In this paper, we study the diffusioosmotic (DOS) transport in a nanochannel grafted with pH-responsive polyelectrolyte (PE) brushes and establish brush-functionalization-driven enhancement in induced nanofluidic electric field and electrokinetic transport. The PE brushes are modeled using our recently developed augmented strong stretching theory. We consider the generation of the DOS transport due to the imposition of a salt concentration gradient along the length of the nanochannel. The presence of the salt concentration gradient induces an electric field that has an osmotic (associated with the flow-driven migration of the ions in the induced electric double layer) and an ionic (associated with the conduction current) component. These two components evolve in a manner such that the electric field in the brush-grafted nanochannel is larger (smaller) in magnitude than that in the brush-less nanochannels for the case where the electric field is positive (negative). Furthermore, we quantify the DOS flow velocity and establish that for most of the parameter choices, the DOS velocity, which is a combination of the induced pressure-gradient-driven chemiosmotic component and the induced electric field driven electroosmotic transport, is significantly larger for the nanochannels grafted with backbone-charged PE brushes (i.e., brushes where the charge is distributed along the entire length of the brushes) as compared to brush-free nanochannels or nanochannels grafted with PE brushes containing charges on their non-grafted ends.

## I. INTRODUCTION

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-workers^{73} 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 study^{69} 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.

## II. THEORY

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 $nH+,\u221e$ 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 $Ldn\u221edx/n\u221e\u226a1$. 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.

### A. Augmented strong stretching theory for pH-responsive PE brushes

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 *γa*^{3} ≠ 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*_{B}*T* 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 paper^{71} 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/*ℓ*^{2} 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 ($nA\u2212$) 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), $Ka\u2032$ = 10^{3}*N*_{A}*K*_{a} (*N*_{A} is the Avogadro number and *K*_{a} is the ionization constant for the reaction *HA* ⇋ *H*^{+} + *A*^{−}), *k*_{B}*T* 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, *ϵ*_{0} 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, $\kappa 2=9\pi 2\omega 8N2a2\nu 2$, $\rho =8a2N23\pi 2$, $\lambda =\u2212\lambda 1\rho =\u2212\lambda 18a2N23\pi 2$ [*λ*_{1} is the Lagrange multiplier for the constraint expressed in Eq. (3)], and $\beta =8N2ea53\pi 2kBT$. 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 *H*_{0}. Finally, Eq. (13) expresses the normalized chain end distribution function that satisfies the condition $\u222b\u2212hH\u2212hg(y\u2032)dy\u2032=1$. 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),

### B. DOS transport in brush-grafted nanochannels

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, $\psi \xaf=e\psi /(kBT)$. Also, *n*_{+,∞} = *n*_{∞}, $n\u2212,\u221e=n\u221e+nH+,\u221e\u2212nOH\u2212,\u221e$, where *n*_{∞} = 10^{3}*N*_{A}*c*_{∞}, $nH+,\u221e=103\u2212pH\u221eNA$, and $nOH\u2212,\u221e=103\u2212pOH\u221eNA$ (*p*OH_{∞} = 14 − *p*H_{∞}). This also implies that $\u2207n\u221e=dn\u221edx=dn+,\u221edx=dn\u2212,\u221edx$. 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, $\u2202p\u2202x$ is the pressure gradient induced due to the employed salt concentration gradient [expressed in Eq. (16)], and $\kappa d=a2/\varphi (y)2=a2H0\sigma a3N\varphi \xaf2$. Here, $\varphi \xaf=\varphi H0\sigma a3N$ 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 Gennes^{100} and Freed and Edwards.^{101} These works^{100,101} showed that the drag coefficient varies as *K*^{2}, 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 Gennes^{100} and Freed and Edwards^{101} 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} ∼ *a*^{2}/*ϕ*^{2}.

In Eq. (18), $\u016b=uU$, $A=2kBTn\u221eh2\eta UL$, $n\xaf1\u2032=Ldn\u221edx/n\u221e$, $U=2kBT\lambda 2\eta (dn\u221edx)$, $\psi \xaf=e\psi kBT$, $\u0112=EE0$, where $E0=kBTeL$ is the scale of electric field, $n\xafi,\u221e=ni,\u221en\u221e$ (i = ±, H^{+}, OH^{−}), $\u0233=yh$, $H\xaf0=H0h$, $\lambda \xaf=\lambda h$, and $\lambda =\u03f50\u03f5rkBT/(2e2(n\u221e+nH+,\u221e))$ is the Debye screening length of the electric double layer (EDL).

In order to solve for $\u016b$ 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 $JOH\u2212$ 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 $\u0112osm=\u0112osm,++\u0112osm,\u2212+\u0112osm,H++\u0112osm,OH\u2212$ 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), $Pe=ULD++D\u2212+DH++DOH\u2212$ is the Peclet number for the flow and $Ri=DiD++D\u2212+DH++DOH\u2212$ is the dimensionless diffusivities of each species *i*, where *i* = ±, H^{+}, OH^{−}. Equation (23) expresses *Ē* in terms of $\u016b$. Therefore, if we use Eq. (23) to replace *Ē* in Eq. (18), we shall eventually get an integro-differential equation in $\u016b$. This resulting equation in $\u016b$ is solved numerically in the presence of the following boundary conditions to obtain $\u016b$:

## III. RESULTS AND DISCUSSIONS

### A. Variation of the diffusioosmotically induced electric field

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: *p*H_{∞} = 3 and *ℓ* = 60 nm, case 2: *p*H_{∞} = 3 and *ℓ* = 10 nm, and case 3: *p*H_{∞} = 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, “*p*H_{∞} = 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 *p*H_{∞} = 3 and *ℓ* = 60 nm. This equality in the surface charge densities is ensured by employing the condition $\sigma c,eq=\u2212e\u222b\u2212h\u2212h+H0\varphi nA\u2212dy$ (where *σ*_{c,eq} is the equivalent surface charge density of the brush-free nanochannels and *ϕ* and $nA\u2212$ 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^{−3}M, for the case of *p*H_{∞} = 4, *ℓ* = 60 nm or for *c*_{∞} = 10^{−2}M, for the case of *p*H_{∞} = 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 $\u0112osmBrush>\u0112osmNo\u2009Brush$ (see Fig. 3). However, this is not true for those particular cases where $(uDOS)Brush<(uDOS)No\u2009Brush$; this happens, for example, for *c*_{∞} = 10^{−3}M, *p*H_{∞} = 4, *ℓ* = 60 nm and *c*_{∞} = 10^{−2}M, *p*H_{∞} = 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 ($c\u221e\u226410\u2212pH\u221e$), *Ē*_{osm,diff} primarily depends on the overall diffusion of the H^{+} ions (dictated by the product of the dimensionless number density $n\xafH+=nH+n\u221e$ 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 $n\xafH+=nH+n\u221e$ 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 $c\u221e>10\u2212pH\u221e$ and $|\psi \xaf|\u226a1$,^{71} we can simplify *Ē*_{osm,diff} as [with $n\xaf+,\u221e=n\u221e/n\u221e=1$, $n\xaf\u2212,\u221e=(n\u221e+nH+,\u221e)/n\u221e=1+n\xafH+,\u221e$, neglecting $n\xafOH\u2212,\u221e$, and $exp(\xb1\psi \xaf)\u22481\xb1\psi \xaf\u22481$]

Therefore, for such larger values of *c*_{∞}, we observe that *Ē*_{osm,diff} becomes constant and does not vary with *c*_{∞}. On increasing *p*H_{∞}, 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 $n\xafH+$, 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 $n\xaf+,\u221e=n\u221e/n\u221e=1$). Using the definitions of *Pe* [see below Eq. (28)] and *U* [see below Eq. (18)] as well as the condition *dn*_{∞}/*dx* = 10^{4}*n*_{∞}, we can express $Pe\u221d11+n\xafH+,\u221e$. Therefore, for small *c*_{∞} (i.e., $c\u221e\u226a10\u2212pH\u221e$), $Pe\u2009\u221d\u2009n\u221e/nH+,\u221e$, implying that *Pe* increases linearly with *n*_{∞}. On the other hand, for much larger *c*_{∞} (i.e., $c\u221e\u226b10\u2212pH\u221e$), $Pe\u2248104\u03f50\u03f5r(kBT)2L\eta e2(D++D\u2212+DH++DOH\u2212)$, 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 $\u016b$. Of course, $|\psi \xaf|$ 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 *p*H_{∞} = 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 ($c\u221e>10\u2212pH\u221e$), i.e., when the effect of $\u016b$ 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^{−4}M to 10^{−3}M (for the case of *p*H_{∞} = 4 and *ℓ* = 60 nm), which is reflected by a steep decrease in *Ē*_{osm,+}, *adv*. As we further move from *c*_{∞} of 10^{−3}M to 10^{−2}M, it can be seen that there is an increase in *Ē*_{osm,+}, *adv* due to a similar increase in $\u016b$. Finally, moving from *c*_{∞} of 10^{−2}M to 10^{−1}M, 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 *p*H_{∞} = 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^{−3}M; subsequently, there is a steep decrease in *Ē*_{osm,+,$adv$} with *c*_{∞} as we move from *c*_{∞} of 10^{−3}M to 10^{−2}M [due to a significantly smaller $\u016b$ at *c*_{∞} = 10^{−2}M, see Fig. 8(b)] and *Ē*_{osm,+,$adv$} increases with *c*_{∞} as we move from *c*_{∞} of 10^{−2}M to 10^{−1}M due to the corresponding enhanced value of $\u016b$ [see Fig. 8(b)]. Finally, for the case of *p*H_{∞} = 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 $\u016b$ profile (as witnessed for cases of *p*H_{∞} = 4 and *ℓ* = 60 nm and *p*H_{∞} = 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^{−1}M {due to a noticeable lowering of $\u016b$ [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^{−1}M for the case of *p*H_{∞} = 3 and *ℓ* = 60 nm [see Fig. 4(a)]. Similarly, for the case of *p*H_{∞} = 3 and *ℓ* = 10 nm, we can easily justify an increase, then a decrease (in the range from *c*_{∞} = 10^{−3}–10^{−2}M), and then again an increase (for *c*_{∞}≥ 10^{−2}M) in *Ē*_{osm,+} from the corresponding variation of *Ē*_{osm,+,$adv$} and *Ē*_{osm,diff} [see Fig. 4(b)]. Finally, for the case of *p*H_{∞} = 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^{−3}M), again increasing (in the range from *c*_{∞} = 10^{−3}–10^{−2}M), and then finally again decreasing (in the range from *c*_{∞} = 10^{−2}–10^{−1}M) with *c*_{∞} [see Fig. 4(c)].

We next study the variation of *Ē*_{osm,−,$adv$} with *c*_{∞} [see Fig. 5(c)]. Given the negative valence of the anion, *Ē*_{osm,−,$adv$} is primarily negative. For this case, $Pe\xd7n\xaf\u2212,\u221e\u221dn\u221en\u221e+nH+,\u221e\xd7n\u221e+nH+,\u221en\u221e\u221d1$ (neglecting $nOH\u2212,\u221e$). Therefore, *Pe* has no role to play in the variation of *Ē*_{osm,−,$adv$}. Accordingly, for smaller *c*_{∞} values, *Ē*_{osm,−,$adv$} remains constant as *c*_{∞} varies. On the other hand, the strong dependence of *Ē*_{osm,−,$adv$} on the velocity profile ($\u016b$) starts to get manifested for larger values of *c*_{∞} (since for such *c*_{∞} values, there are distinct changes in $\u016b$, see Fig. 8). For the case of *p*H_{∞} = 4 and *ℓ* = 60 nm, $\u016b$ significantly decreases, increases, and decreases for the concentration ranges of *c*_{∞} = 10^{−4}–10^{−3}M, *c*_{∞} = 10^{−3}–10^{−2}M, and *c*_{∞} = 10^{−2}–10^{−1}M [see Fig. 8(c)]; accordingly, for the exact same respective concentration ranges, *Ē*_{osm,−,$adv$} decreases, increases, and decreases in magnitude. On the other hand, for *p*H_{∞} = 3 and *ℓ* = 10 nm, $\u016b$ significantly decreases and increases for the concentration ranges of *c*_{∞} = 10^{−3}–10^{−2}M and *c*_{∞} = 10^{−2}–10^{−1}M, respectively [see Fig. 8(b)]; accordingly, for the exact same respective concentration ranges, *Ē*_{osm,−,$adv$} decreases and increases in magnitude. Finally, for the case of *p*H_{∞} = 3 and *ℓ* = 60 nm, there is no such distinctly large increase or decrease of $\u016b$ for any concentration range, except for a noticeable decrease in $\u016b$ for *c*_{∞} = 0.1M [see Fig. 8(a)]; accordingly, *Ē*_{osm,−,$adv$} remains more or less constant for the entire concentration range, except for a slight decrease (in magnitude) for the concentration range of *c*_{∞} = 10^{−2}–10^{−1}M. From this variation of *Ē*_{osm,−,$adv$} and the variation of *Ē*_{osm,diff} [see Fig. 5(a)], we can explain the corresponding non-monotonic variation of *Ē*_{osm,−} in Fig. 4: for *p*H_{∞} = 3 and *ℓ* = 60 nm, *Ē*_{osm,−} monotonically increases (in magnitude) with *c*_{∞} and only decreases slightly for concentration range of *c*_{∞} = 10^{−2}–10^{−1}M [see Fig. 4(a)]; for *p*H_{∞} = 3 and *ℓ* = 10 nm, *Ē*_{osm,−} varies monotonically with *c*_{∞} up to *c*_{∞} = 10^{−3}M and subsequently decreases and increases (in magnitude) for the concentration ranges of *c*_{∞} = 10^{−3}–10^{−2}M and *c*_{∞} = 10^{−2}–10^{−1}M, respectively [see Fig. 4(b)]; and for *p*H_{∞} = 4 and *ℓ* = 60 nm, *Ē*_{osm,−} varies monotonically with *c*_{∞} up to *c*_{∞} = 10^{−4}M and subsequently decreases, increases, and decreases for the concentration ranges of *c*_{∞} = 10^{−4}–10^{−3}M, *c*_{∞} = 10^{−3}–10^{−2}M, and *c*_{∞} = 10^{−2}–10^{−1}M, respectively [see Fig. 4(c)].

We finally study the variation of $\u0112osm,H+,adv$ with *c*_{∞} [see Fig. 5(d)]. For this case, $Pe\xd7n\xafH+,\u221e\u221dn\u221en\u221e+nH+,\u221e\xd7nH+,\u221en\u221e\u221d11+n\u221enH+,\u221e$. Therefore, an increase in *c*_{∞} (or *n*_{∞}) for a given *p*H_{∞} (or $nH+,\u221e$) will lead to a progressive decrease in $\u0112osm,H+,adv$, as evident from Fig. 5(d). For very small *c*_{∞}, the ratio $n\u221enH+,\u221e$ is relatively small implying a weak lowering of $\u0112osm,H+,adv$ with *c*_{∞}. However, as *c*_{∞} increases, this ratio increases making the reduction of $\u0112osm,H+,adv$ more prominent. Of course, for larger *p*H_{∞} (or smaller $nH+,\u221e$), the ratio becomes larger for smaller *c*_{∞} values leading to a much steeper lowering of $\u0112osm,H+,adv$ starting from a much smaller *c*_{∞} value. Of course, the changes in $\u016b$ also contribute to this variation but get overwhelmed by the effect of the ratio $n\u221enH+,\u221e$ at larger *c*_{∞} values. This particular nature of variation of $\u0112osm,H+,adv$ and the corresponding variation of *Ē*_{osm,diff} [see Fig. 5(a)] helps explain the corresponding variation of $\u0112osm,H+$ in Fig. 4: for all combinations of *p*H_{∞} and *ℓ* values, we thus find a monotonic increase of $\u0112osm,H+$ with *c*_{∞} for smaller *c*_{∞} values (for such cases, the decrease in $\u0112osm,H+,adv$ with *c*_{∞} is relatively weaker than the corresponding decrease of *Ē*_{osm,diff} with *c*_{∞}) and a monotonic decrease of $\u0112osm,H+$ with *c*_{∞} for larger *c*_{∞} values (for such cases of larger concentration values, the decrease in $\u0112osm,H+,adv$ with *c*_{∞} governs the variation of $\u0112osm,H+$ since *Ē*_{osm,diff} is nearly constant with *c*_{∞}).

*Ē*_{osm} plotted in Fig. 3 for the different combinations of *p*H_{∞} and *ℓ* is simply the sum of different *Ē*_{osm,i}. The particular variation of *E*_{osm,i}, as described in Fig. 4 and explained in great detail through Fig. 5, leads to this highly non-monotonic variation of *Ē*_{osm} with *c*_{∞} for the different combinations of *p*H_{∞} and *ℓ*, as depicted in Fig. 3.

In Fig. 6, we plot the variation of the *Ē*_{ion} with *c*_{∞}, representing the dimensionless conduction component of the electric field. This is equivalent to the electric field that is generated by the conduction of the mobile EDL ions. Therefore, *Ē*_{ion} depends on the mobility (or the diffusivity) of the ions and the applied ion concentration gradient. In other words, this conduction component of the electric field results from the interplay of this imposed concentration gradient on the EDL counterions and coions, which varies both in diffusivity and the number density. Given that the total surface charge density of the brushes is identical to that of the bare nanochannels, the quantity $\u222b\u2212hh(n+\u2212n\u2212)dy$ should be identical for both the brush-grafted and brush-free nanochannels. However, the ionic diffusivities are different (i.e., *R*_{+} ≠ *R*_{−}). Furthermore, the individual ion number densities for the brush-free and brush-grafted nanochannels are different, i.e., $n\xb1(y)Brush\u2260n\xb1(y)No\u2009Brush$, since the local EDL electrostatic potentials are different for the brush-free and brush-grafted systems. Under such conditions, *Ē*_{p,ion} (i.e., the contribution of the salt cation on the overall ionic current) and *Ē*_{m,ion} (i.e., the contribution of the salt anion on the overall ionic current), with *Ē*_{ion} = *Ē*_{p,ion} −*Ē*_{m,ion} [see Eqs. (26) and (27) for the definition of *Ē*_{p,ion} and *Ē*_{m,ion}], are different between the brush-free and brush-grafted nanochannels (see Fig. 6). This difference manifests as $(\u0112p,ion)No\u2009Brush<(\u0112p,ion)Brush$ and $(\u0112m,ion)No\u2009Brush>(\u0112m,ion)Brush$ for all combinations of grafting density, salt concentrations, and pH. Given that the current due to the migration of the salt anions decreases the overall ionic current, it eventually ensures that $(\u0112ion)Brush>(\u0112ion)No\u2009Brush$ for cases where *E*_{ion} is positive. Interestingly, unlike *Ē*_{osm}, which is always positive, given the fact that the velocity field, which is driving it, is mostly negative causing an accumulation of positive counterions on the left of the nanochannel, *Ē*_{ion} can become negative at different *c*_{∞} ranges depending on the grafting density and pH for both brush-grafted and brush-free nanochannels. Such negative values of *Ē*_{ion} is encountered when *Ē*_{m,ion} > *Ē*_{p,ion}—for such cases, $|(\u0112ion)Brush|<|(\u0112ion)No\u2009Brush|$.

In the inset of each of the subfigures of Fig. 7, we separately plot the variation of $\u0112p,ion,N=n\xaf1\u2032\u222b\u221210R+\u2061exp(\u2212\psi \xaf)d\u0233$ and $\u0112m,ion,N=n\xaf1\u2032\u222b\u221210R\u2212\u2061exp(\psi \xaf)d\u0233$. For very weak salt concentration, the EDL thickness is large causing a large EDL overlap. Accordingly, *ψ* remains uniform. This is manifested as a constant value of *Ē*_{m,ion,N} [which varies linearly with exp(*ψ*)] and *Ē*_{p,ion,N} [which varies linearly with exp(−*ψ*)]. However, a progressive increase in the salt concentration significantly reduces the negative magnitude of *ψ*, which enhances exp(*ψ*) and reduces exp(−*ψ*). Accordingly, *Ē*_{p,ion,N} and *Ē*_{m,ion,N} decrease and increase, respectively.

Such a combination of *Ē*_{osm} and *Ē*_{ion} eventually dictates the overall diffusioosmotically induced electric field (see Fig. 2). It clearly establishes that for all combinations of *c*_{∞}, *ℓ* (quantifying the grafting density), and *p*H_{∞}, the electric field induced inside the brush-grafted nanochannels is larger than that inside the brush-free nanochannels. Accordingly, for conditions where the electric field is positive, *Ē*_{Brush} > *Ē*_{No} _{Brush} and for conditions where the electric field is negative, |*Ē*_{Brush}| < |*Ē*_{No Brush}|. As described above, at different *c*_{∞}, *ℓ*, *p*H_{∞} values, *Ē*_{osm} and *Ē*_{ion} contribute differently to ensure this enhancement.

### B. Variation of the diffusioosmotically induced velocity field

Figure 8 provides the variation of the DOS velocity field in brush-grafted and brush-free nanochannels. The DOS velocity has two contributions: the chemiosmotic (COS) component that is caused by the induced pressure gradient due to the applied salt concentration gradient and the electroosmotic (EOS) component that is triggered by the diffusioosmotically induced electric field. To better understand the impact of these components (COS and EOS) in governing the overall DOS velocity, in Figs. 9–11, we compare the variation of the DOS velocity field (denoted as *u*_{total}) and the COS velocity field (denoted as *u*_{COS}). *u*_{COS} is obtained by switching off the effect of the induced electric field in the Stokes equation, i.e., by solving the equation

Equations (34) and (35) represent the equations for obtaining the dimensionless COS velocity fields in brush-grafted and brush-free nanochannels, respectively. Figures 9–11 provide the variation of *u*_{total} and *u*_{COS} for brush-free and brush-grafted nanochannels for different combinations of *c*_{∞}, *ℓ*, and *p*H_{∞}. For a positive value of the salt concentration gradient, as is the condition for the present case, the pressure gradient is positive enforcing a right-to-left pressure-driven or COS flow field. This is true for both the cases of brush-free and brush-grafted nanochannels for all the different parameter choices (see Figs. 9–11). On the other hand, the direction of the EOS transport, and accordingly whether it aids or reduces the contribution of the COS transport, is determined by the sign of the induced DOS electric field (see Fig. 2). A positive (negative) electric field triggers an EOS component from left-to-right (right-to-left), thereby opposing (augmenting) the effect of the COS transport. For both the brush-grafted and brush-free nanochannels, the induced electric field is negative for *c*_{∞} = 0.1M for *p*H_{∞} = 3, *ℓ* = 10 nm, for *c*_{∞} = 0.01M, 0.1M for *p*H_{∞} = 4, *ℓ* = 60 nm, and for all *c*_{∞} values for *p*H_{∞} = 3, *ℓ* = 60 nm (see Fig. 2). Therefore, for all these cases, the EOS component aids the COS component to enhance the overall DOS transport (see Figs. 9–11). On the other hand, for other values of *c*_{∞}, the induced DOS electric field is positive, and hence, the corresponding EOS component is opposite in direction to the COS component, ensuring a reduction in the DOS transport (see Figs. 9–11). The most interesting facet is that despite the fact that the electric field is comparable for the brush-grafted and brush-free nanochannels, the contribution of the corresponding EOS transport in either aiding or opposing the COS transport is much larger for the case of the brush grafted nanochannels. This stems from our previously hypothesized brush-induced localization of the EDL charge density away from the nanochannel walls enforcing a much larger impact of the EOS body force of similar strengths.^{67,69,70}

The above-described interplay of the effects of different factors eventually dictates the overall DOS velocity in the brush-free and brush-grafted nanochannels (see Fig. 8): depending on whether the EOS transport augments or opposes the COS velocity component, the extent of this augmentation/opposition, and the enhanced effect of the EOS body force due to the localization effect of the brushes, the DOS flow strength might be enhanced or weakened inside the brush-grafted nanochannels as compared to the brush-free nanochannels. For the parameter space studied here, except for a few cases, we find that the DOS flow strength is always larger in the brush-grafted nanochannels.

### C . Comparsion of the DOS transport in two types of brush-grafted nanochannels: Backbone-charged brushes (present study) vs end-charged brushes (Ref. 69)

In a recent study,^{69} we had probed the DOS transport in nanochannels grafted with end-charged PE brushes. It is worthwhile to compare the findings of the present study (DOS transport in nanochannels grafted with backbone-charged PE brushes) with the findings of this previous study. First and foremost, it is critical to point out that in this previous study,^{69} the brushes were described using the simplistic Alexander–de Gennes model that considered a uniform density of the monomers along the length of the brushes. On the other hand, in the present study, we apply a much more rigorous augmented SST model, which accounts for the more appropriate distribution of the monomers (larger concentration at near-wall locations as compared with that away from the wall) along the length of the PE brushes.

In Fig. 12, we compare the variation of the diffusioosmotically induced electric field for these two different cases: nanochannels grafted with end-charged and backbone charged PE brushes. For ensuring that we are considering the identical charge content of the PE brushes for the two cases, the charge density for the end-charged PE brushes is considered to be *σ*_{c,eq} (see Sec. III A for the definition of *σ*_{c,eq}). The comparison reveals that most strikingly, for the present case, the electric field varies non-monotonically with the salt concentration, while for the case of nanochannel with end-charged brushes, the electric field decreases monotonically with the salt concentration. This stems for the non-monotonic variation of the *Ē*_{osm} and *Ē*_{ion} for the present case (see Figs. 3 and 6). The difference in the EDL distribution, dictated by the fact that for the previous study,^{69} the EDL is localized at the non-grafted brush-tip, while for the present case, it is distributed along the brush length (as the charged monomers are distributed along the brush length), led to a specific variation of *Ē*_{p,ion} and *Ē*_{m,ion} that caused this non-monotonic variation of *Ē*_{ion} with *c*_{∞} for the present case (see Fig. 7). Additionally, for the present case, there is the non-monotonic variation in *Ē*_{osm} with *c*_{∞} (see Fig. 3), which is large due to the osmotic contribution of the H^{+} ions (see Fig. 4 and the related discussions). On the other hand, our previous study^{69} did not consider the effect of pH (or the migration of H^{+} ions in dictating the corresponding osmotic component of the diffusioosmotically electric field).

In Fig. 13, we compare the DOS velocity field for these two different cases: nanochannels grafted with end-charged and backbone charged PE brushes. This velocity comparison is the key contribution of this subsection, given that we propose this new design (backbone-charged PE brush grafted nanochannel) to enhance the DOS nanofluidic transport. The central idea is as follows: First, for both the present study and the previous study,^{69} the body forces (induced EOS body force resulting from the induced electric field and the induced COS body force resulting from the induced pressure-gradient) are localized away from the nanochannel wall (i.e., the location of the maximum wall-induced drag force). This leads to an augmented manifestation of both of these driving forces. Second, the previous study considered a uniform monomer distribution along the length of the brush, while the present study considers a more realistic monomer distribution where the monomer concentration is larger concentration at near-wall locations and smaller at locations away from the wall. Therefore, the monomer concentration is much larger (smaller) at the location where the different body forces are localized for the previous (present) study. Given that the drag coefficient varies quadratically with the monomer concentration, such a scenario implies that the PE-brush-induced drag force is much larger (smaller) at the location where the different body forces (COS and EOS) are localized for the previous (present) study. As a consequence, the impact of both the COS and EOS body forces in triggering the corresponding COS and EOS flow components is much smaller (larger) for the previous (present) study. Therefore, for most of the combinations of *c*_{∞} and charge density values, the magnitude of the DOS velocity for the present case (either from left-to-right or right-to-left, depending on the relative direction and strength of the EOS transport with respect to the COS transport) is mostly larger than that for the previous study. This analysis provided in Fig. 13 firmly establishes the novelty of the present study over and above that of our previous study^{69} in terms of significantly enhancing the DOS transport in functionalized nanochannels.

## IV. CONCLUSIONS

In this paper, we developed a theoretical model to quantify the DOS transport in nanochannels grafted with PE brushes modeled using our recently developed augmented SST. The diffusioosmotically induced electric field and water flow characterize this DOS transport, which is generated in the presence of an axially employed salt concentration gradient. The presence of the brushes leads to an enhanced induced electric field for positive values of the electric field. The presence of the brushes further ensures a larger DOS flow velocity as compared to the cases of brush-free nanochannels or nanochannels grafted with end-charged brushes for a major combination of salt concentration, pH, and grafting density values. With respect to the brush-free nanochannels, such an enhancement is contributed to by the brush-induced localization of the EDL charge density away from the nanochannel walls (or *wall-induced drag force*), which in turn leads to a much more enhanced effect of the induced EOS body force caused by the diffusioosmotically induced electric field. On the other hand, with respect to the nanochannel grafted with end-charged brushes, such an enhancement is due to the localization of the *brush-induced drag force* away from the location where the EDL-induced EOS and COS body forces are localized.

## ACKNOWLEDGMENTS

This work was supported by the Department of Energy, Office of Science (Grant No. DE-SC0017741).

### APPENDIX: SIMPLIFICATION OF THE Y-MOMENTUM EQUATION TO OBTAIN THE STARTING POINT OF EQ. (15)

The present study considers a long nanochannel such that *L* ≫ *h*. Under such conditions, one can apply the *Space Charge Theory or SCT* (see Ref. 97 for details) for describing the EDL electrostatic potential, ion number densities, the fluxes and the local flow fields. The critical issue of the SCT is that for this very long and thin nanochannel, one must have local equilibrium in the transverse direction. The condition of this local transverse equilibrium leads to *v*(*x*, *y*) = 0, where *v* is the transverse velocity field (velocity field in the y-direction). We have previously shown (for the case of diffusioosmotic transport in *brush-free* nanochannels), that our problem statement, where we employ a very weak salt concentration gradient along the length of a nanochannel (where *L* ≫ *h*), leads to the case where one can directly apply the SCT (see the supplementary material of Ref. 103). The present problem is the same as that of the previous study, except that we now have the presence of grafted backbone-charged PE brushes. Therefore, for the present case, too, we can consider the existence of local equilibrium in the transverse direction, which in turn will lead to *v*(*x*, *y*) = 0.

On the other hand, the Navier–Stokes y-momentum equation (in the absence of the effect of the gravitational body force, since we consider very small mass of liquid confined in a nanochannel) can be expressed as

In the above equation, the last term on the right hand side represents the body force in the y-direction resulting from the interaction of the net EDL charge density $[en+\u2212n\u2212+nH+\u2212nOH\u2212]$ and the EDL transverse electric field ($\u2212\u2202\psi \u2202y$). Using the condition of *v*(*x*, *y*) = 0 in the above equation leads to

which is the starting point of Eq. (15).