain an analytical solution for the concentration field, the homogenization method^{41} is proposed to derive an expression that allows us to evaluate the convective diffusion equation. Therefore, four distinct time scales are involved in the analysis of colloidal transport, which are as follows:

the scale time due to the angular frequency, $ t 0 \u223c 2 \pi / \omega $,

the convective time, $ t 1 \u223c 2 L / U H S $,

the transversal diffusion time, $ t 2 \u223c 4 h 0 2 / D $,

the longitudinal diffusion time, $ t 3 \u223c 4 L 2 / D $.
In this context, Table I shows current values of these characteristic times involved in this study, considering $ R p = 25 \u2009 nm $,^{7} $ D = 8.58 \xd7 10 \u2212 12 \u2009 m 2 / s $, and $ U H S = 3.7 \xd7 10 \u2212 4 \u2009 m / s $.
Times (s) .  $ O ( 10 \u2212 1 ) $ .  $ O ( 10 1 ) $ .  $ O ( 10 3 ) $ . 

t_{0}  1  ⋯  ⋯ 
t_{1}  2  ⋯  ⋯ 
t_{2}  ⋯  3.8  ⋯ 
t_{3}  ⋯  ⋯  0.8 
Times (s) .  $ O ( 10 \u2212 1 ) $ .  $ O ( 10 1 ) $ .  $ O ( 10 3 ) $ . 

t_{0}  1  ⋯  ⋯ 
t_{1}  2  ⋯  ⋯ 
t_{2}  ⋯  3.8  ⋯ 
t_{3}  ⋯  ⋯  0.8 
Equation (75) was solved with the boundaries conditions Eqs. (72) and (73), using the fourth orderRunge–Kutta method with shooting approximation. We obtain our computational results using Mathematica (version 11.2.0.0), where a grid with a total of 1000 nodes in the xdirection was employed. Crosssectional averages $ A \xaf 1 ( x ) $ and $ A \xaf 2 ( x ) $ are solved using Simpson's 3/8 rule with $ 10 \u2212 3 $ precision, which sets the size of our grid. The results were modified using the transfinite interpolation method^{43} to simulated the geometrical variable shape of the microelectrodes.
III. RESULTS AND DISCUSSION
In Figs. 3, 4, and 5, the variables C_{x} and $ B \omega $ are shown for rectangular, round, and triangular shapes, respectively. The variable $ B \omega $ has been multiplied by the constants ε and Pe ( $ \epsilon P e = 86 $) for better interpretation of the figures. In almost all cases, the product of $ B \omega $ with $ \u2202 C x / \u2202 x $ is at least one order of magnitude higher than C_{x}, allowing the analysis of timeharmonic response with only considering this product. The exception to the aforementioned is shown in Fig. 3(a), where a rectangular shape at β = 0 is one order of magnitude lower than C_{x}, indicating that the timeharmonic concentration field distribution will be linear since $ C x = ( 1 \u2212 x ) / 2 $.
In Figs. 4(d) and 5(d), the cases for β = 0 are shown, in which a high concentration of external particles appears at the center of the microchannel mainly due to the protrusions formed by the microelectrodes shape. These values can be sign changed by the C_{x} gradient and due to the oscillating period. On this subject, the positive magnitude of $ B \omega $ should indicate an abundance of NPs and a negative value a deficit. In both figures, two noticeable effects appear, first that the triangular shape acts as a better NPs attractor than the round shape, which could be associated with sharp edges at the walls. Second, the appearance of two locations where timeharmonic concentration field show a three order of magnitude higher than the rest of the system, appearing at the pressure gradient zeroes as shown in Figs. 4(c) and 5(c).
In Figs. 3(b), 4(e), and 5(e), the effects of considering a firstorder reaction at the wall are shown. The value of the parameter $ \beta = 0.7 $ has been selected from Eq. (57), where $ B \omega $ presents a maximum value when its denominator approaches zero, i.e., $ \beta = 1 / 2 $. The most outstanding result is the position of NPs concentration located at one side of the microchannel walls regardless of its shape. This happens because a chemical reaction creates an $ \pi / 6 $ out of phase response at the reactive wall while lagging the other wall by $ 3 \pi / 2 $, as calculated from the complex argument of Eq. (57). Another important effect is the movement of high order concentration locations from $ x = \xb1 0.41 $ to $ x = \xb1 0.61 $ for the round shape and $ x = \xb1 0.33 $ to $ x = \xb1 0.57 $ for the triangular shape. These values cannot be calculated so easily through the pressure gradient, but can be determined with the roots of $ A 1 \xaf ( x ) , $ crosssectional average of Eq. (68). These high order concentration field locations act as attractors or allegorically as gates that benefit the mixing and reactions at the center of the microchannel, or more precisely where the pressure gradient is negative.
In Figs. 3(c), 4(f), and 5(f), the effects of considering $ \beta \u226b 1 $ are shown. The aspect that stands out is that C_{x} is linear regardless of $ \delta ( x ) $, indicating that when $ \beta \u226b 1 $, the reaction at the wall is faster than the supply of nanoparticles by diffusion. Thus, the mass transport process in the microchannel becomes diffusion controlled. This can be inferred first with the rectangular solution of the firstorder velocity vector U(y), which only presents a gradient near the walls and can be mostly considered U = 1. Neglecting near the wall effects since c = 0 at $ y = \xb1 1 $, the previous assumption of the velocity vector can be seen in Eq. (58), where the effects of convection on the variable $ B \omega $ are purely imaginary. Since Figs. 3(c), 4(f), and 5(f) present similar results, the previous statement is valid for the current results.
The sharp peaks in concentration gradients correspond to the analytical solution, but in actual experiments, these peaks may vary in shape due to electrokinetic interaction between particles or local changes in ionic concentration due to electromigration. Even when the confinement of the gradients is small (i.e., nanoscale) and measurements at those length scales are challenging, further experimental investigation is required to verify the extend of the gradients. Regardless, the previous asymptotic solution for the dimensionless concentration has experimental validation with the work of Dies et al.,^{7} in particular the one sidewall phenomenon that is shown in Fig. 3(c) and an electrochemical reaction that damages the microelectrode located at the roots of $ A \xaf 1 ( x ) $ for an OEF at $ f = 1 \u2009 Hz $. These results indicate that if the transversal diffusion time acts as the long timescale, then there will be enough time for the rate of disappearance β to increase, creating high concentration points at the microelectrodes walls that could possibly damage the opposite microelectrode if reached. This can be deduced from the nondimensional parameter $ \beta = \beta \u0303 h 0 / D $, which is related to the transversal diffusion time if we consider the distance between microelectrodes $ 2 h 0 $. Another observation occurs if the longitudinal diffusion time is the longscale time and considering a reaction at the walls. While high concentrations points could appear at short time span (transversal diffusion time), in the end colloidal transport will be dominated by diffusion but with the special characteristic of symmetric concentration at both walls.
IV. CONCLUSIONS
Colloidal dispersion due to an oscillatory electroosmotic flow between microelectrodes of axially variable shape has been carried out by deriving an analytical expression for the hydrodynamic forces and the silver concentration considering the longtime colloid concentration response. The axially variable shape of the microelectrodes induces a pressure gradient, which significantly modifies the colloidal concentration field in the longitudinal direction with the appearance of high concentration locations that correspond to the pressure gradient zeroes. If an irreversible reaction is catalyzed by the wall, colloids will aggregate at the microelectrodes edges with a preference for wall protrusions, while the highconcentration points will fluctuate to a maximum far from the center of the microchannel. When reaction is faster than mass transport limited regime, most nanoparticles will start reacting preferentially in the previous mentioned highconcentration areas. At this point in time, colloidal transportation is achieved solely by diffusion.
The following key improvements over the literature were obtained:

High concentration locations are a consequence of hydrodynamic forces in combination with $ \delta ( x ) $ and could not be attributed only to the dielectrophoretic force.

An irreversible firstorder reaction catalyzed by the wall, combined with microelectrodes of axially variable shape, could generate a short circuit if the transversal diffusion time acts as the longtime scale.

While the Clausius–Mossotti factor plays an important role in showing the location of NPs, β dictates if the NPs have a symmetric or asymmetric concentration at the microelectrode's walls.
ACKNOWLEDGMENTS
C. Vargas acknowledge the support from DGAPA program for a postdoctoral fellowship at UNAM. C. Escobedo gratefully acknowledge financial support from the Gouvernement du Canada—Natural Sciences and Engineering Research Council of Canada (NSERC) Valid Funder: No. RGPIN20105138, Canada Foundation for Innovation (CFI) Valid Funder: No. 31967, and FEAS Excellence in Research Award Queen's University.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Carlos Vargas: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Federico Méndez: Conceptualization (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Aristides Docoslis: Conceptualization (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Carlos Escobedo: Conceptualization (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding authors upon reasonable request.