Ion-partitioning effect promotes the electroosmotic mixing of non-Newtonian fluids in soft-patterned microchannels

Abstract

We numerically investigate the mixing characteristics of non-Newtonian fluids under the ion-partitioning effect in a micromixer having a built-in patterned soft polyelectrolyte layer (PEL) on its inner wall surfaces. We show that the mixing phenomenon is greatly modulated by the migration of counter-ions triggered by the Born energy difference caused by the electrical permittivity differences between the PEL and bulk electrolyte. We demonstrate counter-ion concentration field, flow velocity variation, species concentration distribution, mixing efficiency and neutral species dispersion by varying the electrical permittivity ratio and rheological parameters. In contrast to the scenario of no ion-partitioning, results show that a decrease in counter-ions in the PEL permits a greater prediction of the induced potential field therein by the ion-partitioning effect. This phenomenon results in a higher electrical body force in the PEL at a lower permittivity ratio when the ion-partitioning effect is considered. Notably, for a lower permittivity ratio (= 0.2), the ion-partitioning effect results in an electrical body force that is significantly higher than that in the no ion-partition case. Consequently, when the ion-partitioning effect is present, we find that flow velocity and recirculation strength are an order of magnitude higher than those in the no ion-partitioning case. Furthermore, we revealed that because of the ion-partitioning effect, higher vortex strength at lower permittivity ratios leads to better species homogeneity and mixing efficiency. Thus, mixing efficiency surpasses 90% for lower permittivity ratio values. Neutral species dispersion is faster owing to the ion-partitioning effect, especially for higher Carreau numbers. Utilizing the ion-partitioning effect, the results of this study can be utilized to design and develop efficient micromixers intended for the mixing of non-Newtonian fluids for diagnostic applications.

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1. Introduction

Practical interest in electrically actuated mixing^1–3 of fluids is overwhelming, ranging from DNA analysis, biotechnology tasks including protein folding research, and lab-on-a-chip (LOC) devices for enhanced chemical reactions to sample mixing in biochemical analysis.^4–7 Researchers have extensively explored the potential and applications of electroosmotic mixing across these diverse fields, propelling advancements in micro- and nanofluidic technology.^8–10 When a fluidic confinement subjected to an electric field develops a net charge density,^11–18 electroosmotic flow occurs. Electroosmotic mixing outperforms pressure-driven mixing in microfluidics due to its precise control, better dispersion, low sample volume requirements, and a simple, minimally invasive technique.^10,19–22 These advantages make it more efficient for achieving uniform mixing in several sensitive biological, biomedical and biochemical applications.

Electroosmotic flow (EOF) is initiated by the generation of an electrical force through the interaction of an applied external electric field acting on the electrolytic liquid mass within the charged micro-confinements and the induced electric field due to the electrical double layer effect. Thus, the application of a pulsating electric field regulates mass mobility inside a micro-cylinder. It is reported that a larger mass flow rate is produced by a pulsating wave with a higher frequency.²³ Additionally, the combined effect of electroosmotic actuation and an applied pressure gradient has been found to influence several fluidic functionalities, including enhanced mass transfer,²⁴ regulating surface adsorption–desorption reactions in microchannels.²⁵ Note that the generation of vortices in micro-confinements, which can be achieved by a heterogeneous surface charge profile, has been crucial for providing convective mixing strength for species in micromixers.²⁶ It may be mentioned here that blood, serum, saliva, and other bio-samples are non-Newtonian in nature and exhibit intricate rheological behavior.^27–29 Notably, many biochemical activities and pathological diagnoses depend on the efficient mixing of biofluids.^30–36 In order to replicate the rheology of these biofluids, researchers have used both elastic and inelastic models, as reported in the literature.^37–40 It should be noted that attempts have been made to develop efficient micromixers by utilizing external fields^22,41 and geometrically adjusting the flow configuration.^42–45 The electroosmotic mixing characteristics of non-Newtonian fluids in the heterogeneously charged micromixer were investigated by Haque et al.⁴⁶ They reported that reducing the liquid's shear-thinning properties improves mixing efficiency. The electroosmotic mixing characteristics of non-Newtonian fluids flowing via non-uniformly charged nanochannels containing polyelectrolyte layers (PELs) were investigated in a recent study.⁴⁷ The authors revealed that a weakly permeable PEL allows for overall flow modulation to obtain improved mixing efficiency owing to the strong impact of rotational flow at the cost of reduced flow rate. Increasing the normalized time parameter for non-Newtonian Carreau fluids results in an increasing-decreasing trend of mixing efficiency for a non-uniformly charged wavy micromixer with a 180° phase lag of surface potential.⁴⁸ Reported inferences suggest that a surface charge-dependent slip length allows for improved mixing efficiency for non-Newtonian liquids in a non-uniformly charged wavy micro-mixer compared to the zero-slip scenario.⁴² As witnessed in the reported study, a hydrophobic patch has the potential to induce electroosmotic vortices in a small scale system.⁴⁹ The aforementioned discussion signifies that electroosmotic vortex production is an appropriate method for convective mixing of non-Newtonian liquids in micromixers.

It is important to note that a soft polymeric layer (PL) grafted onto the walls of lab-on-a-chip (LOC) devices/systems has already demonstrated remarkable potential to protect biofluids from possible degradation while in contact with the solid surface during transportation.^50–53 Usually referred to as the polyelectrolyte layer, or PEL for short, this polymeric layer (PL) in contact with the electrolytic solution functions as a charged soft layer and captures specific kinds of ions that are different from the mobile electrolyte ions.^54,55 Using the patterned soft PEL layer, the non-Newtonian Carreau model has been employed to explore electroosmotic mixing of shear-thinning fluids. It has been found that mixing efficiency exceeds 90% when both PEL length and thickness are increased alongside the higher values of the Carreau number. According to recent research, the pH-induced protonic exchange within the PEL considerably affects the vortex formation mechanism and mixing quality. Since the PEL fixed charge can generate vortices that improve convective mixing strength in the presence of an electric field,^56–59 we can infer that employing the PEL in the paradigm of convection-dominated electroosmotic mixing is beneficial.

The distinction in electrostatic force at the interface between PEL and bulk electrolyte under electroosmotic flow may be caused by an electrical permittivity difference.^60–63 As a result, this mechanism may produce Born energy⁶¹ and a unique ionic distribution near the interface. This phenomenon is referred to as the ion partitioning effect.^64–66 The inequitable buildup of ions caused by variations in charge densities and dielectric characteristics affects the mobility of charged species during electroosmotic flow. This leads to a disruption of the expected fluid flow patterns and ionic transport behavior in microfluidic systems. Recognizing and accounting for these effects are critical for accurate modeling and control of electroosmotic flow behavior, which ultimately influence the entire operation and capabilities of such systems.

Based on a thorough literature review, we found that the electroosmotic vortices formed by the PEL may boost convective mixing strength, thereby improving mixing quality. In addition, the ion-partitioning effect between the polyelectrolyte layer and the bulk electrolyte is significant owing to the electroosmotic interaction caused by the permittivity difference. However, the influence of the ion partitioning effect^{17,18,67–69} on the electroosmotic mixing characteristics of non-Newtonian fluids has not been studied in the open literature until this endeavor. Previous studies did not contribute to the PEL-induced ion partitioning effect on the underlying mixing of non-Newtonian fluids/solutes, which demands a separate analysis. Thus, our main goal is to investigate how the ion partitioning affects the underlying mixing of non-Newtonian fluids. Also, we have considered the Carreau model to describe the fluid rheology of non-Newtonian fluids, accounting for a few distinctive features of this model, including the capability to predict viscosity at very low and high shear rates.⁴⁸ In addition, we investigated the effect of ion-partitioning on the temporal dispersion of neutral species^70–75 within the PEL-grafted microchannels, which enhances the impact of the current work. It is important to mention that the solute dispersion has potential diagnostic applications, particularly in protein analysis.⁷⁶

2. Mathematical modeling

The schematic diagram of the micromixer is depicted in Fig. 1. The soft negatively charged PEL and polymeric layers (PLs) are grafted onto both walls of the micromixer. The flow of the candidate liquids (cf.Fig. 1) through the micromixer is driven by an applied pressure gradient along the x-direction. It is noted that the negatively charged PEL is able to generate the electroosmotic vortices under the influence of an applied potential difference, ΔV in the context of pressure-driven primary flow. The height of the micromixer is taken as 2H, while the length is taken as L = (25/3)H. The length of the PEL and PL patch is taken as which is equal to .^54,55 The thickness of the PEL and PL patches is taken as . To investigate the mixing characteristics of the non-Newtonian fluids, the pure non-Newtonian liquid (liquid 1) is injected at the upper part of the micromixer inlet (y* > H), while the liquid with a tracer particle having the same physical properties (liquid 2) is inserted at the lower half of the inlet (y* < H). We assumed that the flow of the non-Newtonian fluid is steady and incompressible. Also, we do not consider the finite ionic size and the Joule heating effect in this study for the considered range of surface potential and external electric field.⁴⁸ It is evident that the ion transport in microfluidic channels is mainly dependent on the diffusion due to smaller ionic Péclet number (Pe_i ≪ 1).⁴⁸ Hence, assuming the Boltzmann ionic distribution, the dimensionless form of the Poisson–Boltzmann equation⁷⁷ is written as follows:^54,55

Ωε_r(∇²ψ) = [κ² sinh(ψ)]f_i − Θκ_p² (1)

Here, the dimensionless potential is Ψ, normalized with the scale, . The dimensionless Debye parameter κ can be expressed as H/λ_D and ; here ε₀, ε_rf, k_B, T, Z, e and n₀ are electrical permittivity of free space, relative permittivity of electrolytic liquid, Boltzmann constant, absolute temperature, ionic valence, elemental charge on a single electron and ionic bulk concentration, respectively. The term f_i appearing in eqn (1) is the ion portioning coefficient, as elaborated next. The value of Θ is taken as 0 for PEL-free region and 1 for PL-PEL region, respectively. Moreover, the dimensionless Debye parameter for PEL can be written as κ_p = H/λ_P, here ; Z and N are the polyelectrolyte valence and the total number density of polyelectrolyte ions, respectively. The permittivity ratio of PEL to bulk electrolyte is represented by ε_r(= ε_r,PEL/ε_rf). Here, Ω is taken as 1/ε_r and 1 for the PEL-free region and the PL-PEL region, respectively. To account for the effect of ion-partitioning effect due to the electrical permittivity difference between the PEL and the bulk electrolyte in the underlying analysis, the contribution of Born energy (w_i) has been considered. The expression for the Born energy can be written as follws:^64,78–80In eqn (1), r_i is the average ionic radius of the ionic species. The value of r_i is taken as 3.3 × 10⁻¹⁰ m, corresponding to ionic radius of K⁺ and Cl⁻ ion.⁶² By considering eqn (2), the ion partitioning coefficient appearing in eqn (1) is expressed as follows:^64,78–80

f_i = e^{(−Δw_i/k_BT)} (3)

As we have taken the average ionic radius of the species, the Born energy difference is denoted as Δw instead of Δw_i afterwards.

Fig. 1 Schematic of the micromixer with a negatively charged polyelectrolyte layer (PEL) and polymeric soft layer (PL). The flow is actuated by the pressure-driven flow and modulated by the electroosmotic flow generated by an external potential difference.

In this study, we have ignored the effect of finite ionic size because of the diluted salt solution. Note that the finite ionic size effect can be taken into account by the steric effect which is expressed as:⁴⁸υ = (4/3)π(2r_i)³N_A(1000M_c). Here, N_A and M_c are the Avogadro number and molar concentration, respectively. For dilute salt solution with the order of M_c from ∼10⁻⁶ to 10⁻³, the range of υ can be obtained between ∼10⁻⁶ and ∼10⁻³. The finite ionic size effect will be significant for ionic distribution when υ ∼ 10⁻¹.^81,82 Therefore, we can ignore the effect of ionic size on the ionic distribution because the order of υ ≪ 1 in the present case. As a result, even though the magnitude of the EDL potential field near the wall is very high (greater than the Debye–Hückel limit), the smaller value of υ diminishes the effect of finite ion size on the ionic distribution.

To estimate the flow-field in the micromixer, we use the dimensionless form of the continuity and momentum equations, which are written as follows:^54,55

∇·u⃑ = 0 (4)

Re(u⃑·∇)u⃑ = ∇p + ∇·[small tau, Greek, vector] + κ² sinh(ψ)∇ϕ − ΛF_d²u⃑ (5)

Here, is the dimensionless velocity vector normalized by the scale, . Note that is the external reference electric field and is expressed as . Here, the deviatoric stress tensor, [small tau, Greek, vector], can be expressed as [small tau, Greek, vector] = ([small mu, Greek, macron]_∞ + (1 − [small mu, Greek, macron]_∞)[1 + (Cu[small gamma, Greek, dot above])²]^0.5(n−1))(∇u⃑ + (∇u⃑)^T), the dimensionless form of second invariant of deformation tensor and strain rate tensor can be written as and S⃑ = 0.5[∇u⃑ + (∇u⃑)^T], respectively. In the previous term, the dimensionless apparent viscosity is expressed as ([small mu, Greek, macron]_∞ + (1 − [small mu, Greek, macron]_∞)[1 + (Cu[small gamma, Greek, dot above])²]^0.5(n−1)) according to Carreau model.⁴⁸ The Carreau model, typically used to describe the rheology of inelastic non-Newtonian fluids, has a few distinctive features, including the prediction of the apparent viscosity of the fluids even at very low and high shear rates and its ability to accurately predict the rheology of biological fluids.⁴⁸ Furthermore, [small mu, Greek, macron]_∞, Cu and n are dimensionless infinite shear rate viscosity normalized with zero shear rate viscosity (μ₀), Carreau number and flow behaviour index, respectively; λ is the time parameter for Carreau model.

The term Λ in the last term of the eqn (5) represents the fluid fraction within porous PEL and PL, and F_d is the frictional drag coefficient. The value of Λ is taken as 0 for the PEL-PL free region and as 1 for the PEL-PL region. Pertaining to the present study, the Reynolds number can be expressed as , where ρ is the fluid density.

To estimate the external electric field, the dimensionless external potential field ϕ can be computed by solving the Laplace equation as follows:⁶⁰

∇²ϕ = 0 (6)

Moreover, the tracer species transport can be described by the dimensionless form of the convection-diffusion equation, which is mathematically represented as follows:^54,55

Pe(u⃑·∇C) = ∇²C (7)

Here, C is the normalized species concentration scaled by the tracer fluid inlet concentration. The diffusive Péclet number can be expressed as , where D is the diffusion coefficient of the uncharged tracer species.

The following boundary conditions are imposed to solve the aforementioned governing equations. At the inlet: n⃑·∇ψ = 0, u_in = 0.1, ϕ = 25/3, C = 0 for 1 ≤ y ≤ 2 and C = 1 for 0 ≤ y ≤ 1; at the outer wall: n⃑·∇ψ = 0, n⃑·∇ϕ = 0, u⃑ = 0 and n⃑·∇C = 0; at the outlet: n⃑·∇ϕ = 0, p = 0, n⃑·∇C = 0; at bulk electrolyte and PEL interface, at the PEL and PL interface and at the PL and bulk electrolyte interface, the continuity of transport variables and their flux is maintained.

The mixing quality at the micromixer outlet can be expressed in terms of mixing efficiency as follows:^54,55

Here, C_∞ = 0.5, and C₀ = 0 or 1, representing the fully mixed and unmixed state, respectively.

Further, to examine the temporal dispersion of neutral species inside the PEL-grafted microchannel, we employ the time-dependent convection-diffusion equation as follows:⁸³

Here, t is the normalized time and is expressed as ; . For solving eqn (9), a plug of C = 1 is placed at 0.1 ≤ x ≤ 0.5 and set as C = 0 for x < 0.1 and x > 0.5 when t = 0 (initial condition). The following boundary conditions are also used: at the walls and at the outlet: n⃑·∇C = 0 and C = 0 at the inlet.

3. Numerical methodology and model verification

The governing transport equations used in this study to estimate the underlying mixing of two miscible fluids [eqn (1)–(7)] are solved using COMSOL Multiphysics 5.6, a finite element-based numerical solver.⁸⁴ The micromixer's computational region is divided into subdomains utilizing non-uniform triangular mesh elements. To ensure precise prediction of the transport variables, a highly dense mesh is deployed in the following regions as follows: at the inlet, around the interfaces, the walls, and the PEL patch region. The linear shape function for the velocity, pressure, and species concentration fields is used to convert the differential equation into a system of linear equations. For the external and EDL potential fields, the quadratic shape function is used. The multifrontal massively parallel sparse direct solver (MUMPS) is utilized to solve the EDL and external potential fields in the stationary solver. Following the generation of these potential fields, the flow, pressure, and species concentration fields are computed using the parallel direct sparse solver (PARDISO). Also, the simulation setup, including the methodology details used for solving the transport equations using the framework of COMSOL Multiphysics, is provided in Appendix A. We also conducted the grid independence test by calculating the mixing efficiency at the micromixer's outlet, as shown in Table 1. It is clear from the grid test that the use of mesh type M5, which has 230 872 meshes, leads to a very small change in mixing efficiency (<0.1%). Following this analysis, we used mesh type M5 to simulate all the cases considered in this study.

Table 1 Grid sensitivity analysis by computing the outlet mixing efficiency for different mesh systems when n = 0.3, u_in = 0.1, d_p = 0.2, κ_p = 10, κ = 20, f_d = 10 and Pe = 1000

Mesh type	Number of elements	Mixing efficiency at the outlet	% error in mixing efficiency with respect to mesh, M5
M1	20 954	96.31111	19.913
M2	52 292	92.96534	15.748
M3	90 971	82.40016	2.593
M4	177 627	80.37890	0.077
M5	230 872	80.31695	—

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To confirm the accuracy of our developed modeling framework, we compare the results obtained from this endeavor with the reported numerical and experimental results for a similar configuration, as shown in Fig. 2. First, we undertake an effort to benchmark the species concentration field in an electroosmotically actuated flow with the results of Gaikwad et al.,⁸⁵ as depicted in Fig. 2(a). This benchmarking analysis is conducted for the limiting case of n = 1, while the other parameters are taken as u_in = 0.1, d_p = 0.2, κ_p = 10, κ = 15, f_d = 10 and Pe = 500. Next, we compare the EDL potential field developed in the microchannel under the consideration of the ion-partitioning effect (Δw ≠ 0) with the result of Sin,⁷⁷ as shown in Fig. 2(b). The values of the other parameters are taken as follows: d_p = 0.3, κ/κ_p = 0.7, 1/κ = 0.3, ε_r = 1 with the ionic radius taken as 0.33 nm. Furthermore, we undertake an effort to compare the electroosmotic vortex formation under the non-uniformly charged microchannel surface with the experimental result of Stroock et al.,⁸⁶ as illustrated in Fig. 2(c). For this validation, we consider n = 1 with no PEL, u_in = 0.025, κ = 250 and electroosmotic mobility as 1.9 μm s⁻¹ V⁻¹ cm⁻¹). Also, a quantitative comparison of the electroosmotic flow field with the reported results by Hsieh et al.⁸⁷ is depicted in Fig. 2(d) for different strengths of the external electric field. The other parameters considered for this plot are as follows: the zeta potential as −55.1 32 mV, 10 mM NaCl salt solution, and n = 1. In Fig. 2(e), we undertook an effort to verify the normalized species concentration field developed in electroosmotic flow using the experimental results of Biddiss et al.,⁸⁸ shown for two different values of electric field intensities of 70 V cm⁻¹ and 280 V cm⁻¹, respectively. For this comparison, we have used a 25 mM salt solution with a viscosity of 0.001 Pa-s, a density of 1000 kg m⁻³, a species diffusion coefficient of 4.37 × 10⁻¹⁰ m² s⁻¹ and n = 1. The zeta potential has been determined based on the electroosmotic mobility of PDMS as −5.9 × 10⁻⁴ cm² V⁻¹ s⁻¹.

Fig. 2 (a) Comparison of dimensionless species concentration at the section x = 5 in the electroosmotic micromixer having a PEL with the numerical work of Gaikwad et al.⁸⁵ at n = 1, u_in = 0.1, d_p = 0.2, κ_p = 10, κ = 15, f_d = 10, Pe = 500 and Δw = 0. (b) Comparison of dimensionless potential in the microchannel having a PEL with the numerical work of Sin⁷⁷ at d_p = 0.3, κ/κ_p = 0.7, 1/κ = 0.3, ε_r = 1, Δw ≠ 0, ionic radius = 0.33 nm, and the limiting case n = 1 (c) Validation of electroosmotic vortex formation with the experimental results of Stroock et al.⁸⁶ for the limiting case n = 1 with no PEL. Parameters: u_in = 0.025, κ = 250 and electroosmotic mobility as 1.9 μm s⁻¹ V⁻¹ cm⁻¹. (d) Comparison of the experimental electroosmotic velocity profile with the results of Hsieh et al.,⁸⁷ shown for different values of external electric field strengths. Other parameters for this comparison: zeta potential = −55.1 32 mV, 10 mM NaCl salt solution, n = 1 and no grafted PEL. (e) Comparison of the normalized species concentration profile with the experimental results of Biddiss et al.,⁸⁸ depicted for two different external electric field strengths of 70 V cm⁻¹ and 280 V cm⁻¹. Other parameters considered: n = 1 and no PEL.

The benchmarking results, as depicted in Fig. 2, endorse that the predictions made by the present model are consistent with the reported results. This benchmarking endeavor suggests that the modeling framework developed in this study is capable of producing physically consistent results.

4. Results and discussion

In this work, we have investigated the electroosmotic mixing characteristics of non-Newtonian fluids in a micromixer with a built-in negatively charged soft polyelectrolyte layer (PEL) considering the ion-partitioning effect. In doing so, we have systematically investigated the induced potential field, streamlines, recirculation flow velocity (u_r), species concentration field and mixing efficiency (η), which have been explored systematically by varying the permittivity ratio of the PEL and electrolyte (ε_r), Carreau number (Cu) and flow behaviour index (n) in the range of 0.2 ≤ ε_r ≤ 1, 0.1 ≤ Cu ≤ 5 and 0.3 ≤ n ≤ 0.9, respectively, with the diffusive Péclet number of the order of 1000.^{54,55,64,78–80}

4.1. Counter-ion concentration field

First, Fig. 3 illustrates how the normalized counter-ion concentration field is influenced by the ion-partitioning effect. The ion-partitioning effect acts strongly for the permittivity ratio of ε_r = 0.2, whereas it has no effect when the permittivity ratio is unity (ε_r = 1). It is readily apparent that the ion-partitioning effect causes a substantial reduction in the number of counter-ions in the PEL region. The decreased electrical permittivity in the PEL compared to the free electrolytic region is the cause of the increased electrostatic energy. Consequently, the permittivity difference compels counter-ions to migrate to the low electrostatic energy region. This phenomenon results in a lower counter-ion concentration in the PEL region forε_r = 0.2 as compared to the case of ε_r = 1. Following the above-mentioned observation, it may be conjectured that the EDL potential and flow field may be significantly modulated by the notable decrease in counter-ions in the PEL.

Fig. 3 Contours of the normalized counter-ion concentration for ε_r = 0.2 and 1, when d_p = 0.2, κ_p = 10, κ = 20 and Δw ≠ 0.

4.2. Induced potential field

The variation of the dimensionless induced potential field (consistent with the profile in the middle cross-section of the micromixer) is depicted in Fig. 4 at different ε_r with (Δw ≠ 0) and without (Δw = 0) considering the ion-partitioning effect. It is observed that the ion-partitioning effect allows a higher magnitude of induced potential to develop in the PEL when ε_r ≠ 1. This is attributed to the higher Born energy in the PEL at smaller permittivity (see eqn (2)), which inhibits electrolytic ions from penetrating the PEL. Therefore, the smaller number of counter-ions (see Fig. 3) allows a weaker shielding effect at the smaller permittivity ratio. Hence, the counter-ions experience a higher strength of electrostatic force from the charged PEL, which in turn, results in a higher magnitude of induced potential in the PEL at a smaller permittivity ratio. Interestingly, this effect unrealistically underestimates the induced potential field in the PEL when the Born energy difference is not taken into account (Δw = 0) (see inset of Fig. 4). The ongoing discussion suggests that the influence of Born energy on the induced potential fields cannot be trivially ignored for a PEL with a smaller permittivity compared to the bulk electrolyte.

Fig. 4 Variation in the dimensionless potential field with (Δw ≠ 0) and without (Δw = 0) the ion-partitioning effect for different permittivity ratios, when d_p = 0.2, κ_p = 10 and κ = 20.

4.3. Flow field

With (Δw ≠ 0) and without (Δw = 0) considering the ion-partitioning effect, the streamline contours at various ε_r are depicted in Fig. 5. The vortices are seen to form in the region close to the PEL. The interaction of the external electric field and the cations in the negatively charged PEL leads to the development of vortices in the flow pathway. The electric body force is directed from right to left in this situation. This enables electroosmotic actuation in that direction as well. Vortices near the PEL are formed simultaneously accounting for the viscous shearing effect of two flow streams: the left-to-right directed pressure-driven flow and the right-to-left directed electroosmotically actuated flow. Interestingly, the vortex size and strength are found to be significantly larger in the presence of the ion-partitioning effect (Δw ≠ 0). In contrast to the case of zero ion-partitioning effect, the higher space charge density resulting from the larger induced potential magnitude in the PEL at smaller ε_r (see Fig. 4) permits a greater electrical force (see eqn (5)) on the fluid element. As a result, when (Δw ≠ 0), the high electroosmotic flow strength permits a stronger vortex with a lower ε_r compared to the case of Δw = 0. Furthermore, when the value of ε_r increases, the vortex size and strength decrease. Because of the drop in the PEL Born energy caused by the higher ε_r, the number of PEL counter-ions becomes larger and the magnitude of the induced potential in the PEL is decreased. Because of the decrease in space charge density caused by the decrease in ψ magnitude with ε_r (see Fig. 4), the electrical body force thus exhibits a decreasing trend. Consequently, as ε_r increases, the reverse electroosmotic strength and vortex size decrease owing to the decreased electrical body force. Notably, for the cases with Δw = 0, the permittivity ratio has an insignificant effect on vortex size and strength (see right side of Fig. 5). This finding implies that when analyzing PEL-induced vortex production, the ion-partitioning effect cannot be disregarded.

Fig. 5 Contours of streamlines with the ion-partitioning effect (left side, (Δw ≠ 0)) and without the ion-partitioning effect (right side, (Δw = 0)) at a specific Carreau number (Cu = 1) other parameters are as follows: n = 0.3, u_in = 0.1, d_p = 0.2, κ_p = 10, κ = 20, and f_d = 10.

To understand the ion-partitioning effect-modulated flow physics of non-Newtonian fluids in the chosen fluidic pathway, we present the velocity profile for different values of the flow behavior index and permittivity ratio in Fig. 6. Due to the onset of electroosmotic flow in regions closer to the PEL in the opposite direction to the primary flow stream, flow reversal is observed close to the PEL, as indicated by the negative flow velocity therein. It is seen that the magnitude of the reverse flow velocity in the PEL becomes larger as the shear-thinning nature of the fluid increases. This is due to a decrease in apparent viscosity caused by the increase in the shear-thinning nature of the liquid (smaller n). To satisfy the mass conservation constraint, this flow reversal in the PEL increases the flow velocity in the PEL-free zone with increasing shear-thinning nature of the fluid. Interestingly, when the ion-partitioning effect is involved (Δw ≠ 0, ε_r = 0.2) [see Fig. 6(a)], the magnitude of the flow velocity (either in the PEL or the PEL-free region) is substantially higher (by an order of unity) than in the no-ion partitioning scenario (Δw = 0, ε_r = 1) [see Fig. 6(b)]. This observation is attributed to the amplified electrical body force caused by the ion-partitioning effect, which increases the reversed flow velocity, and hence strengthens the magnitude of flow velocity in the PEL-free region as well. Furthermore, the higher gradient of counter-ion distribution or induced potential field at the interface of the PEL and the PEL-free region results in the existence of a crossing-point [see Fig. 6(a)] near that interface. When the ion-partitioning effect is ignored, this crossing-point is located far from the interface [see Fig. 6(b)]. Based on the foregoing discussion, we can infer that the ion-partitioning effect substantially modulates the local flow field, which is deemed adequate to alter the mixing dynamics, as discussed in the forthcoming sections.

Fig. 6 Variation in the flow velocity profile (u) at the middle cross-section for different values of the flow behaviour index (n) and Cu = 1 when (a) ε_r = 0.2 and (b) ε_r = 1.

We also investigated the recirculation strength in terms of the maxima magnitude of reverse flow velocity, referred to as the recirculation velocity (u_r),⁴⁸ with changes in rheological parameters (the flow behavior index and the Carreau number, Cu) considering the ion-partitioning impact. We have presented the variation of recirculation velocity with the permittivity ratio at various Cu values, both with and without the ion-partitioning effect, in Fig. 7(a). It is clear that as Cu increases, so does the recirculation velocity. This increase is attributed to a decrease in viscous resistance caused by the reduction in the apparent viscosity of the fluids with increasing Cu values.

Fig. 7 (a) Variation in recirculation velocity (u_r) with the permittivity ratio at different Carreau numbers (Cu = 0.1, 1, 1.5). (b) Variation in recirculation velocity (u_r) with the permittivity ratio at different flow behaviour indices (n = 0.3, 0.5, 0.9) with (Δw ≠ 0) and without (Δw = 0) considering the ion-partitioning effect when u_in = 0.1, d_p = 0.2, κ_p = 10, κ = 20, and f_d = 10. (c) Variation in normalized PEL electrical body force with change in the permittivity ratio with (Δw ≠ 0) and without (Δw = 0) considering the ion-partitioning effect. (d) Variation in percentage enhancement in PEL electrical body force due to the ion-partitioning effect as compared to the zero ion-partitioning effect with change in the permittivity ratio.

Note that increasing the magnitude of permittivity weakens the flow reversal, attributed primarily to the decrease in the magnitude of the electrical body force (see Fig. 7(c)) for Δw ≠ 0. As a result, the recirculation velocity decreases as the permittivity ratio increases for Δw ≠ 0. Interestingly, when the ion-partitioning effect is considered (for Δw ≠ 0), the rate of decrease in recirculation velocity with the permittivity ratio is shown to be significant compared to the zero-ion partitioning scenario (Δw = 0). This is attributed to the substantial reduction in the magnitude of reverse flow velocity at higher values of the permittivity ratio (see Fig. 6), which reduces the electrical body force in the PEL (see Fig. 7(c)). On the other hand, when the ion-partitioning effect is ignored, the trivial change in the electrical body force due to a change in the permittivity ratio (see Fig. 7(c)) results in a negligible change in recirculating velocity. Moreover, the higher magnitude of the electrical body force at Δw ≠ 0 compared to Δw = 0 (∼3800% higher, see Fig. 7(d)) at lower permittivity ratios leads to significant changes in recirculation velocity. In Fig. 7(b), we also show how the recirculation velocity varies with the flow behavior index and permittivity ratio. We found that increasing the flow behavior index reduces the magnitude of the reversed flow velocity [see Fig. 6], resulting in a decrease in recirculation velocity.

4.4. Mixing of non-Newtonian fluids

The contours of the dimensionless species concentration field can help visualize the underlying mixing quality. We show in Fig. 8 the dimensionless species concentration contours for different values of the permittivity ratio, with and without considering the ion-partitioning effect. The other parameter considered is Pe = 1000.⁴⁸ In addition, Fig. 8 depicts the average deviation (|C − 0.5|_avg) in concentration across the fully mixed condition (C → 0.5). It is observed that the ion-partitioning effect at a lower permittivity ratio results in a highly uniform concentration field in the downstream of the micromixer. This observation is attributed to the presence of augmented convective mixing intensity because of the larger size and strength of vortices [see Fig. 5 and 7(a), (b)]. As a result, we obtain a smaller concentration deviation (|C − 0.5|_avg) compared to the fully mixed state for a smaller permittivity ratio. Furthermore, increasing the permittivity ratio results in a relatively less uniform species concentration field in the downstream. Because of a decrease in recirculation strength with the permittivity ratio [see Fig. 7(a) and (b)], the convective mixing strength declines, allowing the concentration uniformity to decrease. Hence, when the ion-partitioning effect is taken into account, the value of |C − 0.5|_avg becomes higher as the permittivity ratio increases. Note that for the cases with a zero ion-partitioning effect (Δw = 0), the insignificant change in recirculation strength [see Fig. 7] with the permittivity ratio results in a negligible change in the species concentration field. Therefore, we may infer that the ion-partitioning effect because of the permittivity difference between the PEL and the electrolyte solution improves species homogeneity for both adequate and efficient mixing. Moreover, it is established that the ion-partitioning effect plays an important role in convection-driven species transport for smaller values of the permittivity ratio.

Fig. 8 Contours of dimensionless species concentration field with patch thickness, d_p = 0.2 at different Carreau numbers (Cu = 0.1, 1, 5) when L_p = 1.2, u_in = 0.1, κ_p = 10, κ = 20, f_d = 10 and Pe = 1000 with (left-side, (Δw ≠ 0)) and without (right-side, (Δw = 0)) considering the ion-partitioning effect.

Fig. 9(a) depicts the variation in mixing efficiency (η) versus permittivity ratio for various values of Cu, obtained with and without taking the ion-partitioning effect into account. When the ion-partitioning effect is considered, the mixing efficiency is shown to be quite high (η > 90%) at smaller values of the permittivity ratio. On the other hand, when the ion-partitioning effect is disregarded, the mixing efficiency is significantly underestimated at the smaller permittivity ratios. These findings can be attributed to an underestimation of convective mixing intensity, which is primarily due to the underestimation of recirculation strength [see Fig. 7] resulting from ignoring the ion-partitioning effect. The value of mixing efficiency improves when the flow behavior index decreases [see Fig. 9(b)] and the Carreau number increases. Because of the increase in the shear-thinning nature of the fluid caused by changes in these rheological parameters, convective mixing strength increases as recirculation velocity increases [see Fig. 7]. Hence, the mixing efficiency follows an increasing trend with an increase in the Carreau number and a decrease in the flow behaviour index, as shown in Fig. 9.

Fig. 9 (a) Variation in mixing efficiency (η) with the permittivity ratio at different Carreau numbers (Cu = 0.1,1,5) when n = 0.3. (b) Variation in mixing efficiency (η) with the permittivity ratio at different flow behaviour indices(n = 0.3, 0.5, 0.9), Cu = 0.1, with (Δw ≠ 0) and without (Δw = 0) considering the ion-partitioning effect, with u_in = 0.1, d_p = 0.2, κ_p = 10, κ = 20, and f_d = 10.

4.5. Uncharged solute dispersion

We have shown the ion-partitioning effect on the uncharged solute species dispersion in Fig. 10(a), obtained at different temporal instances. Note that the underlying solute dispersion is evaluated using eqn (9). We can see that the plug of species initially moves forward due to the convection effect. As the plug moves near the PEL patches, the presence of electroosmotic vortices tries to mix the species, which can be attributed to the convective mixing strength provided by the vortices developed therein. Also, the base of the plug near the wall almost gets attached because of the very low strength of the flow field in the polymeric layer. The pore structure of the polymeric region offers greater viscous resistance to the underlying flow and reduces the magnitude of the flow velocity. Hence, near the wall region of the plug, the transport of species is relatively very slow compared to the free region. Furthermore, due to the ion-partitioning effect, the larger vortex strength (see Fig. 5) allows faster transportation of the plug (composed of species) when Δw ≠ 0 compared to the caseΔw = 0, as seen in Fig. 10(a). Also, due to the stronger vortex strength (see Fig. 4), the ion-partitioning effect enables relatively higher trapping of species, as seen in Fig. 10(b).

Fig. 10 (a) Temporal contours of the plug of normalized species concentration with (Δw ≠ 0) and without (Δw = 0) the ion-partitioning effect, when n = 0.3, Cu = 0.1, u_in = 0.1, d_p = 0.2, κ_p = 10, κ = 20, and f_d = 10. (b) Normalized species concentration profile with and without the ion-partitioning effect at t = 0.03 for the section x = 4.2, when n = 0.3, Cu = 0.1, u_in = 0.1, d_p = 0.2, κ_p = 10, κ = 20, and f_d = 10.

In Fig. 11(b), we also show how the rheological parameter Cu affects uncharged solute dispersion while the ion-partitioning effect is considered. We found that plug transport is faster in the core region for higher values of the Carreau number (Cu = 5). This observation is attributed to the greater decrease in apparent viscosity at higher Cu because of the greater shear-thinning nature, which increases the core flow velocity for Cu = 5 compared to Cu = 0.10. As a result, the core region plug moves quickly with increasing Cu. As observed in Fig. 11(b), the faster movement of the plug in the core allows for greater species concentration intensity. In contrast, near the wall, we observe a higher reverse flow velocity for Cu = 5 than for Cu = 0.1, allowing for a lower value of species concentration. Thus, we can deduce that fluid rheology has a major impact on neutral species dispersion under the convection-driven species transport.

Fig. 11 (a) Temporal contours of the plug of normalized species concentration with (Δw ≠ 0) the ion-partitioning effect for the rheological parameter, Cu = 0.1 and 5, when n = 0.3, u_in = 0.1, d_p = 0.2, κ_p = 10, κ = 20, and f_d = 10. (b) Normalized species concentration profile for Cu = 0.1 and 5 at t = 0.03 for the section x = 4.2, when n = 0.3, u_in = 0.1, d_p = 0.2, κ_p = 10, κ = 20, and f_d = 10.

5. Conclusions

In this study, we numerically investigated the effect of ion-partitioning on the vortex-driven mixing characteristics of non-Newtonian fluids in a microchannel with patterned soft PEL grafted onto the inner wall surfaces. We have considered the ion-partitioning effect – a physically realistic phenomenon that results from a contrast in the permittivity of the grafted soft PEL and the bulk electrolytic solution. We have shown that the viscous shearing effect resulting from two different flow streams – the left-to-right directed pressure-driven flow and the right-to-left directed electroosmotically actuated flow – leads to the formation of electroosmotic vortices in the regions closer to the PEL, which, in turn, promotes the mixing of two components. We have calculated the mixing efficiency of the constituent components by systematically investigating the normalized counter-ion concentration field, induced potential field, streamlines, recirculation velocity, and species concentration field. We have estimated the aforementioned parameters for a chosen window of the PEL to bulk electrolyte permittivity ratio, Carreau number, and flow-behaviour index in their physically justified range. To benchmark the developed modeling framework, under limiting conditions, we have verified our numerical results with the reported theoretical and experimental results. We found that the counter-ion concentration in the PEL becomes extremely low due to the ion-partitioning effect at low permittivity ratios. When ion-partitioning is ignored, this effect causes an overestimation of the induced potential field within the PEL at lower permittivity ratios. As a result, when the ion-partitioning effect is present, the magnitude of the flow velocity (whether in the PEL or the PEL-free region) becomes significantly higher (by an order of unity) than in the no ion-partitioning scenario. We have demonstrated that in the case of lower permittivity ratios, the intensity of the vortex is greatly underestimated when the ion-partitioning effect is not considered. The strength of the vortex is found to decrease significantly with increasing permittivity ratio due to a reduction in the electrical body force caused by the ion-partitioning effect. Notably, we found that increased vortex strength at lower permittivity ratios results in excellent species homogeneity and mixing efficiency owing to the ion-partitioning effect. Consequently, we have shown that for lower permittivity ratio values, the mixing efficiency exceeds 90%. In addition, we have shown that increasing the magnitude of the Carreau number and decreasing the flow behaviour index improve mixing efficiency following the higher strength of flow recirculation (the vortex strength becomes higher), primarily due to the reduction in the apparent viscosity of the fluid. We found that the dispersion of neutral species is triggered by the ion-partitioning effect, and the underlying effect is enhanced at higher Carreau numbers. The inferences of this study seem to be important for the design of on-chip devices involving the mixing of species in a background non-Newtonian liquid, which is largely used for diagnostic applications.

Author contributions

Sumit Kumar Mehta: conceptualization (equal); formal analysis (equal); investigation (equal); methodology (equal); software (equal); validation (equal); writing – original draft (equal). Prateechee Padma Behera: conceptualization (equal); investigation (equal); methodology (equal); writing – original draft (equal). Abhishek Dutta: conceptualization (equal); investigation (equal); methodology (equal); writing – original draft (equal). Bhashkar Jyoti Sharma: conceptualization (equal); investigation (equal); methodology (equal); writing – original draft (equal). Anubhab Gaurav Borah: conceptualization (equal); investigation (equal); methodology (equal); writing – original draft (equal). Pragyan Bora: conceptualization (equal); investigation (equal); methodology (equal); writing – original draft (equal). Subhrajit Borah: conceptualization (equal); investigation (equal); methodology (equal); writing – original draft (equal). Pranab Kumar Mondal: conceptualization (equal); methodology (equal); project administration (equal); supervision (equal); writing – review & editing (equal). Somchai Wongwises: project administration (equal); resources (equal); writing – review & editing (equal).

Conflicts of interest

The authors have no conflicts to disclose.

Data availability

The data that support the findings of this study are available in the manuscript.

Appendix

A. Simulation method in COMSOL multiphysics

Fig. 12 shows the simulation setup used in COMSOL Multiphysics. This method is used to evaluate the external and induced potential fields, the flow field, and the species concentration field, as described below.

Fig. 12 Schematic illustrating the simulation methods used in COMSOL Multiphysics.

Acknowledgements

S. K. M. acknowledges the financial support provided by the National Post-Doctoral Fellowship (N-PDF), SERB, with Reference No. PDF/2023/002072. P. K. M. gratefully acknowledges the financial support provided by the SERB (DST), India, through project no. CRG/2022/000762. S. W. acknowledges Thailand Science Research and Innovation (TSRI) and National Science, Research and Innovation Fund (NSRF) Fiscal year 2025 Grant number (FRB680074/0164). S. K. M. also acknowledges KMUTT for the financial support provided for numerical simulations. P. K. M. also wishes to acknowledge KMUTT for visiting professorship.

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Footnote

† Authors have equal contribution.

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