Electrokinetic cells powered by osmotic gradients: an analytic survey of asymmetric wall potentials and hydrophobic surfaces

Nowadays, the fabrication of microelectromechanical systems has given rise to several studies whose main purpose is to obtain the greater benefit of micro-nano scales, putting special interest in the improvement of the design of such devices. One of several applications is harvesting energy due to electrokinetic phenomena, more specifically, streaming potential. Nonetheless, there is a lack of theoretical studies encompassing coupled asymmetries in both slip conditions and electric potentials (these being associated with the chemical and physical characteristics of the surfaces). By virtue of the previous explanation, ideal assumptions based on the symmetry of some variables must be reconsidered, especially when manufacturing symmetric flat surfaces on a tiny scale is quite difficult to achieve. This work presents a theoretical study in power generation, exploiting streaming potentials produced by an asymmetric membrane which in turn prompts a flux inside a microchannel made of two flat parallel surfaces. The driving force in this electrokinetic battery is the osmotic gradient on both sides of the membrane. The model uses the Debye–Hückel approximation together with the appropriate asymmetric boundary conditions for both slips and potentials on the surfaces. The main variables of interest, such as the dimensionless horizontal velocity component, the pressure field, and the average streaming potential, were estimated.

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Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Osmotic pressure of draw bulk (Pa) P os F,b Osmotic pressure of feed bulk ( Electric charge density ( C m −3 ) σ os D,b Electrolyte conductivity in the draw bulk ( S m −1 ) σs Electric charge density ( C m −2 ) ϕ Streaming potential (V) ϕ c Characteristic streaming potential Φ Dimensionless streaming potential ψ Electric double layer potential (V) Ψ Dimensionless electric double layer potential

Introduction
Sustainable energy conversion attracted attention several decades ago due to the search for other possibilities for power sources. An alternative that has been presented is the salinity gradient energy, which goes back to the 1950s with the work of Pattle (1954) where an experimental setup (hydroelectric pile) was made which was capable of producing electric energy utilizing fresh and salt water, obtaining up to 3.1 V. Researchers continued developing several works for exploiting salt concentrations in the following decades. Norman (1974), through thermodynamical reasoning highlights that salination of water provides a huge amount of available osmotic energy, namely, the seawater in contact with freshwater, where it is remarked that the potnetial of this energy is equal in magnitude to hydroelectric power. A converter was theoretically proposed, where the osmotic energy would be collected as a pressure head, which in turn would move a waterwheel connected to a generator (this would become the principle of pressure retarded osmosis (PRO)). Olsson et al (1979) applied vapor compression desalination in reverse to extract power from the pressure difference between two water reservoirs with different concentrations, where they concluded that the vapor pressure difference increases with temperature, suggesting a better performance in heat sources, such as geothermal heat or waste heat (very common in power plants). In addition,a small model was built at 27 • C that showed an efficiency of 23%. Jiao et al (2022) made a compilation of the power generation based on membrane technologies for harvesting energy with the osmosis process, where the following classification was proposed: First class is the membrane-based indirect technology, including pressure retarded osmosis, forward-osmosis electrokinetic (FO-EK), which use different submodules (the osmosis module, and the electric module). Second class is the membrane-based direct technology, which includes reverse electrodialysis, nanofluidic electrodialysis, and electric double layer (EDL) capacitors among others. These techniques not only induce the movement of the fluid but also the displacement of ions to generate electric current or electric potential. Finally, the third class category contains technologies that combine osmotic gradients with other properties, such as the vapor pressure difference technique that exploits evaporation rates of electrolyte solutions with different concentrations to indirectly harvest electricity. This compilation focuses on providing a review of the classical PRO technology and the innovative FO-EK technology. The latter has been studied experimentally due to the growing interest in renewable energies that rely on FO-EK modules (Hon et al 2012 andJiao et al 2014): the forward osmosis module which pumps an electrolyte into the electrokinetic module that makes possible the generation of the electric potential between these extremes, namely, the streaming potential. The aforementioned modules were made of semi-permeable membranes and a porous medium that acquires a surface charge when entering into contact with an aqueous medium, giving as a result, a streaming potential up to several hundred millivolts, ranging from 400 to 600 mV. A similar work was developed by Yang et al (2003) where an electrokinetic battery was analyzed that consists of an array of microchannels, allowing the addition of all the streaming currents, giving as a result, the conversion of hydrostatic pressure into electrical power. Their analytical results agreed with experiments, yielding currents of the order of 1 − 2µA. It must be highlighted that these FO-EK modules pretend to benefit from an apparently inexhaustible source: seawater.
Streaming potential is a phenomenon that takes special relevance in fluid mechanics when the scale of work is of the order of 10 −6 to 10 −9 meters. The applicability of this phenomenon has been oriented mainly to material characterization (Gu andLi 2000, Sides andPrieve 2016). Nonetheless, besides the aforementioned purpose, the streaming potential has taken relevance in microelectromechanical systems (MEMS) and power harvesting in micro and nanofluidics. Pennathur et al (2007) carried out a comprehensive review of the state-of-the-art related to microscale energy generation where it was identified that theoretical conversion efficiencies are reached using slip flow conditions, although they could not conclude which kind of microdevices perform the best. Ren and Stein (2008) investigated theoretically that hydrodynamic slip can enhance the conversion of hydrostatic energy to electrical power through streaming potential and streaming current, where they reported efficiencies for nanodevices ranging from 40%-70% (the latter for extraordinary slip lengths that exceed 10 nm); Bazant et al (2009) reviewed the experimental and analytical works in electrokinetics for large voltages in concentrated solutions where specific adsorption of ions is relevant with hydrophobic surfaces. These hydrophobic surfaces are characterized by the slip length b which can reach tens of nanometers or even the magnitude of microns for super-hydrophobic surfaces. In addition, Sánchez and Méndez (2022) obtained semi-analytical solutions for an electrokinetic battery considering large wall potentials and slip conditions, which in certain cases are able to diminish the viscoelectric effects and even produce saturation values for progressively large slip coefficients in induced streaming potentials.  presented a model for an electrokinetic process in rectangular nanochannels, considering a chemical model between surface and electrolyte showing a relationship between the size of the EDL and the channel size, whose results were calculated in terms of streaming conductance and electrical conductance. This study was conducted with the classical non-slip assumption for the surfaces. Daiguji et al (2004) developed a numerical study of an electromechanical battery at the nanoscale that works by promoting a flux inside a nanochannel. This work unveiled that the streaming potential and streaming current that are generated depend on surface charge density and bulk properties. It was also determined that if the Debye length of the EDL is approximately half of the channel height, the maximum efficiency is achieved. A similar study carried out by Chang and Yang (2010) presented a Poisson-Nernst-Planck model obtained from Ohm's law and from Kirchhoff's current law taking into account the reservoir resistance on power generation, showing a great impact on the efficiency of the system. Works, such as that of Sánchez and Méndez (2021), for power generation using forward osmosis as a driving force inside a microchannel made of silica, showed that considering viscoelectric effects for high wall potentials (that is, greater or equal than) can diminish the power harvested, suggesting the existence of optimal points of operation for such microsystems.
After determining the capacity of inducing electric potentials from experimental batch configurations for studies focused on the microscale and its phenomenological behavior, boundary conditions related to wall potentials and wall slip velocity become transcendental, especially those that account for asymmetries. In practical applications, the walls of a microchannel are normally made of dissimilar materials, which strongly influences the flow and the induced electric potentials. This is very important during the design and fabrication of cutting-edge microdevices such as micromotors, microbearings, MEMS sensors, microvalves and lab-on-achip systems for drug delivery, among others, which normally include microchannels where electrokinetic and microfluidics phenomena rule the physical domain (Soong andWang 2003, Mukhopadhyay 2009). In addition, there is a growing trend to make miniaturized devices such as miniaturized heat engines, micro-batteries, micro supercapacitors, micro gas turbines and micro reactors; and specifically for micro and nanofluids, power generation takes advantage of electrokinetic phenomena (Penathur et al 2007), which are focused primarily on portable applications such as batteries and mobile electronics (cell phones and laptops). Works attending this problem, such as that made by Soong and Wang (2003) where a microchannel with parallel plates with different wall potentials and imposed heat fluxes are assumed, where a fluid flow is provoked by an axial pressure gradient. The streaming potential was analytically computed using the Debye-Hückel approximation, where the boundary conditions despite being asymmetrical are not general enough: simple velocity conditions are assumed on the surface (one plate is immobile and the other one is moving in the axial direction, inherently invoking the no-slip condition). Other works focused on electro-osmotic flow (Mukhopadhyay et al 2009) with similar thermal, flow, and electrokinetic conditions, suggesting that velocity profiles are very sensitive to asymmetric boundary conditions.
A more recent study conducted by Sailaja et al (2019), considered a slit microchannel with asymmetric surfaces where electroviscous effects were considered and the Debye-Hückel approximation was employed when possible. Also, a power law fluid is invoked for Newtonian and shear thinning and thickening fluids. Streaming potential was determined by optimization methods. Analytical and numerical results showed that streaming potential has a strong dependence on zeta potential asymmetries and Newtonian fluids have a maximum value that depends on the dimensionless Debye-Hückel length: the higher the zeta potential, the smaller the microchannel height should be.
The exploration of this alternative energy demands the knowledge of several approaches: from the theoretical with exhaustive and complex models attending mixed phenomena to the experimental with sophisticated setups and a lack of consistency in determining physical parameters by several methods. Based on previous research, the present work obtains analytical solutions for an electrokinetic battery made of parallel plates that is driven by an osmotic flow (stepping aside from an a priori pressure-driven flow). The Debye-Hückel approximation is used for the calculation of the EDL potential coupled with asymmetric hydrodynamic slippage. The results shown here differ from previous works by explicitly considering both asymmetries in the Navier slip lengths and the zeta potentials, in order to assess the conjugate interplay of these boundary conditions. Additionally, the explicit relation of the streaming potential with the osmotic flow, the electrolyte bulk properties, and the physicochemical properties of silica surfaces with low wall potentials are demonstrated. Finally, the results are able to recover the classical symmetric conditions, such as the non-slip velocity at the surfaces and the equal wall potential is determined.

Physical model and mathematical description
The physical model is sketched in figure 1. An asymmetric membrane separates two electrolytes with different molar concentrations on both sides, promoting an osmotic flow inside a microchannel whose surfaces (silica surfaces) create an electrokinetic region where a normal distribution of charges is stimulated. These charges accumulate downstream as a consequence of the flow, this being the mechanism of generation of electric potential in the axial direction: the streaming potential. The aforementioned potential is responsible for harvesting power in the microchannel in stationary conditions. The coordinate system origin is located at the entrance of the microchannel and at the center of itself, being the domain of Microchannel with an electrokinetic region, whose parallel plates are made of silica which give rise to a normal distribution of ions due to an electric double layer. An osmotic flow prompted by an asymmetric membrane drags free ions downstream, giving rise to the streaming potential between the extremes of the microchannel.
In the following subsections, we explain in detail the governing equations that describe the power generation cell in a microchannel compound with parallel flat surfaces by invoking the Debye-Hückel approximation for a symmetric electrolyte ζ ⩽ k B T/e, specifically for room temperature T = 298 K, that is ζ ⩽ 25 mV. Moreover, the model accounts for the most general case: both asymmetric slip conditions and asymmetric surface potentials, which are represented by the boundary conditions as discussed in the following subsections.

EDL and streaming potential
In aqueous systems charged surfaces give rise to a distribution of ions. The free ions are repelled or attracted from the surface depending on the surface charge. Such reorganization of ions is called EDL, which is composed of two layers: the Stern layer (inner layer) and the diffuse layer (outer layer). The first one is composed of ions with the opposite charge from the surface, whereas the second one has ions that are free to move.
In our case, the charged surface arises from the interaction between silicon dioxide (SiO 2 ) with an electrolyte compound of water and sodium chloride (H 2 O + NaCl) which motivates the formation of silanol groups (SiO − ), provoking a negatively charged surface. The EDL potential ψ is mathematically described by the Poisson-Boltzmann equation for a symmetric electrolyte as follows (Karnadiakis et al 2005) where ρ f ,ε, C os D,b and z are the free electric charge density, the permittivity, the molar concentration in the electrokinetic region, and the valence of the suspended particles in the aqueous solution, respectively, whereas F is the Faraday constant. This EDL potential is responsible for the distribution of free charges which subsequently is vital in the formation of an electric potential downstream in the microchannel. Such potential is caused when is flowing the electrolyte in a parallel direction of the surface (in this work, parallel to the walls of the microchannel). This potential arises when free charges that are found dispersed in an electrolyte (specifically in the diffuse layer) are dragged downstream, which originates a stationary electric potential φ , best known as the streaming potential. For low surface potentials, that is, ψ ≪ RT/zF equation (1) can be linearized as follows: where λ is defined the Debye length defined as this length quantifies the span of the EDL near the surfaces. Properly, the linearization given in equation (2) is better known as the Debye-Hückel approximation, which is widely used in electrokinetic phenomena to obtain analytical results.

Boundary conditions of Poisson-Boltzmann equation.
Solving equation (2) requires the boundary conditions associated with the zeta potential at each surface, which are the following: The conditions given above for y = −l and y = l are related to the physical parameters of the lower and upper plates of the microchannel, respectively. This perspective is far from the classical symmetrical boundary conditions where ζ 1 = ζ 2 . Even more, the value of both potentials is a priori unknown, nonetheless, each of them is determined by the following transcendent system of equations (Behrens andGrier 2001, Wang et al 2010): and where the subindex i = 1, 2 denotes the corresponding lower and upper surfaces. The parameters k B , T, e, and pH are the Boltzmann constant, the absolute temperature, the elementary charge, and the hydrogen potential of the bulk, respectively. Besides pK i , Γ i , C Stern i , σ s i and ζ i are the dissociation constant, the total chargeable sites, the capacitance of the Stern layer, the electric charge density, and the zeta potential. All of these properties are given for the ith surface. In addition, the ionic concentration of the bulk in the electrokinetic region η os D,b is directly related to the molar concentration by means of η os D,b = N A C os D,b , with N A being the Avogadro's number. Both equations have the following meaning: equation (5) relates the surface charge density with the wall potential ζ and the electrolyte ionic concentration, whereas equation (6) is formulated from the creation of silanol groups through the reaction SiOH ⇌ SiO − + H + , which in turn comes about from the chemical reaction between the electrolyte H 2 O + NaCl with the silicon dioxide SiO 2 . The system mentioned above and compounded by equations (5) and (6) must be solved for each surface, explicitly giving the boundary conditions (4). Consequently, these values make possible the solution of equation (2) giving, as a result, the EDL potential in the microchannel, thus,

Osmotic flow
As it was already mentioned, the streaming potential is induced with the aid of a specific flow pattern. A passive mechanism for promoting the water flux J w (m/s) through the difference of concentration inside the microchannel via an asymmetric osmotic membrane whose equation is written as follows (Helfer et al 2014), where R s (s m −1 ), k w (m Pa −1 s −1 ), k s (m s −1 ) are the solute resistivity, the water permeability and the mass transport coefficient inside the membrane, respectively. Also, P os D,b , P os F,b are the osmotic pressure at the bulk of the draw and feed solution, respectively, which in turn are determined by the Van't Hoof equation (Probstein 2003) where i defines the number of active particles in the electrolyte (in the case of NaCl i = 2), is the universal gas constant and T is the absolute temperature. In addition, the mass transfer coefficient is obtained from the Sherwood number Sh as where D is the coefficient of mass diffusion and the hydraulic diameter of the microchannel is given by d h = 4A/S, with A and S being the area and the perimeter of the microchannel, respectively. For laminar flow inside a rectangular microchannel (Mulder 1996) Sh = 1.85 being Re = J w ρd h /µ the Reynolds number, and Sc = µ/ρD the Schmidt number.

Nernst-Planck equation and electric current
The relationship between the osmotic flow and the streaming potential is contained in the Nernst-Planck equation (Masliyah and Bhattacharjee 2006): where ⃗ j i (mol m −2 s −1 ) is the ionic flux for the ith specie, and the subindex i = 1, 2 denotes the species of the solvated Na + and Cl − , respectively. In addition, the electric potential in stationary conditions is the superposition of the streaming potential and the potential of the EDL, in other words, Φ (x, y) = φ (x) + ψ (y). The components of equation (12) are shown as follows, and Neglecting the variations of the concentration along the microchannel because of H ≫ l (that is, ∂C/∂x ≈ 0) and given that there is no permeation flux to the parallel plates (v = 0), equations (13) and (14) become into, and In turn, the ionic flux of species causes an electric current whose definition is the following In steady-state conditions, the net current between ⃗ I and the ionic flux ⃗ j i are zero due to the electrolyte properties do not change outside the electrokinetic region. Therefore, making use of equations (15)-(17) becomes: and The molar concentration C i is computed from the differential equation obtained from equation (16) for j i,y = 0, giving as a result The coefficient A is determined through the conditions just in the center of the microchannel C i (0) = C os D,b (bulk concentration of draw solution) and ψ (0) = 0 (the electric potential in the EDL is assumed as negligible far away from the plates). Correspondingly, the molar concentration of the ith specie results in the well-known Boltzmann distribution, Finally, for a symmetric electrolyte z 1 = −z 2 = z, C 1 (0) = C 2 (0) = C os D,b , (solvated ions have exactly the same concentration) and D 1 ≈ D 2 = D. Integrating equation (18) in the interval (−l, l) in y-direction, provides the following net current equation: This equation can be simplified by invoking once again the Debye-Hückel approximation where the second integral is reduced to 2l. Thus, equation (22) adopts the following form: where ρ f = ∑ 2 i =1 Fz i C i is the same free charge density given in equation (1) and σ os D,b = 2F 2 z 2 DC os D,b /RT is the electrolyte conductivity in the bulk.

Navier-Stokes and mass conservation equations
The streaming potential phenomenon is coupled with the velocity inside the microchannel. For this reason, the mass conservation equation for an incompressible fluid together with the Navier-Stokes equations for a Newtonian fluid in steady-state for describing the hydrodynamic is presented in their bidimensional version as follows: and where the electric body forces that appear in equations (25) and (26) are associated with the electric potential via the electric field force ⃗ (25) and (26) can be further simplified by substituting the electric charge density in terms of the EDL potential from equation (2), thus, giving, as a result, the following modification in the Navier-Stokes equations: and Immediately afterward, we apply the change of variable p ′ = P − ( ε/2λ 2 ) ψ 2 (Ajdari 1995) producing the final simplification in equations (28) and (29), thus, and The viscous terms associated with x-direction were neglected due to l ≪ H. The previous modification to the Navier-Stokes allows us to obtain the lubrication theory for the variable p ′ in the following sections, making it possible to reduce the problem of solving a set of partial differential equations into a single integrodifferential one. In addition to the stated equations, the osmotic effect due to the pressure difference between both sides of the membrane yields a known volumetric flow rate, which can be written with the aid of the following relationship, where A = 2 w · l is the perpendicular area to the flow, being the depth of the microchannel w ∼ 10 −2 m. We must remark that this equation is essential to determine the pressure gradient along the axial direction, which is inherently associated with the velocity field. Before proceeding with the solution of the proposed equations (23), (24) and (30)-(32) for u, v, p ′ , and φ are necessary the following boundary conditions: whereb 1 andb 2 are the Navier slip lengths associated with lower and upper surfaces, respectively; and being of the order ofb ∼ 10 −9 to 10 −6 meters (Choi et al 2003, Henry et al 2004, Jimenez Bolaños and Vernescu 2017. For some novel applications in electroosmosis and streaming potential, silica can be endowed with physicochemical treatments which in turn decrease the chemical activity and the wall potentials (Zhao et al 2011) or be made of fumed silica (Barthel et al 1995). The condition for ϕ is due to the zero accumulation of free charges at the beginning of the microchannel, and for V are because of no filtration towards the walls (that is, there is no mass exchange between the plates of the microchannel). The condition for p ′ is because of the osmotic pressure at the entrance of the electrokinetic region (it will be shown in the next section that this fictional pressure is exclusively dependent on the variable x by applying lubrication theory).

Dimensionless set of equations
Determining the solution for the streaming potential, the velocity field, and the pressure requires further manipulation, which resides in becoming the previously mentioned equations in their dimensionless version with the following change of variables: The majority of the parameters that appear in the denominator of some of these variables are clear to elucidate and are straightforward given the geometrical and physical characteristics of the microchannel, whereas the rest of them are derived from the following scale analysis: v c ∼ J w l/H is obtained from the mass conservation equation (24) from assuming that the viscous terms and the electric body force in equation (28) are similar between them (beingζ = (ζ 1 + ζ 2 ) /2 the average value of the wall potentials). Finally, substituting (34) into equations (7), (23), (24) and (30)-(32) and using the rearranged change of variable proposed by Ajdari (1995) ψ 2 gives as result the corresponding equations: Reξ 2 Reξ 2 and where Writing equations (5), (6) and (8) in terms of the dimensionless variables, the resulting equations depend, among others, on Re and D ζ and considering that these parameters are unknown, these equations must be first iteratively solved.
In order to solve the coupled equations (35)-(41) it is essential to evaluate the dimensionless parameters given in (42). For this purpose, we use the values shown in table 1, which are based on previous works, such as those of Scales et al (1992), Behrens and Grier (2001), Berli et al (2003), van der Heyden (2005), , Wang and Revil (2010) and Sánchez and Méndez (2022), giving, as a result, the magnitude of the dimensionless parameters presented in table 2. As a consequence, some non-linear terms in the set of equations (35)-(41) are discarded, and explicitly substituting the EDL potential (35) into (36), (38) and (41) provides the following equations: ε = 6.954 × 10 −10 CV −1 m µ = 0.891 × 10 −3 Pa · s D = 1.312 × 10 −9 m 2 s −1 µ = 1000 kg/m 3 T = 298 K z = 1 C os D,b = 0.5 M C os F,b = 0.005 M pH = 7.0 pK 1 = 7.9, pK 2 = 7.9 C Stern1 = 0.10 Fm −2 , C Stern2 = 0.15 Fm −2 Γ 1 = 2 sites nm −2 , Γ 2 = 3 sites nm −2 kw = 0.65 × 10 −12 m Pa −1 s −1 Rs = 6.9 × 10 5 s m −1 H = 10 −3 m l = 10 −8 m and We must point out that since ∂Π /∂Y = 0, we recover the lubrication theory for the dimensionless fictional pressure Π , giving as a result, an exclusive dependency on the axial direction. Along with equations (43)-(48), the theoretical model is completed with the dimensionless boundary conditions obtained from equation (33), where the dimensionless Navier lengths for the lower and the upper plates are b 1 =b 1 /l and b 2 =b 2 /l, respectively. It is important to mention that U,Φ and Π R , apart from their dependency on X and Y, are functions D ζ , κ, δ, Π D , Π E , b 1 and b 2 where we define for simplicity's sake Π D = αξ 2 . In addition, it will be shown in the following section that V = 0 inside the microchannel due to the nature of the boundary conditions indicated in equation (49).

Analytical solution
It is of primal importance to determine the dimensionless velocity and the pressure field inside the microchannel. For this reason, we obtain a set of equations for determining U and Π integrated by equations (45) and (47) where dΦ /dX is explicitly substituted by equation (43) giving consequently Also, equations (50) and (47) are simplified by the change of variable providing the following Fredholm integrodifferential equation and the magnitude of the fictional pressure gradient Notice that the integral in equation (52) is a constant that only depends on δ, i.e.: Therefore, equation (52) becomes, Integrating respect to Y once and again The constants A 1 and A 2 are computed from the boundary conditions F (−1) = b 1 dF (−1) /dY and F (1) = −b 2 dF (−1) /dY, which provides the following results: and . (59) Consequently, the parameter ω is found through the substitution of equation (57) into (54), thus, To sum up, equations (59) and (60) allow one to determine completely the function F given in (57). The succeeding step is to compute the magnitude of the fictional pressure gradient via equation (53) to give rise to The field of velocity in X-direction is finally obtained from equation (51) through the product of equations (57) and (61): Notice that the solution has only dependence on the transverse coordinate due to the absence of the variable X in the fictional pressure gradient (61). From this equation, we can appreciate that the pressure gradient is uniform and therefore with the aid of equation (62), the longitudinal velocity component, given by equation (62), is only dependent on the transverse coordinate Y. Using the continuity equation (44), then we can obtain The above equation can be integrated and using the last of the boundary conditions (49), it is easily determined that V = 0.
After that, fictional pressure is calculated by integrating equation (61) with the boundary condition Π (0) = 1, producing the following result: The last expression is substituted in equation (48) to determine the dimensionless physical pressure inside the microchannel: In the final stage, the streaming potential is straightforwardly computed from the substitution of equation (62) into equation (43), using the non-accumulation of free charge boundary conditionΦ (0) = 0, thus: dY . (66) Before proceeding to discuss the results, it must be emphasized that the solution of equation (66) despite being completely analytic, requires the evaluation of a huge number of terms, and it is necessary to use symbolic calculations provided by Wolfram Mathematica.

Results
Before showing the results for the dimensionless velocity field, the physical pressure, and the streaming potential, we define for simplicity the average dimensionless streaming potential as follows:  which gives more valuable information about the net-induced voltage between the extremes of the microchannel. The following discussion attends to the main consequences of the asymmetries on the surfaces, specifically slippage and wall potentials that are lower than ζ i ≪ RT/zF = 25 mV. Figure 2 shows how the symmetric slippage at the surfaces distorts the velocity profile, the greater the magnitudes of b 1 and b 2 the more it tends to a plug flow. For figure 3, the profile is distorted due to the asymmetric slippage, which in turn displaces the maximum value towards the more hydrophobic surface, in this case, the upper one. It is worth mentioning that not only do slippage conditions play an important role in the perturbation of the velocity profiles, but also the parameter D ζ (figure 4) which encodes the degree of asymmetry of the potentials, being positive for larger potentials at the upper surface in comparison with the lower one and negative in the opposite case. In summary, both asymmetries (slippage and wall potentials) modify the velocity field inside the microchannel, which is an important result because both strongly depend on the physicochemical properties of the surface. For example, for fumed silica, hydrophobic behavior is larger for surfaces  with lower chemical activity, and inherently lower potentials (Berthel et al 1995). In addition, in figure 5, it is shown that the parameter κ (responsible for the induced streaming potential) propitiates a delay in the flow near the surfaces for high values of it, being this caused by the electromigration flow produced by the electric fieldΦ which is mainly present inside the EDL.
The average streaming potential in the microchannel reveals a multiparametric dependency, especially on the parameters κ, δ, D ζ and the dimensionless slip coefficients b 1 and b 2 . Firstly, as is shown in figure 6, the dimensionless streaming potential is enhanced by increasing the slippage conditions and the parameter κ, where the higher magnitude of electric potential is harvested for symmetrical conditions. Secondly, figure 7 exhibits that the parameter δ = l/λ provides greater potentials for small Debye lengths, that is, the smaller the EDL the higher the electric potentials harvested. This is caused due to a large quantity of free ions being available to be dragged when the EDL is several times smaller in comparison with the width of the microchannel, which in turn contributes to a larger electromigration of free ions, giving rise to a greater downstream accumulation, where this behavior is  enhanced with progressive increments in slip conditions both symmetrical and asymmetrical (this conclusion is similar to the obtained from Sánchez and Méndez 2022). Finally, as is shown in figures 8 and 9, the parameter D ζ which measures the asymmetry of the electric potential indicates that the highest voltage is induced for symmetric wall potentials (D ζ = 0) and tends to decrease for absolute values greater than zero; in other words, the streaming potential is inversely proportional to the difference of wall potentials. Nonetheless, for high values of D ζ , there exist asymmetric slip conditions that make greater streaming potentials possible that vary linearly, even exceeding those achieved from symmetric wall potentials. This suggests that power generation on a microscale can be collected even in conditions where perfect procedures of fabrication cannot be warranted, in particular, for batch configurations such as those proposed by Jiao et al (2014) whose electrokinetic battery based on heterogeneous capillaries fit better with experimental results. Moreover, these results are fundamental so that the design of MEMS takes into account these asymmetries, which in most cases are impossible to avoid due to the unwieldy process of fabrication of these devices.  To summarize, in general, hydrophobic surfaces prompt a better performance in the mechanism of streaming potential and the asymmetric wall potentials modify the optimal point of operation. Figure 10 depicts the physical pressure inside the electrokinetic region in the microchannel. The osmotic pressure due to the concentration of the electrolyte rules almost the entire region; however, there is a slight pressure drop in the axial direction due to dissipative terms associated with the viscosity and a surge pressure near the plates (caused by the ion accumulation in the EDL). These results arise from the interaction between the hydraulic and osmotic pressure, the first due to the osmotic flow and the latter from the chemical reaction between the bulk and the surfaces. The asymmetry of potentials is noticeable in the increase in pressure at the upper plate due to the positive value of D ζ . Changing some crucial parameters such as the size of the channel, the electrostatic terms in the EDL (Π E ) and the value of D ζ permit one to obtain the pressure field shown in figure 11. This plot instances a more marked surge in the surfaces as well as the disparity of wall potentials. The information obtained from these results could improve some processes of micro-nano mixing in pharmaceutical applications, especially those based on microdevices and  the characterization of materials with hydrophobic surfaces and different surface potentials.

Conclusions
This work presents a model that was solved analytically for an electrokinetic battery driven by an osmotic flow prompted by an asymmetric membrane, whose main aim was the determination of the streaming potential between the extremes of the microchannel. The model included transcendental equations for an asymmetric membrane and the wall potentials of fumed silica, in addition to the use of the Debye-Hückel approximation. The calculations were effectuated considering slippage due to the hydrophobicity of fumed silica.
The involved equations were treated to obtain an integrodifferential Fredholm equation which was solved through the direct computation method. The results were analytical and evaluated through symbolic calculations for the streaming potential, velocity, and pressure fields. The main points of the research are enumerated as follows: (1) Streaming potential can be collected regardless of the inherent asymmetries of the surfaces (slippage and low wall potentials).
(2) The performance of the electrokinetic battery is sharpened by the slippage at the surfaces regardless of whether the slip coefficients are symmetric or not.
(3) For symmetric slippage, a larger streaming potential is generated when the wall potentials are identical. Nonetheless, for big differences between them, that is, |D ζ | ∼ 1, an increasing linear streaming potential could be propitiated, even reaching larger potentials than in the symmetric case. (4) For small sizes of the EDL concerning the microchannel height (δ ≫ 1), greater induced electric potentials are attained (especially for progressively larger slip conditions). The same argument applies to the parameter κ, which quantifies the effect of the driving force of the streaming potential in the electrokinetic battery, which in turn produces a distortion in the velocity field along with the parameter D ζ . (5) The physical pressure shows a homogeneous distribution apart from the surfaces where the increase is noticeably larger, due to the ionic concentration due to the EDL. Additionally, the parameter D ζ dictates which surface presents the larger electrostatic pressure.

Data availability statement
The data cannot be made publicly available upon publication because no suitable repository exists for hosting data in this field of study. The data that support the findings of this study are available upon reasonable request from the authors.