Streaming potential and electroviscous effects in soft nanochannels beyond Debye–Hückel linearization

Chemiosomotic flow in a soft conical nanopore: harvesting enhanced blue energy

The salinity gradient energy or the 'blue energy' is one of the most promising inexpensive and abundant sources of clean energy, having immense capabilities to serve modern-day society. In this article, we overlay an extensive analysis of reverse electrodialysis (RED) for harvesting salinity gradient energy in a single conical nanochannel, grafted with a pH-tunable polyelectrolyte layer (PEL) on the inner surfaces. We primarily focus on the distinctiveness of the solution pH of the connecting reservoirs. In spite of acquiring a maximum power density of ∼1.2 kW m^-2 in the chosen configuration, we notice a counter-intuitive patterning of the ion transport for a certain span of pH, leading to diminishing power. To this end, we discuss the possible strategic avenues essentially to achieve a higher amount of power density. In order to achieve a desirable outcome within that pH zone, we employ two separate approaches intending to counter the underlying physics. Results reveal a great enhancement in the power density as well as in the efficiency even under the framework of both strategies proposed herein. Moreover, as shown, the window of solution pH has increased by three times, implicating the maximum power density mentioned above. We expect that the strategic procedure of augmented energy harvesting as discussed in this analysis can be of importance from the perspective of fabricating state-of-the-art nanodevices aimed at blue energy harvesting.

Surface and zeta potentials of charged permeable nanocoatings

An electrokinetic (zeta) potential of charged permeable porous films on solid supports generally exceeds their surface potential, which often builds up to a quite high value itself. Recent work provided a quantitative understanding of zeta potentials of thick, compared to the extension of an inner electrostatic diffuse layer, porous films. Here, we consider porous coatings of thickness comparable to or smaller than that of the inner diffuse layer. Our theory, which is valid even when electrostatic potentials become quite high and accounts for finite hydrodynamic permeability of the porous materials, provides a framework for interpreting the difference between values of surface and zeta potentials in various situations. Analytic approximations for the zeta potential in the experimentally relevant limits provide a simple explanation of transitions between different regimes of electro-osmotic flows and also suggest strategies for its tuning in microfluidic applications.

Charge Properties and Electric Field Energy Density of Functional Group-Modified Nanoparticle Interacting with a Flat Substrate

Functionalized nanofluidics devices have recently emerged as a powerful platform for applications of energy conversion. Inspired by biological cells, we theoretically studied the effect of the interaction between the nanoparticle and the plate which formed the brush layer modified by functional zwitterionic polyelectrolyte (PE) on the bulk charge density of the nanoparticle brush layer, and the charge/discharge effect when the distance between the particle and the plate was changed. In this paper, The Poisson-Nernst-Planck equation system is used to build the theoretical model to study the interaction between the nanoparticle and the plate modified by the PE brush layer, considering brush layer charge regulation in the presence of multiple ionic species. The results show that the bulk charge density of the brush layer decreases with the decrease of the distance between the nanoparticle and the flat substrate when the interaction occurs between the nanoparticle and the plate. When the distance between the particle and the plate is about 2 nm, the charge density of the brush layer at the bottom of the particle is about 69% of that at the top, and the electric field energy density reaches the maximum value when the concentration of the background salt solution is 10 mm.

Electrokinetic ion transport and fluid flow in a pH-regulated polymer-grafted nanochannel filled with power-law fluid

In this article, we have discussed extensively electrokinetic ion transport and fluid flow through a slit polymer-grafted nanochannel filled with power-law fluid. The rigid walls of the channel are coated with the ion and fluid penetrable polymer layer containing a pH-regulated zwitterionic functional group (e.g., succinoglycan). The mathematical model is based on the non-linear Poisson-Boltzmann equation for electric double layer potential and the flow field within the polymer layer is governed by a modified Darcy-Brinkman equation; the Cauchy momentum equation governs the fluid flow outside of the polymer layer along with the equation of continuity for incompressible fluid. In order to consider a wide range of pertinent parameters, we adopt a finite difference based numerical tool to solve the coupled set of governing equations. We have analyzed several interesting features of electrokinetic transport phenomena through such a polymer-grafted nanochannel for a wide range of electrostatic and hydrodynamic properties of the polymer layer, parameters describing the non-Newtonian rheology of the background fluid, and the pH and concentration of the bulk electrolyte. In addition, we have also illustrated the ionic current across the undertaken nanochannel and observed that it can be either cation selective, anion selective or non-selective, depending on the critical choice of the pertinent parameters.

Efficient electroosmotic mixing in a narrow-fluidic channel: the role of a patterned soft layer

We propose a novel and efficient mixing technique in a soft narrow-fluidic channel under the influence of electrical forcing. We show that a grafted polyelectrolyte layer (PEL) added as a patch to the channel wall modulates the electrical double layer (EDL) so that an applied electric field initiates a local electroosmotic flow (EOF) at the patched section. This EOF develops in the opposite direction to the primary pressure-driven flow. This localized EOF leads to the formation of Lamb vortices at the patched sections through the phenomenon of momentum exchange with the primary stream and promotes the mixing therein. Our study, consistent with the stream-function/vorticity approach, primarily focuses on the numerical analysis of the mixing phenomena. Through a quantitative description, we reveal the effect of different patterns on the underlying mixing phenomena in the convective mixing regime. We also discuss the impact of key parameters on the mixing efficiency, the onset of the recirculation zone, variation in the mixing length, and the shear-driven aggregation kinetics in soft matter systems. Finally, considering the practicability of the present problem, we unveil the values of several design parameters for which the mixing efficiency in the channel reaches the maximum.

The Effect of Streaming Potential and Viscous Dissipation in the Heat Transfer Characteristics of Power-Law Nanofluid Flow in a Rectangular Microchannel

The non-Newtonian nanofluid flow becomes increasingly important in enhancing the thermal management efficiency of microscale devices and in promoting the exploration of the thermal-electric energy conversion process. The effect of streaming potential and viscous dissipation in the heat transfer characteristics of power-law nanofluid electrokinetic flow in a rectangular microchannel has been investigated to assist in the development of an energy harvesting system. The electroviscous effect caused by the streaming potential influences the hydrodynamical and thermal characteristics of flow. With the change in constitutive behavior of power-law nanofluid, the viscous dissipation effect is considered. The Poisson–Boltzmann equation, the modified Cauchy momentum equation, and the energy equation were solved. The temperature and heat transfer rate were analytically expressed for Newtonian nanofluid and numerically obtained for power-law nanofluid. The interactive influence of streaming potential, viscous dissipation, and hydrodynamical features of power-law nanofluid on the heat transfer characteristics were studied. The presence of streaming potential tends to reduce the dimensionless bulk mean temperature. The introduction of nanoparticles augments dimensionless temperature difference between channel wall and bulk flow, which decreases the heat transfer rate. The shear thinning nanofluid is more sensitive to the above effects. The temperature is a weak function of the flow behavior index.

Mass transfer of a neutral solute in polyelectrolyte grafted soft nanochannel with porous wall

A soft nanochannel involves a soft interface that contains a polyelectrolyte layer (PEL) sandwiched between a rigid surface and a bulk electrolyte solution. Mass transfer of a neutral solute in a combined electroosmotic and pressure driven flow through a polyelectrolyte grafted charged nanochannel with porous wall is presented in this work. Assuming the PEL as fixed charged layer and PEL-electrolyte interface as a semi-penetrable membrane, analytical solutions were obtained for potential distributions (for small wall potential). Velocity profiles were also derived in the same domains, for both inside and outside the PEL. Convective-diffusive species balance equation was semi-analytically solved inside the PEL. Expression of length averaged Sherwood number was also obtained and effects of different parameters, namely, drag parameter (α), Debye parameter ( κ ¯ ) , and PEL thickness were studied in detail. The variation of permeate concentration and permeation flux across the porous wall was obtained.

Soft interface design for electrokinetic energy conversion

The exploitation and utilization of renewable clean energy is of great significance to the sustainable development of society. Electrokinetic energy conversion (EKEC) based on micro/nanochannels is expected to provide immense potential for ocean energy harvesting, self-powered micro/nanodevices, and small portable power supplies through converting environmental energy into electrical energy. Herein, aiming to get a deeper understanding of EKEC based on micro/nanochannels, several classic theoretical models and corresponding calculation equations are introduced briefly. For high efficiency energy conversion, it is essential to clearly discuss the interface properties between the inner surface of the channel and the bulk electrolyte solution. Therefore, we put forward soft interface designs of solid-liquid and liquid-liquid interfaces, and summarize their recent progress. In addition, the different applications of EKEC, harvesting from environmental energy, are further discussed. We hope that this review will attract more scientists' attention to transform the experimental results of EKEC systems in the lab into available products on shelves.

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

In this paper, we provide a theory to quantify the ionic current ( i ion ) in nanochannels grafted with pH-responsive polyelectrolyte (PE) brushes. We consider the PE brushes to be modeled by our recently proposed augmented strong stretching theory (SST) model that improves the existing SST models by incorporating the effects of excluded volume interactions and an extended mass action law. Use of such augmented SST for this problem implies that this is the first study on computing i ion in PE brush-grafted nanochannels accounting for the appropriate coupled configuration-electrostatic description of the PE brushes. i ion is obtained as functions of PE brush grafting density, medium pH and salt concentration ( c ∞ ), and the density of polyelectrolyte chargeable sites (PECS). For large c ∞ , i ion increases linearly with c ∞ (as for such c ∞ , i ion becomes independent of the PE charge and is dominated by the bulk mobility and number density of the electrolyte ions), whereas i ion is independent of c ∞ at small c ∞ (where the electric double layer electrostatics and the total number of ions in the system is dominated by the hydrogen ions). We further witness an enhancement of i ion for smaller pH and larger grafting density at low and moderate c ∞ , while there is little to no effect of the PECS density on the ionic current except for weakly grafted brushes at low c ∞ . We anticipate that this study will serve as a theoretical foundation for a large number of applications that are based on the brush-induced modification of the ionic current in a nanochannel.

Electrokinetic energy conversion in nanochannels grafted with pH-responsive polyelectrolyte brushes modelled using augmented strong stretching theory

In this paper, we develop a theory to quantify the electrokinetic energy conversion in electrolyte-filled nanochannels grafted with pH-responsive polyelectrolyte (PE) brushes. A pressure-driven flow drives the mobile electrolyte ions of the electric double layer (EDL) supported by the charged PE brushes leading to the generation of a streaming current, a streaming electric field and eventually an electrical energy. The salient feature of this study is that the brushes are described using our recently developed augmented Strong Stretching Theory (SST) model. In all the previous theoretical studies on liquid transport in PE-brush-grafted nanochannels, the brushes have either been assumed to be of constant height (independent of salt concentration or pH) or modelled using the Alexander-de-Gennes model that considers uniform monomer distribution along the brush height. Such simplifications have meant that the salt and the pH dependence of the brush height, the monomer distribution, and the resulting electrostatics have not been appropriately accounted for in the transport calculations. This paper addresses these limitations and provides a much more detailed description of the brushes while capturing the corresponding electrokinetic energy conversion. The results establish that the presence of the PE brushes ensures a localization of the average EDL charge density away from the grafting surface, thereby enabling the migration of the EDL ions with a larger background flow velocity; as a consequence, there is an enhancement of the streaming current, streaming electric field, and the resulting electrical energy generation under certain grafting densities of the PE brushes.

Influence of salt valence on the rectification behavior of nanochannels

Taking account of the influence of electroosmotic flow, the behavior of the ion current rectification of a charged conical nanochannel is studied theoretically focusing on the effect of ionic valence. A continuum-based model comprising coupled Poisson-Nernst-Planck (PNP) equations for the ionic mass transport and Navier-Stokes equations for the hydrodynamic field is adopted. We show that if the bulk salt concentration is fixed, the behavior of the current-voltage curve depends highly on the ionic valence, which arises from the difference in ionic strength and ion diffusivity. As the bulk salt concentration varies, the rectification factor shows a local maximum, and the bulk salt concentration at which it occurs depends upon the salt valence: the higher the valence the lower that concentration. However, regardless of the salt valence, the ionic strength at which that local maximum occurs is essentially the same, implying that the thickness of electric double layer is the key factor. Due to the difference in ionic diffusivity, the magnitude of the rectification factor depends upon the type of salt. For example, the rectification factor of KCl is larger than that of KNO₃. The qualitative behavior of the ion current rectification of a positively charged conical nanochannel is similar to that of a negatively charged nanochannel.

Efficient electrochemomechanical energy conversion in nanochannels grafted with end-charged polyelectrolyte brushes at medium and high salt concentration

We develop a theory to study the generation of the streaming potential and the resulting electrochemomechanical energy conversion (ECMEC) in the presence of pressure-driven transport in nanochannels grafted with end-charged polyelectrolyte (PE) brushes. Our theory gives a thermodynamically self-consistent coupled description of the PE-brush and the electrostatics of the electric double layer (EDL) induced by the PE charges. The end-charged brushes localize the maximum EDL charge density away from the wall, thereby enabling a larger magnitude of pressure-driven transport to stream the ions downstream. This effect is retarded by the drag force imparted by the brushes as well as by the enhanced electroosmotic transport in a direction opposite to the pressure-driven transport. An interplay of these three issues leads to highly non-trivial electrohydrodynamic transport that eventually allows us to converge on appropriate properties of the brushes (e.g., grafting density and the number of monomers) that lead to the generation of a significantly larger streaming potential and a much improved efficiency of the ECMEC as compared to the brush-free nanochannels particularly at medium and high salt concentrations.

Softness Induced Enhancement in Net Throughput of Non-Linear Bio-Fluids in Nanofluidic Channel under EDL Phenomenon

In this article, we describe the electro-hydrodynamics of non-Newtonian fluid in narrow fluidic channel with solvent permeable and ion-penetrable polyelectrolyte layer (PEL) grafted on channel surface with an interaction of non-overlapping electric double layer (EDL) phenomenon. In this analysis, we integrate power-law model in the momentum equation for describing the non-Newtonian rheology. The complex interplay between the non-Newtonian rheology and interfacial electrochemistry in presence of PEL on the walls leads to non-intuitive variations in the underlying flow dynamics in the channels. As such, we bring out the variations in flow dynamics and their implications on the net throughput in the channel in terms of different parameters like power-law index (n), drag parameter (α), PEL thickness (d) and Debye length ratio (κ/κ_PEL) are discussed. We show, in this analysis, a relative enhancement in the net throughput through a soft nanofluidic channel for both the shear-thinning and shear-thickening fluids, attributed to the stronger electrical body forces stemming from ionic interactions between polyelectrolyte layer and electrolyte layer. Also, we illustrate that higher apparent viscosity inherent with the class of shear-thickening fluid weakens the softness induced enhancement in the volumetric flow rate for the shear-thickening fluids, since the viscous drag offered to the f low f ield becomes higher for the transport of shear-thickening fluid.

Theory of diffusioosmosis in a charged nanochannel

We probe the diffusioosmotic transport in a charged nanofluidic channel in the presence of an applied tangential salt concentration gradient. Ionic salt gradient driven diffusioosmosis or ionic diffusioosmosis (IDO) is characterized by the generation of an induced tangential electric field and a diffusioosmotic velocity (DOSV) that is a combination of an electroosmotic velocity (EOSV) triggered by this electric field and a chemiosmotic velocity (COSV) triggered by an induced tangential pressure gradient. We explain that unlike the existing theories on IDO, it is more appropriate to apply the zero net current conditions (formulation F2) and not more restrictive zero net local flux conditions (formulation F1) particularly for the case where one considers a nanochannel connected to two reservoirs. We pinpoint limitations in the existing literature in correctly predicting the diffusioosmotic behavior even for the case where formulation F1 is used. We address these limitations and establish that (a) the induced electric field is an interplay of the differences in ionic diffusivity, the EDL-induced imbalance in ion concentrations, and the advection effects, (b) formulation F1 may overpredict or underpredict the electric field and the EOSV leading to an overprediction/underprediction of the DOSV and (c) formulation F2 demonstrates remarkable fluid physics of localized backflows owing to a dominant local influence of the COSV, which is missed by formulation F1. We anticipate that our theory will provide the first rigorous understanding of nanofluidic IDO with applications in multiple areas of low Reynolds number transport such as biofluidics, microfluidic separation, and colloidal transport.

Ion size effect on electrostatic and electroosmotic properties in soft nanochannels with pH-dependent charge density

We report a theoretical study of the ion size effect on various properties in a soft nanochannel with pH-dependent charge density.

Highly enhanced liquid flows via thermoosmotic effects in soft and charged nanochannels

This paper proposes a massively augmented thermoosmotic transport in nanochannels grafted with end-charged polyelectrolyte brushes.

Electro-osmosis of nematic liquid crystals under weak anchoring and second-order surface effects

Advent of nematic liquid crystal flows has attracted renewed attention in view of microfluidic transport phenomena. Among various transport processes, electro-osmosis stands as one of the efficient flow actuation mechanisms through narrow confinements. In the present study, we explore the electrically actuated flow of an ordered nematic fluid with ionic inclusions, taking into account the influences from surface-induced elasticity and electrical double layer (EDL) phenomena. Toward this, we devise the coupled flow governing equations from fundamental free-energy analysis, considering the contributions from first- and second-order elastic, dielectric, flexoelectric, charged surface polarization, ionic and entropic energies. The present study focuses on the influence of surface charge and elasticity effects in the resulting linear electro-osmosis through a slit-type microchannel whose surfaces are chemically treated to display a homeotropic-type weak anchoring state. An optical periodic stripe configuration of the nematic director has been observed, especially for higher electric fields, wherein the Ericksen number for the dynamic study is restricted to the order of unity. Contrary to the isotropic electrolytes, the EDL potential in this case was found to be dependent on the external field strength. Through a systematic investigation, we brought out the fact that the wavelength of the oscillating patterns is dictated mainly by the external field, while the amplitude depends on most of the physical variables ranging from the anchoring strength and the flexoelectric coefficients to the surface charge density and electrical double layer thickness.

Transient alternating current electroosmotic flow of a Jeffrey fluid through a polyelectrolyte-grafted nanochannel

Comparison of the velocity amplitude for Newtonian, Maxwell and Jeffrey fluids in PE-grafted and rigid nanochannels.

Electrokinetics in polyelectrolyte grafted nanofluidic channels modulated by the ion partitioning effect

The effects of ion partitioning on the electrokinetics in a polyelectrolyte grafted nanochannel, which is the representative of a soft nanochannel, are analyzed. Earlier studies in this regard have considered low polyelectrolyte layer (PEL) grafting density at the rigid nanochannel wall and, hence, an equal permittivity inside and outside the grafted layer. In order to overcome this shortcoming, the concept of Born energy is revisited. The coupled system of the modified Poisson-Boltzmann and Navier-Stokes equation is solved numerically, going beyond the widely employed Debye-Hückel linearization and low PEL densities. The complex interplay between the hydrodynamics and charge distribution, modulated by the ion partitioning effect, along with their consequent effects on the streaming potential and electrokinetic energy conversion efficiency (EKEC) have been systemically investigated. It has been observed that the ion partitioning effect reduces the EKEC in comparison to the case with equal permittivity up to a certain electrical double layer thickness after which it increases the EKEC. For a high concentration of mobile charges within the PEL, the net gain in the maximum EKEC due to the ion partitioning effect is about 10 fold that of the case when the ion partitioning effect is not considered. We delve into the various scaling regimes in the streaming potential and intriguingly point out the exact location of peaks in efficiency. The present study also reveals the possibility of improvement in streaming potential mediated energy conversion by the use of polyelectrolyte materials, which possess substantially lower dielectric permittivity than the bulk electrolyte.

Surface conductivity in electrokinetic systems with porous and charged interfaces: Analytical approximations and numerical results

We derive an approximate analytical representation of the conductivity for a 1D system with porous and charged layers grafted onto parallel plates. Our theory improves on prior work by developing approximate analytical expressions applicable over an arbitrary range of potentials, both large and small as compared to the thermal voltage (RTF). Further, we describe these results in a framework of simplifying nondimensional parameters, indicating the relative dominance of various physicochemical processes. We demonstrate the efficacy of our approximate expression with comparisons to numerical representations of the exact analytical conductivity. Finally, we utilize this conductivity expression, in concert with other components of the electrokinetic coupling matrix, to describe the streaming potential and electroviscous effect in systems with porous and charged layers.

Polyelectrolyte brushes: theory, modelling, synthesis and applications

Polyelectrolyte (PE) brushes are a special class of polymer brushes (PBs) containing charges. Polymer chains attain "brush"-like configuration when they are grafted or get localized at an interface (solid-fluid or liquid-fluid) with sufficiently close proximity between two-adjacent grafted polymer chains - such a proximity triggers a particular nature of interaction between the adjacent polymer molecules forcing them to stretch orthogonally to the grafting interface, instead of random-coil arrangement. In this review, we discuss the theory, synthesis, and applications of PE brushes. The theoretical discussion starts with the standard scaling concepts for polymer and PE brushes; following that, we shed light on the state of the art in continuum modelling approaches for polymer and PE brushes directed towards analysis beyond the scaling calculations. A special emphasis is laid in pinpointing the cases for which the PE electrostatic effects can be de-coupled from the PE entropic and excluded volume effects; such de-coupling is necessary to appropriately probe the complicated electrostatic effects arising from pH-dependent charging of the PE brushes and the use of these effects for driving liquid and ion transport at the interfaces covered with PE brushes. We also discuss the atomistic simulation approaches for polymer and PE brushes. Next we provide a detailed review of the existing approaches for the synthesis of polymer and PE brushes on interfaces, nanoparticles, and nanochannels, including mixed brushes and patterned brushes. Finally, we discuss some of the possible applications and future developments of polymer and PE brushes grafted on a variety of interfaces.

Ion transport and selectivity in biomimetic nanopores with pH-tunable zwitterionic polyelectrolyte brushes

Inspired by nature, functionalized nanopores with biomimetic structures have attracted growing interests in using them as novel platforms for applications of regulating ion and nanoparticle transport. To improve these emerging applications, we study theoretically for the first time the ion transport and selectivity in short nanopores functionalized with pH tunable, zwitterionic polyelectrolyte (PE) brushes. In addition to background salt ions, the study takes into account the presence of H(+) and OH(-) ions along with the chemistry reactions between functional groups on PE chains and protons. Due to ion concentration polarization, the charge density of PE layers is not homogeneously distributed and depends significantly on the background salt concentration, pH, grafting density of PE chains, and applied voltage bias, thereby resulting in many interesting and unexpected ion transport phenomena in the nanopore. For example, the ion selectivity of the biomimetic nanopore can be regulated from anion-selective (cation-selective) to cation-selective (anion-selective) by diminishing (raising) the solution pH when a sufficiently small grafting density of PE chains, large voltage bias, and low background salt concentration are applied.

Combined electroosmotically and pressure driven flow in soft nanofluidics

The present study is devoted to the analysis of mixed electroosmotic and pressure driven flows through a soft charged nanochannel considering boundary slip and constant charge density on the walls of the slit channel. The sources of the fluid flow are the pressure gradient along the channel axis and the electrokinetic effects that trigger an electroosmotic flow under the influence of a uniformly applied electric field. The polyelectrolyte layer (PEL) is denoted as a fixed charge layer (FCL) and the electrolyte ions can be present both inside and outside the PEL i.e., the PEL-electrolyte interface acts as a semi-penetrable membrane. The Poisson-Boltzmann equation is solved assuming the Debye-Hückel linearization for the low electric potential to provide us with analytical closed form solutions for the conservation equations. The conservation equations are solved to obtain the electric potential and velocity distributions in terms of governing dimensionless parameters. The results for the dimensionless electric potential, the dimensionless velocity and Poiseuille number are presented graphically and discussed in detail.