Ionic diffusioosmotic transport in nanochannels grafted with pH-responsive polyelectrolyte brushes modeled using augmented strong stretching theory

Liquid and ion transport in nanochannels and nanopores^1–6 is central to a variety of emerging applications in energy research,^7,8 development of ionic sensors, biosensors, and ionic gating,^9–12 and fabrication of novel biomedical and drug delivery platforms,^13,14 as well as our endeavor to better understand a myriad of biological systems for fabricating different biomimetic systems.^15,16 Several strategies have been devised to enhance the electrohydrodynamic (EHD) fluxes in nanochannels and nanopores. These strategies involve tuning the external forces (e.g., electric field and magnetic field) driving the transport and/or functionalizing the walls of these nanochannels/nanopores with entities that interplay with the liquid and ions that are being transported. For example, in a recent study, an enhanced electro-osmotic (EOS) flux was reported in a charged nanochannel with alternating slipping surface.¹⁷ Other recent related studies report enhancement in flow by controlling the zeta potential, wall slip coefficient,¹⁸ and aspect ratio of the pH-regulated nanochannel.¹⁹ Functionalizing micro-nanochannels by grafting their inner walls with environmental-stimuli sensitive polymer and polyelectrolyte (PE) molecules has emerged as an extremely popular strategy that has found applications in a large number of disciplines ranging from ion sensing and biosensing^20,21 to fabrication of diodes and current rectifiers.^22–25 For instance, in a recent study, Sadhegi discussed the manipulation of EOS flow velocity by varying thickness of the polyelectrolyte layer (PEL) coating in a microchannel.²⁶ These polymer and PE molecules are often grafted densely enough so that they stretch out away from the grafting surface forming “brush”-like configurations.^27–34 These brushes change their configurations as functions of the ion concentration and pH of the solution and accordingly enable the micro-nanochannels to be employed for such a wide variety of applications.

While there has been extensive research on probing the behavior of the PE brushes and their responses to different factors (e.g., salt concentration, pH, and solvent quality),^35–50 significantly less has been done in probing the electrohydrodynamic (EHD) transport in such PE-brush-functionalized nanochannels. The initial group of studies effectively considered a decoupled problem: they probed the EHD transport in such nanochannels assuming a constant (pH and salt concentration independent) brush height and monomer distribution.^51–66 The brush height and the monomer distribution severely affect the fluid flow. Disregard of the dependence of these parameters on the salt concentration and pH meant that the appropriate dependence of the fluid flow on salt concentration and pH also got disregarded. In order to address these lacunae, Das and co-workers in a series of recent papers^67–70 considered for the first time a simplistic yet coupled EHD model on the PE-brush-grafted nanochannel where the salt concentration dependence of the brush height was accounted for. The model was simplistic in the sense that it considered end-charged PE brushes with the brushes being modeled using the Alexander–de Gennes model^30,31 (where one considers a uniform monomer density along the length of the brush). In a couple of recent studies, Das and co-workers extended their theory to probe the EHD transport in such PE-brush-grafted nanochannels with the brushes being modeled by the augmented Strong Stretching Theory (SST).^71,72 This augmented SST was recently developed by Das and co-workers⁷³ and improved the well-known SST for the PE brushes^45–48 (used for modeling the thermodynamics, configuration, and electrostatics of the PE brushes) by accounting for the influence of the excluded volume interactions and a more complete form of the mass action law. These studies,^71,72 therefore, represent the most comprehensive theoretical analysis of the problem of EHD transport in brush-grafted nanochannels until date, where the brushes have been appropriately modeled and an appropriate connection between the brush configuration and the EHD transport has been considered.

In this paper, we study the diffusioosmotic (DOS) transport in nanochannels grafted with the pH-responsive PE brushes modeled using our recently developed augmented SST.^71–73 The DOS transport, which is a form of induced electro-soluto-hydrodynamic transport, is triggered here by applying a salt concentration gradient along the length of the nanochannel. The presence of this gradient interplays with the charge distribution and the charge imbalance of the electric double layer (EDL). As a consequence, there is a generation of an induced electric field. This electric field interacts with the EDL charge density and induces (also gets influenced by) an electroosmotic (EOS) transport. Additionally, the imposed salt concentration gradient induces a pressure-gradient and there is a resulting pressure-driven transport, which is also referred to as a chemiosmotic (COS) transport. Therefore, the DOS effect is a manifestation of three things: generation of an induced electric field, generation of a pressure gradient, and generation of liquid transport that is a combination of the EOS and COS transport. There has been extensive previous studies probing the DOS transport either by the imposition of a salt concentration gradient^74–85 or an uncharged solute gradient.^86–93 In addition to giving rise to the highly intriguing fluid mechanics of induced electro-soluto-hydrodynamic DOS transport, such DOS transport has been extensively employed for a large number of applications ranging from triggering microfluidic and interfacial transport^88,91 to designing novel strategies for sensing,⁹³ phase separation,⁹⁴ and particle manipulation.⁹⁵ In a recent study, Hoshyargar et al. studied the diffusioosmotic flow of an analyte solution in a charged microchannel and discussed its potential application in the separation of analytes.⁹⁴ In fact, in our recent study,⁶⁹ we probed the DOS transport in nanochannels grafted with the end-charged PE brushes, with the brushes being described by the simplistic Alexander–de Gennes model.^30,31 In the present paper, we consider a much more realistic system, where the brushes contain charges along their entire backbone and these charges are pH-responsive. More importantly, the brushes are modeled using the augmented SST. Such a description ensures that we are considering the most advanced description until date of an induced EHD transport (namely DOS transport) in a pH-responsive PE-brush-grafted nanochannel.

Our results first establish that the presence of the PE brush grafting significantly enhances (with respect to the brush-free nanochannels having identical charge content as the brush-grafted nanochannels) the diffusioosmotically induced electric field for all the different choices of the salt concentration, pH, and grafting density values. This induced electric field is a combination of the osmotic component (electric field developed due to the downstream migration of the EDL ions in the presence of the background DOS transport) and the ionic component (electric field that is generated due to the conduction current). We perform detailed analysis to show the relative variation of these individual components (osmotic and ionic) of the electric field as functions of the strength of the DOS transport, localization of the EDL charge density by the brushes away from the nanochannel wall enforcing a larger magnitude of the background flow to be responsible for the downward osmotic migration of the ions,⁷¹ and the varying diffusivity of the positive and negative ions (dictating different strengths of the cationic and anionic conduction). These analyses explain such augmented electric field generation by the brush-grafted nanochannels, establishing brush-grafting as a novel mechanism for inducing (through the facile route of applying a salt concentration gradient) energetically favorable scenarios. The second key finding of this study is the discovery that the DOS water transport is significantly enhanced in the presence of the brush grafting for most of the parameter combination. We explain that the DOS transport is a combination of the induced pressure-driven transport (triggered by the pressure gradient induced by the applied salt concentration gradient) also known as the chemiosmotic (COS) transport and the induced electroosmotic (EOS) transport triggered due to the induced electric field. For the choice of the positive value of the salt concentration gradient, the induced pressure-gradient is negative, triggering a COS liquid transport from right-to-left in the nanochannel. The EOS transport, therefore, augments (retards) the COS transport for cases where the induced electric field is negative (positive), triggering an EOS transport from right-to-left (left-to-right) in the nanochannel. Such an understanding, coupled with the knowledge that the presence of the brushes localizes the EOS body force away from the nanochannel wall and ensures a larger manifestation of the effect of the EOS body force, helps explain the overall DOS velocity profiles in brush-grafted and brush-free nanochannels. For this case too, the brushes emerge as an enabler of a significant increase in the overall DOS nanofluidic water transport. Finally, we provide a thorough comparison between the present study and our previous study.⁶⁹ Both the studies probe the DOS transport in the brush grafted nanochannel; however, while the present study considers the brushes to be backbone-charged (i.e., containing charges distributed along their entire backbone), our previous study⁶⁹ considered brushes that contain charges only at their non-grafted ends. The comparison reveals that for most of the parameter choices, the strength of the DOS velocity is significantly larger for the present case: we associate such an occurrence to the significantly smaller brush-induced drag force at the location where the different driving forces (EOS and COS body forces) are localized, as compared with that of the present study. Therefore, the present study establishes that functionalizing nanochannels with backbone-charged PE brushes indeed leads to a significantly enhanced (diffusioosmosis) form of induced electrokinetic nanofludic transport as compared to either brush-free nanochannels or nanochannels grafted with end-charged PE brushes.

We consider the ionic DOS transport in a nanochannel of half height h and length L and grafted with backbone-charged, pH-responsive PE brushes [see Fig. 1(b)]. The nanochannel walls do not contain any charge. This implies that σ, which is the net charge density at the wall, is zero; given that σ ∝ dψ/dy (where ψ is the EDL electrostatic potential), at the nanochannel wall, dψ/dy = 0. On the other hand, all the charges are present on the brushes. The nanochannel is connected to microfluidic reservoirs (not shown in the schematic). n_∞ and $nH+,∞$ are the bulk number densities of the electrolyte salt ions and H⁺ ions inside these reservoirs. We model the PE brushes using our recently developed augmented SST.^71–73 Use of such a model decides the brush height, the monomer distribution along the length of the brush, and the corresponding EDL electrostatic potential distribution in a thermodynamically self-consistent fashion. Consequently, the model for the DOS transport includes the thermodynamically self-consistent description of the brush height, monomer distribution, and the ion distribution. We consider that the ionic DOS transport is generated by employing a constant axial concentration gradient of the salt ions, namely, ∇n_∞ = dn_∞/dx. Here, dn_∞/dx is so chosen that $Ldn∞dx/n∞≪1$⁠. The presence of this dn_∞/dx implies that the brush height, the monomer distribution along the brush height, the corresponding drag coefficient that depends on the monomer distributions, and the electrostatic potential and the ion distributions within the brush-induced EDL will all have a weak gradient in the axial direction. The DOS transport is quantified by the corresponding diffusioosmotically induced electric field and the DOS velocity field. The DOS velocity is a combination of the COS flow (generated by the induced pressure-gradient) and the electroomsotic (flow) (generated by the diffusioosmotically induced electric field) [see Fig. 1(b)]. In this paper, such a diffusioosmotically induced electric field and the DOS velocity field for the brush-grafted nanochannel are compared with those in brush-free nanochannels having identical surface charge as that of the brush-grafted nanochannels.

In this section, we shall first re-visit the key steps of our augmented SST for the pH-responsive PE brushes. The detailed derivation of this theory has already been provided in our previous papers.^71–73 Here, we first summarize the key steps for the sake of completion. Subsequently, we shall provide the theory for the DOS transport in such brush-grafted nanochannels.

The augmented SST, described in details in our previous papers,^71–73 extends the existing SST for the PE brushes^45–48 by accounting for the contributions of (a) the excluded volume effects and (b) an expanded form of the mass action law (characterized by the consideration of a more generic value of γ with γa³ ≠ 1, where γ is the maximum density of PE chargeable sites and a is the Kuhn length). The key findings of this augmented SST are summarized below.

We first consider the free energy functional (F) of a given PE molecule, which is expressed as

where k_BT is the thermal energy, and F_els, F_EV, F_elec, F_EDL, and F_ion are the elastic (entropic), excluded volume, electrostatic, EDL, and ionization free energies (per PE molecule), respectively. Our previous paper⁷¹ provides a detailed expression for each of these energy terms.

A variational minimization of the above energy functional is conducted in the presence of the following constraints:

In the above equations, σ ∼ 1/ℓ² is the grafting density (where ℓ is the distance of separation between two adjacent grafting sites), N is the PE size (i.e., the number of monomers per PE chain), E(y, y′) = dy/dn represents the local stretching of the PE chain at a distance y from the grafting surface with y′ denoting the distance of the end of the PE chain, ϕ(y) is the dimensionless monomer distribution, and H is the brush height. This minimization eventually provides the equations (summarized below) that govern the brush thermodynamics,

Equation (4) provides the number density (⁠$nA−$⁠) of the A⁻ ion—the PE brush undergoes acid-like dissociation (HA ⇋ H⁺ + A⁻) and acquires negative charge by producing A⁻ ions on their backbone. In Eq. (4), $Ka′$ = 10³N_AK_a (N_A is the Avogadro number and K_a is the ionization constant for the reaction HA ⇋ H⁺ + A⁻), k_BT is the thermal energy, e is the electronic charge, and ψ is the EDL electrostatic potential. Equation (5) is the Poisson equation providing distribution of the EDL electrostatic potential ψ for the bottom half of the nanochannel (−h ≤ y ≤ 0). Also, in Eq. (5), H is the brush height, ϵ₀ is the permittivity of free space, ϵ_r is the relative permittivity of the electrolyte solution (both inside and outside the brush layers), and n_i is the number density of ion i (i = ±, H⁺, OH⁻). Equations (6)–(8) relate the ion number densities n_i to the corresponding bulk number densities n_i,∞ (i = ±, H⁺, OH⁻) through the Boltzmann distributions. In Eq. (9), which provides the monomer distribution profile, ν and ω are the virial coefficients quantifying the excluded volume effects, $κ2=9π2ω8N2a2ν2$⁠, $ρ=8a2N23π2$⁠, $λ=−λ1ρ=−λ18a2N23π2$ [λ₁ is the Lagrange multiplier for the constraint expressed in Eq. (3)], and $β=8N2ea53π2kBT$⁠. Equation (10) expresses the profile of the local stretching. Equation (11), which is based on the net unbalanced charge q_net in the system [see Eq. (12)], quantifies the equilibrium brush height H₀. Finally, Eq. (13) expresses the normalized chain end distribution function that satisfies the condition $∫−hH−hg(y′)dy′=1$⁠. The brush configuration and electrostatics (which eventually provide the brush height, monomer distribution, and the brush-induced EDL distribution) are finally obtained by solving Eqs. (4)–(13) in the presence of the following boundary condition for the EDL electrostatics (assuming an uncharged grafting surface),

The flow in the nanochannel is triggered by applying a salt concentration (or salt number density) gradient along the length of the channel, as discussed earlier. The resulting flow is considered to be steady, fully developed, and unidirectional. The DOS transport of the electrolyte in the nanochannel is governed by the following Navier–Stokes (NS) equations. The pressure field is obtained from the NS equation in the y-direction and by using the Boltzmann distributions [see Eqs. (6)–(8)] for the ion number densities,

In the above equation, $ψ¯=eψ/(kBT)$⁠. Also, n_+,∞ = n_∞, $n−,∞=n∞+nH+,∞−nOH−,∞$⁠, where n_∞ = 10³N_Ac_∞, $nH+,∞=103−pH∞NA$⁠, and $nOH−,∞=103−pOH∞NA$ (pOH_∞ = 14 − pH_∞). This also implies that $∇n∞=dn∞dx=dn+,∞dx=dn−,∞dx$⁠. The starting point of Eq. (15) comes from the simplification of the y-momentum conservation equation by considering the Space Charge Theory (SCT)^96–99 (see the  Appendix for more details). Consequently, in the presence of the applied axial gradients in electrolyte ion concentrations, we can write

This pressure-gradient, as can be seen, is dictated by the imposed gradient in the salt concentration.

The x-momentum equation, on the other hand, for the bottom half of the nanochannel can be expressed as (considering a steady and fully developed flow)

In Eq. (17), u is the velocity profile, η is the dynamic viscosity of the electrolyte, E is the induced electric field, $∂p∂x$ is the pressure gradient induced due to the employed salt concentration gradient [expressed in Eq. (16)], and $κd=a2/ϕ(y)2=a2H0σa3Nϕ¯2$⁠. Here, $ϕ¯=ϕH0σa3N$ is the normalized profile for the monomer distribution ϕ [expressed in Eq. (9) using the augmented SST calculations]. Equation (17) establishes that there are two driving forces for the DOS transport: the induced pressure gradient that drives an induced chemiosmotic transport and the induced electric field that drives an induced electroosmotic transport. κ_d, which is inversely related to the drag coefficient, can be obtained from the works of de Gennes¹⁰⁰ and Freed and Edwards.¹⁰¹ These works^100,101 showed that the drag coefficient varies as K², where K⁻¹ is the length that screens the flow inside the polymer coil in a semi-dilute polymer solution. Given that for this problem too, the flow inside the brushes is significantly lowered as compared with that outside the brushes, we can use this theory of de Gennes¹⁰⁰ and Freed and Edwards¹⁰¹ to express the drag coefficient in terms of K⁻¹. As κ_d varies inversely as the drag coefficient, we can write κ_d ∼ K⁻². Furthermore, K ∼ ϕ/a. Therefore, κ_d ∼ a²/ϕ².

Using Eq. (16), we can rewrite Eq. (17) in the dimensionless form as

In Eq. (18), $ū=uU$⁠, $A=2kBTn∞h2ηUL$⁠, $n¯1′=Ldn∞dx/n∞$⁠, $U=2kBTλ2η(dn∞dx)$⁠, $ψ¯=eψkBT$⁠, $Ē=EE0$⁠, where $E0=kBTeL$ is the scale of electric field, $n¯i,∞=ni,∞n∞$ (i = ±, H⁺, OH⁻), $ȳ=yh$⁠, $H¯0=H0h$⁠, $λ¯=λh$⁠, and $λ=ϵ0ϵrkBT/(2e2(n∞+nH+,∞))$ is the Debye screening length of the electric double layer (EDL).

In order to solve for $ū$ from Eq. (18), we need to first obtain the dimensionless electric field Ē. The electric field E is obtained from the condition

where J₊ and J₋ are the fluxes of the electrolyte cation and anion and $JH+$ and $JOH−$ are the ionic fluxes of the H⁺ and OH⁻ ions. We can express these fluxes as follows:

In the above equations, D_i are the diffusivities of species i (i = ±, H⁺, OH⁻).

Using Eqs. (20)–(22) as well as Eqs. (6)–(8) in Eq. (19), we can eventually obtain the dimensionless electric field as

where $Ēosm=Ēosm,++Ēosm,−+Ēosm,H++Ēosm,OH−$ with Ē_osm,i being the osmotic contribution associated with ion i and Ē_ion = Ē_p,ion − Ē_m,ion (with Ē_p,ion and Ē_m,ion being the ionic components of the electric field associated with the positive and negative salt ions). We can also express these different components as (with z_i being the valence of the ion of type i)

In Eq. (23), $Pe=ULD++D−+DH++DOH−$ is the Peclet number for the flow and $Ri=DiD++D−+DH++DOH−$ is the dimensionless diffusivities of each species i, where i = ±, H⁺, OH⁻. Equation (23) expresses Ē in terms of $ū$⁠. Therefore, if we use Eq. (23) to replace Ē in Eq. (18), we shall eventually get an integro-differential equation in $ū$⁠. This resulting equation in $ū$ is solved numerically in the presence of the following boundary conditions to obtain $ū$⁠:

We first study the variation of the diffusioosmotically induced dimensionless electric field Ē with salt concentration c_∞ in the presence of an axially employed salt number density gradient dn_∞/dx (see Fig. 2). Results are shown for six different cases: three different cases of DOS transport in brush grafted nanochannels and three more cases of the DOS transport in the brush-free nanochannel, having the same surface charge density as a given brush grafted nanochannel. For example, the three cases of the brush-grafted nanochannels are the following—case 1: pH_∞ = 3 and ℓ = 60 nm, case 2: pH_∞ = 3 and ℓ = 10 nm, and case 3: pH_∞ = 4 and ℓ = 60 nm. The three cases for the brush-free nanochannels will be the cases where the nanochannels have an equivalent surface charge density as that of a given brush-grafted nanochannel case: for example, “pH_∞ = 3 and ℓ = 60 nm (No Brush)” in the legend of Fig. 2, as well as in the legend of the subsequent figures, implies that we are considering the DOS transport in a brush-free nanochannel having the same surface charge density as the brush-grafted nanochannel with pH_∞ = 3 and ℓ = 60 nm. This equality in the surface charge densities is ensured by employing the condition $σc,eq=−e∫−h−h+H0ϕnA−dy$ (where σ_c,eq is the equivalent surface charge density of the brush-free nanochannels and ϕ and $nA−$ are defined in Sec. II A).

The diffusioosmotically induced electric field is a combination of the osmotic (Ē_osm) and the ionic (Ē_ion) contributions [i.e., Ē = Ē_osm + Ē_ion, see Eqs. (23)–(28)]. The osmotic contribution to the electric field, Ē_osm, is due to the downstream migration of the mobile ions of the electric double layer (EDL) in the presence of the diffusioosmotically induced velocity field. We shall discuss later in detail the variation of this DOS velocity field. Figure 3 compares Ē_osm − vs − c_∞ variation for the different cases for the brush-free and brush-grafted nanochannels. For the majority of the c_∞ values, Ē_osm is larger for the brush-grafted nanochannel. There are two interrelated factors that ensure such enhanced Ē_osm for brush-grafted nanochannels. First, for brush-grafted nanochannels, the DOS velocity field is significantly enhanced, as compared with that in the corresponding brush-free nanochannels across wide ranges of salt concentration and pH values (see Figs. 8–11 below). Only for a very few conditions, this might not be true: for example, for c_∞ = 10⁻³M, for the case of pH_∞ = 4, ℓ = 60 nm or for c_∞ = 10⁻²M, for the case of pH_∞ = 3, ℓ = 10 nm. Second, the presence of the brushes localizes the net charge of the EDL away from the wall. The strength of a velocity field is much larger at locations away from the nanochannel wall. Accordingly, the contribution of the background flow that drives the EDL charges, thereby leading to the development of Ē_osm, gets enhanced. These two effects interplay to dictate the final value of Ē_osm. Accordingly, we mostly find $ĒosmBrush>ĒosmNoBrush$ (see Fig. 3). However, this is not true for those particular cases where $(uDOS)Brush<(uDOS)NoBrush$⁠; this happens, for example, for c_∞ = 10⁻³M, pH_∞ = 4, ℓ = 60 nm and c_∞ = 10⁻²M, pH_∞ = 3, ℓ = 10 nm. The contribution of the osmotic migration of the different ions (±, H⁺, OH⁻) to Ē_osm has been provided in Fig. 4 [also see Eq. (25)]. The osmotic migration associated with the H⁺ ions primarily contributes to Ē_osm for small salt concentrations. An increase in the salt concentration progressively increases the contribution associated with the osmotic migration of both the salt cation and anion. These contributions negate each other. Such behaviors are true for both the brush-free and brush-grafted nanochannels. Under these circumstances, we eventually obtain a Ē_osm distribution that first increases and then decreases with c_∞ for both the brush-grafted and brush-free nanochannels. In Fig. 4, we do not show the contribution associated with the osmotic migration of OH⁻ ions as it is very small.

Here, we first try to attempt to understand the significantly non-monotonic variation of Ē_osm,i with c_∞. As evident from Eq. (25), one can express that

where

and

In Fig. 5, we plot the variation of Ē_osm,diff and Ē_osm,i,$adv$ (for i = ±, H⁺) with c_∞. Given that Ē_osm,i is simply the ratio of Ē_osm,i,$adv$ and Ē_osm,diff, all we have to do to shed light on the variation of Ē_osm,i with c_∞ (see Fig. 4) is to better understand the corresponding variation of Ē_osm,i,$adv$ and Ē_osm,diff. It can be seen that Ē_osm,diff decreases monotonically with an increase in the salt concentration [see Fig. 5(a)]. In the lower salt concentration regime (⁠$c∞≤10−pH∞$⁠), Ē_osm,diff primarily depends on the overall diffusion of the H⁺ ions (dictated by the product of the dimensionless number density $n¯H+=nH+n∞$ and the dimensionless diffusivity $RH+$⁠), given that H⁺ ions have significantly larger diffusivity among all the ions. Therefore, as the salt concentration increases, i.e., n_∞ increases (in this weak salt concentration regime), a progressive lowering of the ratio $n¯H+=nH+n∞$ is encountered, leading to a steep decrease in Ē_osm,diff. This decrease is further augmented by the fact that the EDL potential decreases in magnitude monotonically with an increase in salt concentration,⁷¹ resulting in a smaller value of $nH+$⁠. On the other hand, for much larger values of the salt concentration, i.e., when $c∞>10−pH∞$ and $|ψ¯|≪1$⁠,⁷¹ we can simplify Ē_osm,diff as [with $n¯+,∞=n∞/n∞=1$⁠, $n¯−,∞=(n∞+nH+,∞)/n∞=1+n¯H+,∞$⁠, neglecting $n¯OH−,∞$⁠, and $exp(±ψ¯)≈1±ψ¯≈1$]

Therefore, for such larger values of c_∞, we observe that Ē_osm,diff becomes constant and does not vary with c_∞. On increasing pH_∞, when all other parameters are kept constant, Ē_osm,diff decreases owing to the decrease in the bulk hydrogen ion number density. On the other hand, an increase in the grafting density (decreasing ℓ) leads to an increased magnitude of the EDL potential due to the larger overall charge of the PE brush. This in turn increases $n¯H+$⁠, which has the highest diffusivity among all the given ions, which enhances the Ē_osm,diff.

We next study the variation of the advection-based field Ē_osm,+,$adv$ [see Fig. 5(b)]. From Eq. (31), we find that Ē_osm,+,$adv$ depends on the Peclet number (Pe) (or characteristic velocity U), EDL potential, and the fluid velocity (with $n¯+,∞=n∞/n∞=1$⁠). Using the definitions of Pe [see below Eq. (28)] and U [see below Eq. (18)] as well as the condition dn_∞/dx = 10⁴n_∞, we can express $Pe∝11+n¯H+,∞$⁠. Therefore, for small c_∞ (i.e., $c∞≪10−pH∞$⁠), $Pe∝n∞/nH+,∞$⁠, implying that Pe increases linearly with n_∞. On the other hand, for much larger c_∞ (i.e., $c∞≫10−pH∞$⁠), $Pe≈104ϵ0ϵr(kBT)2Lηe2(D++D−+DH++DOH−)$⁠, i.e., Pe does not vary with salt concentration. Accordingly, for smaller salt concentrations, variation of Ē_osm,+,$adv$ with c_∞ is mostly dictated by the corresponding variation of Pe (which varies linearly with c_∞ for such small concentration values, see above), while for larger salt concentration, where Pe no longer varies with c_∞ (see above), variation of Ē_osm,+,$adv$ with c_∞ is mostly dictated by the corresponding variation of the dimensionless diffusioosmotic velocity profile $ū$⁠. Of course, $|ψ¯|$ monotonically decreases with c_∞⁷¹ and that also contributes to the overall variation of Ē_osm,+,$adv$ with c_∞. Under such circumstances, for the case pH_∞ = 4 and ℓ = 60 nm, Ē_osm,+,$adv$ increases monontonically with c_∞ for smaller c_∞ values reflecting the dominant influence of Pe. At larger concentrations (⁠$c∞>10−pH∞$⁠), i.e., when the effect of $ū$ starts to dominate the variation of Ē_osm,+, adv, we see a very large decrease in velocity [see Fig. 8(c)] as we move from c_∞ of 10⁻⁴M to 10⁻³M (for the case of pH_∞ = 4 and ℓ = 60 nm), which is reflected by a steep decrease in Ē_osm,+, adv. As we further move from c_∞ of 10⁻³M to 10⁻²M, it can be seen that there is an increase in Ē_osm,+, adv due to a similar increase in $ū$⁠. Finally, moving from c_∞ of 10⁻²M to 10⁻¹M, the velocity decreases slightly, which is reflected by a slight decrease in Ē_osm,+,$adv$. On the other hand, if we consider the case of pH_∞ = 3 and ℓ = 10 nm, we find this monotonic increase in Ē_osm,+,$adv$ with c_∞ (due to the corresponding increase in Pe with c_∞) for up to c_∞ = 10⁻³M; subsequently, there is a steep decrease in Ē_osm,+,$adv$ with c_∞ as we move from c_∞ of 10⁻³M to 10⁻²M [due to a significantly smaller $ū$ at c_∞ = 10⁻²M, see Fig. 8(b)] and Ē_osm,+,$adv$ increases with c_∞ as we move from c_∞ of 10⁻²M to 10⁻¹M due to the corresponding enhanced value of $ū$ [see Fig. 8(b)]. Finally, for the case of pH_∞ = 3 and ℓ = 60 nm, we do find a monotonic increase in Ē_osm,+,$adv$ with c_∞ for smaller c_∞ values; however, no drastic variation in the corresponding $ū$ profile (as witnessed for cases of pH_∞ = 4 and ℓ = 60 nm and pH_∞ = 3 and ℓ = 10 nm) implies that for larger salt concentrations (i.e., where Pe does not vary with c_∞), Ē_osm,+,$adv$ does not show significant variation with c_∞ except for a slight decrease in the range from c_∞ = 10⁻²–10⁻¹M {due to a noticeable lowering of $ū$ [see Fig. 8(a)]}.We can use these information on Ē_osm,diff [see Fig. 5(a)] and Ē_osm,+,$adv$ [see Fig. 5(b)] for interpreting the variation of Ē_osm,+ for the three different cases. For example, using these variations, we can straightaway explain the steep increases in Ē_osm,+ for the majority of salt concentration and small decrease in the range from c_∞ = 10⁻²–10⁻¹M for the case of pH_∞ = 3 and ℓ = 60 nm [see Fig. 4(a)]. Similarly, for the case of pH_∞ = 3 and ℓ = 10 nm, we can easily justify an increase, then a decrease (in the range from c_∞ = 10⁻³–10⁻²M), and then again an increase (for c_∞≥ 10⁻²M) in Ē_osm,+ from the corresponding variation of Ē_osm,+,$adv$ and Ē_osm,diff [see Fig. 4(b)]. Finally, for the case of pH_∞ = 4 and ℓ = 60 nm, this studied variation for Ē_osm,+,$adv$ and Ē_osm,diff helps justify Ē_osm,+ first increasing, then decreasing (in the range from c_∞ = 10⁻⁴–10⁻³M), again increasing (in the range from c_∞ = 10⁻³–10⁻²M), and then finally again decreasing (in the range from c_∞ = 10⁻²–10⁻¹M) with c_∞ [see Fig. 4(c)].