Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel

Physica A xxx (xxxx) xxx

Contents lists available at ScienceDirect

Physica A journal homepage: www.elsevier.com/locate/physa

Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel ∗

Sandip Sarkar a , , Suvankar Ganguly b a b

Department of Mechanical Engineering, Jadavpur University, Kolkata, 700 032, India TATA Global R&D Division, India

article

info

Article history: Received 5 November 2018 Received in revised form 9 February 2019 Available online xxxx Keywords: Narrow-channel Magnetic field Streaming potential Energy conversion efficiency Hartmann number Hydrophobicity

a b s t r a c t In the present study, we attempt to analyse the consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel. In contrast to the traditional slip length based theoretical approach; we use a thermodynamically consistent phase-field based order parameter formalism to account for the near-wall surface-hydrophobicity mediated depletion layer formation. The research questions based on the knowledge gap we have addressed in this study are: Whether the consequences of magnetohydrodynamic forcing and surface wettability alter the development of streaming potential? To what extent the energy conversion efficiency gets affected for such microfluidic system? Towards this, we derive an expression for the induced streaming potential by considering nonlinear interaction between wall hydrophobicity, and superimposed magnetic field in the transverse direction of the flow. Notably, we have shown that hydrophobicity induced interfacial kinetics can contribute in remarkable enhancement of the induced streaming potential field. Our analysis also revealed that the magnetohydrodynamic effect may result in augmentations in the energy transfer efficiency under appropriate conditions. The analytical and numerical results presented in this work are expected to provide valuable guidelines towards optimizing the overall energy transfer performance of micro- and nano-systems by judicious employment of magnetohydrodynamic flows and hydrophobic substrate, and could be useful in further developments of theory, simulation, and experimental work. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Interfacial electrokinetic phenomena such as electroosmosis and streaming potential have far-ranging scientific and technological consequences. In recent years, due to the rapid advancements in micro- and nano-fabrication technologies, application of electrokinetics in controlling and manipulating liquid flows through narrow fluidic confinements have become immensely popular [1]. The origin of electrokinetic phenomena lies in the development of an electrical double layer (EDL), which is formed as a result of the interaction of ionized solution with static charges on dielectric surfaces [2]. The particular phenomenon of streaming potential arises when a pressure-driven fluid flow transports ions in the double layer by advection. The transfer of ions gives rise to an electric current (streaming current) to flow, and the resultant accumulation of ions across the channel establishes an induced electrical field (streaming potential). Several ∗ Corresponding author. E-mail address: [email protected] (S. Sarkar). https://doi.org/10.1016/j.physa.2019.123450 0378-4371/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.

2

S. Sarkar and S. Ganguly / Physica A xxx (xxxx) xxx

technologically relevant applications have utilized the generation of streaming potential to convert hydrostatic energy into useful electrical power, thereby providing efficient energy transfer protocols in miniaturized systems [3–6]. In the recent past, research interest on the performances of electrokinetic energy conversion devices has intensified, giving rise to novel design and use of micro- and nano-fluidic devices [6–8]. It can be noted in this context that surface characteristics of the channel walls play an important role towards dictating the interfacial hydrodynamics at micro/nanoscale. Presence of hydrodynamic slip along a solid surface has significant effect on the electrokinetic transport, and can lead to an increase in the hydro-electric energy conversion efficiency [6,9,10]. Several experimental and theoretical studies have been carried out to develop fundamental understanding in this regard [11–18]. Experiments [11–14] on variety of interfaces and physical conditions provide evidence of slip lengths ranging from 100 nm to as high as 33 mm, which is primarily attributed to the presence of nano-bubbles trapped on the surface, leading to the formation of an effective smoothing layer on the surface. The performance of a streaming potential based energy conversion system has also been shown to be strongly enhanced by injecting gas-bubbles into a liquid filled channel regard. In a theoretical investigation, Ren and Stein [6] predicted a potentially practical 40% efficiency for a moderate 30 nm slip length in a 10nm-high channel. Effects of oscillating liquid flow on the electroviscous effects in the presence of slip in microchannels have been studied by Yang and Kwok [15]. Goswami and Chakraborty [16] theoretically investigated the energy transfer through streaming effects induced by time-periodic pressure-driven nano-channel flows with interfacial slip. Seshadri and Baier [17] carried out numerical investigation to analyse the effect of electrokinetic flow on energy conversion on superhydrophobic surfaces. Yan et al. [18] used asymptotic and numerical methods to study the influence of slip length and Ohmic resistance on the energy conversion efficiency of pressure-driven electrolyte flow through a charged nanopore. Furthermore, there exist few analytical and numerical studies exploring the effects of substrate wettability variations on the electrohydrodynamics and thermofluidic transport in narrow fluidic confinements [19–21]. However, all the above mentioned references have considered the substrate wettability in an implicit manner, through introduction of a slipbased conceptual paradigm. The theoretical proposition of modelling the hydrodynamics at the interface through the use of a Navier slip coefficient is traditionally based on the existence of a wall-adjacent depleted layer formed due to hydrophobic interactions. It may be mentioned here that although from a continuum viewpoint, consideration of preimposed slip parameter in a two-layer model [19] presents a convenient mathematical framework; this approach fails to depict any explicit relationship between the contact angle and electromechanics over interfacial scales. Moreover, such considerations are limited by the fact that the relationship between slip lengths, the depletion layer thickness and surface wettability are not based on any fundamental thermodynamic and fluid dynamic considerations, thereby simplifying the overall physical paradigm of electrohydrodynamic interactions. Accordingly, attempts have been made to accommodate the implications of hydrodynamic interactions over disparate physical scales with specified wettability characteristics in a thermodynamically consistent manner [22–24]. However, in all such cases, solutions are obtained under classical approximations, such as mean field assumption, condition of nonoverlapping EDLs, assumption of EDL ions as point charges, etc. On the other hand, in order to derive maximum energy transfer efficiency, several electrokinetic energy harvesting devices operate in nanoscopic spatial scales where strong EDL interactions eventually lead to overlapped EDL with high surface charge density conditions. Recognizing the scientific importance and technological relevance of the transport processes outlined as above, several studies in the past have underscored the significance of various coexisting physical phenomena such as EDL interactions, steric effects and hydrodynamic slippage [25–27] in influencing the overall electrokinetic transport. However, until now, no studies have considered the simultaneous effects of EDL overlap and substrate wettability induced slippage in the presence of more novel flow actuation mechanisms, such as externally applied magnetic field, on the streaming potential and energy conversion efficiency in narrow fluidic confinements. On the other hand, in recent years, use of magnetic field as flow actuation mechanism has assumed great importance in wide spectrum of micro/nanofluidic applications [28– 30]. There is a plethora of applications of such flow actuation mechanism, e.g., flow augmentation in micropumps [31], magnetophoresis [32–34], separation of biological and chemical moieties [28,35], magnetohydrodynamic flow control [36], to name a few. It has been experimentally established [31] that the average flow rates in micropumps can be substantially augmented by employing low-magnitude magnetic fields. Tso and Sundaravadivelu [37] have examined the influence of electromagnetic fields on the flow characteristics in a parallel plate microchannel. In their study [37], authors have considered a transverse electric field interacting with externally applied magnetic field acting in the longitudinal vertical plane of flow, thereby producing a net driving axial body force to enhance the flow rates. However, electrical double layer effects were not considered in their study. In another study, Chakraborty and Paul [36] demonstrated the possibility of utilizing electrical fields, together with a transverse magnetic field to optimize the microfluidic transport. Munshi and Chakraborty [38] investigated the combined influences of axial pressure gradients and transverse magnetic fields on hydroelectrical energy conversion mechanisms in narrow fluidic confinements [39]. However, no theoretical studies have yet been reported to elucidate the combined consequences of magnetohydrodynamic forces and hydrophobic influences on streaming potential mediated flow in a micro/nanochannel. Also, possibilities of hydroelectric energy conversion have also not been extensively explored in these investigations. Therefore the need for a more general study arises, so as to conceptualize a new paradigm of miniaturized energy conversion protocol with magnetic field as an independent tuning parameter. Accordingly, in this study, we present a theoretical analysis of streaming potential mediated flow and electrokinetic energy conversion efficiency in narrow fluidic confinements under the influences of external magnetic field. Considering the consequences of EDL overlap and substrate wettability variations, we invoke considerations for estimation Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.

S. Sarkar and S. Ganguly / Physica A xxx (xxxx) xxx

3

Fig. 1. Schematic depiction of the physical domain under consideration.

of streaming potential and the resultant energy transfer in the microchannel under the effect of a transverse magnetic field, which have not yet been addressed in the literature in related contexts. An explicit accounting of the substrate wettability effect on interfacial electromechanics is provided by employing a free-energy-based formalism, without taking resort to the traditional slip-length-based approach. In particular, we model the interfacial phenomena governed by the substrate contact angle through the introduction of an order parameter, which enables us to capture the evolution of the relative phase distributions in a thermodynamically consistent manner, coupled with electromagnetohydrodynamic considerations. Our findings may bear far-ranging scientific and technological implications towards design, optimization and operation of miniaturized energy conversion devices of practical relevance. 2. Mathematical formulation 2.1. Problem description We examine the pressure–driven fully developed flow of a binary electrolyte through a slit–type parallel plate narrowfluidic confinement of height 2a, with the origin set at the bottom wall. The X coordinate represents the direction parallel to the walls and Y is the transverse coordinate (being the channel centreline at Y = a). The flow is subjected to a superimposed transverse magnetic field of strength BY along the Y − direction. A schematic representation of the present problem under consideration is illustrated in Fig. 1. The counterions present in the diffuse layer of EDL streams along the pressure-driven fluid flow, resulting a competing advection–electromigration mechanism in the strengths of the ionic species. Consequently, parallel to the flow direction an electrical potential difference sets up, viz. the streaming potential (EX ). This induced streaming potential culminates in a back electrokinetic transport, so as to oppose the pressure– gradient–driven flow to which the forwarding motion of the ionic charges is due. Further, we consider the formation of depletion layer (see Fig. 1) adjacent to the wall such that there are two fluid regions in the channel. The depleted phase renders a cushioning effect for the smooth sailing of the bulk liquid over it, which, as a result gives rise to the apparent slip, being characterized by the substrate wettability at the three-phase contact line. We employ order parameter based phase-field model to capture the variation in electrical permittivity and the dynamic viscosity (because of the depletion in the liquid adjacent to the hydrophobic wall) along the transverse (Y −) direction. In the following sections, we present a mathematical model as appropriate to the governing electromagnetohydrodynamic mechanism in the narrow fluidic confinement under consideration. 2.2. Order parameter based phase-field model We address the hydrodynamics associated with the depletion layer formation through the evolution of an order parameter. In an effort to track the spatial variation of the composition of this binary mixture, we use an order parameter variable [40] φ = (n1 − n2 ) /(n1 + n2 ), where ni are the number densities of the two separating phases. The bulk liquid and depleted phases are represented by φ = −1 and φ = 1 respectively, whereas the average interfaces location between the two phases is located at a position φ = 0 [40]. Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.

4

S. Sarkar and S. Ganguly / Physica A xxx (xxxx) xxx

Following the assumptions of flow invariance on order parameter distribution, the demixing thermodynamics is introduced by considering Ginzburg–Landau free energy functional in terms of a phase–field parameter for a binary mixture [40]:

∆F (φ) =

∞

∫

[ ( k

dφ

2

dY

0

where,

k 2

(

dφ dY

)2

]

)2

+ ∆f (φ) dY + Φs

(1)

is the penalty parameter for the presence of the interfacial gradient, Φs is the surface energy parameter

accounting the interactions between the fluid and substrate, and ∆f (φ) is the free energy parameter being represented by a double-well potential [40], B

∆f (φ) =

√ ]2 [

[

A

φ−

4

√ ]2 A

φ+

B

(2)

B

where, A, B, are two positive constants satisfying A, B ∼ kB TC , with TC being the critical temperature signifying liquid– ∆F vapour coexistence. By minimizing the free energy functional (Eq. (1)), δδφ = 0, one may obtain governing description of the interfacial profile as [40] d

[

dY

{(

k d 2 dφ ′

dφ

)2 }]

d∆f

−

dY

=0

dφ

(3)

The dependency of the substrate wettability on the interfacial electromagnetohydrodynamics may be established via interfacial order parameter variation with the contact angle θ . Accordingly, the equilibrium free energy can be obtained by the minimum value of the functional defined in Eq. (1) as: χ = Φs +

∫ φ0 √ φs

2k∆f (φ) dφ , where φs is the order parameter

at the channel substrate, and φ0 is the order parameter at the reservoir respectively. Furthermore, one may obtain the surface free energies of the solid–liquid (χsl ), solid–vapour (χsv ), liquid–vapour (χlv ) interfaces [15]. Thus, the equilibrium contact angle θ at the contact line may be determined by Young’s law: cos θ =

χsv −χsl χlv

=

φs3 −3φs β 2 , 2β 3

where, β =

√

A/B.

Substituting Eq. (2) in Eq. (3), and after some simplifications, one can obtain a dimensionless form of the order parameter evolution equation as [40] d2 φ dy2

−

1 (

) φ 3 − φβ 2 = 0

Cn2

(4)

The parameter Cn, in Eq. (4) is the Cahn number and is defined as Cn = ξint /a, and y = Y /a. Eq. (4) is subjected to the following boundary conditions: dφ/dy|y=1 = 0 (channel centreline), φ|y=0 = φs (channel wall). To obtain the order parameter solution φ (y) (Eq. (4)) along with the corresponding set of boundary conditions, we have used the finite volume based numerical technique [41]. We employ TDMA (tri-diagonal matrix algorithm) solver for the numerical solution of the resultant discretized equations [41]. Having obtained the order parameter profile, we assume a linearized functional dependence of the interfacial viscosity (µ), permittivity (ε), and density (ρ) variation with φ as [40],

µ = µBulk

ε = εBulk

ρ = ρBulk

(

(

1−φ

)

2 1−φ

)

2

(

1−φ 2

)

+ µDep

+ εDep

(

(

+ ρDep

1+φ

) (5a)

2 1+φ

) (5b)

2

(

1+φ

) (5c)

2

where the subscripts Bulk and Dep denote the bulk and depleted regions respectively. 2.3. Ionic charge density and EDL potential distribution The EDL potential distribution for a slit-type narrow-fluidic confinement (present situation) is obtained from the Poisson–Boltzmann formalism with considerations of permittivity (ε) variation as a function of transverse coordinate (ε (Y )) and an explicit influence of depletion layer (on the electrochemical ) potential. The latter is captured through local fluid density dependent extra potential term [24]

Ωext = −kB T ln ρ ρ

Bulk

, and modifies the electrochemical potential as

Λ± = kB T ln n± ± ez Ψ + Ωext , where e is the protonic charge, n± is the number density of the ionic species, T is the absolute temperature, Ψ is the EDL potential, z is the valency of the ionic species, and kB is the Boltzmann constant. Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.

S. Sarkar and S. Ganguly / Physica A xxx (xxxx) xxx

5

Furthermore, we assume equilibrium conditions between the channel-to-connecting reservoir and for a z: z symmetric electrolyte we obtain a modified version of the dimensionless Poisson–Boltzmann equation from this particular form of the electrochemical potential as [24]

( ) ( a )2 ∂ψ ∂ ε sinh (ψ) =ρ ∂y ∂y λ

(6)

We consider the dimensionless parameters [24]: ψ = ez Ψ /kB T , ε = ε/εBulk , ρ = ρ/ρBulk , and λ = εBulk kB T /2n0 e2 z 2 being the Debye screening length based on the bulk fluid permittivity value. Here, n0 is the bulk ionic concentration at the reservoir. It is apparent that the solution of Eq. (6) can be obtained if the proper specifications of the boundary conditions at the channel walls as well as at the channel centreline are known. To accomplish this, we assume one representative case where the bare silica glass surface is used as the channel walls. Thus, through the dissociation of terminal silanol groups (as a result of the chemical reaction between bare silica, added cations, and the hydrogen ions) the surface zeta potential is governed by the buffer pH, and the bulk ionic concentration, n0 . Under the conditions mentioned above, a functional dependence of the zeta potential (ζ ) with the surface charge density (σ ) may be obtained as [42]:

√

ζ =

kB T e

( ln

−σ eΓ + σ

) −

kB T e

(pH − pKa ) ln 10 −

σ

(7)

CStern

where Γ is the fraction of dissociated chargeable sites, pKa is the dissociation constant, and CStern is the Stern layer capacitance. The surface charge density can be expressed as [43]

σ = ±ε

(

dΨ dY

) (8) w all

The solution methodology for Eq. (6) goes by first assuming a dimensionless zeta potential ζ ζ = (ez ζ ) /(kB T ) . For

[

]

a given set of parameters, we compute ψ (y) by numerically integrate Eq. (6) under the conditions: ψ|y=0 = ζ and [∂ψ/∂ y]y=1 = 0. We perform the numerical solution of Eq. (6) along with the corresponding boundary conditions by utilizing finite volume based method [41]. The discretized algebraic equations are then solved by using TDMA solver [41]. Thereafter, we obtain dimensionless charge density σ (through Eq. (8), in dimensionless form σ = ±ε [dψ/dy]y=0 , where σ = (ezaσ ) /(εBulk kB T )) with the provisional value ψ (y). Now, we use the current value of σ in Eq. (7) to calculate a new value of ζ . We continue these iteration steps until convergence is achieved to all of the variables, ζ , σ , and ψ [43]. 2.4. Governing transport equation for electromagnetohydrodynamic flow Incorporating the additional body force terms due to induced streaming potential field, superimposed magnetic field, and pressure field, the governing transport equation for electromagnetohydrodynamic velocity distribution can be obtained by solving the Navier momentum equation of the following form,

( − →)

D ρU

}] [ {( − → − →) ( − →)T + FB = −∇ P + ∇ · µ ∇ U + ∇ U

Dt

− →

(9)

− →

where U is the velocity vector, P is the hydrostatic pressure, and FB represents the net body force per unit volume. Considering low Reynolds number (Re ≪ 1) hydrodynamics typical to nano-fluidic confinements, we neglect the inertial term in Eq. (20), so that [44,45]

[ {( }] − →) ( − →)T − → ∇U + ∇U + FB

0 = −∇ P + ∇ · µ

(10)

It is worth mentioning here that the net body force acting on the fluid is essentially composed of induced electrical force, − → together with superimposed magnetic field induced Lorentz force. Thus, the general expression for FB is [45,46]

− →

− →

− →

− →

FB = ρe E + J × B

(11)

− →

where the electric current density vector J is expressed from the Ohm’s law as [44,46]

− →

J = σe

(− →

− →

− →)

E + U × B

(12)

− →

− →

Here E is the electric field vector, B magnetic field vector, and σe is the electrical conductivity of the fluid. It is extremely important to mention here that because of the presence of EDL the conductivity (σe ) would actually vary across the channel. Therefore, in the present investigation we consider the following expressions for the conductivity [2]:

σe =

z 2 e2 f

(n+ + n− ) =

2z 2 e2 n0 f

( cosh

ez Ψ kB T

) (13)

Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.

6

S. Sarkar and S. Ganguly / Physica A xxx (xxxx) xxx

where f =

2n0 z 2 e2

σB

, is the ionic friction factor and is assumed to be identical for both anionic and cationic species; σB is

the bulk ionic conductivity [2]. We assume that the flow is steady and hydrodynamically fully developed, and therefore the velocity gradient along the X − direction is neglected. Further, we consider that the magnetic Reynolds number is very small, which is typical to the particular flow geometry under consideration. With the above assumptions coupled with the equation of mass conservation, the dimensionless form of the momentum transport equation (Eq. (10)) for electromagnetohydrodynamic velocity distribution may be written as [24,45–47]

( ) ( ) ∂u E ∂ ∂ψ ∂ 2 µ − cosh (ψ) Ha u = ε −1 ∂y ∂y ∂y ζ ∂y

(14)

where the velocity{ has as [45] u = U / − a2 /µBulk (dP /dX ) , the dimensionless streaming ( 2been nondimensionalized ) } potential E = EX / − a /εBulk ζ (dP /dX ) , the dimensionless interfacial viscosity µ = µ/µBulk , and Ha is the Hartmann

{ (

)

}

√(

σB B2Y a2 /µBulk . Here, number indicating the strength of the superimposed magnetic field, being defined as Ha = −dP /dX is the applied pressure gradient. ( ) ( The relevant) boundary conditions for Eq. (14) are no slip walls u|y=0 = 0 and channel centreline symmetry du/dy|y=1 = 0 [43]. It is important to note here that the parametric value of the dimensionless streaming electric field E remains unknown and is an implicit function of velocity field; therefore the governing electromagnetohydrodynamic velocity distribution Eq. (14) is not mathematically closed. A closure relationship may be obtained through an overall electroneutrality constraint, being described in the following section.

)

2.5. Overall electroneutrality constraint and the streaming potential, energy conversion efficiency At each sections of the channel, the induced streaming current due to the downstream advection of the ions must be balanced out by the net conduction current due to electromigration through the bulk and the stern layer. Therefore, the net ionic current (Inet ) over the channel cross section can be written as [2] Inet = Istream + Icond + Istern

(15)

where, Istream is the streaming current, Icond is the bulk conduction current passing through the ‘‘mobile’’ fluid layers, and Istern is the conduction current passing through the ‘‘immobilized’’ stern layer. Considering the ions are moving with the local fluid velocity in the EDL [2], a

∫

z e (n+ − n− ) U (Y ) dY

Istream =

(16a)

−a

Icond =

z 2 e2 EX

∫

f

a

(n+ + n− ) dY

(16b)

−a

Istern = 2σstern EX

(16c)

where σstern is the stern layer conductivity [2]. In the absence of applied electric field, to satisfy electroneutrality, the sum total of all currents in the channel must be zero [2], Inet = 0

(17)

We substitute Eq. (16) in Eq. (17), and after nondimensionalization, the electroneutrality constraint reads 1

∫

sinh (ψ) u (y) dy + 0

where α = − 2ε

µBulk σB

Bulk n0 (kB T )

αE ζ

1

∫

cosh (ψ) dy + 0

αE ζ

Du = 0

, a dimensionless conductivity parameter; Du =

(18) σstern , aσB

Dukhin number.

From Eq. (18) we get the final expression for streaming potential E=

−ζ I1 α (I2 + Du)

where the integrals, I1 =

(19)

∫1 0

sinh (ψ) u (y) dy, and I2 =

∫1 0

cosh (ψ) dy, respectively.

The method of obtaining the velocity solution u (y) (Eq. (14)) along with the streaming potential (Eq. (19)) involves an iterative technique. Towards this, with a self-consistent approximation of the streaming potential (E ), we solve Eq. (14) together with the boundary conditions by employing the finite volume based numerical method [41]. The TDMA solver has Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.

S. Sarkar and S. Ganguly / Physica A xxx (xxxx) xxx

7

been employed for the numerical solution [41]. Now, with the provisional solution of u (y), we check for electroneutrality as given in Eq. (18), and correct the new guess value of E from Eq. (19). We continue these iteration steps till the convergence is achieved to the variables u (y), and E (by satisfying the electroneutrality condition, Inet ≤ 10−12 ). It may be noted here that the induced streaming potential field is immensely consequential in converting hydraulic energy into an electrical power generation in narrow fluidic devices. The ratio of the amount of extractable energy induced by the streaming potential field to the input power required for the flow to happen is defined as the energy conversion efficiency η, and is expressed as [38]

η=

Istream |ES |

⏐( ) ⏐ ⏐ dP ⏐ dX eff ⏐

(20)

Q⏐

where ES is the streaming potential field, Q is the volume flow rate, and (dP /dX )eff is the effective pressure gradient as a combined consequences of the applied pressure gradient and the magnetic field. The value of (dP /dX )eff may be estimated by comparing flow rates with and without invoking the external magnetic field BY and setting dP /dX = (dP /dX )eff , for the situation BY = 0 [38]. 3. Results and discussions In an effort to validate the present model, Fig. 2 compares the variation in the dimensionless streaming potential E with the Hartmann number (Ha) for the magnetohydrodynamic flows in a microchannel to the available data in the literature [48]. The other relevant parameters used in comparison are given in the caption. As revealed in Fig. 2, the prediction from our model matches fairly well with the published data of Zhao et al. [48]. For demonstrating our key results in a physical system, we consider typical fluid properties at a reference temperature of T = 300 K as [43]: µBulk ≈ 10−3 kg/ms, ρBulk ≈ 103 kg/m3 , εBulk ≈ 702.24 × 10−12 C2 /J m. For a z: z symmetric electrolyte, the electrochemical constants may be taken as, σB ≈ 7.2 mS/m, f ≈ 10−12 N s/m [43]. Together with a bulk ionic concentration (n0 ) ranging from millimolarity to molarity, we assume representative reaction parameters as follows [43]: CStern = 0.3 F/m2 , Γ = 8 nm−2 , pH = 8, pKa = 7.5. Furthermore, conforming to these reaction parameters, the magnitude of the dimensional zeta potential turns out to be in the tune of 10 mV–50 mV, the magnitude of dimensional surface charge density is of the order 4 mC/m2 , while the characteristic Stern layer conductivity value ranging from 0 to 72 nS/cm, and the corresponding Dukhin number ranging from 0 to 100. The other pertinent parameters used in the present investigation are Hartmann number (Ha), and the half channel height (a). We assume the range of Hartmann number between 0–5, with a between 10–1000 nm, which falls within the practically realizable range [30]. We consider the dimensionless conductivity parameter, |α| ≈ 10, µDep /µBulk = 1/3, εDep /εBulk = 0.8, and ρDep /ρBulk = 10−3 [24]. We choose the dimensionless parameters β and Cn2 , same as that one used for comparison in [24]. It is of interest to study the variation of channel-centreline potential (ψc ) as a result of the substrate wettability (θ) effects. To elucidate this, we plot the magnitude of the channel-centreline potential (|ψc |) as a function of the contact angle (θ) for differenta/λ ratios in Fig. 3. The general trend observed in Fig. 2 and all the insets is that with stronger hydrophobic effects, i.e. with increasing value of the contact angle, |ψc | follows a rapid monotonic increase. This can be attributed to the fact that more effective electrokinetic pumping of fluid takes place within the interfacial layer with higher degrees of hydrophobicity. It is also noted that as the a/λ ratio increases, the magnitude of the channel-centreline potential decreases for the entire regime of the contact angle θ . In general, the ‘‘penetration’’ of the EDL into the channel is manifested by the a/λ ratio. A lower a/λ represents the case where the channel is much ‘‘thinner’’ than the EDL and that the penetration of the EDL is greatly significant. Therefore, thicker the EDL, the influence of electrochemical perturbation from the wall gets propagated (in sync with relative phase distributions through a continuously varying viscosity, consistent with the substrate wettability) into the bulk fluid. This effect exhibits a dramatic reduction in the |ψc | value as a/λ ratio is increased [49]. Fig. 4(a) depicts the variation of the dimensionless streaming potential field (E ) as a function of the contact angle (θ), with variations in the Hartmann number (Ha). Here, we have not considered the effect of Stern layer conductivity (Du = 0), and a/λ ratio is kept fixed at a/λ = 3. This figure essentially brings out the combined implications of surface wettability and magnetic field on the streaming potential magnitude. As the contact angle (θ) is increased progressively in Fig. 4(a), it is seen that the magnitude of streaming potential increases. From a physical standpoint, the extent of hydrophobicity is dictated by the surface wettability (θ) constraints. Such hydrophobicities are not thermodynamically favoured to form hydrogen bonds with the water molecules, which give rise to barred volume region enveloping the locations bounded with rapidly weakening number density of water molecules [40]. This in effect retracts liquid molecules, thereby forming a depletion layer where the small scale fluctuations in density and viscosity are found to exist. As a consequence, stronger hydrophobicity (characterized by higher values of contact angles) enhances the strength of the depletion having higher values of φs at the substrate, leading to the lower viscosity region of the depletion to be more conspicuous within its dimension [24]. Reduction in effective viscosity along the depleted region gives rise to an apparent slip effect, which is, as a result of balance between the shear stress continuity, compensated through the enhancement in local shear strain rate. This results in reduction of overall resistances to the bulk fluid motion. The indicated effect causes augmentation in the pressure driven flow and therefore triggers streaming current of the ions. Higher the streaming Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.

8

S. Sarkar and S. Ganguly / Physica A xxx (xxxx) xxx

Fig. 2. Variation in the dimensionless streaming potential E with the Hartmann number, Ha for a/λ = 30 and volume fraction, ϕ = 0%.

Fig. 3. Variation of the magnitude of the channel-centreline potential (|ψc |) as a function of the contact angle (θ ) for different a/λ ratios.

current, larger is the streaming potential, and therefore higher is the streaming potential value with increasing values of surface contact angles manifested through the degree of hydrophobicity [24,40]. Nevertheless, it can be observed from Fig. 4(a) that for all values of θ , streaming potential reduces as the strength of the superimposed magnetic field is increased. This can be attributed to the increase in opposing electromagnetic forces due to the increase in strength of the applied magnetic field (Ha), which tends to reduce the effective driving pressure gradient in the flow. As a result, the advective transport of the ionic species gets retarded, and hence the streaming potential field. We next proceed to examine how the a/λ ratio variations would affect the streaming potential field in the present context. Fig. 4(b) shows the variation of the dimensionless streaming potential field (E ) as a function of the contact angle (θ), with variations in the a/λ ratio. The results are analysed for a particular value of Hartmann number, Ha = 1, and Dukhin number, (Du = 0). We observe that as the values of the a/λ ratio enhances, streaming potential field becomes progressively smaller. It is important to mention in this context that the relative ‘‘thickness’’ of the EDL becomes comparable to channel half height when a/λ = 1. Such situations may encompass the development of a ‘‘thick’’ interfacial layer through overlapping EDLs [2,49]. The resulting consequence reduces the resistance faced by the ion, implying increase in the bulk conductivity of the flow. For an enhanced bulk conductivity, to satisfy the same streaming current, a higher streaming electric field must be induced so as to have a corresponding conduction current to satisfy electroneutrality. As a result, smaller value of a/λ the ratio is always accompanied by the larger streaming potential magnitude. We observe that for all the values of a/λ studied, the maximum magnitude of E occurs at a/λ = 1. It may be of interest to explore the possibilities of employing magnetic field in the presence of multiple physical effects toward electrokinetic energy conversion in narrow fluidic confinements having extremely small characteristic Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.

S. Sarkar and S. Ganguly / Physica A xxx (xxxx) xxx

9

Fig. 4. Variation of the dimensionless streaming potential field (E ) as a function of the contact angle (θ) with (a) variations in the Hartmann number (Ha), and at a/λ = 3, Du = 0; (b) variations in the a/λ ratio, and at Ha = 1, Du = 0.

scales. For the purpose of examining the combined consequence of surface wettability variations of hydrophobic channel walls and influences of external magnetic field, we study the variation in energy conversion efficiency corresponding to the cases described above. In Fig. 5(a), we plot the energy conversion efficiency (η) as a function of the contact angle (θ), for different values of the Hartmann number Ha and for fixed values of a/λ = 3, Dukhin number Du = 0. Fig. 5(a) essentially corresponds to the case presented in Fig. 4(a). It is seen that with increase in Hartmann number, the energy conversion efficiency increases, the effect is more prominent ] value of magnetic field (Ha = 5). This can be [ for higher attributed to the fact that the effective pressure gradient (dP /dX )eff gets lowered at the higher magnitudes of Ha, accompanied with lower magnitudes of the induced streaming potential. The rate of decrement of effective external pressure gradient, however, is greater than rate of decrement of the streaming potential. Since the energy conversion efficiency essentially is a ratio of these two parameters (Eq. (20)), with increasing Ha, the efficiency increases [38]. It can also be observed from Fig. 5(a) that there is an increasing trend in the energy conversion efficiency η, with increasing contact angle. This can be explained on the basis of corresponding streaming potential variation with contact angle (Fig. 3). It is seen from Fig. 4 that the magnitude of E increases sharply with increasing θ and attains its peak value at θ = 140◦ . This essentially determines the nature of variation of the energy conversion efficiency, which shows an increasing trend with increase in θ (Fig. 5(a)). However, η always increases with enhanced strengths of the imposed magnetic field, for Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.

10

S. Sarkar and S. Ganguly / Physica A xxx (xxxx) xxx

Fig. 5. Variation of the energy conversion efficiency (η) as a function of the contact angle (θ ) with (a) variations in the Hartmann number (Ha), and at a/λ = 3, Du = 0; (b) variations in the a/λ ratio, and at Ha = 1, Du = 0.

the entire regime of the contact angles. Such behaviour brings out the combined implications of imposed magnetic field and substrate wettability variations on the overall energy transfer performance, as a parametric function of the Hartmann number as well as the contact angle. Therefore, the combination of these physical effects is not merely a linear superimposition of all effects considered individually, but turns out to be a complicated and interconnected function of interfacial electromagnetohydrodynamics over small scales, as considered in the present study [50]. Fig. 5(b) displays the variations in the energy conversion efficiency (η), as a function of the contact angle (θ), with variations in a/λ ratio, for constant values of Ha = 1, Du = 0. The overall conclusion from Fig. 5(b) is that reduction in the a/λ ratios strongly augments the energy conversion efficiency. This can be attributed to the fact that decrement in the a/λ ratios leads to thick EDL limits, for which the Debye length is significantly higher than the characteristic dimension of the channel. The other band of the characteristics pertains to thin EDL limits where the a/λ ratio is steadily increased. An important point here is that thick EDLs are common to very dilute ionic solutions (e.g. λ = 0.3 nm for 1M NaCl solution, whereas, λ = 30 nm for 10−4 M of the same solution) [38]. Such situation leads to an enhancement in the EDL thickness and therefore stretching of the EDL along the transverse direction (y−) of the channel [38]. The combined influences of this condition along with the characteristic length scale of the channel finally culminate in augmenting the energy conversion efficiency with reducing a/λ ratios. Indeed, increasing surface wettability always plays an add-on effect in intensifying η. Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.

S. Sarkar and S. Ganguly / Physica A xxx (xxxx) xxx

11

Fig. 6. Energy conversion efficiency (η) as a function of the Hartmann number (Ha) and the contact angle (θ), at a/λ = 3, Du = 0.

Towards inspecting the exact nature of variations in the energy conversion efficiency, we plot η as a function of the Hartmann number and the substrate wettability in Fig. 6; the values of other relevant parameters are given in the caption. We attempt to demonstrate that the surface wettability and the magnitudes of superimposed magnetic field play a vital role towards modulating the energy conversion efficiency. It is seen that as the values of Ha progressively increased, the magnitude of η for both the high and low θ increases by several orders of magnitude. If the nature of the variations in η with θ is compared against Ha, we see that the percentage enhancement in η with the mediation of the substrate wettability is higher in comparison to that for magnetohydrodynamic influences, especially at θ = 140◦ . Therefore, we conclude that for every choice of Ha, there is a certain value of contact angle, for which the energy conversion efficiency is maximum. In other words, it is possible to maximize the energy conversion efficiency by judiciously choosing the optimum magnetic field strength and substrate wettability distribution. 4. Conclusions To summarize, we have investigated, the combined effects of substrate wettability and magnetohydrodynamic forces toward altering streaming potential and hydroelectrical energy conversion in nano-fluidic confinements. In sharp contrast with the traditional slip-length-based formalism, we employ a phase-field model to capture the surface hydrophobicity induced depletion layer formation in the wall-adjacent region. Our theoretical calculations and analyses demonstrated that the magnetic field may be used as an important tuning parameter for modulating the energy conversion efficiency. It is revealed that, energy transfer efficiency can also be effectively optimized by employing optimal combination of magnetic field and surface hydrophobicity. This is one of the key findings from our research. We believe that the outcome of our study will be an important milestone in the design of miniaturized micro/nano-fluidic devices for energy conversion, such as ‘‘electrokinetic microchannel battery’’ as an alternative energy source to rival wind and solar power, where water is pumped through a glass filter riddled with tiny holes of micro/nano metre dimensions, which has an ability to actuate microscopic gears, switches, and electronic devices. Present theory is expected to throw light on the intricate interaction mechanism between magnetohydrodynamic effects, and surface hydrophobicity leading towards optimization of energy transfer performance in futuristic fluidic devices of small dimension. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

R.F. Probstein, Physicochemical Hydrodynamics, Wiley, New York, 1994. R.J. Hunter, Zeta Potential in Colloid Science: Principles and Applications, Academic Press, London, 1981. J. Yang, F. Lu, L.W. Kostiuk, D.Y. Kwok, J. Micromech. Microeng. 13 (2003) 963. H. Daiguji, P. Yang, A.J. Szeri, A. Majumdar, Nano Lett. 4 (2004) 2315. W. Olthuis, B. Schippers, J. Eijkel, A. van den Berg, Sensors Actuators B 111–112 (2005) 385. Y. Ren, D. Stein, Nanotechnology 19 (2008) 195707. F.A. Morrison, J.F. Osterle, J. Chem. Phys. 43 (6) (1965) 2111. D. Burgreen, F.R. Nakache, J. Appl. Mech. 32 (3) (1965) 675. C.H. Choi, C.J. Kim, Phys. Rev. Lett. 96 (1965) 066001. S. Pennathur, J.C.T. Eijkel, A. van den Berg, Lab Chip 7 (2007) 1234. P. Huang, J.S. Guasto, K.S. Breuer, J. Fluid Mech. 566 (2006) 447. E. Lauga, M.P. Brenner, Phys. Rev. E 70 (2004) 026311.

Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.

12 [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

S. Sarkar and S. Ganguly / Physica A xxx (xxxx) xxx J.K. Holt, H.G. Park, Y.M. Wang, M. Stadermann, A.B. Artyukhin, C.P. Grigoropoulos, A. Noy, O. Bakajin, Science 312 (2006) 1034. M. Majumder, N. Chopra, R. Andrews, B.J. Hinds, Nature 438 (2005) 44. J. Yang, D.Y. Kwok, Langmuir 19 (2003) 1047. P. Goswami, S. Chakraborty, Langmuir 26 (2010) 581. G. Seshadri, T. Baier, Phys. Fluids 25 (2013) 042002. Yu. Yan, Q. Sheng, C. Wang, J. Xue, H.-C. Chang, J. Phys. Chem. C 117 (2013) 8050. D.C. Tretheway, C.D. Meinhart, Phys. Fluids 16 (2004) 1509. G.C. Shit, A. Mondal, A. Sinha, P.K. Kundu, Physica A 462 (2016) 1040. D. Andrienko, B. Dunweg, O.I. Vinogradova, J. Chem. Phys. 119 (2003) 13106. V.E. Badalassi, H.D. Ceniceros, S. Banerjee, J. Comput. Phys. 190 (2003) 371. R. Borcia, I. Borcia, M. Bestehorn, Phys. Rev. E 78 (2008) 066307. J. Chakraborty, S. Chakraborty, Phys. Rev. E 88 (2013) 043007. E. Lauga, M.P. Brenner, H.A. Stone, Handbook of Experimental Fluid Dynamics, Springer, Berlin, 2007. M. Sbragaglia, R. Benzi, L. Biferale, S. Succi, F. Toschi, Phys. Rev. Lett. 97 (2006) 204503. A. Steinberger, C. Cottin-Bizonne, P. Kleimann, E. Charlaix, Nature Mater. 6 (2007) 665. N. Pamme, Lab Chip - Miniaturisation Chem. Biol. 6 (2006) 24. M.C. Weston, M.D. Gerner, I. Fritsch, Anal. Chem. 82 (2010) 3411. N.T. Nguyen, Microfluid Nanofluid 12 (2012) 1. J. Jang, S.S. Lee, Sensors Actuators A 80 (2000) 84. T.B.Jones, Electromechanics of Particles, Cambridge University Press, New York, 1995. E.P. Furlani, J. Appl. Phys. 99 (2006) 024912. S. Das, S. Chakraborty, S.K. Mitra, Microfluid Nanofluid 13 (2012) 799. J.S. Andreu, J. Camacho, J. Faraudo, M. Benelmekki, C. Rebollo, L.M. Martinez, Phys. Rev. E 84 (2011) 021402. S. Chakraborty, D. Paul, J. Phys. D 39 (2006) 5364. C.P. Tso, K. Sundaravadivelu, J. Phys. D 34 (2001) 3522. F. Munshi, S. Chakraborty, Phys. Fluids 21 (2009) 122003. S. Chakraborty, Anal. Chim. Acta 605 (2007) 175. S. Chakraborty, Phys. Rev. Lett. 99 (2007) 094504. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Taylor & Francis, Kundli, India, 2004. S.H. Behrens, D.G. Grier, J. Chem. Phys. 115 (2001) 6716. A. Bandopadhyay, S. Chakraborty, Langmuir 27 (2011) 12243. R.J. Moreau, Magnetohydrodynamics, Springer, 1990. S. Sarkar, S. Ganguly, S. Chakraborty, Microfluid Nanofluid 21 (2017) 56. S. Sarkar, S. Ganguly, P. Dutta, Int. J. Heat Mass Transfer 100 (2016) 451. A. Bandopadhyay, S. Chakraborty, Phys. Rev. E 85 (2012) 056302. G. Zhao, Y. Jian, F. Li, J. Magn. Magn. Mater. 407 (2016) 75. U. Ghosh, S. Mandal, S. Chakraborty, Phys. Rev. Fluids 2 (6) (2017) 064203. M. Abdulhameed, D. Vieru, R. Roslan, Physica A 484 (2017) 233.

Please cite this article as: S. Sarkar and S. Ganguly, Consequences of substrate wettability on the hydro-electric energy conversion in electromagnetohydrodynamic flows through microchannel, Physica A (2019) 123450, https://doi.org/10.1016/j.physa.2019.123450.