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 long-time colloid concentration response is derived using the homogenization method together with multiple-scale 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.

Topics
Colloids, Asymptotic analysis, Nanoparticle, Electrophoresis, Microchannel, Electroosmosis, Microfluidics, Hydrodynamic dispersion, Microelectrode arrays

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, surface-enhanced 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 single-molecule 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 rectangular10 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, cross-sectional 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 cross-sectional 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.

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 ∗ ⁠. Since observing the electrical deposition between microelectrodes is of interest and considering the symmetry of the quadrupolar electrode array regarding the electrokinetic phenomena, this close-up will be our main domain for the analytical solution. Its length is defined as L = 2 R e h 0 ⁠, where Re is the radius of the microelectrodes, and h0 is the minimum gap half-height. The variable shape of the microelectrodes walls is defined with the aid of the function h ( x ̃ ) ⁠, e.g., for a round shape h ( x ̃ ) = ± h 0 ( 1 + x ̃ 2 2 R e h 0 ) ⁠, where · ̃ indicates that the variable has dimensional units. For convenience, the origin of the coordinate system x ̃ , y ̃ is placed at the center between the gap of microelectrodes, as shown in Fig. 1(b), where x ̃ and y ̃ are longitudinal and transverse coordinates, respectively.

FIG. 1.
Schematic representation of the studied physical model top-down view (not to scale). (a) Quadrupolar Au microelectrodes connected to an AC source are used to guide the assembly of colloidal silver nanoparticles. (b) Close-up to the domain where silver nanoparticles assemble.
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Schematic representation of the studied physical model top-down view (not to scale). (a) Quadrupolar Au microelectrodes connected to an AC source are used to guide the assembly of colloidal silver nanoparticles. (b) Close-up to the domain where silver nanoparticles assemble.

FIG. 1.
Schematic representation of the studied physical model top-down view (not to scale). (a) Quadrupolar Au microelectrodes connected to an AC source are used to guide the assembly of colloidal silver nanoparticles. (b) Close-up to the domain where silver nanoparticles assemble.
View largeDownload slide

Schematic representation of the studied physical model top-down view (not to scale). (a) Quadrupolar Au microelectrodes connected to an AC source are used to guide the assembly of colloidal silver nanoparticles. (b) Close-up to the domain where silver nanoparticles assemble.

Close modal
The governing equations that allow determining the starting point for studying the OEF are the following continuity equation:
∇ · u ̃ = 0 ,
(1)
and the Navier–Stokes equations are given by
ρ m [ ∂ u ̃ ∂ t ̃ + ( u ̃ · ∇ ) u ̃ ] = − ∇ P ̃ + ∇ · τ ̃ − ρ e ∇ Φ ̃ .
(2)
In the above-mentioned equations, ρm is the fluid density and is considered a constant, u ̃ = ( u ̃ , v ̃ ) is the velocity vector, t ̃ is the time, P ̃ is the pressure, ∇ is the Nabla operator defined as ∇ ≡ ( ∂ / ∂ x ̃ , ∂ / ∂ y ̃ ) ⁠, and τ ̃ is the stress tensor for a Newtonian fluid, given by
τ ̃ = μ ( ∇ u ̃ + ∇ u ̃ T ) ,
(3)
where μ is the dynamic viscosity of the fluid. The hydrodynamic boundary conditions related to the impermeability and no-slip condition at the microelectrodes walls are given by
u ̃ · n = 0 ,
(4)
and
u ̃ − ( u ̃ · n ) n = 0 ,
(5)
respectively. n represents the unit vector normal to the microelectrode surface pointing toward the fluid. The pressure is P0 at x ̃ = ± L ⁠. The electrical body force in Eq. (2) consists of the electric charge density ρe and the total electrical potential Φ ̃ ⁠. The former follows the Boltzmann distribution,33,
ρ e = − 2 z e n ∞ sinh ( z e ψ ̃ k B T ) ,
(6)
where z is the valence of the electrolyte, e represents the fundamental charge of an electron, n ∞ is the ionic number in the concentration in the bulk solution, kB is the Boltzmann constant, and T is the absolute temperature. The total electrical potential Φ ̃ is governed by the following Poisson equation:
∇ 2 Φ ̃ = − ρ e ε m ,
(7)
where εm denotes the dielectric permittivity of the medium. Here, Φ ̃ is split into the induced non-uniform equilibrium potential in the EDL, ψ ̃ ( x ̃ , y ̃ ) ⁠, and the potential describing the external electric field γ ( x ̃ , t ̃ ) = − x ̃ E x   sin   ( ω t ̃ ) ⁠, where E x = ϕ 0 / 2 L ,   ϕ 0 is the voltage provided by the generator, and ω is the angular frequency of the electrical source. Regarding the external electric field component in the y ̃ coordinate, it is known that when an AC signal is applied to the electrodes, a fraction of the potential is dropped across the EDL, an effect that is referred to as electrode polarization.34 The effect is frequency dependent because the most dropped of the applied voltage occurs across the EDL at low frequencies.29 Thus, at low frequency, Ey = 0 can be considered. The boundary condition for ψ ̃ at the microelectrode walls is
ψ ̃ = ζ   at   y ̃ = ± h ( x ̃ ) ,
(8)
where ζ is the zeta potential. The concentration field consists of external NPs, which are electrically neutral, governed by the convective diffusion equation,11,
∂ c ̃ ∂ t ̃ + u ̃ · ∇ c ̃ = D ∇ 2 c ̃ ,
(9)
where c ̃ is the concentration of the diffusing NPs, and D is the molecular diffusion coefficient estimated using the well-known Stokes–Einstein equation,35,
D = k B T 6 π μ R p ,
(10)
where Rp is the radius of the NPs. The concentration field is subject to an oxidation-reduction reaction at the microelectrode's walls, while a constant finite NPs concentration is fed from the left side and all stay constraint within our domain. These conditions are stipulated as follows:
D ∂ c ̃ ∂ y ̃ + β ̃ c ̃ = 0   at   y = ± h ( x ̃ ) ,
(11)
c ̃ = C ∗   at   x ̃ = − L ,
(12)
c ̃ = 0   at   x ̃ = L ,
(13)
and
c ̃ = C ∗ f ( x ̃ )   at   t ̃ = 0 ,
(14)
where β ̃ is the rate of disappearance of solute due to an irreversible first-order reaction catalyzed by the wall,36,37 and C ∗ is the initial concentration in mol / m 3 units.
The governing equations in Sec. II A, together with their corresponding boundary conditions, can be written in a non-dimensional form by introducing the dimensionless variables: x = x ̃ / L ,   y ′ = y ̃ / h 0 ,   u = u ̃ / U H S ,   v = v ̃ L / U H S h 0 ,   p ′ = ( P ̃ − P 0 ) h 0 2 / μ U H S L ,   t = t ̃ ω , τ = τ ̃ h 0 / U H S μ ,   ψ = ψ ̃ z e / k B T ,   Φ = Φ ̃ / ϕ 0 ⁠, and c = c ̃ / C ∗ ⁠. Here, U H S = − ε m ζ E x / μ is the Helmholtz–Smoluchowski velocity. When defining the non-dimensional pressure p ′ ⁠, we have introduced the useful definition P = p ′ − α k 2 ψ 2 ⁠,38 where α = ζ / Φ 0 ,   k = h 0 / λ D ⁠, and λ D = ( ε m k B T / 2 e 2 z 2 n ∞ ) 1 / 2 ⁠. In addition to the non-dimensional form, and because the microelectrodes have a variable shape, a simple transformation related to the dimensionless y ´ coordinate is proposed as y = y ′ / δ ( x ) ⁠,39 where δ ( x ) = 1 + x 2 ⁠. Therefore, the expanded non-dimensional forms of the hydrodynamic and electrical governing Eqs. (1), (2), and (7) are as follows:
∂ u ∂ x + ∂ u ∂ y ∂ y ∂ x + ∂ v ∂ y ∂ y ∂ y ′ = 0 ,
(15)
R e ω ∂ u ∂ t + ε R e L ( u ∂ u ∂ x + u ∂ u ∂ y ∂ y ∂ x + v ∂ u ∂ y ∂ y ∂ y ′ ) = − ∂ P ∂ x − ∂ P ∂ y ∂ y ∂ x + k 2 ψ   sin   ( t ) + ∂ u ∂ y ∂ 2 y ∂ y ′ 2 + ∂ 2 u ∂ y 2 ( ∂ y ∂ y ′ ) 2 +  2 ε [ ∂ 2 u ∂ x 2 + 2 ∂ 2 u ∂ x ∂ y ∂ y ∂ x + ∂ u ∂ y ∂ 2 y ∂ x 2 + ∂ 2 u ∂ y 2 ( ∂ y ∂ x ) 2 ] +  ε [ ∂ 2 v ∂ x ∂ y ∂ y ∂ y ′ + ∂ 2 v ∂ y 2 ∂ y ∂ y ′ ∂ y ∂ x + ∂ v ∂ y ∂ 2 y ∂ x ∂ y ′ ] ,
(16)
ε R e ω ∂ v ∂ t + ε 2 R e L ( u ∂ v ∂ x + u ∂ v ∂ y ∂ y ∂ x + v ∂ v ∂ y ∂ y ∂ y ′ ) = − ∂ P ∂ y ∂ y ∂ y ′ + ε [ ∂ 2 u ∂ x ∂ y ∂ y ∂ y ′ + ∂ 2 u ∂ y 2 ∂ y ∂ x ∂ y ∂ y ′ + ∂ u ∂ y ∂ 2 y ∂ x ∂ y ′ + 2 ∂ v ∂ y ∂ 2 y ∂ y ′ 2 + 2 ∂ 2 v ∂ y 2 ( ∂ y ∂ y ′ ) 2 ] ε 2 [ ∂ 2 v ∂ x 2 + 2 ∂ 2 v ∂ x ∂ y ∂ y ∂ x + ∂ v ∂ y ∂ 2 y ∂ x 2 + ∂ 2 v ∂ y 2 ( ∂ y ∂ x ) 2 ] ,
(17)
and
ε [ ∂ 2 ψ ∂ x 2 + 2 ∂ 2 ψ ∂ x ∂ y ∂ y ∂ x + ∂ ψ ∂ y ∂ 2 y ∂ x 2 + ∂ 2 ψ ∂ y 2 ( ∂ y ∂ x ) 2 ] + ∂ 2 ψ ∂ y 2 ( ∂ y ∂ y ′ ) 2 + ∂ ψ ∂ y ∂ 2 y ∂ y ′ 2 = k 2 ψ .
(18)
In Eqs. (15)–(18), ε = ( h 0 / L ) 2 ,   R e ω = ρ m ω h 0 2 / μ ⁠, and R e L = ρ m U H S L / μ are the angular and longitudinal Reynolds numbers, respectively. It should be noted that the Debye–Hückel approximation was considered in the Boltzmann distribution, Eq. (6). The dimensionless boundary conditions of Eqs. (15)–(18) are
u = v = 0   at   y = ± 1 ,
(19)
P = 0   at   x = ± 1 ,
(20)
ψ = 1   at   y = ± 1.
(21)
In microfluidics systems, typical values of the parameter ε (⁠ h 0 = 9 μ m , R e = 90 μ m ,   L = 40 μ m ⁠, and ε = 0.05 ⁠) and the longitudinal Reynolds number are very small (⁠ U H S = 3.7 × 10 − 4   m / s and R e L = 1.5 × 10 − 2 ⁠). Therefore, a regular perturbation technique40 can be used to solve the set of mentioned equations for small values of the parameter ε. Thus, a regular expansion is proposed for each dependent variable (say, X) in the following form:
X = X 0 + ε X 1 + O ( ε 2 ) ,
(22)
where X = u , v , P ⁠. Substituting the expansion Eq. (22) in the non-dimensional hydrodynamic and electrical Eqs. (15)–(18), and collecting terms of O ( 1 ) ⁠, we obtain the following problems:
∂ u 0 ∂ x − y δ ( x ) d δ d x ∂ u 0 ∂ y + 1 δ ( x ) ∂ v 0 ∂ y = 0 ,
(23)
R e ω ∂ u 0 ∂ t = − ∂ P 0 ∂ x + y δ ( x ) d δ d x ∂ P 0 ∂ y + 1 δ ( x ) 2 ∂ 2 u 0 ∂ y 2 + k 2 ψ 0   sin   ( t ) ,
(24)
∂ P 0 ∂ y = 0 ,
(25)
and
∂ 2 ψ 0 ∂ y 2 = k 2 δ ( x ) 2 ψ 0 .
(26)
From Eq. (25), P 0 = P 0 ( x , t ) and should be determined as a part of the hydrodynamics problem together with u0 and v0. The solution of the Poisson–Boltzmann Eq. (26), considering the O ( 1 ) boundary condition Eq. (21), is given by
ψ 0 = cosh [ k δ ( x ) y ] cosh [ k δ ( x ) ] .
(27)
To solve the previous hydrodynamic equations at O ( 1 ) ⁠, a periodic solution that ignores the transient stage is proposed, specifying that the hydrodynamic dependence on time is carried out through an oscillating behavior. An oscillatory solution for each dependent variable is assumed of the following form:
( u 0 , v 0 , P 0 ) = Im [ ( U ( x , y ) , V ( x , y ) , P ( x ) ) exp   ( i t ) ] ,
(28)
where Im [ X 0 ] represents the imaginary part of the complex quantity X0, and i = − 1 is the imaginary number. Substituting Eq. (28) in Eqs. (23) and (24) yields
∂ U ∂ x − y δ ( x ) d δ d x ∂ U ∂ y + 1 δ ( x ) ∂ V ∂ y = 0 ,
(29)
1 δ ( x ) 2 ∂ 2 U ∂ y 2 − i R e ω U = d P d x − k 2 ψ 0 .
(30)
The associated boundary conditions of Eqs. (29) and (30) are
U = V = 0   at   y = ± 1 ,
(31)
P = 0   at   x = ± 1.
(32)
The general solution of Eq. (30) is obtained by substituting Eq. (27) in the momentum equation and integrating the result twice with respect to y, taking into account the no-slip boundary condition [Eq. (31)],
U = cosh [ ( 1 + i ) R e ω 2 δ ( x ) y ] cosh [ ( 1 + i ) R e ω 2 δ ( x ) ] ( 1 − i R e ω d P d x ) + i R e ω d P d x − k 2 ( k 2 − i R e ω ) cosh [ k δ ( x ) y ] cosh [ k δ ( x ) ] .
(33)
Equation (33) can be simplified when R e ω ≪ 1 ⁠, i.e., low frequencies (f = 10 Hz, R e ω = 2 π × 10 − 3 ⁠, see  Appendix). Therefore, taking into account a low frequency source, U is simplified as
U = δ ( x ) 2 2 d P d x ( y 2 − 1 ) + 1 − cosh [ k δ ( x ) y ] cosh [ k δ ( x ) ] .
(34)
At this point, the pressure gradient is unknown and can be obtained using the continuity equation, Eq. (29). Substituting Eq. (34) in Eq. (29), integrating the result in transverse direction, and applying the impermeability condition V = 0 at y = 1 yield
V = δ ( x ) 3 2 d 2 P d x 2 ( y − y 3 3 − 2 3 ) + δ ( x ) 2 d δ d x d P d x ( y − 1 ) − tanh [ k δ ( x ) ] d δ d x ( sinh [ k δ ( x ) y ] cosh [ k δ ( x ) ] − tanh [ k δ ( x ) ] ) .
(35)
When applying the boundary condition V = 0 at y = −1 to Eq. (35), a second-order ordinary differential equation is obtained to determine the pressure field, given by
d 2 P d x 2 + 3 δ ( x ) d δ d x d P d x = 3 tanh [ k δ ( x ) ] 2 δ ( x ) 3 d δ d x .
(36)
Equation (36) is solved using integrating factors and the transformation z ( x ) = d P / d x ⁠; therefore, a first integration of Eq. (36) yields that
d P d x = 3 δ ( x ) 2 − 3 tanh [ k δ ( x ) ] k δ ( x ) 3 + A δ ( x ) 3 ,
(37)
where A is a constant. Equation (37) cannot be analytically integrated with respect to x; thus, Simpson's 3/8 rule is used to solve Eq. (37) applying the boundary conditions Eq. (32), yielding a value of A = − 3.5 for a round shape (⁠ δ ( x ) = 1 + x 2 ⁠) and A = −4 for a triangular shape (⁠ δ ( x ) = 1 + | x | ⁠). V is simplified by substituting Eq. (36) in Eq. (35),
V = d δ d x { δ ( x ) 2 2 d P d x ( y 3 − y ) + tanh [ k δ ( x ) ] 2 2 ( 3 y − y 3 ) − tanh [ k δ ( x ) ] sinh [ k δ ( x ) y ] cosh [ k δ ( x ) ] } .
(38)
The next step is to evaluate the solutions at O ( ε ) for u1, v1, and P1. However, at this order, the solution seeks time-averaged effects, defined for any function f as ⟨ f ⟩ = 1 2 π ∫ 0 2 π f d t ⁠, where the angular brackets ⟨ · ⟩ will indicate the time average of the quantity inside. This analysis results in ⟨ u 1 ⟩ = ⟨ v 1 ⟩ = ⟨ P 1 ⟩ = 0 since these variables present a linear time solution, where only at O ( ε 2 ) the products of periodic functions will present a non-zero time average solution.

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 method41 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 ∼ 2 π / ω ⁠,

  • the convective time, t 1 ∼ 2 L / U H S ⁠,

  • the transversal diffusion time, t 2 ∼ 4 h 0 2 / D ⁠,

  • the longitudinal diffusion time, t 3 ∼ 4 L 2 / D ⁠.

In this context, Table I shows current values of these characteristic times involved in this study, considering R p = 25   nm ⁠,7  D = 8.58 × 10 − 12   m 2 / s ⁠, and U H S = 3.7 × 10 − 4   m / s ⁠.

TABLE I.

Time scale analysis of colloidal transport.

Times (s) O ( 10 − 1 ) O ( 10 1 ) O ( 10 3 )
t0  1  ⋯  ⋯ 
t1  2  ⋯  ⋯ 
t2  ⋯  3.8  ⋯ 
t3  ⋯  ⋯  0.8 
Times (s) O ( 10 − 1 ) O ( 10 1 ) O ( 10 3 )
t0  1  ⋯  ⋯ 
t1  2  ⋯  ⋯ 
t2  ⋯  3.8  ⋯ 
t3  ⋯  ⋯  0.8 
View Large
From Table I, the following ratios are obtained:
( 2 π ω ∼ L U H S ) : 4 h 0 2 D : 4 L 2 D = 2 π ω ( 1 : 1 ε : 1 ε 2 ) .
(39)
Introduce the four time coordinates
t 0 = t 1 = t ,   t 2 = ε t ,   t 3 = ε 2 t ,
(40)
and use Eq. (22) to expand for the dimensionless concentration
c = c 0 + ε c 1 + ε 2 c 2 + O ( ε 3 ) ,
(41)
where c i = c i ( x , y , t , t 2 , t 3 ) ⁠. The original time derivative becomes, according to the chain rule
∂ ∂ t → ∂ ∂ t + ε ∂ ∂ t 2 + ε 2 ∂ ∂ t 3 .
(42)
Substituting Eqs. (41) and (42) in the dimensionless version of Eq. (9) yields
( ∂ ∂ t + ε ∂ ∂ t 2 + ε 2 ∂ ∂ t 3 ) ( c 0 + ε c 1 + ε 2 c 2 ) +  ε P e ( u 0 ∂ ( c 0 + ε c 1 ) ∂ x + v 0 ∂ ( c 0 + ε c 1 ) ∂ y ∂ y ∂ y ′ ) = ε ∂ 2 ( c 0 + ε c 1 ) ∂ x 2 + ∂ 2 ( c 0 + ε c 1 + ε 2 c 2 ) ∂ y 2 ( ∂ y ∂ y ′ ) 2 + ∂ ( c 0 + ε c 1 + ε 2 c 2 ) ∂ y ∂ 2 y ∂ y ′ 2 ,
(43)
where P e = U H S L / D is the Péclet number. At order O ( 1 ) ⁠, the governing equation is given by
∂ c 0 ∂ t = 1 δ ( x ) 2 ∂ 2 c 0 ∂ y 2 ,
(44)
subject to the dimensionless boundary condition [from Eq. (11)],
1 δ ( x ) ∂ c 0 ∂ y + β c 0 = 0   at   y = ± 1 ,
(45)
where β = β ̃ h 0 / D ⁠. The general solution for the leading order, using Fourier method, is
c 0 = C x ( x , t 2 , t 3 ) + ∑ n = 1 ∞ B n   exp   [ − t ( π ( 2 n − 1 ) 2 δ ( x ) ) 2 ] × [ sin   ( y π ( 2 n − 1 ) 2 ) + 2 β δ ( x ) π ( 2 n − 1 ) cos   ( y π ( 2 n − 1 ) 2 ) ] ,
(46)
where Bn is obtained from Eq. (14),
B n = 2 β δ ( x ) ( δ 2 ( x ) − C x ) ( − 1 ) n + 1 ,
(47)
where δ 2 ( x ) = ( 1 − x ) / 2 is the non-dimensional form of f ( x ̃ ) ⁠. For the case when β = 0, Bn = 0, which indicates that c0 is independent of y, which can be inferred from the boundary condition [Eq. (45)]. The procedure that determines the function Cx is given lines below. Limiting ourselves to the behavior long after periodicity is completed, i.e., t ∼ O ( 1 / ε ) ⁠, we shall omit the series part in Eq. (46)41 and take the solution to be
c 0 = C x ( x , t 2 , t 3 ) ,
(48)
where Cx does not depend on t or y. Taking the O ( ε ) from Eq. (43) and considering Eq. (28) yields the governing equation for c1,
∂ C x ∂ t 2 + ∂ c 1 ∂ t + P e ( Im [ U ( x , y )   exp   ( i t ) ] ∂ C x ∂ x ) = ∂ 2 C x ∂ x 2 + 1 δ ( x ) 2 ∂ 2 c 1 ∂ y 2 ,
(49)
with the boundary condition
1 δ ( x ) ∂ c 1 ∂ y + β c 1 = 0   at   y = ± 1.
(50)
As previously stated in the hydrodynamic section, the transient stage is ignored, seeking the time-harmonic response after the initial transient relative to the shortest timescale. Therefore, the next step is to evaluate the time average of Eq. (49) over a full period as follows:
∂ C x ∂ t 2 = ∂ 2 C x ∂ x 2 + 1 δ ( x ) 2 ∂ 2 ⟨ c 1 ⟩ ∂ y 2 ,
(51)
where the time average on scale t does not affect the mathematical operators of Cx since these do not depend on time t. The cross-sectional average of any function f is defined as f ¯ = 1 2 ∫ − 1 1 f d y ⁠, where · ¯ will indicate the averaged function. Thus, the cross-sectional average of Eq. (51) is
∂ C x ∂ t 2 = ∂ 2 C x ∂ x 2 ,
(52)
where the second right-hand term in Eq. (51) vanishes since ⟨ c 1 ¯ ⟩ does not depend on y; therefore, its derivative with respect to y is zero. Subtracting Eq. (52) from Eq. (49) yields
∂ c 1 ∂ t + P e ( Im [ U ( x , y )   exp   ( i t ) ] ∂ C x ∂ x ) = 1 δ ( x ) 2 ∂ 2 c 1 ∂ y 2 .
(53)
Considering the linearity of Eq. (53), the solution c1 can be expressed as
c 1 = P e ∂ C x ∂ x { Im [ B ω ( x , y )   exp   ( i t ) ] } ,
(54)
and the substitution of Eqs. (53) and (50) leads to a second-order ordinary differential equation for B ω as follows:
i B ω + U ( x , y ) = 1 δ ( x ) 2 ∂ 2 B ω ∂ y 2 ,
(55)
with the boundary condition
1 δ ( x ) ∂ B ω ∂ y + β B ω = 0   at   y = ± 1 ,
(56)
where U(x, y) is obtained from Eq. (34). Solving Eq. (55) yields
B ω = 2 sinh [ δ ( x ) i ] ( i δ ( x ) d P d x − k tanh [ k δ ( x ) ] ( k 2 − i ) ) × (   exp   [ − δ ( x ) i y ] β − i −   exp   [ δ ( x ) i y ] β + i ) − 2 β cosh [ δ ( x ) i ] × ( d P d x + i − 1 k 2 − i ) (   exp   [ − δ ( x ) i y ] β − i +   exp   [ δ ( x ) i y ] β + i ) + d P d x [ i δ ( x ) 2 2 ( y 2 − 1 ) + 1 ] + i − 1 ( k 2 − i ) cosh [ k δ ( x ) y ] cosh [ k δ ( x ) ] .
(57)
While Eq. (57) could be used for different values of β, one can simplify the boundary condition Eq. (11) when β ≫ 1 ⁠. This special case represents the phenomenon when all the external nanoparticles have reacted at the walls, making the boundary condition c = 0 at y = ± 1 ⁠. With this new boundary condition, another solution for Eq. (55) when β ≫ 1 is obtained as follows:
B ω = cosh [ δ ( x ) i y ] cosh [ δ ( x ) i ] ( 1 k 2 − i + i + d P d x ) + i d P d x [ i δ ( x ) 2 2 ( y 2 − 1 ) + 1 ] − 1 k 2 − i cosh [ k δ ( x ) y ] cosh [ k δ ( x ) ] .
(58)
The O ( ε 2 ) from Eq. (43) is given by
∂ C x ∂ t 3 + ∂ c 1 ∂ t 2 + ∂ c 2 ∂ t + P e ( u 0 ∂ c 1 ∂ x + v 0 δ ( x ) ∂ c 1 ∂ y ) = ∂ 2 c 1 ∂ x 2 + 1 δ ( x ) 2 ∂ 2 c 2 ∂ y 2 ,
(59)
where
∂ c 1 ∂ x = P e { ∂ 2 C x ∂ x 2 Im [ B ω   exp   ( i t ) ] + ∂ C x ∂ x Im [ ∂ B ω ∂ x exp   ( i t ) ]   + ∂ C x ∂ x Im [ ∂ B ω ∂ y ∂ y ∂ x exp   ( i t ) ] } ,
(60)
∂ c 1 ∂ y = P e ∂ C x ∂ x I m [ ∂ B ω ∂ y exp   ( i t ) ]
(61)
and
∂ 2 c 1 ∂ x 2 = P e { ∂ 3 C x ∂ x 3 I m [ B ω   exp   ( i t ) ] + 2 ∂ 2 C x ∂ x 2 I m [ ∂ B ω ∂ x exp   ( i t ) ]   +  2 ∂ 2 C x ∂ x 2 I m [ ∂ B ω ∂ y ∂ y ∂ x exp   ( i t ) ] + ∂ C x ∂ x I m [ ∂ 2 B ω ∂ x 2 exp   ( i t ) ] +  2 ∂ C x ∂ x I m [ ∂ 2 B ω ∂ x ∂ y ∂ y ∂ x exp   ( i t ) ] + ∂ C x ∂ x I m [ ∂ B ω ∂ y ∂ 2 y ∂ x 2 exp   ( i t ) ] + ∂ C x ∂ x I m [ ∂ 2 B ω ∂ y 2 ( ∂ y ∂ x ) 2   exp   ( i t ) ] } .
(62)
Substituting Eqs. (52), (54), (60), and (61) in Eq. (59) returns
∂ C x ∂ t 3 + ∂ c 2 ∂ t + P e ∂ 3 C x ∂ x 3 Im [ B ω   exp   ( i t ) ] + P e 2 { Im [ U   exp   ( i t ) ] ( ∂ 2 C x ∂ x 2 Im [ B ω   exp   ( i t ) ] + ∂ C x ∂ x Im [ ∂ B ω ∂ x exp   ( i t ) ] + ∂ C x ∂ x Im [ ∂ B ω ∂ y ∂ y ∂ x exp   ( i t ) ] ) + 1 δ ( x ) Im [ V   exp   ( i t ) ] ∂ C x ∂ x Im [ ∂ B ω ∂ y exp   ( i t ) ] } = ∂ 2 c 1 ∂ x 2 + 1 δ ( x ) 2 ∂ 2 c 2 ∂ y 2 .
(63)
The next step is taking the time average of Eq. (63) regarding the timescale t. The expression for the time average products in complex notation can be written as
x 0 = | X |   exp   ( i θ ) ,
(64)
where x0 is any oscillatory solution, | · | is the modulus of the complex number, and θ is the angle phase of each dependent variable. The time average of the product of two functions K1 and K2 is defined as
⟨ K 1 ( t ) K 2 ( t ) ⟩ ≡ | K 1 | | K 2 | 2 π ∫ 0 2 π I m {   exp   [ i ( θ k 1 + t ) ] } Im {   exp   [ i ( θ k 2 + t ) ] } d t = | K 1 | | K 2 | 2 cos   ( θ k 1 − θ k 2 ) .
(65)
It should be mentioned that U and V are real-valued functions at low frequencies [e.g., compare Eq. (33) with Eq. (34)], i.e., θ u , θ v = 0 ⁠. Therefore, taking the time average of Eq. (63), using the procedure shown in Eq. (65), the following governing equation is obtained:
∂ C x ∂ t 3 + P e 2 ( ∂ 2 C x ∂ x 2 A 1 ( x , y ) + ∂ C x ∂ x A 2 ( x , y ) ) = 1 δ ( x ) 2 ∂ 2 ⟨ c 2 ⟩ ∂ y 2 ,
(66)
where
∂ 2 ⟨ c 1 ⟩ ∂ x 2 = 0 ,
(67)
A 1 ( x , y ) = U | B ω | 2 cos   ( θ B ω )
(68)
and
A 2 ( x , y ) = U | ∂ B ω ∂ x | 2 cos   ( θ B ω x ) − y δ ( x ) d δ d x U | ∂ B ω ∂ y | 2 cos   ( θ B ω y )     + 1 δ ( x ) V | ∂ B ω ∂ y | 2 cos   ( θ B ω y ) .
(69)
Taking the cross-sectional average of Eq. (66) returns
∂ C x ∂ t 3 = − P e 2 ( ∂ 2 C x ∂ x 2 A 1 ¯ ( x ) + ∂ C x ∂ x A 2 ¯ ( x ) ) .
(70)
Finally, Eq. (52) is added to Eq. (70), where the artifice of three times is no longer needed and can be removed,41,
∂ C x ∂ t = − P e 2 ( ∂ 2 C x ∂ x 2 A 1 ¯ ( x ) + ∂ C x ∂ x A 2 ¯ ( x ) ) + ∂ 2 C x ∂ x 2 .
(71)
The initial and boundary conditions of Eq. (71) are taken from Eqs. (12) to (14), as follows:
C x = 1   at   x = − 1 ,
(72)
C x = 0   at   x = 1 ,
(73)
and
C x = δ 2 ( x )  at t = 0.
(74)
A common approach to solve Eq. (71) is by using Chatwin's linear approximation14,42 for rectangular microchannels. Therefore, considering δ ( x ) = 1 and taking into account that the terms A 2 ̃ = 0 while A 1 ̃ is a constant, gives the solution for Cx as C x = ( 1 − x ) / 2 ⁠. Substituting the aforementioned solution in Eq. (71) returns that Cx does not depend on t in rectangular channels. For h ( x ) ≠ a ⁠, where a is any constant, Eq. (71) can be simplified when P e ≫ 1 ⁠. Thus, as a first approximation, we get the following governing equation:
∂ 2 C x ∂ x 2 = − A 2 ¯ ( x ) A 1 ¯ ( x ) ∂ C x ∂ x .
(75)

Equation (75) was solved with the boundaries conditions Eqs. (72) and (73), using the fourth order-Runge–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 x-direction was employed. Cross-sectional averages A ¯ 1 ( x ) and A ¯ 2 ( x ) are solved using Simpson's 3/8 rule with 10 − 3 precision, which sets the size of our grid. The results were modified using the transfinite interpolation method43 to simulated the geometrical variable shape of the microelectrodes.

In Secs. II A–II C, the non-dimensional velocity vector, pressure gradient, and concentration field of silver colloids that are transported in a hydrodynamic flow field were calculated. To estimate the values of dimensionless parameters involved in the analysis, we consider values of physical and geometrical parameters that have been reported in previous work:7  h 0 = 9   μ m ,   L = 40   μ m ,   R e = 90   μ m ,   R p = 25   nm ,   T = 293   K ,   ω = 20 π   rad / s ,   ε m = 7.8 × 10 − 10   C / Vm ,   ρ m = 997   kg / m 3 , μ = 1 × 10 − 3   kg / ms ,   ϕ 0 = 3   Vpp ,   ζ = − 25.4   mV ⁠, z = 1, n ∞ = 1.2 × 10 24   m − 3 ,   λ D = 71   Å ,   U HS = 3.7 × 10 − 4   m / s ⁠, and D = 8.6 × 10 − 12 m 2 / s ⁠. With the previous physical domain, the dimensionless parameters for the calculations assume the following values: ε = 0.05 , R e ω = 2 π × 10 − 3 ,   R e L = 1.5 × 10 − 2 ⁠, k = 1260, Pe = 1729, and α = 8.5 × 10 − 3 ⁠. For the analytical process, we consider two different microelectrodes shapes represented by the non-dimensional function δ ( x ) ⁠, a round δ ( x ) = 1 + x 2 and a triangular δ ( x ) = 1 + | x | functions, respectively. The latter was modeled using Fourier series seeking a function that can be differentiated with respect to x and continuous at x = 0,
δ ( x ) = 3 2 − ∑ n = 1 ∞ 4   cos   [ π ( 2 n − 1 ) x ] π 2 ( 2 n − 1 ) 2 .
(76)
Time-harmonic non-dimensional concentration field is governed by the following equation:
c = C x + ε P e ∂ C x ∂ x I m [ B ω ( x , y )   exp   ( i t ) ] + O ( ε 2 ) ,
(77)
where the variables Cx and B ω ( x , y ) dictate its behavior. Both variables are dependent on x direction, where the dependence of B ω comes from the pressure gradient and this in turn depends on whether the function δ ( x ) is differentiable with respect to x as shown in Eq. (36). Therefore, for a rectangular shape, there will be no induced pressure field and B ω = B ω ( y ) ⁠. In Figs. 2(a) and 2(b), the pressure field, gradient and Laplacian, for round and triangular shapes is shown, respectively. The triangular configuration presents a higher magnitude for the pressure gradient and its Laplacian, along with a reduction in the interval where the pressure gradient is negative from − 0.33 ≤ x ≤ 0.33 ⁠; in contrast, the round shape pressure gradient roots appear at x = ± 0.41 ⁠. This effect can be calculated from Eq. (37), where the pressure gradient presents a high dependency on the microelectrode shape δ ( x ) [ d P / d x ∝ δ ( x ) − 3 ] and through the constant value A. Concerning this constant, if A = –3, the pressure gradient will be positive from − 1 ≤ x ≤ 1 and if A = −6, it will present only negative values.
FIG. 2.
First order non-dimensional pressure field, gradient, and Laplacian of (a) δ ( x ) = 1 + x 2 and (b) δ ( x ) = 1 + | x | .
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First order non-dimensional pressure field, gradient, and Laplacian of (a) δ ( x ) = 1 + x 2 and (b) δ ( x ) = 1 + | x | ⁠.

FIG. 2.
First order non-dimensional pressure field, gradient, and Laplacian of (a) δ ( x ) = 1 + x 2 and (b) δ ( x ) = 1 + | x | .
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First order non-dimensional pressure field, gradient, and Laplacian of (a) δ ( x ) = 1 + x 2 and (b) δ ( x ) = 1 + | x | ⁠.

Close modal

In Figs. 3, 4, and 5, the variables Cx and B ω are shown for rectangular, round, and triangular shapes, respectively. The variable B ω has been multiplied by the constants ε and Pe (⁠ ε P e = 86 ⁠) for better interpretation of the figures. In almost all cases, the product of B ω with ∂ C x / ∂ x is at least one order of magnitude higher than Cx, allowing the analysis of time-harmonic 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 Cx, indicating that the time-harmonic concentration field distribution will be linear since C x = ( 1 − x ) / 2 ⁠.

FIG. 3.
Real part of the non-dimensional variable B ω in a rectangular microchannel at (a) β = 0, (b) β = 0.7 , and (c) β ≫ 1 .
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Real part of the non-dimensional variable B ω in a rectangular microchannel at (a) β = 0, (b) β = 0.7 ⁠, and (c) β ≫ 1 ⁠.

FIG. 3.
Real part of the non-dimensional variable B ω in a rectangular microchannel at (a) β = 0, (b) β = 0.7 , and (c) β ≫ 1 .
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Real part of the non-dimensional variable B ω in a rectangular microchannel at (a) β = 0, (b) β = 0.7 ⁠, and (c) β ≫ 1 ⁠.

Close modal
FIG. 4.
(a) First-order Cx in a curved microchannel for different values of parameter β. (b) Close-up of Cx gradient and (c) Cx gradient in a curved microchannel for different values of parameter β. Real part of the non-dimensional variable B ω in a curved microchannel at (d) β = 0, (e) β = 0.7 , and (f) β ≫ 1 .
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(a) First-order Cx in a curved microchannel for different values of parameter β. (b) Close-up of Cx gradient and (c) Cx gradient in a curved microchannel for different values of parameter β. Real part of the non-dimensional variable B ω in a curved microchannel at (d) β = 0, (e) β = 0.7 ⁠, and (f) β ≫ 1 ⁠.

FIG. 4.
(a) First-order Cx in a curved microchannel for different values of parameter β. (b) Close-up of Cx gradient and (c) Cx gradient in a curved microchannel for different values of parameter β. Real part of the non-dimensional variable B ω in a curved microchannel at (d) β = 0, (e) β = 0.7 , and (f) β ≫ 1 .
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(a) First-order Cx in a curved microchannel for different values of parameter β. (b) Close-up of Cx gradient and (c) Cx gradient in a curved microchannel for different values of parameter β. Real part of the non-dimensional variable B ω in a curved microchannel at (d) β = 0, (e) β = 0.7 ⁠, and (f) β ≫ 1 ⁠.

Close modal
FIG. 5.
(a) First-order Cx in a triangular microchannel for different values of parameter β. (b) Close-up of Cx gradient and (c) Cx gradient in a triangular microchannel for different values of parameter β. Real part of the non-dimensional variable B ω in a triangular microchannel at (d) β = 0, (e) β = 0.7 , and (f) β ≫ 1 .
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(a) First-order Cx in a triangular microchannel for different values of parameter β. (b) Close-up of Cx gradient and (c) Cx gradient in a triangular microchannel for different values of parameter β. Real part of the non-dimensional variable B ω in a triangular microchannel at (d) β = 0, (e) β = 0.7 ⁠, and (f) β ≫ 1 ⁠.

FIG. 5.
(a) First-order Cx in a triangular microchannel for different values of parameter β. (b) Close-up of Cx gradient and (c) Cx gradient in a triangular microchannel for different values of parameter β. Real part of the non-dimensional variable B ω in a triangular microchannel at (d) β = 0, (e) β = 0.7 , and (f) β ≫ 1 .
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(a) First-order Cx in a triangular microchannel for different values of parameter β. (b) Close-up of Cx gradient and (c) Cx gradient in a triangular microchannel for different values of parameter β. Real part of the non-dimensional variable B ω in a triangular microchannel at (d) β = 0, (e) β = 0.7 ⁠, and (f) β ≫ 1 ⁠.

Close modal

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 Cx gradient and due to the oscillating period. On this subject, the positive magnitude of B ω 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 time-harmonic 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 first-order reaction at the wall are shown. The value of the parameter β = 0.7 has been selected from Eq. (57), where B ω presents a maximum value when its denominator approaches zero, i.e., β = 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 π / 6 out of phase response at the reactive wall while lagging the other wall by 3 π / 2 ⁠, as calculated from the complex argument of Eq. (57). Another important effect is the movement of high order concentration locations from x = ± 0.41 to x = ± 0.61 for the round shape and x = ± 0.33 to x = ± 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 ¯ ( x ) , cross-sectional 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 β ≫ 1 are shown. The aspect that stands out is that Cx is linear regardless of δ ( x ) ⁠, indicating that when β ≫ 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 first-order 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 = ± 1 ⁠, the previous assumption of the velocity vector can be seen in Eq. (58), where the effects of convection on the variable B ω 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 ¯ 1 ( x ) for an OEF at f = 1   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 non-dimensional parameter β = β ̃ 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 long-scale 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.

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 long-time 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 high-concentration 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 high-concentration 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 δ ( x ) and could not be attributed only to the dielectrophoretic force.

  • An irreversible first-order 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 long-time 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.

In summary, the induced pressure gradient plays a major role in colloidal transportation, from highly affecting NPs distribution to some mass transport phenomena not yet fully explained by electrophoresis. Further studies on colloidal transportation are recommended, from an experimental study that examines different microelectrodes shapes to theoretical analysis, such as considering Navier-slip boundary condition, Joule heating if E x ≥ 100   V / m ⁠,44 and considering that the product of ε P e ∼ O ( 1 ) that transforms the zero-order dimensionless concentration at O ( 1 ) as follows:
∂ c 0 ∂ t + u 0 ( ∂ c 0 ∂ x 2 + ∂ c 0 ∂ y ∂ y ∂ x ) + v 0 δ ( x ) ∂ c 0 ∂ y = 1 δ ( x ) 2 ∂ 2 c 0 ∂ y 2 .
(78)

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. RGPIN-201-05138, Canada Foundation for Innovation (CFI) Valid Funder: No. 31967, and FEAS Excellence in Research Award Queen's University.

The authors have no conflicts to disclose.

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).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

In this  Appendix, Eq. (34) is deduced from the velocity component U shown in Eq. (33) when R e ω ≪ 1 ⁠,
U = cosh [ ( 1 + i ) R e ω 2 δ ( x ) y ] cosh [ ( 1 + i ) R e ω 2 δ ( x ) ] ( 1 − i R e ω d P d x ) + i R e ω d P d x − k 2 ( k 2 − i R e ω ) cosh [ k δ ( x ) y ] cosh [ k δ ( x ) ] .
(A1)
Rewriting the first right-hand complex term in Eq. (A1) in the form of c = a + i b ⁠, where a and b represent the real and imaginary parts of the complex function c, respectively, returns
U = ( 2   cos   [ 2 r ( x ) ] + cosh [ 2 r ( x ) ] ) ( 1 − i R e ω d P d x ) × ( cos   [ r ( x ) y ] cosh [ r ( x ) y ] + i   sin   [ r ( x ) y ] sinh [ r ( x ) y ] ) × ( cos   [ r ( x ) ] cosh [ r ( x ) ] − i   sin   [ r ( x ) ] sinh [ r ( x ) ] ) + i R e ω d P d x − k 2 ( k 2 − i R e ω ) cosh [ k δ ( x ) y ] cosh [ k δ ( x ) ] ,
(A2)
where
r ( x ) = δ ( x ) R e ω 2 .
(A3)
Using Taylor series for the expanded complex function and considering only the first term in the power series yields
U = ( 1 − i R e ω d P d x ) [ 1 + i r ( x ) 2 ( y 2 − 1 ) ]  + i R e ω d P d x − cosh [ k δ ( x ) y ] cosh [ k δ ( x ) ] + O ( R e ω ) .
(A4)
Finally, taking the first right-hand product and introducing Eq. (A3) into Eq. (A4) returns Eq. (34),
U = δ ( x ) 2 2 d P d x ( y 2 − 1 ) + 1 − cosh [ k δ ( x ) y ] cosh [ k δ ( x ) ] + O ( R e ω ) ,
(A5)
where the imaginary terms are of O ( R e ω ) ⁠.
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