The Effect of Hydrodynamic Slip on Membrane ... - ACS Publications

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The Effect of Hydrodynamic Slip on Membrane-Based SalinityGradient-Driven Energy Harvesting Daniel Justin Rankin and David Mark Huang* Department of Chemistry, The University of Adelaide, Adelaide, SA 5005, Australia

Langmuir 2016.32:3420-3432. Downloaded from pubs.acs.org by UNIV OF SUNDERLAND on 11/05/18. For personal use only.

S Supporting Information *

ABSTRACT: The effect of hydrodynamic slip on salinitygradient-driven power conversion by the process of reverse electrodialysis, in which the free energy of mixing of salt and fresh water across a nanoporous membrane is harnessed to drive an electric current in an external circuit, is investigated theoretically using a continuum fluid dynamics model. A general one-dimensional model is derived that decouples transport inside the membrane pores from the effects of electrical resistance at the pore ends, from which an analytical expression for the power conversion rate is obtained for a perfectly ion-selective membrane as a function of the slip length, surface charge density, membrane thickness, pore radius, and other membrane and electrolyte properties. The theoretical model agrees quantitatively with finite-element numerical calculations and predicts significant enhancementsup to several timesof salinity-gradient power conversion due to hydrodynamic slip for realistic systems.

■

INTRODUCTION An enormous amount of free energy is “lost” when fresh and salt water mix where rivers meet the sea. It has been estimated that, if harnessed, this clean renewable power source could supply 1−2 TW of power globally,1,2 or up to about 10% of global energy needs. Methods for harnessing so-called salinitygradient power or osmotic power have been known for more than 40 years1−3 and include processes such as pressureretarded osmosis,4 reverse electrodialysis,5 and capacitive mixing.6−8 Nevertheless, the first prototype salinity-gradient power plant (in Norway), using pressure-retarded osmosis, was not opened until 20099 and it was only in 2014 that the first pilot reverse-electrodialysis plant was opened (in The Netherlands).10 A major barrier to more widespread adoption of salinity-gradient power is the low power-conversion rates, which must improve for salinity-gradient power to become economically competitive with other forms of energy production.11 On a totally different scale, salinity-gradient power has been suggested as an effective means for powering nanoscale systems such as biomedical devices operating in saline fluid environments.12 The most widely used technologies for harnessing salinitygradient power, pressure-retarded osmosis, and reverse electrodialysis are based on fluid or ion transport across a porous membrane.1,2 Optimizing the membrane for these transport processes is key to improving power conversion and requires a comprehensive understanding of electrolyte transport at the nano scale. For example, reverse electrodialysis, which is the direct conversion of the free energy in a salinity gradient across a porous membrane into electricity, relies on the net transfer of ionic charge across the membrane in response to the salt © 2016 American Chemical Society

gradient (Figure 1). Oxidation and reduction at electrodes on either side of the membrane maintain electroneutrality and

Figure 1. Schematic of direct electrical energy harvesting from a salinity gradient (reverse electrodialysis) using a charged membrane interposed between high and low salt-concentration reservoirs.

produce an electron flow in an external circuit that can be used for electrical power. Efficient ionic charge transfer across the membrane requires ions with the same charge as the membrane pores (co-ions) to be expelled by the membrane while oppositely charged counterions pass freely.13 Because electrostatic interactions in salt water are typically screened over distances of nanometers, efficient membranes must have nanosized pores.14 Nanoscale electrolyte transport differs fundamentally from macroscopic transport,15,16 but most previous theoretical predictions of salinity-gradient power conversion efficiencies Received: February 3, 2016 Revised: March 18, 2016 Published: March 18, 2016 3420

DOI: 10.1021/acs.langmuir.6b00433 Langmuir 2016, 32, 3420−3432

Article

Langmuir and rates have used macroscopic models.14,17 However, the standard macroscopic theory of electrokinetics18 sometimes fails to describe the transport processes that govern energy conversion unless additional adjustable parameters are used.19 A key area where the standard theory fails is in its treatment of the solid−liquid interface, where a nonslip boundary condition (equivalent to infinite solid−liquid friction) is assumed. Recent experiments20 and molecular simulations21−23 have shown that interfacial hydrodynamic slip, which is sensitive to molecular surface properties such as roughness, charge density, and hydrophobicity,24 can have a dramatic impact on electrokinetic transport.20−23,25 Slip has major implications for electrokinetic power conversion. For electrical energy generation using pressure-driven electrolyte flow, a process closely related to salinity-gradient power, efficiency gains of an order of magnitude are predicted when low-friction (high-slip) membranes26,27 are used instead of nonslip surfaces.13 Slip is also potentially important for salinity-gradient power, since ion transport due to fluid flow, and not just ion diffusion and electromigration, has been shown to play a significant role in power conversion. For example, huge power densities (∼kW m−2 cf. W m−2 for conventional reverse-electrodialysis membranes2) measured for reverse electrodialysis using a membrane made of a single boron-nitride nanotube were attributed to the large diffusioosmotic flow induced by the anomalously high apparent charge on the nanotube inner surface.28 Diffusioosmosisfluid flow induced by a solute concentration gradienthas been shown theoretically to be enhanced considerably by interfacial hydrodynamic slip,23,29 so it stands to reason that slip could be beneficial for salinitygradient power conversion. Indeed, one computational study,30 in which the coupled Poisson, Nernst−Planck, and Navier− Stokes (PNP-NS) equations for continuum electrolyte transport were solved numerically for a rectangular slit pore connecting two reservoirs, showed that interfacial slip could enhance power conversion rates by 44% for physically reasonable parameters. However, this study provides little general theoretical insight into how power conversion depends on membrane and electrolyte parameters. The effects of nanoscale electrolyte transport phenomena and in particular hydrodynamic slipon salinity-gradient power conversion have not been extensively investigated theoretically. Besides the aforementioned study, only a handful of works31−33 have accounted for the effect of fluid flow on power conversion, but none of these considered slip. Osterle and co-workers31,32 developed a continuum theory for the power conversion rate and efficiency of a number of electrokinetic energy conversion processes, including reverse electrodialysis, based on approximations to the PNP-NS equations for electrolyte transport in a cylindrical pore without slip and ignoring pore end effects. To the best of our knowledge, all other computational studies of salinity-gradient power conversion have ignored fluid flow by solving only the coupled Poisson and Nernst−Planck (PNP) equations without the Navier−Stokes equation, investigating numerically the effects on power conversion of parameters such as the membrane pore size,14,34,35 surface charge,34,36 pore length,34 salinity gradient,14,34−36 pore asymmetry,36,37 and ion type.36 None of these studies derived analytical expressions for the scaling of the salinity-gradient power conversion rate with these membrane parameters, which would provide useful physical insight into the process and offer a simple means to optimize power conversion. In this work, we focus on deriving such

relationships for the effect of interfacial hydrodynamic slip and other membrane properties on salinity-gradient power conversion, accounting for the effect of electrical resistance at the membrane pore ends, which can significantly impact power conversion as the internal electrical resistance of the pore declines for high-slip surfaces. Improving understanding of the dynamics of nanoconfined electrolytes has wider significance, as it underlies important processes in a diverse array of natural systems and technological applications. Biological cells rely on the regulation of water and ions across cell membranes via nanosized peptide channels.38,39 The efficacy of polymer-electrolyte-membrane fuel cells for next-generation energy storage and conversion40 and membranes for chemical separations41 and desalination42 depends on the relative fluxes of water and ions in nanoporous materials. The translocation of biomolecules such as DNA across nanopores in response to a salt gradient has been proposed as a method of single-molecule sensing and characterization.43 Many of these processes can be described by the same fundamental physics used to understand salinity-gradient power conversion. We begin the paper by introducing the numerical model for continuum fluid and ion transport that is solved to study salinity-gradient power conversion using a porous membrane. From this model, we derive approximate one-dimensional equations for electrolyte transport across a long narrow cylindrical membrane pore, from which we obtain closedform analytical expressions for a perfectly ion-selective pore. We then present an equivalent-circuit analysis to derive the maximum power output in terms of the intrapore transport coefficients and the access electrical resistance of the pore ends. We go on to compare the theory with finite-element numerical solutions to the full continuum model and finally make predictions about the limits of salinity-gradient power conversion for realistic systems. Continuum fluid dynamics has previously been shown to successfully predict the experimental scaling with reservoir salt concentration and channel width of reverse-electrodialysis power output and efficiency in nonslip silica nanochannels.14 Furthermore, continuum models accounting for slip have been found to accurately describe experimental results for a subset of the transport phenomena studied here, including pressure-driven fluid flow44 and electroosmosis.20

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THEORETICAL METHODS 2D Numerical Model. To model the process of salinitygradient-driven power conversion, continuum hydrodynamics for fluid and ion transport was assumed for a symmetric z:z electrolyte across a cylindrical pore of length L and radius a connecting two large fluid reservoirs (Figure 2). Continuum hydrodynamics has previously been shown to describe fluid flow and electrolyte transport accurately for pore dimensions down to a few nanometers.21,22,45,46 Assuming low Reynoldsnumber flow (generally accurate on the nano scale47) and a dilute electrolyte, the governing equations for the electrical potential ϕ, ion flux density ji of species i (i = + or −), and fluid velocity u are the Poisson, Nernst−Planck, and Stokes equations,48 respectively,

3421

−ϵϵ0∇2 ϕ = ze(c+ − c −)

(1)

∇·ji = ∇·(ci u − zieμi ci∇ϕ − Di∇ci) = 0

(2)


Article

Langmuir Table 1. Calculation Parametersa parameter

Figure 2. Schematic of the axisymmetric computational domain for finite-element calculations, comprising a pore of radius a and length L connecting two reservoirs of width and radius w. Solid lines denote solid−liquid boundaries and dashed lines liquid boundaries. The radial and axial directions are along the r and x axes, respectively. Due to cylindrical symmetry, only half of the system is considered.

η∇2 u = ∇p + ze(c+ − c −)∇ϕ

(3)

together with the continuity equation for incompressible flow48 (4)

∇·u = 0

symbol

value

ϵ T η z+ z− D+ D− a L w b bm σ

78.46 298 K 0.894 mPa s 1 −1 1.96 × 10−9 m2 s−1 2.03 × 10−9 m2 s−1 1−25 nm (5 nm) 20−1000 nm (200 nm) 0.5−8 μm (4 μm) 0−500 nm 0 −(1−30) mC m−2 (−10 mC m−2)

σm

0 or σ

cL cH

0.1−30 mmol L−1 (0.1 mmol L−1) 10cL

dielectric constant temperature fluid viscosity cation valence anion valence cation diffusivity anion diffusivity pore radius pore length reservoir radius/width slip length (poreb) slip length (membranec) surface charge density (poreb) surface charge density (membranec) low-salt reservoir conc. high-salt reservoir conc.

Where a range of values is given, the parameter was fixed at the value in parentheses unless otherwise indicated. bRefers to pore surface. c Refers to membrane outer surface. a

The quantity μi is the mobility of species i, Di is the diffusion coefficient (which is assumed to be related to the mobility by D the Einstein relation,48 μi = k Ti ), zi is the valency of species i

1D Intrapore Transport Model. Assuming the pore is much longer than it is wide (L ≫ a), the equations for fluid and ion transport inside the pore can be simplified considerably. The derivation below of analytical approximations to the transport equations generalizes that by Fair and Osterle32 for reverse electrodialysis in a pore with zero slip to the case of a nonzero slip length b. Writing the electrical potential in the pore as the sum of a part that depends only on the axial coordinate x and a part measuring departures from electroneutrality that depends on x and the radial coordinate r

B

(z+ = −z− = z for a z:z electrolyte), e is the elementary charge, η is the fluid viscosity, ϵ0 is the vacuum permittivity, ϵ is the dielectric constant of the fluid, p is the pressure, and ci is the concentration of species i. The pore surface (line DE in Figure 2) was uniform with surface charge density σ and slip length b. The outer surfaces of the membrane (lines CD and EF in Figure 2) were uniform with surface charge density σm and slip length bm. The bulk salt concentration far from the pore was cH in reservoir I (high salt) and cL in reservoir II (low salt). The electrical potential far from the pore was ϕH in reservoir I and ϕL in reservoir II. The fluid pressure far from the pore was zero in both reservoirs. The transport equations were solved numerically using the finiteelement method with COMSOL Multiphysics (version 4.3a),49 as described in detail in the Supporting Information. Model parameters are defined and their values given in Table 1. This treatment makes a few simplifying assumptions, namely, that (1) the fluid is a continuum medium and ions are uncorrelated point particles that interact only via electrostatic forces; (2) the membrane consists of monodisperse cylindrical channels connecting high- and low-salt concentration reservoirs; (3) the effects of interactions between neighboring pores can be ignored, so that only a single pore must be considered; (4) the pore and outer membrane surfaces are uniform; and (5) electrode reactions are instantaneous and go to completion, so that conservation of charge dictates that the electrical current in the external circuit equals the ion current across the membrane (Figure 1). We note that anomalous molecular effects on the transport phenomena studied here have been observed in molecular dynamics simulations of confined electrolytes at high surface charge densities and salt concentrations (higher than studied here), which are not captured by our continuum model.50,51 Nevertheless, our approach should be adequate for the purposes of deriving general scaling relationships for salinity-gradient power conversion and developing a semiquantitative understanding of the process that can be refined in the future by relaxing the aforementioned assumptions.

ϕ(x , r ) = ϕ ̅ (x) + ψ (x , r )

(5)

the boundary condition for the electrical potential at the pore surface is18 ⎛ ∂ψ ⎞ σ = ϵϵ0⎜ ⎟ ⎝ ∂r ⎠r = a

(6)

and the slip boundary condition for the fluid velocity at the pore surface is24 ⎛ ∂u ⎞ (ux)r = a = −b⎜ x ⎟ ⎝ ∂r ⎠r = a

(7)

If the pore length is much larger than its radius (L ≫ a), the radial solute flow in the Nernst−Planck equations will be small (ji,r ≈ 0), giving32

⎛ z eψ ⎞ ci = c(x) exp⎜ − i ⎟ ⎝ kBT ⎠

(8)

where c(x) is the value of ci where ψ = 0. Note that c(x) is only equal to the salt concentration at the centerline of the pore when there is no electric double layer overlap in the pore, which is not a necessary condition for this theory to be valid.32 Rather, c(x) can be considered the concentration of a “virtual” electroneutral ion reservoir in local thermodynamic equilibrium with the ions in the pore cross section at position x. This assumption of local thermodynamic equilibrium at each cross 3422


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Langmuir section should be reasonable provided that L ≫ a.52 Similarly, assuming no radial fluid flow (ur ≈ 0) in the Stokes equation gives32 p = p0 (x) + π

⎡Q ⎤ ⎡ k11 k12 ⎢ ⎥ ⎢ ⎢ Js ⎥ = ⎢ k 21 k 22 ⎢ ⎥ ⎢ ⎣ I ⎦ ⎣ k 31 k 32

(9)

where

π = kBT (c+ + c −)

(10)

is the local osmotic pressure and p0 is the local solvent partial pressure. Assuming also that the axial gradient in the fluid velocity can be ignored, the axial component of the Stokes equation simplifies to

k11 =

(11)

∫0

drr {μ + exp[−Ψ(r )] + μ− exp[Ψ(r )]}

∫0

a

⎧ drr cosh[Ψ(r )]⎨ ⎩

∫r

cosh[Ψ(r″)] +

1 ∂ ⎛ ∂ψ ⎞ ⎜r ⎟ = ze(c − c ) + − r ∂r ⎝ ∂r ⎠

(12)

k 33 = 2π (ze)2 c

from which eq 8 for the ion concentration gives ⎤ 1 ∂⎡ ∂ 2 ⎢⎣r Ψ(x , r )⎥⎦ = [κ(x)] sinh[Ψ(x , r )] r ∂r ∂r

(13)

∫0

b a

∫0

a

a

dr ′ r′

∫0

r′

dr ″ r ″

⎫ dr″r″ cosh[Ψ(r″)]⎬ ⎭

drr {μ + exp[−Ψ(r )] + μ− exp[Ψ(r )]}

∫0

a

⎛ σ ⎞2 ⎫ ⎡ ∂ψ (r ) ⎤2 ⎪ + drr ⎢ ab ⎜ ⎟⎬ ⎥ ⎪ ⎣ ∂r ⎦ ⎝ ϵϵ0 ⎠ ⎭

where

(23)

ze Ψ(x , r ) ≡ ψ (x , r ) kBT

k12 = k 21 =

(14)

πc η

and ⎛ 2z 2e 2c(x) ⎞1/2 1 =⎜ κ (x ) = ⎟ λD(x) ⎝ ϵϵ0kBT ⎠

k13 = k 31 = −

a − r + 2ab ⎛ dp0 ⎞ 2c ⎧ ⎜− ⎟+ ⎨ 4η η⎩ ⎝ dx ⎠ 2

2

∫r

cosh Ψ + 2zec ⎧ ⎨ − η ⎩

b a a

∫0

a

dr ′ r′ ⎫⎛ dϕ ̅ ⎞ sinh Ψ⎬⎜ − ⎟ ⎭⎝ dx ⎠

∫r

Js = 2π

∫0 ∫0

I = 2πze

a

dr ′ r′

∫0

= 2πzec

r′

dr ″ r ″

−

r′

a

drr[ψ (a) − ψ (r )] + a 2b

dr″r″ sinh Ψ +

b a

∫0

σ ⎫ ⎬ ϵϵ0 ⎭

∫0

a

4πϵϵ0c η

drr {μ + exp[−Ψ(r )] − μ− exp[Ψ(r )]}

∫0

a

⎡ σ ⎤ drr cosh[Ψ(r )]⎢ψ (a) − ψ (r ) + b ⎥ ϵϵ0 ⎦ ⎣

These local transport coefficients kij depend only on the axial coordinate x. (NB: the dependence of kij, ψ, Ψ, and c on x is implicit in the above equations.) Calculating kij requires solving the coupled 1D ordinary differential equations in eq 20 for p0(x), c(x), ϕ̅ (x), and ψ(x, r), subject to the boundary conditions at the pore ends. By assuming local thermodynamic equilibrium between the solution just inside the pore and the eletroneutral bulk solution just outside the pore, the “virtual” quantities p0(x), c(x), and ϕ̅ (x) at the pore ends can be shown to be equal to the corresponding measurable quantities (solvent partial pressure, salt concentration, and electrical potential) in the electroneutral bulk solution just outside the pore.32 By requiring that the fluxes across the channel, Q, Js, and I, in eq 20 be constant, p0(x), c(x), and ϕ̅ (x) (and hence ψ(x, r) and kij) can be obtained by solving the boundary-value problem for these variables with Q, Js, and I as parameters.32 Limiting Case: Ion-Selective Pore. Closed-form expressions can be derived for kij in several limiting cases. For a perfectly ion-selective pore, in which only ions of one type can be found

a

dr ″ r ″

(16)

drrux

(17)

drr(j+ + j− )

(18)

a

a

drr(j+ − j− )

∫0

(26)

a

∫0

πϵϵ0 ⎧ ⎨2 η ⎩

(24)

k 23 = k 32

The flow rate Q, solute flux Js, and electrical current I can be calculated by integrating the fluid velocity in eq 16 and solute flux density in eq 2 over the pore cross section using Q = 2π

drr(a2 − r 2 + 2ab) cosh[Ψ(r )]

(25)

⎫⎛ d ln c ⎞⎟ dr″r″ cosh Ψ⎬⎜ −kBT ⎝ ⎭ dx ⎠

∫0

∫0

a

(15)

is the inverse of the local Debye screening length, λD(x). Integrating eq 11 subject to the slip boundary condition and eq 8 for the ion concentrations gives the axial fluid velocity as ux =

(22)

a

⎧ 2π (ϵϵ0)2 ⎪ ⎨ ⎪ η ⎩

+

(21)

a

8πc 2 η

+

and the Poisson equation reduces to −ϵϵ0

πa 4 ⎛ 4b ⎞ ⎜1 + ⎟ 8η ⎝ a ⎠

k 22 = 2πc

dϕ ̅ ze (c+ − c −) η dx

(20)

where

kT 1 ∂ ⎛ ∂ux ⎞ 1 dp0 d ln c ⎜r ⎟= + B (c+ + c −) ⎝ ⎠ r ∂r ∂r dx η dx η +

⎤ ⎡ dp ⎥ ⎢− 0 ⎥ k13 ⎤⎢ dx ⎥⎢ d ln c ⎥ k 23 ⎥⎢−kBT ⎥ dx ⎥ ⎥⎢ k 33 ⎦⎢ ⎥ ⎥ ⎢ − dϕ ̅ ⎦ ⎣ dx

(19)

which gives 3423


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Langmuir in the pore, an analytical solution for the dimensionless potential Ψ exists, from which the following closed-form analytical expressions can be derived for the local transport coefficients (see the Supporting Information) k 22 =

2 ⎛ a ⎞ 8π ⎛ ϵϵ0kBT ⎞ 1 ⎡ ⎢ ⎜ ⎟ + Γ (1 ) ⎟ ⎜ η ⎝ ze ⎠ (ze)2 ⎢⎣ ⎝ SGC ⎠

⎛ a ⎞ ⎛ a ⎞⎛ b ⎞⎤ − 2 ln⎜1 + ⎟+⎜ ⎟⎜ ⎟⎥ 2SGC ⎠ ⎝ SGC ⎠⎝ SGC ⎠⎥⎦ ⎝

(27) Figure 3. Equivalent circuit for a pore with electrical resistance Rp and access electrical resistances RaI and RaII connected to an external load RL. I is the output current, Ic is the conduction current driven by the potential difference ϕII − ϕI, and Id is the “osmotic” current driven by the salt concentration difference between the two sides of the pore. ϕH and ϕL are the electrical potentials in the reservoirs far from the pore.

k12 = k 21 =

2SGC ⎞ ⎛ 2πa 2 ϵϵ0kBT ⎡⎛ a ⎞ ⎢ ⎜ ⎟ + + 1 ln 1 ⎜ ⎟−1 η (ze)2 ⎢⎣⎝ a ⎠ ⎝ 2SGC ⎠ ⎛ b ⎞⎤ +⎜ ⎟⎥ ⎝ SGC ⎠⎥⎦

k 33 = (ze)2 k 22 ,

⎡ k ̅ k ̅ k ̅ ⎤⎡−Δp ⎤ ⎡Q ⎤ 0 ⎢ 11 12 13 ⎥⎢ ⎥ ⎢ ⎥ 1⎢ k 21 ̅ k 22 ̅ k 23 ̅ ⎥⎢−kBT Δln c ⎥ ⎢ Js ⎥ = ⎥⎢ L⎢ ⎥ ⎢ ⎥ ⎣I ⎦ ⎢⎣ k 31 ⎥⎦ ̅ k 32 ̅ k 33 ̅ ⎥⎦⎢⎣−Δϕ

(28)

k13 = k 31 = zek12 ,

k 23 = k 32 = zek 22

(32)

(29)

where kij̅ are global transport coefficients related by kij̅ = kji̅ in

where

SGC

2ϵϵ0kBT = ze|σ |

the linear response regime according to the Onsager reciprocal relations.32,54 These global transport coefficients are in general different from the local transport coefficients kij derived above. However, kij̅ = kij in the case of a perfectly ion-selective pore,

(30)

is the Gouy−Chapman length, which is the distance at which the thermal energy balances the ion−surface interaction energy, and Γ=

as discussed above. For reverse electrodialysis, the current is driven by a salt concentration difference between the reservoirs (Δln c ≠ 0), while the pressure is the same in the two reservoirs (Δp = 0). In this case, the difference in solvent partial pressure between the reservoirs is Δp0 = −2kBTΔc for a symmetric z:z electrolyte from eqs 9 and 10. Thus, the output electrical current I is

η(ze)2 μ+ 2ϵϵ0kBT

(31)

is a measure of the ratio of the counterion mobilities due to electromigration and electroosmosis in the absence of slip.53 Note that these local transport coefficients kij are independent of x for all i and j and are therefore global properties of the pore. Thus, the fluid and ion fluxes can be obtained trivially by integrating eq 20 across the length of the pore. Equations are also given in the Supporting Information for the cases of thin electric double layers (a ≫ λD) and for the Debye−Hückel approximation (|zeψ(x, r)| ≪ kBT). Salinity-Gradient Power Output. Assuming that the system responds linearly to changes in the applied potential, which will be shown later to be accurate for the range of parameters used in the numerical model studied, the electrical characteristics of the reverse electrodialysis system in Figure 1 can be described by the equivalent circuit shown in Figure 3, in which the variable external load resistor RL is assumed to incorporate the electrical resistance of the electrodes. In addition to the electrical resistance of the pore, Rp, this model accounts for the access electrical resistances of the pore where it connects to reservoirs I and II, RaI and RaII.27 Assuming a linear response of fluxes across the pore to the applied fields, the flow rate Q, solute flux Js, and electrical current I are related to the difference between the solvent partial pressure p0, salt concentration c, and electric potential ϕ on either end of the pore by32

I=

k 31 k̅ k̅ ̅ (2kBT Δc) + 32 ( −kBT Δln c) + 33 ( −Δϕ) L L L (33)

where Δϕ = ϕII − ϕI is the potential drop across the pore. The electrical resistance of the pore, Rp, is related to the transport coefficients by Rp =

L k 33 ̅

(34)

From the equivalent circuit in Figure 3, the potential drop Δϕ̃ = ϕL − ϕH across the external load RL is Δϕ ̃ =

⎡ k 31 ⎤ k̅ ̅ θ ⎢ (2kBT Δc) + 32 (−kBT Δln c)⎥ 1 + β + θ ⎣ k 33 k 33 ̅ ̅ ⎦ (35)

where θ ≡ RL/Rp is the ratio of the load resistance to the pore resistance and β = Ra/Rp is the ratio of the total access resistance, Ra ≡ RaI + RaII, to the pore resistance. The output power Pout is 3424


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Langmuir Pout = I Δϕ ̃ =

=

(Δϕ)̃ 2 RL

The maximum power output Pmax for reverse electrodialysis in eq 38 is completely specified by the global intrapore transport coefficients kij̅ , which determine the open-circuit voltage Δϕoc and pore electrical resistance Rp, and the total access electrical resistance Ra = RaI + RaII. Thus, at least in the case of a perfectly ion-selective pore, the closed-form expressions for kij̅ = kij in eqs 27−29 and for Ra in eqs 41 and 42 define a simple closed-form analytical theory for Pmax.

(36)

θ R p(1 + β + θ )2 ⎡ k 31 ⎤2 k 32 ̅ ̅ ⎢ (2kBT Δc) + (−kBT Δln c)⎥ k 33 ̅ ̅ ⎣ k 33 ⎦

(37)

■

The maximum power output occurs when θ = 1 + β, i.e., RL = Rp + Ra, and is Pmax =

(Δϕoc)2 4(R p + R a)

=

RESULTS AND DISCUSSION As described in Table 1, unless otherwise indicated, all calculations used a pore radius of a = 5 nm, pore length of L = 200 nm, reservoir length/radius of w = 4 μm, low-salt reservoir concentration of cL = 0.1 mmol L−1, high-salt reservoir concentration of cH = 10cL, and pore surface charge density of σ = −10 mC m−2. This choice of parameters corresponds to an almost completely cation-selective pore, since for the high-salt reservoir concentration both the Debye length (λHD ≈ 10 nm), which measures the electric double layer width at the pore surface, and the Dukhin length59 (S H Du ≈ 40 nm), which characterizes the ratio of surface to bulk conductivity in the pore in the absence of slip and thus the predominance of counterions over co-ions, are significantly larger than the pore radius. The Debye length was computed using eq 15, while the Dukhin length was calculated using the approximate equation, SDu = |σ|γ/(2zecs), where γ = −SGC/λD + (SGC/λD)2 + 1 and cs is the reservoir salt concentration, which has been shown to be very accurate for typical experimental surface charge densities.59 The effect of relaxing the condition of ion selectivity is investigated later. The maximum power output in the 2D finite-element calculations was computed from current−voltage curves obtained from a series of simulations in which the electrical potential ϕH in the high-salt reservoir was fixed at 0 mV while the electrical potential ϕL in the low-salt reservoir was varied in increments of 10 mV between 0 and 70 mV. The applied voltage was thus Δϕ̃ = ϕL − ϕH (the theoretical maximum kT open-circuit voltage is zeB ln(c H/c L) ≈ 59 mV for cH = 10cL). The electrical current I was computed from the integral of the ionic charge flux over the circular cross section of the pore at its midpoint using eq 19 (and was verified to be independent of the axial position x along the pore at which the integral was computed). Typical current−voltage curves from the finiteelement calculations for an uncharged outer membrane surface are shown in Figure 4. Slight deviations from linear (Ohmic) behavior are evident in the current−voltage curves, most likely due to external concentration polarization at high ion fluxes,27,60,61 which is most pronounced at high slip lengths. Although nonlinear, the curves are almost perfectly quadratic, and so the maximum power output Pmax (maximum value of Pout(Δϕ̃ ) = IΔϕ̃ ) and open-circuit voltage Δϕoc (intercept with the I = 0 axis) were determined from the quadratic fit. Nevertheless, because deviations from linearity were small, discrepancies between Pmax computed from a linear or a quadratic fit were less than 12% in all cases. For calculations in which the outer membrane surface was charged (σm = σ), the current−voltage curves were essentially perfectly linear, possibly because of a reduced concentration polarization effect due to enhanced ion concentrations near the charged outer membrane surface.

IscΔϕoc 4

(38)

where Δϕoc =

k 31 k̅ ̅ (2kBT Δc) + 32 (−kBT Δln c) k 33 k 33 ̅ ̅

(39)

and Isc =

⎡ k 31 ⎤ k̅ ̅ 1 ⎢ (2kBT Δc) + 32 (−kBT Δln c)⎥ R p + R a ⎣ k 33 k 33 ̅ ̅ ⎦

(40)

are the open-circuit voltage and short-circuit current, respectively. The current and potential difference at maximum power are I = Isc/2 and Δϕ̃ = Δϕoc/2, respectively. The power-conversion efficiency of reverse electrodialysis is discussed in detail in the Supporting Information. Although a simple expression for the efficiency cannot be derived in the general case, if the dissipated power due to fluid flux, |Q(2kBTΔc)|, is small compared with that due to salt flux, |Js(kBTΔln c)|, which is accurate for most of the systems studied here, for an ion-selective pore the maximum efficiency can be shown to be 100% (but for zero power output), while the efficiency at maximum power is always 50% and is thus independent of any pore parameters. Access Resistance. Electrical resistance as an electrolyte enters or exits a nanoporous membrane arises mainly due to focusing of electric-field lines and concentration polarization at the pore ends.27 An analytical expression that has been widely applied to modeling ion transport in nanopores and membranes27,55−57 has previously been derived58 to account for the effect of electric-field focusing. The access resistance is taken to be the resistance between a conducting disk of radius equal to the pore radius a embedded in an insulator and a hemispherical conductor far away, giving the access resistance at a single aperture as58 ρj R aj = (41) 4a where ρj is the resistivity of the uniform medium between the two conductors. The resistivity is generally taken to be the resistivity of the bulk electrolyte outside the pore outside pore end j, which for a z:z electrolyte with a uniform concentration cj can be computed from eq 2 to be ρj =

kBT 2

2

cje (z+ D+ + z −2D−)

assuming the Einstein relation, μi =

(42) Di , kBT

for the ion mobility

and diffusivity. 3425


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Figure 4. Typical current−voltage curves from numerical calculations for various pore slip lengths b for an uncharged outer membrane surface (σm = 0). The dashed and solid lines are linear and quadratic fits, respectively, indicating slight deviations from linearity but almost perfectly quadratic behavior.

Thus, no difference was found for Pmax computed from a linear or quadratic fit in this case. Reservoir Size Dependence. Since computational expense precludes numerical simulation of high- and low-salt reservoirs of the size used experimentally, the minimum reservoir size needed to obtain converged power outputs was determined. As shown in Figure S1 of the Supporting Information, the maximum power output converged to a constant value for a reservoir width/radius of w ≳ 4 μm for a system with an uncharged outer membrane surface (similar convergence was observed for a charged outer membrane surface). Thus, w = 4 μm was used in all subsequent calculations. Access Resistance. The total access electrical resistance, Ra = RaI + RaII, in the 2D finite-element calculations was estimated as a function of reservoir salt concentration (with cH = 10cL) and pore surface charge density σ, for a single system geometry (pore radius a = 5 nm and pore length L = 200 nm) to be the total electrical resistance, Rt = Rp + Ra, in the limit of large pore slip lengths, b → ∞ (i.e., Rp → 0). Due to the nonlinearity of the current−voltage curves, R t could not be defined unambiguously. We defined Rt in terms of the maximum power output Pmax and open-circuit voltage Δϕoc in eq 38 obtained from a quadratic fit to the current−voltage curve; the discrepancy between the resistance defined in this way and that obtained from the slope of a linear fit to the current−voltage curve was ≲12%. The total electrical resistance Rt is plotted versus the pore slip length b for uncharged and charged outer membrane surfaces in Figures S2 and S3, respectively, in the Supporting Information. The total access electrical resistance Ra obtained from the high-slip limit of Rt is shown in Figure 5 and Figure S4 in the Supporting Information, respectively, for uncharged and charged outer membrane surfaces, where it is compared with the predictions of the analytical theory given by eq 41 with the electrolyte concentration cj in eq 42 taken to be the bulk salt concentration, cL or cH, in the respective reservoirs. The access resistance from the analytical theory is within an order of magnitude of the numerical values for all conditions studied in Figure 5 for an uncharged outer membrane surface, with the relative error largest at high salt concentrations and very low surface charge densities. For a charged outer membrane surface,

Figure 5. Total access resistance Ra for an uncharged outer membrane surface (σm = 0) as a function of (a) bulk salt concentration cL in the low salt reservoir (with σ = −10 mC m−2) and (b) pore surface charge density σ (with cL = 0.1 mmol L−1) from numerical simulations (symbols) and theory (lines). The insets show the ratio of the numerical to the theoretical resistance.

the discrepancies are larger, with the analytical theory overestimating the access resistance by a factor of 10 at the lowest salt concentration and by a factor of 40 at the highest surface charge density studied, as shown in Figure S4. The shortcomings of the simple analytical theory for the access resistance given by eqs 41 and 42 have been discussed previously,55,57 and the results presented here are quantitatively consistent with these previous findings. The theory assumes (1) a homogeneous conducting medium with properties of the bulk reservoir and (2) a pore entrance that is an equipotential surface (i.e., a conductor). Thus, discrepancies are expected to increase with increasing surface charge density (with greater discrepancies if both the pore and outer membrane surface are charged), as previously verified numerically,55 due to the breakdown of both assumptions. Specifically, increasing surface charge density increases the local ion concentration near the pore entrance relative to the bulk concentration, decreasing the access resistance compared with the analytical predictions. This effect should be most pronounced at high salt concentrations, for which the relative change in the local ion concentration will be greatest, which is consistent with Figures 5 and S4. Furthermore, variations in the electrostatic potential at the pore entrance are expected to increase with surface charge density. These variations should also increase with salt concentration due to the decreasing Debye length.57 The discrepancy found at very low surface-charge densities in Figures 5 and S4 may be 3426


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model, which decouples the effects of pore and access resistance and computes intrapore transport coefficients analytically, quantitatively describes reverse-electrodialysis power conversion. Good agreement is also found for a charged outer membrane surface, as shown in Figures S5−S8 of the Supporting Information. The theory gives insight into the scaling of the output power density with the various parameters studied. For the relatively low reservoir salt concentrations used, according to eq 39 , the open-circuit voltage for an ion-selective pore is approximately

due to external concentration polarization in the numerical simulations (as suggested previously27) at the high currents in the high-slip limit used to calculate the access resistance, particularly due to the low salt concentrations used in these calculations. Progress has been made toward understanding the scaling of the access resistance with surface charge59 but in the regime of small double-layer overlap that is not relevant to most of our calculations. Despite these discrepancies between the analytical theory and numerical calculations, for typical salt concentrations (∼1 mmol L−1 for the low-salt (e.g., freshwater) reservoir, with the resistance of the high-salt (e.g., seawater) reservoir negligible) and surface charge densities of interest (∼10 mC m−2), the analytical theory is very close to the numerical results in Figures 5 and S4. Thus, the analytical theory should be sufficient for semiquantitative predictions of salinity-gradient power conversion under typical experimental conditions, which is indeed demonstrated in Figure S12 in the Supporting Information. Although a fully quantitative analytical theory of the access resistance would be desirable, to demonstrate the accuracy of the intrapore transport coefficients and the 1D theory for salinity-gradient power conversion derived here, in the powerconversion results presented below, we have used the numerical values for the access resistance given in Figures 5 and S4, assuming (as given in eq 41) that the access resistance scales with a−1 to extrapolate the numerical values obtained for a single pore radius a to other radii. Power Conversion. Ion-Selective Pores. We have compared the maximum salinity-gradient power output Pmax predicted by the 1D transport theory that we have derived with the 2D finite-element simulations under conditions for which the membrane pore is approximately ion-selective. As mentioned above, a closed-form expression for Pmax exists in this case, given by eq 38 with the open-circuit voltage Δϕoc (eq 39) and pore resistance Rp related to the pore transport coefficients in eqs 27−29. The maximum output power density (maximum power output per unit pore cross-sectional area), Pmax/(πa2), from the theory and simulations is compared in Figures 6−9 as a function of the pore slip length b and as a function of the pore length L, pore radius a, and surface charge density σ for various slip lengths for an uncharged outer membrane surface. The theory and numerical simulations agree very well, indicating that the approximate one-dimensional

Δϕoc ≈

kBT ze

cH cL

( ) and thus independent of pore properties.

ln

The analytical theory for the access resistance (eqs 41 and 42) suggests that the only pore property on which the access resistance depends is the pore radius a, as Ra ∼ a−1, but the numerical simulations (Figures 5 and S4) indicate that it also depends on the surface charge density as Ra ∼ 1/(aσ0.56) and Ra ∼ 1/(aσ4/3) for uncharged and charged outer membrane surfaces, respectively, under the conditions studied. However, Ra is independent of the slip length b. From eqs 27 and 29, the L pore resistance scales approximately as R p ∼ aσ for small slip L

lengths and as R p ∼ abσ 2 for large slip lengths b. Thus, Pmax, which depends on the inverse of Rp in eq 38, increases with slip length b, since Rp decreases with b−1 at large slip lengths, and converges to a limit as b → ∞ determined by the approximately constant open-circuit voltage and access resistance when Rp becomes negligible (Figure 6). At least for the range of physically relevant parameters studied in this work, Pmax saturates at a slip length b on the order of 100 nm, suggesting that little is gained in reverse electrodialysis using more highly slipping surfaces. On the other hand, Pmax decreases as a hyperbolic function of the pore length L, since Rp is proportional to L, but the rate of decline is diminished by increasing slip due to the proportionality of Rp with b−1 (Figure 7). Pmax is proportional to the pore radius a (and thus the maximum power density Pmax/(πa2) decreases as a−1) because both Rp and Ra are proportional to a−1 (Figure 8). Deviations from this relationship are evident at large radii, where the assumption L ≫ a becomes inaccurate and the assumption of ion selectivity of the pore breaks down as the pore radius becomes larger than the Debye length λHD ≈ 10 nm and

Figure 6. Maximum output power density Pmax/(πa2) versus pore slip length b from numerical simulations (points) and theory for an ionselective pore (lines) for an uncharged outer membrane surface (σm = 0). The inset shows the same data on a logarithmic horizontal scale.

Figure 7. Maximum output power density Pmax/(πa2) versus pore length L from numerical simulations (points) and theory for an ionselective pore (lines) for various pore slip lengths b for an uncharged outer membrane surface (σm = 0). 3427


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high-salt reservoir concentration, S H Du ≈ 0.7 nm, is much smaller than the pore radius). As a result, the pore resistance computed from the ion-selective theory deviates at low charge densities from the value computed from the 2D finite-element simulations as the difference between the total resistance Rt and access resistance Ra (Figure S9). This discrepancy is masked in the plotted maximum power density in Figure 9 because the access resistance is much larger than the pore resistance under these low charge-density conditions. Deviations between the theory and numerical simulations for Pmax also appear for short pores when the outer membrane surface is charged (Figure S6), probably due to the breakdown of the assumption of high-aspect-ratio pores (L ≫ a) used to derive the theory. This discrepancy is particularly evident when the outer membrane surface is charged due to the negligible access electrical resistance. It should be noted that the theory for ion-selective pores is expected to be accurate even in the absence of electric double layer overlap (λD > a), provided that the number of counterions in the pore greatly exceeds the number of co-ions, which is the case in the “surface-charge-governed” regime where the Dukhin length is much larger than the pore radius (SDu ≫ a). Although the separation between the Debye and Dukhin lengths in most of our calculations is not significantly large to test this hypothesis, we have verified its accuracy by comparing the local transport coefficients kij computed using the general 1D intrapore transport model (eqs 22−26) and that for an ionselective pore (eqs 27−29) under conditions in which there is no double layer overlap (salt concentration c = 1 mmol L−1 and pore radius a = 100 nm, for which λD ≈ 10 nm ≪ a) as a function of increasing surface charge density σ (and thus increasing Dukhin length SDu ). We find that the transport coefficients k13 = k31, k23 = k32, and k33 that determine the maximum power output Pmax converge to within 10% of the ion-selective values for σ ≳ 100mC m−2 (SDu ≳ 500 nm), i.e., at Dukhin lengths significantly larger than the pore radius, as shown in Figure S13 in the Supporting Information. Non-Ion-Selective Pores. Although closed-form expressions for the global intrapore transport coefficients kij̅ do not exist in the case of non-ion-selective pores, the open-circuit voltage Δϕoc and pore electrical resistance Rp can be computed numerically by solving the general 1D intrapore transport model for the fluxes in eq 20 using the local transport coefficients kij defined by eqs 21−26 for several different applied voltages and using the same procedure described earlier to fit the current−voltage curves from the full 2D finite-element calculations. In this case, the pore resistance is the total resistance (Rp = Rt), since the 1D model neglects pore-end effects. The maximum power output can then be obtained from eq 38, given the value of the access electrical resistance Ra. While this procedure requires numerical calculations, it is much more efficient than solving the full 2D numerical model. Figure 10 compares the maximum output power density Pmax from the full 2D numerical simulations and from the simplified 1D transport model for various slip lengths for an uncharged outer membrane surface. The agreement is very good, and demonstrates once again that transport within the pore and at the pore ends can effectively be treated separately, significantly simplifying the analysis of salinity-gradient power conversion. To emphasize this point, the pore resistance Rp, computed from the 2D finite-element calculations by subtracting the access resistance Ra from the total resistance Rt, can be seen in Figures

Figure 8. Maximum output power density Pmax/(πa2) versus pore radius a from numerical simulations (points) and theory for an ionselective pore (lines) for various pore slip lengths b for an uncharged outer membrane surface (σm = 0).

approaches the Dukhin length lDuH ≈ 40 nm in the high-salt reservoir. Nevertheless, the theory and numerical simulations agree to within ∼50% even when the pore radius is twice the Debye length. The variation of Pmax with surface charge density σ is somewhat more complex, since Rp and Ra scale differently with σ but are of similar magnitude under the conditions in Figure 9,

Figure 9. Maximum output power density Pmax/(πa2) versus pore surface charge density σ from numerical simulations (points) and theory for an ion-selective pore (lines) for various pore slip lengths b for an uncharged outer membrane surface (σm = 0).

although the access resistance dominates. Since both Rp and Ra decrease with σ, Pmax increases monotonically with σ, scaling roughly with σ0.56 as determined by the access resistance. The effect of slip on Pmax increases with σ, due to the scaling of Rp with 1/(bσ2) at large slip lengths, except at the highest surface charge densities, where Rp is negligible compared with Ra. The interpretation of the scaling of Pmax with surface charge density is simpler for a charged outer membrane surface, shown in Figure S8: for small slip lengths, the pore resistance Rp dominates the access resistance Ra, so Pmax ∼ σ, whereas, at large slip lengths, the pore resistance is negligible compared with the access resistance, so Pmax ∼ σ4/3. Although not evident in Figure 9, the simulated pores deviate from ion selectivity at the smallest charge densities, since the surface charge density is not sufficient to induce a significant excess of counterions over co-ions (the Dukhin length for the 3428


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pore. Although a simple exact expression for this scaling cannot be derived, assuming a small concentration gradient (i.e., c approximately constant, so Δc/c ≈ Δln c) in the intrapore transport coefficients for thin electric double layers given in the Supporting Information, Δϕoc ≈

μσkBT Δln c a(ze)2 cμ + bσ 2/η

(43)

which decreases with slip length b. The open-circuit voltage is also independent of the access resistance, which is confirmed by the numerical results in Figures S10 and S11. The pore resistance Rp also decreases monotonically with reservoir salt concentration, as shown in Figure 11. The maximum output power is approximately independent of slip for low reservoir salt concentration because the total electrical resistance in this regime is dominated by access resistance, which is independent of slip. For intermediate salt concentrations, slip has a pronounced effect on power output as access resistance diminishes. At high salt concentrations, the effect of slip once again becomes small as bulk electrolyte transport, which does not depend on slip, dominates interfacial transport. Effect of Outer Membrane Charge. The effect of adding charge to the outer membrane surface on salinity-gradient power can be seen by comparing Figure 12, which shows the

Figure 10. Maximum output power density Pmax/(πa2) versus low salt reservoir concentration cL from numerical simulations (symbols) and theory (lines) for various pore slip lengths b for an uncharged outer membrane surface (σm = 0).

11 and Figure S9 to be roughly independent of the charge on the outer membrane surface, and thus independent of the

Figure 11. Pore resistance Rp versus low salt reservoir concentration cL from numerical simulations for uncharged (filled symbols) and charged (empty symbols) outer membranes and from theory (lines) for various pore slip lengths b.

Figure 12. Maximum output power density Pmax/(πa2) versus low salt reservoir concentration cL from numerical simulations (symbols) and theory (lines) for various pore slip lengths b for a charged outer membrane surface (σm = σ).

access resistance Ra (which differs by up to an order of magnitude for charged and uncharged surfaces in the systems studied). Figure 11 also shows that the pore resistance computed from the 1D transport model agrees well with the full 2D simulations. Figure 10 shows that Pmax varies nonmonotonically with reservoir salt concentration, which is due to the competing effects of the decreasing open-circuit voltage and decreasing pore and access resistance, which appear respectively in the numerator and denominator of eq 38 for Pmax, with reservoir salt concentration. The open-circuit voltage (plotted versus reservoir salt concentration in Figures S10 and S11 for uncharged and charged outer membrane surfaces, respectively) decreases monotonically with increasing reservoir salt concentration due to decreasing ion selectivity of the pore with decreasing Debye and Dukhin lengths, as discussed previously;34,62 in the limit of a completely nonselective membrane, in which one anion is transported across the membrane for each cation, the open-circuit voltage is zero. The open-circuit voltage also decreases with slip length for a non-ion-selective

maximum output power density Pmax for a charged outer membrane surface, with Figure 10 for an uncharged surface. Although Pmax is essentially the same in the two cases at high reservoir salt concentrations, it is significantly higher for the charged outer membrane surface at low salt concentrations, with the effect particularly substantial for large pore slip lengths. For the lowest salt concentrations, Pmax is boosted by a factor of 8 by adding charge to the outer membrane surface. This can be explained by the reduction in access resistance with added outer membrane surface charge, which can be seen by comparing Figures 5 and S4. The effect is greatest for low reservoir salt concentrations, where the access resistance is highest in the absence of outer membrane surface charge and dominates pore resistance, and at high pore slip lengths, where pore resistance is negligible. These results indicate that a charged outer membrane surface is highly beneficial for salinity-gradient power conversion. Given that the outer surface of a porous 3429


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on membrane properties, which could be exploited to design better membranes. Of particular note, we showed that the effect of slip on power conversion is amplified by surface charge and that adding charge to the outer membrane surface can significantly improve power conversion by reducing access resistance. We also showed that, for experimentally feasible system parameters, high-slip membranes could potentially be used to enhance power output compared with what is currently achieved with conventional membranes.

membrane is likely to consist of the same material as the pore, this characteristic is likely to occur automatically in real systems but can also be achieved by chemically functionalizing the pore mouth with charged chemical groups. Figure 12 also shows that the simple 1D transport model, while giving slightly poorer agreement than for the case of the uncharged outer membrane surface, agrees well with the 2D numerical simulations. Our results are consistent with previous numerical calculations of reverse-electrodialysis power conversion with hydrodynamic slip in a rectangular slit channel.30 These calculations used high and low salt reservoir concentrations 5 times higher than those employed here but comparable pore diameters and lengths for most calculations, resulting in only partial ion selectivity. This work did not consider the effect of reservoir salt concentration nor did it treat charged outer membrane surfaces, for which access resistance is substantially reduced, and thus only found a maximum slip-induced enhancement of salinity-gradient power output of 44%, considerably less than that found here. Experimental Implications. Having verified the accuracy of our theoretical model by comparison with numerical simulation, we can examine the model’s predictions of achievable salinity-gradient power for experimentally feasible systems. With modern nanofabrication techniques, macroscopic membranes can now be produced with highly ordered onedimensional nanoscopic pores, using materials such as carbon nanotubes,63,64 mesoporous carbon,65 and anodic aluminum oxide.66 Many such surfaces can be chemically modified to change properties such as surface wettability67,68 that govern interfacial hydrodynamic slip.15 For water in carbon nanotubes, slip lengths of hundreds of nanometers and above have been measured, although there is some controversy with some of these measurements.69 Assuming an electrolyte consisting exclusively of sodium chloride (which makes up 86% of all ions in seawater 70 ) and low- and high-salt reservoir concentrations, cL and cH , of 5 and 500 mmol L−1 , corresponding roughly to the salinity of river and seawater,1 respectively, and using experimentally feasible values63,65,66 for the pore radius a of 5 nm, pore length L of 1 μm, pore fractional area in the membrane of 1%, and pore and outer membrane surface charge densities σ = σm of −10 mC m−2, we predict a maximum output power density Pmax that increases by almost a factor of 3 from 6 to 17 W m−2 as the slip length b of the pore increases from 0 to 100 nm. The latter value is higher than that currently achieved (up to 2.4 W m−2) in reverse electrodialysis with standard polymer membranes.2,71 Thus, using high-slip surfaces could have a significant impact on power conversion.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b00433. Derivation of equations for local membrane transport coefficients and power-conversion efficiency and additional results for the total electrical resistance, access electrical resistance, maximum power density, opencircuit voltage, pore resistance, and transport coefficients (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +61 (0)8 8313 5580. Fax: +61 (0)8 8313 4380. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank Lydéric Bocquet for fruitful discussions. This research was undertaken with the assistance of resources from the National Computational Infrastructure (NCI), which is supported by the Australian Government.

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REFERENCES

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CONCLUSIONS In summary, we have shown that interfacial hydrodynamic slip can have a significant effect on membrane-based salinitygradient-driven power conversion using reverse electrodialysis. We have demonstrated that the power conversion rate of the process, and its dependence on parameters such as the slip length, pore radius, pore length, surface charge density, and reservoir salt concentrations, can be quantitatively predicted using a simple one-dimensional transport model and that the effects of pore and access electrical resistance can be decoupled in predicting power conversion. In the case of a perfectly ionselective pore, we derived simple closed-form expressions for the intrapore transport coefficients, which we used to obtain analytical scaling laws for the dependence of power conversion 3430


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