The Effect of Hydrodynamic Slip on Membrane-Based Salinity

Subscriber access provided by Mount Allison University | Libraries and Archives

Article

The Effect of Hydrodynamic Slip on MembraneBased Salinity-Gradient-Driven Energy Harvesting Daniel Justin Rankin, and David Mark Huang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b00433 • Publication Date (Web): 18 Mar 2016 Downloaded from http://pubs.acs.org on March 25, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

The Eect of Hydrodynamic Slip on Membrane-Based Salinity-Gradient-Driven Energy Harvesting Daniel Justin Rankin and David Mark Huang

∗

Department of Chemistry, The University of Adelaide, SA 5005, Australia E-mail: [email protected]

Phone: +61 (0)8 8313 5580. Fax: +61 (0)8 8313 4380

Abstract The eect of hydrodynamic slip on salinity-gradient-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 uid dynamics model. A general one-dimensional model is derived that decouples transport inside the membrane pores from the eects of electrical resistance at the pore ends, from which an analytical expression for the power conversion rate is obtained for a perfectly ionselective 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 nite-element numerical calculations and predicts signicant enhancements up to several times of salinity-gradient power conversion due to hydrodynamic slip for realistic systems. ∗

To whom correspondence should be addressed 1

ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Page 2 of 40

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 12 TW of power globally, 1,2 or up to about 10% of global energy needs. Methods for harnessing so-called salinity-gradient power or osmotic power have been known for more than 40 years 13 and include processes such as pressure-retarded osmosis, 4 reverse electrodialysis, 5 and capacitive mixing. 68 Nevertheless, the rst prototype salinity-gradient power plant (in Norway), using pressure-retarded osmosis, was not opened until 2009 9 and it was only in 2014 that the rst 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 dierent scale, salinity-gradient power has been suggested as an eective means for powering nano-scale systems such as biomedical devices operating in saline uid environments. 12 The most widely used technologies for harnessing salinity-gradient power, pressure-retarded osmosis and reverse electrodialysis, are based on uid 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 gradient (Figure 1). Oxidation and reduction at electrodes on either side of the membrane maintain electroneutrality and produce an electron ow in an external circuit that can be used for electrical power. Ecient 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, ecient membranes must have nano-sized pores. 14 2


Page 3 of 40

electric current load

e-

–

+ – – + ––––––––– + – + + – + + + + – + ion + – + + + + – + current – + + – + + + ––––––––– – – + + membrane pore –

–

high conc.

cathode

e-

anode

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

low conc.

Figure 1: Schematic of direct electrical energy harvesting from a salinity gradient (reverse electrodialysis) using a charged membrane interposed between high and low saltconcentration reservoirs. Nano-scale electrolyte transport diers fundamentally from macroscopic transport, 15,16 but most previous theoretical predictions of salinity-gradient power conversion eciencies and rates have used macroscopic models. 14,17 However, the standard macroscopic theory of electrokinetics 18 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 solidliquid interface, where a non-slip boundary condition (equivalent to innite solidliquid friction) is assumed. Recent experiments 20 and molecular simulations 2123 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. 2023,25 Slip has major implications for electrokinetic power conversion. For electrical energy generation using pressure-driven electrolyte ow, a process closely related to salinity-gradient power, eciency gains of an order of magnitude are predicted when low-friction (high-slip) membranes 26,27 are used instead of non-slip surfaces. 13 Slip is also potentially important for salinity-gradient power, since ion transport due to uid ow, and not just ion diusion and electromigration, has been shown to play a signicant role in power conversion. For example, huge power densities ( ∼kW m−2 cf. W m−2 for conventional reverse-electrodialysis membranes 2 ) measured for reverse electrodialysis using

3


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

a membrane made of a single boron-nitride nanotube were attributed to the large diusioosmotic ow induced by the anomalously high apparent charge on the nanotube inner surface. 28 Diusioosmosis uid ow 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 benecial for salinity-gradient power conversion. Indeed, one computational study, 30 in which the coupled Poisson, NernstPlanck, and NavierStokes (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 eects of nano-scale electrolyte transport phenomena and in particular hydrodynamic slip on salinity-gradient power conversion has not been extensively investigated theoretically. Besides the aforementioned study, only a handful of works 3133 have accounted for the eect of uid ow on power conversion, but none of these considered slip. Osterle and co-workers 31,32 developed a continuum theory for the power conversion rate and eciency 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 eects. To the best of our knowledge, all other computational studies of salinity-gradient power conversion have ignored uid ow by solving only the coupled Poisson and NernstPlanck (PNP) equations without the NavierStokes equation, investigating numerically the eects on power conversion of parameters such as the membrane pore size, 14,34,35 surface charge, 34,36 pore length, 34 salinity gradient, 14,3436 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 oer a simple means to optimize power conversion. In this work,

4


Page 4 of 40

Page 5 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

we focus on deriving such relationships for the eect of interfacial hydrodynamic slip and other membrane properties on salinity-gradient power conversion, accounting for the eect of electrical resistance at the membrane pore ends, which can signicantly impact power conversion as the internal electrical resistance of the pore declines for high-slip surfaces. Improving understanding of the dynamics of nano-conned electrolytes has wider signicance, 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 nano-sized peptide channels. 38,39 The ecacy of polymer-electrolytemembrane fuel cells for next-generation energy storage and conversion 40 and membranes for chemical separations 41 and desalination 42 depends on the relative uxes of water and ions in nano-porous 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 uid 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 closed-form 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 intra-pore transport coecients and the access electrical resistance of the pore ends. We go on to compare the theory with nite-element numerical solutions to the full continuum model and nally make predictions about the limits of salinity-gradient power conversion for realistic systems. Continuum uid dynamics has previously been shown to successfully predict the experimental scaling with reservoir salt concentration and channel width of reverse-electrodialysis power output and eciency in non-slip silica nanochannels. 14 Furthermore, continuum models accounting for slip have been found to accurately describe experimental results for a subset of the transport

5


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Page 6 of 40

phenomena studied here, including pressure-driven uid ow 44 and electroosmosis. 20

Theoretical Methods 2D Numerical Model

To model the process of salinity-gradient-driven power conversion, continuum hydrodynamics for uid and ion transport was assumed for a symmetric z :z electrolyte across a cylindrical pore of length L and radius a connecting two large uid reservoirs (Figure 2). Continuum hydrodynamics has previously been shown to describe uid ow and electrolyte transport accurately for pore dimensions down to a few nanometers. 21,22,45,46 Assuming low ReynoldsB

C

F

reservoir I r x A

w

G

reservoir II

w

D E pore a L

H

Figure 2: Schematic of axisymmetric computational domain for nite-element calculations, comprising a pore of radius a and length L connecting two reservoirs of width and radius w. Solid lines denote solidliquid 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 the system is considered. number ow (generally accurate on the nano scale 47 ) and a dilute electrolyte, the governing equations for the electrical potential φ, ion ux density ji of species i (i = + or −), and uid velocity u are the Poisson, NernstPlanck, and Stokes equations, 48 respectively,

−0 ∇2 φ = ze (c+ − c− ) ,

(1)

∇ · ji = ∇ · (ci u − zi eµi ci ∇φ − Di ∇ci ) = 0 ,

(2)

η∇2 u = ∇p + ze (c+ − c− ) ∇φ ,

(3)

6


Page 7 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

together with the continuity equation for incompressible ow, 48 (4)

∇ · u = 0.

The quantity µi is the mobility of species i, Di is the diusion coecient (which is assumed to be related to the mobility by the Einstein relation, 48 µi =

Di ), kB T

zi is the valency of species

i (z+ = −z− = z for a z : z electrolyte), e is the elementary charge, η is the uid viscosity, 0 is the vacuum permittivity, is the dielectric constant of the uid, 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 uid pressure far from the pore was zero in both reservoirs. The transport equations were solved numerically using the nite-element method with COMSOL Multiphysics (version 4.3a), 49 as described in detail in the Supporting Information. Model parameters are dened and their values given in Table 1. This treatment makes a few simplifying assumptions, namely that (1) the uid 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 eects 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 eects on the transport phenomena studied here have been observed in molecular dynamics simulations of conned electrolytes at high surface charge densities and salt concentrations (higher

7


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Table 1: Calculation parameters: where a range of values is given, the parameter was xed at the value in parentheses unless otherwise indicated. Parameter Symbol Value Dielectric constant 78.46 Temperature T 298 K Fluid viscosity η 0.894 mPa s Cation valence z+ 1 Anion valence z− −1 Cation diusivity D+ 1.96 × 10−9 m2 s−1 Anion diusivity D− 2.03 × 10−9 m2 s−1 Pore radius a 125 nm (5 nm) Pore length L 201000 nm (200 nm) Reservoir radius/width w 0.58 µm (4 µm) a Slip length (pore ) b 0500 nm Slip length (membrane b ) bm 0 a −2 Surface charge density (pore ) σ −(130) mC m (−10 mC m−2 ) b Surface charge density (membrane ) σm 0 or σ Low-salt reservoir conc. cL 0.130 mmol L−1 (0.1 mmol L−1 ) High-salt reservoir conc. cH 10cL a Refers to pore surface; b Refers to membrane outer surface. 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 semi-quantitative understanding of the process that can be rened in future by relaxing the aforementioned assumptions.

1D Intra-Pore Transport Model

Assuming the pore is much longer than it is wide ( L a), the equations for uid and ion transport inside the pore can be simplied considerably. The derivation below of analytical approximations to the transport equations generalizes that by Fair and Osterle 32 for reverse electrodialysis in a pore with zero slip to the case of a non-zero 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

8


Page 8 of 40

Page 9 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

x and the radial coordinate r, ¯ φ(x, r) = φ(x) + ψ(x, r) ,

(5)

the boundary condition for the electrical potential at the pore surface is 18

σ = 0

∂ψ ∂r

(6)

r=a

and the slip boundary condition for the uid velocity at the pore surface is 24

(ux )r=a = −b

∂ux ∂r

(7)

. r=a

If the pore length is much larger than its radius ( L a), the radial solute ow in the NernstPlanck equations will be small ( ji,r ≈ 0), giving 32

zi eψ ci = c(x) exp − kB T

(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-section should be reasonable provided that

L a. 52 Similarly, assuming no radial uid ow ( ur ≈ 0) in the Stokes equation gives 32

p = p0 (x) + π ,

(9)

π = kB T (c+ + c− )

(10)

where

9


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Page 10 of 40

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

1 ∂ r ∂r

∂ux r ∂r

=

1 dp0 kB T d ln c ze dφ¯ + (c+ + c− ) + (c+ − c− ) η dx η dx η dx

(11)

and the Poisson equation reduces to

1 ∂ −0 r ∂r

∂ψ r = ze (c+ − c− ) , ∂r

(12)

from which eq 8 for the ion concentration gives

1 ∂ ∂ r Ψ(x, r) = [κ(x)]2 sinh [Ψ(x, r)] , r ∂r ∂r

(13)

where

Ψ(x, r) ≡

ze ψ(x, r) kB T

and

1 κ(x) = = λD (x)

2z 2 e2 c(x) 0 kB T

(14)

1/2

(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 uid velocity as

dp0 a2 − r2 + 2ab − ux = 4η dx (Z ) Z Z 0 a dr0 r 00 00 b a 00 00 2c d ln c + dr r cosh Ψ + dr r cosh Ψ −kB T η r0 0 a 0 dx r (Z ) Z Z 0 a 2zec dr0 r 00 00 b a 00 00 dφ¯ dr r sinh Ψ + − dr r sinh Ψ − . η r0 0 a 0 dx r

(16)

The ow rate Q, solute ux Js , and electrical current I can be calculated by integrating

10


Page 11 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

the uid velocity in eq 16 and solute ux density in eq 2 over the pore cross-section using

Z

a

Q = 2π

drrux ,

(17)

drr(j+ + j− ) ,

(18)

0

Z

a

Js = 2π 0

Z

a

(19)

drr(j+ − j− ) ,

I = 2πze 0

which gives









 Q   k11 k12 k13     J = k  s   21 k22 k23    I k31 k32 k33

0 − dp dx

    −k T d ln c B  dx  ¯ − ddxφ

   ,  

(20)

where

k11

πa4 = 8η

Z

4b 1+ a

(21)

,

a

drr {µ+ exp[−Ψ(r)] + µ− exp[Ψ(r)]} (Z ) Z Z 0 Z a dr0 r 00 00 b a 00 00 8πc2 a 00 00 drr cosh[Ψ(r)] dr r cosh[Ψ(r )] + dr r cosh[Ψ(r )] , + η r0 0 a 0 r 0

k22 = 2πc

0

(22)

k33 = 2π(ze)2 c

Z

a

drr {µ+ exp[−Ψ(r)] + µ− exp[Ψ(r)]} (Z 2 2 ) a 2π(0 )2 ∂ψ(r) σ + drr + ab , η ∂r 0 0 Z πc a drr a2 − r2 + 2ab cosh[Ψ(r)] , = η 0 Z a π0 2 σ = − 2 drr [ψ(a) − ψ(r)] + a b , η 0 0 Z a = 2πzec drr {µ+ exp[−Ψ(r)] − µ− exp[Ψ(r)]} 0 Z 4π0 c a σ − drr cosh[Ψ(r)] ψ(a) − ψ(r) + b . η 0 0 0

k12 = k21 k13 = k31 k23 = k32

(23) (24) (25)

(26)

These local transport coecients kij depend only on the axial coordinate x. (NB: the depen11


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Page 12 of 40

dence of kij , ψ , Ψ, and c on x is implicit in the above equations.) Calculating kij requires

¯ , and solving the coupled 1D ordinary dierential equations in eq 20 for p0 (x), c(x), φ(x) 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 φ(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 uxes across the channel, Q, Js , and I , in eq 20 be

¯ constant, p0 (x), c(x), and φ(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 ionselective pore, in which only ions of one type can be found 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 coecients (see Supporting Information):

k22

8π = η

0 kB T ze

2

1 a a a b (1 + Γ) − 2 ln 1 + + , (ze)2 `GC 2`GC `GC `GC (27)

2πa2 0 kB T 2`GC a b = (1 + ) ln 1 + −1+ , η (ze)2 a 2`GC `GC

k12 = k21

k33 = (ze)2 k22 , , k13 = k31 = zek12 , , k23 = k32 = zek22 ,

(28) (29)

where

`GC =

20 kB T ze|σ|

12


(30)

Page 13 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

is the GouyChapman length, which is the distance at which the thermal energy balances the ionsurface interaction energy, and

Γ=

η(ze)2 µ+ 20 kB T

(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 coecients kij are independent of x for all i and j and are therefore global properties of the pore. Thus, the uid and ion uxes 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 DebyeHückel approximation ( |zeψ(x, r)| kB T ).

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 reservoir I and II, RaI and RaII . 27 Assuming linear response of uxes across the pore to the applied elds, the ow rate

Q, solute ux Js , and electrical current I are related to the dierence between the solvent partial pressure p0 , salt concentration c, and electric potential φ on either end of pore by 32







¯ ¯ ¯  Q   k11 k12 k13     J  = 1  k¯  s  L  21 k¯22 k¯23    I k¯31 k¯32 k¯33

13





−∆p0     −k T ∆ ln c B   −∆φ

  ,  


(32)

Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Page 14 of 40

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 dierence φII − φI and Id is the "osmotic" current driven by the salt concentration dierence between the two sides of the pore. φH and φL are the electrical potentials in the reservoirs far from the pore. where k¯ij are global transport coecients related by k¯ij = k¯ji in the linear response regime according to the Onsager reciprocal relations. 32,54 These global transport coecients are in general dierent from the local transport coecients kij derived above. However, k¯ij = kij in the case of a perfectly ion-selective pore, as discussed above. For reverse electrodialysis, the current is driven by a salt concentration dierence between the reservoirs (∆ ln c 6= 0), while the pressure is the same in the two reservoirs ( ∆p = 0). In this case, the dierence in solvent partial pressure between the reservoirs is ∆p0 = −2kB T ∆c for a symmetric z : z electrolyte from eqs 9 and 10. Thus, the output electrical current I is

I=

k¯32 k¯33 k¯31 (2kB T ∆c) + (−kB T ∆ ln c) + (−∆φ) , L L L

(33)

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

L Rp = ¯ k33

(34)

From the equivalent circuit in Figure 3, the potential drop ∆φ˜ = φL − φH across the external

14


Page 15 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

load RL is

¯ ¯32 k θ k 31 (2kB T ∆c) + ¯ (−kB T ∆ ln c) , ∆φ˜ = 1 + β + θ k¯33 k33

(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

˜2 (∆φ) Pout = I∆φ˜ = RL θ = Rp (1 + β + θ)2

(36)

2 ¯ k¯32 k31 (2kB T ∆c) + ¯ (−kB T ∆ ln c) . k¯33 k33

(37)

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

Pmax =

Isc ∆φoc (∆φoc )2 = , 4 (Rp + Ra ) 4

(38)

where

k¯31 k¯32 ∆φoc = ¯ (2kB T ∆c) + ¯ (−kB T ∆ ln c) k33 k33

(39)

and

1 Isc = Rp + Ra

¯ k31 (2kB T ∆c) + k¯33

k¯32 (−kB T ∆ ln c) k¯33

(40)

are the open-circuit voltage and short-circuit current, respectively. The current and potential dierence at maximum power are I = Isc /2 and ∆φ˜ = ∆φoc /2, respectively. The power-conversion eciency of reverse electrodialysis is discussed in detail in the Supporting Information. Although a simple expression for the eciency cannot be derived in the general case, if the dissipated power due to uid ux, |Q (2kB T ∆c) |, is small compared with that due to salt ux, |Js (kB T ∆ ln c) |, which is accurate for most of the systems studied here, for an ion-selective pore the maximum eciency can be shown to be 100% (but for zero power output), while the eciency at maximum power is always 50% and is thus independent of any pore parameters. 15


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Page 16 of 40

Access Resistance

Electrical resistance as an electrolyte enters or exits a nanoporous membrane arises mainly due to focusing of electric-eld lines and concentration polarization at the pore ends. 27 An analytical expression that has been widely applied to modelling ion transport in nanopores and membranes 27,5557 has previously been derived 58 to account for the eect of electric-eld 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 as 58

Raj =

ρj , 4a

(41)

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, which for a

z : z electrolyte with a uniform concentration cj can be computed from eq 2 to be

ρj = assuming the Einstein relation, µi =

kB T 2 2 cj e (z+ D+ + Di , kB T

2 z− D− )

,

(42)

for the ion mobility and diusivity.

The maximum power output Pmax for reverse electrodialysis in eq 38 is completely specied by the global intra-pore transport coecients k¯ij , which determine the open-circuit voltage ∆φoc 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 k¯ij = kij in eqs 2729 and for Ra in eqs 41 and 42 dene a simple closed-form analytical theory for Pmax .

16


Page 17 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

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 (λH D ≈ 10 nm), which measures the electric double layer width at the pore surface, and the Dukhin length 59 (`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 counter-ions over co-ions, are signicantly larger than the pore radius. The Debye lengh was computed using eq 15, while the Dukhin length was calculated using the approximate equation in ref p 59, `Du = |σ|γ/(2zecs ), where γ = −`GC /λD + (`GC /λD )2 + 1 and cs is the reservoir salt concentration, which has been shown to be very accurate for typical experimental surface charge densities. The eect of relaxing the condition of ion selectivity is investigated later. The maximum power output in the 2D nite-element calculations was computed from currentvoltage curves obtained from a series of simulations in which the electrical potential

φH in the high-salt reservoir was xed 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 ∆φ˜ = φL − φH (the theoretical maximum open-circuit voltage is

kB T ze

ln(cH /cL ) ≈ 59 mV

for cH = 10cL ). The electrical current I was computed from the integral of the ionic charge ux over the circular cross-section of the pore at its midpoint using eq 19 (and was veried to be independent of the axial position x along the pore at which the integral was computed). Typical currentvoltage curves from the nite-element calculations for an uncharged outer membrane surface are shown in Figure 4. Slight deviations from linear (Ohmic) behavior are evident in the currentvoltage curves, mostly likely due to external concentration polarization at high ion uxes, 27,60,61 which is most pronounced at high slip lengths. Although nonlinear, the curves are almost perfectly 17


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Figure 4: Typical currentvoltage 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 ts, respectively, indicating slight deviations from linearity but almost perfectly quadratic behavior.

˜ = I∆φ˜) and quadratic, and so the maximum power output Pmax (maximum value of Pout (∆φ) open-circuit voltage ∆φoc (intercept with the I = 0 axis) were determined from the quadratic t. Nevertheless, because deviations from linearity were small, discrepancies between Pmax computed from a linear or a quadratic t were less than 12% in all cases. For calculations in which the outer membrane surface was charged ( σm = σ ), the currentvoltage curves were essentially perfectly linear, possibly because of a reduced concentration polarization eect due to enhanced ion concentrations near the charged outer membrane surface. Thus, no dierence was found for Pmax computed from a linear or quadratic t 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. 18


Page 18 of 40

Page 19 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

Access Resistance

The total access electrical resistance, Ra = RaI + RaII , in the 2D nite-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 currentvoltage curves,

Rt could not be dened unambiguously. We dened Rt in terms of the maximum power output Pmax and open-circuit voltage ∆φoc in eq 38 obtained from a quadratic t to the currentvoltage curve; the discrepancy between the resistance dened in this way and that obtained from the slope of a linear t to the currentvoltage 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 to 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, 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 ndings. The theory assumes (1) a homogeneous conducting medium with properties of the bulk reservoir and (2) a pore entrance that is an equipotential 19


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

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.

20


Page 20 of 40

Page 21 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

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 veried numerically, 55 due to the breakdown of both assumptions. Specically, 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 eect 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 due to external concentration polarization in the numerical simulations (as suggested previously 27 ) 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 towards understanding the scaling of the access resistance with surface charge, 59 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 sucient for semi-quantitative 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 intra-pore transport coecients and the 1D theory for salinity-gradient power conversion derived here, in the power-conversion results presented below we have used the numerical values for the access resistance given in Figures 5 and S4,

21


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

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 nite-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 ∆φoc (eq 39) and pore resistance Rp related to the pore transport coecients in eqs 2729. The maximum output power density (maximum power output per unit pore cross-sectional area), Pmax /(πa2 ), from the theory and simulations are compared in Figures 69 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 model, which decouples the eects of pore and access resistance and computes intra-pore transport coecients analytically, quantitatively describes reverse-electrodialysis power conversion. Good agreement is also found for a charged outer membrane surface, as shown in Figures S5S8 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 H kB T eq 39 the open-circuit voltage for an ion-selective pore is approximately ∆φoc ≈ ze ln ccL and thus independent of pore properties. 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/(aσ 0.56 ) and Ra ∼ 1/(aσ 4/3 ) for uncharged and charged outer membrane surfaces, respectively, under the conditions studied. 22


Page 22 of 40

Page 23 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

Figure 6: Maximum output power density Pmax /(πa2 ) versus pore slip length b from numerical simulations (points) and theory for an ion-selective 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 ion-selective pore (lines) for various pore slip lengths b for an uncharged outer membrane surface ( σm = 0).

23


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Figure 8: Maximum output power density Pmax /(πa2 ) versus pore radius a 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).

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).

24


Page 24 of 40

Page 25 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

But Ra is independent of the slip length b. From eqs 27 and 29, the pore resistance scales approximately as Rp ∼

L aσ

for small slip lengths and as Rp ∼

L 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 λH D ≈ 10 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 dierently with σ but are of similar magnitude under the conditions in Figure 9, although the access resistance dominates. Since both Rp and Ra decrease with σ ,

Pmax increases monotonically with σ , scaling roughly with σ 0.56 as determined by the access resistance. The eect of slip on Pmax increases with σ , due to the scaling of Rp with 1/(bσ 2 ) 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 ∼ σ 4/3 .

25


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

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 sucient to induce a signicant excess of counter-ions over co-ions (the Dukhin length for the high-salt reservoir concentration, `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 nite-element simulations as the dierence 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 highaspect-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 be accurate even in the absence of electric double layer overlap ( λD < a), provided that the number of counter-ions 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 (`Du a). Although the separation between the Debye and Dukhin lengths in most of our calculations is not signicantly large to test this hypothesis, we have veried its accuracy by comparing the local transport coecients kij computed using the general 1D intra-pore transport model (eqs 2226) and that for an ion-selective pore (eqs 2729) 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 `Du ). We nd that the transport coecients k13 = k31 ,

k23 = k32 , and k33 that determine the maximum power output Pmax converge to within 10% of the ion-selective values for σ & 100 mC m−2 (`Du & 500 nm), i.e. at Dukhin lengths signicantly larger than the pore radius, as shown in Supporting Information Figure S13.

26


Page 26 of 40

Page 27 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

Non-Ion-Selective Pores Although closed-form expressions for the global intra-pore transport coecients k¯ij do not exist in the case of non-ion-selective pores, the open-circuit voltage ∆φoc and pore electrical resistance Rp can be computed numerically by solving the general 1D intra-pore transport model for the uxes in eq 20 using the local transport coecients kij dened by eqs 2126 for several dierent applied voltages and using the same procedure described earlier to t the currentvoltage curves from the full 2D nite-element calculations. In this case, the pore resistance is the total resistance ( Rp = Rt ) since the 1D model neglects pore-end eects. 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 ecient 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 simplied 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 eectively be treated separately, signicantly simplifying the analysis of salinity-gradient power conversion. To emphasize this point, the pore resistance Rp , computed from the 2D nite-element calculations by subtracting the access resistance Ra from the total resistance Rt , can be seen in Figures 11 and S9 to be roughly independent of the charge on the outer membrane surface, and thus independent of the access resistance Ra (which diers 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 non-monotonically with reservoir salt concentration, which is due to the competing eects 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 27


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

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).

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

28


Page 28 of 40

Page 29 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

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 non-selective 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 pore. Although a simple exact expression for this scaling cannot be derived, assuming a small concentration gradient (i.e. c approximately constant, so ∆c/c ≈ ∆ ln c) in the intra-pore transport coecients for thin electric double layers given in the Supporting Information,

∆φoc ≈

µσkB T ∆ ln 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 conrmed 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 eect on power output as access resistance diminishes. At high salt concentrations, the eect of slip once again becomes small as bulk electrolyte transport, which does not depend on slip, dominates interfacial transport.

Eect of Outer Membrane Charge The eect of adding charge to the outer membrane surface on salinity-gradient power can be seen by comparing Figure 12, which shows the 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 signicantly higher for the charged outer membrane surface at low salt concentrations, with the eect

29


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

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 eect 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 benecial for salinity-gradient power conversion. Given that the outer surface of a porous 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.

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 ) . 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 ve times higher than those employed here, 30


Page 30 of 40

Page 31 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

but comparable pore diameters and lengths for most calculations, resulting in only partial ion selectivity. This work did not consider the eect 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 found here.

Experimental Implications

Having veried 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 one-dimensional nanoscopic pores, using materials such as carbon nanotubes, 63,64 mesoporous carbon, 65 and anodic aluminium oxide. 66 Many such surfaces can be chemically modied to change properties such as surface wettability 67,68 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 sea water 70 ) and lowand high-salt reservoir concentrations, cL and cH , of 5 and 500 mmol L−1 , corresponding roughly to the salinity of river and sea water, 1 respectively, and using experimentally feasible values 63,65,66 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 three from 6 W m−2 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 signicant impact on power conversion.

31


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Conclusions In summary, we have shown that interfacial hydrodynamic slip can have a signicant eect on membrane-based salinity-gradient-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 eects of pore and access electrical resistance can be decoupled in predicting power conversion. In the case of a perfectly ion-selective pore, we derived simple closed-form expressions for the intra-pore transport coecients, which we used to obtain analytical scaling laws for the dependence of power conversion on membrane properties, which could be exploited to design better membranes. Of particular note, we showed that the eect of slip on power conversion is amplied by surface charge and that adding charge to the outer membrane surface can signicantly 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.

Acknowledgement

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.

Supporting Information Available

Derivation of equations for local membrane transport coecients and power-conversion efciency and additional results for the total electrical resistance, access electrical resistance, maximum power density, open-circuit voltage, pore resistance, and transport coecients. 32


Page 32 of 40

Page 33 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

This material is available free of charge via the Internet at http://pubs.acs.org/ .

References

(1) Ramon, G. Z.; Feinberg, B. J.; Hoek, E. M. V. Membrane-based production of salinitygradient power. Energy Environ. Sci.

2011, 4, 44234434.

(2) Logan, B. E.; Elimelech, M. Membrane-based processes for sustainable power generation using water. Nature

2012, 488, 313319.

(3) Norman, R. S. Water Salination: A Source of Energy. Science (4) Loeb, S.; Norman, R. S. Osmotic Power Plants. Science

1974, 186, 350352.

1975, 189, 654655.

(5) Weinstein, J. N.; Leitz, F. B. Electric Power from Dierences in Salinity: The Dialytic Battery. Science

1976, 191, 557559.

(6) Brogioli, D. Extracting Renewable Energy from a Salinity Dierence Using a Capacitor.

Phys. Rev. Lett. 2009, 103, 058501. (7) Sales, B. B.; Saakes, M.; Post, J. W.; Buisman, C. J. N.; Biesheuvel, P. M.; Hamelers, H. V. M. Direct Power Production from a Water Salinity Dierence in a MembraneModied Supercapacitor Flow Cell. Environ. Sci. Technol.

2010, 44, 56615665.

(8) Brogioli, D.; Zhao, R.; Biesheuvel, P. M. A prototype cell for extracting energy from a water salinity dierence by means of double layer expansion in nanoporous carbon electrodes. Energy Environ. Sci.

2011, 4, 772777.

(9) Statkraft, http://www.statkraft.com . (10) REDstack, http://www.redstack.nl . (11) Gerstandt, K.; Peinemann, K. V.; Skilhagen, S. E.; Thorsen, T.; Holt, T. Membrane processes in energy supply for an osmotic power plant. Desalination 33


2008, 224, 6470.

Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Page 34 of 40

(12) Guo, W.; Cao, L.; Xia, J.; Nie, F.-Q.; Ma, W.; Xue, J.; Song, Y.; Zhu, D.; Wang, Y.; Jiang, L. Energy Harvesting with Single-Ion-Selective Nanopores: A ConcentrationGradient-Driven Nanouidic Power Source. Adv. Funct. Mater.

2010, 20, 13391344.

(13) van der Heyden, F. H. J.; Bonthuis, D. J.; Stein, D.; Meyer, C.; Dekker, C. Electrokinetic Energy Conversion Eciency in Nanouidic Channels. Nano Lett.

2006, 6, 22322237.

(14) Kim, D.-K.; Duan, C.; Chen, Y.-F.; Majumdar, A. Power generation from concentration gradient by reverse electrodialysis in ion-selective nanochannels. Microuid. Nanouid.

2010, 9, 12151224. (15) Bocquet, L.; Charlaix, E. Nanouidics, from bulk to interfaces. Chem. Soc. Rev.

2010,

39, 10731095. (16) Bocquet, L.; Tabeling, P. Physics and technological aspects of nanouidics. Lab. Chip

2014, 14, 31433158. (17) Post, J. W.; Veerman, J.; Hamelers, H. V. M.; Euverink, G. J. W.; Metz, S. J.; Nymeijer, K.; Buisman, C. J. N. Salinity-gradient power: Evaluation of pressure-retarded osmosis and reverse electrodialysis. J. Membrane Sci.

2007, 288, 218230.

(18) Hunter, R. J. Foundations of Colloid Science ; Oxford University Press, 2001. (19) van der Heyden, F. H. J.; Bonthuis, D. J.; Stein, D.; Meyer, C.; Dekker, C. Power Generation by Pressure-Driven Transport of Ions in Nanouidic Channels. Nano Lett.

2007, 7, 10221025. (20) Bouzigues, C. I.; Tabeling, P.; Bocquet, L. Nanouidics in the Debye Layer at Hydrophilic and Hydrophobic Surfaces. Phys. Rev. Lett.

2008, 101, 114503.

(21) Huang, D. M.; Cottin-Bizonne, C.; Ybert, C.; Bocquet, L. Ion-specic anomalous electrokinetic eects in hydrophobic nanochannels. Phys. Rev. Lett.

34


2007, 98, 177801.

Page 35 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

(22) Huang, D. M.; Cottin-Bizonne, C.; Ybert, C.; Bocquet, L. Aqueous electrolytes near hydrophobic surfaces: Dynamic eects of ion specicity and hydrodynamic slip. Langmuir

2008, 24, 14421450. (23) Huang, D. M.; Cottin-Bizonne, C.; Ybert, C.; Bocquet, L. Massive amplication of surface-induced transport at superhydrophobic surfaces. Phys. Rev. Lett.

2008, 101,

064503. (24) Bocquet, L.; Barrat, J. L. Flow boundary conditions from nano- to micro-scales. Soft

Matter 2007, 3, 685693. (25) Joly, L.; Ybert, C.; Trizac, E.; Bocquet, L. Hydrodynamics within the Electric Double Layer on Slipping Surfaces. Phys. Rev. Lett.

2004, 93, 257805.

(26) Ren, Y.; Stein, D. Slip-enhanced electrokinetic energy conversion in nanouidic channels. Nanotechnology

2008, 19, 195707.

(27) Yan, Y.; Sheng, Q.; Wang, C.; Xue, J.; Chang, H.-C. Energy Conversion Eciency of Nanouidic Batteries: Hydrodynamic Slip and Access Resistance. J. Phys. Chem. C

2013, 117, 80508061. (28) Siria, A.; Poncharal, P.; Biance, A.-L.; Fulcrand, R.; Blase, X.; Purcell, S. T.; Bocquet, L. Giant osmotic energy conversion measured in a single transmembrane boron nitride nanotube. Nature

2013, 494, 455458.

(29) Ajdari, A.; Bocquet, L. Giant Amplication of Interfacially Driven Transport by Hydrodynamic Slip: Diusio-Osmosis and Beyond. Phys. Rev. Lett.

2006, 96, 186102.

(30) Kang, B.; Yoo, J.; Kim, H.; Kim, D.-K. Slip-enhanced reverse electrodialytic power generation in ion-selective nanochannels. J. Therm. Sci.

2013, 22, 3641.

(31) Gross, R. J.; Osterle, J. F. Membrane Transport Characteristics of Ultrane Capillaries.

J. Chem. Phys. 1968, 49, 228234. 35


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Page 36 of 40

(32) Fair, J. C.; Osterle, J. F. Reverse Electrodialysis in Charged Capillary Membranes. J.

Chem. Phys. 1971, 54, 33073316. (33) Kang, B. D.; Kim, H. J.; Lee, M. G.; Kim, D.-K. Numerical study on energy harvesting from concentration gradient by reverse electrodialysis in anodic alumina nanopores.

Energy 2015, 86, 525538. (34) Kim, D.-K. Numerical study of power generation by reverse electrodialysis in ionselective nanochannels. J. Mech. Science Technol.

2011, 25, 510.

(35) Kim, J.; Kim, S. J.; Kim, D.-K. Energy harvesting from salinity gradient by reverse electrodialysis with anodic alumina nanopores. Energy

2013, 51, 413421.

(36) Yeh, H.-C.; Chang, C.-C.; Yang, R.-J. Reverse electrodialysis in conical-shaped nanopores: salinity gradient-driven power generation. RSC Adv.

2014, 4, 27052714.

(37) Gao, J.; Guo, W.; Feng, D.; Wang, H.; Zhao, D.; Jiang, L. High-Performance Ionic Diode Membrane for Salinity Gradient Power Generation. J. Am. Chem. Soc.

2014,

136, 1226512272. (38) Sui, H. X.; Han, B. G.; Lee, J. K.; Walian, P.; Jap, B. K. Structural basis of waterspecic transport through the AQP1 water channel. Nature

2001, 414, 872878.

(39) Hunte, C.; Screpanti, E.; Venturi, M.; Rimon, A.; Padan, E.; Michel, H. Structure of a Na+ /H+ antiporter and insights into mechanism of action and regulation by pH. Nature

2005, 435, 11971202. (40) Rolison, D. R.; Long, R. W.; Lytle, J. C.; Fischer, A. E.; Rhodes, C. P.; McEvoy, T. M.; Bourga, M. E.; Lubers, A. M. Multifunctional 3D nanoarchitectures for energy storage and conversion. Chem. Soc. Rev.

2009, 38, 226252.

(41) Bowen, W. R.; Williams, P. M. Quantitative predictive modelling of ultraltration

36


Page 37 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

processes: Colloidal science approaches. Adv. Colloid Interface Sci.

2007, 134-135,

314. (42) Elimelech, M.; Phillip, W. A. The Future of Seawater Desalination: Energy, Technology, and the Environment. Science

2011, 333, 712717.

(43) Wanunu, M.; Morrison, W.; Rabin, Y.; Grosberg, A. Y.; Meller, A. Electrostatic focusing of unlabelled DNA into nanoscale pores using a salt gradient. Nat. Nanotechnol.

2010, 5, 160165. (44) Joseph, P.; Cottin-Bizonne, C.; Benoît, J. M.; Ybert, C.; Journet, C.; Tabeling, P.; Bocquet, L. Slippage of Water Past Superhydrophobic Carbon Nanotube Forests in Microchannels. Phys. Rev. Lett.

2006, 97, 156104.

(45) Thomas, J. A.; McGaughey, A. J. H. Water Flow in Carbon Nanotubes: Transition to Subcontinuum Transport. Phys. Rev. Lett.

2009, 102, 184502.

(46) Hansen, J. S.; Dyre, J. C.; Daivis, P. J.; Todd, B. D.; Bruus, H. Nanoow hydrodynamics. Phys. Rev. E

2011, 84, 036311.

(47) Sparreboom, W.; van den Berg, A.; Eijkel, J. C. T. Transport in nanouidic systems: a review of theory and applications. New J. Phys.

2010, 12, 015004.

(48) Probstein, R. F. Physicochemical Hydrodynamics: An Introduction , 2nd ed.; WileyInterscience: Hoboken, 1994. (49) Comsol 4.3a, http://www.comsol.com. (50) Yoshida, H.; Mizuno, H.; Kinjo, T.; Washizu, H.; Barrat, J.-L. Molecular dynamics simulation of electrokinetic ow of an aqueous electrolyte solution in nanochannels. J.

Chem. Phys. 2014, 140, 214701. (51) Yoshida, H.; Mizuno, H.; Kinjo, T.; Washizu, H.; Barrat, J.-L. Generic transport coefcients of a conned electrolyte solution. Phys. Rev. E 37


2014, 90, 052113.

Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Page 38 of 40

(52) Yaroshchuk, A. E. Transport properties of long straight nano-channels in electrolyte solutions: A systematic approach. Adv. Colloid Interface Sci.

2011, 168,

278 291,

Surface forces and thin liquid lms. (53) Chang, C.; Yang, R. Electrokinetic energy conversion eciency in ion-selective nanopores. Appl. Phys. Lett.

2011, 99, 083102.

(54) de Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics ; Dover, 1984. (55) Aguilella-Arzo, M.; Aguilella, V.; Eisenberg, R. Computing numerically the access resistance of a pore. Eur. Biophys. J.

2005, 34, 314322.

(56) Mao, M.; Ghosal, S.; Hu, G. Hydrodynamic ow in the vicinity of a nanopore induced by an applied voltage. Nanotechnology

2013, 24, 245202.

(57) Luan, B. Numerically testing phenomenological models for conductance of a solid-state nanopore. Nanotechnology

2015, 26, 055502.

(58) Hall, J. E. Access resistance of a small circular pore. J. Gen. Physiol. 1975, 66, 531532. (59) Lee, C.; Joly, L.; Siria, A.; Biance, A.-L.; Fulcrand, R.; Bocquet, L. Large Apparent Electric Size of Solid-State Nanopores Due to Spatially Extended Surface Conduction.

Nano Lett. 2012, 12, 40374044. (60) Chang, H.-C.; Yossifon, G. Understanding electrokinetics at the nanoscale: A perspective. Biomicrouidics

2009, 3, 012001.

(61) Chang, H.-C.; Yossifon, G.; Demekhin, E. A. Nanoscale Electrokinetics and Microvortices: How Microhydrodynamics Aects Nanouidic Ion Flux. Annu. Rev. Fluid Mech.

2011, 44, 401426. (62) Lakshminarayanaiah, N. Transport Phenomena in Membranes ; Academic Press: New York, 1969. 38


Page 39 of 40

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Langmuir

(63) Holt, J. K.; Park, H. G.; Wang, Y. M.; Stadermann, M.; Artyukhin, A. B.; Grigoropoulos, C. P.; Noy, A.; Bakajin, O. Fast mass transport through sub-2-nanometer carbon nanotubes. Science

2006, 312, 10341037.

(64) Fornasiero, F.; Park, H. G.; Holt, J. K.; Stadermann, M.; Grigoropoulos, C. P.; Noy, A.; Bakajin, O. Ion exclusion by sub-2-nm carbon nanotube pores. Proc. Natl. Acad. Sci.

USA 2008, 105, 1725017255. (65) Lee, J.; Kim, J.; Hyeon, T. Recent Progress in the Synthesis of Porous Carbon Materials.

Adv. Mater. 2006, 18, 20732094. (66) Jani, A. M. M.; Losic, D.; Voelcker, N. H. Nanoporous anodic aluminium oxide: Advances in surface engineering and emerging applications. Prog. Mater. Sci.

2013, 58,

636704. (67) Stein, A.; Wang, Z.; Fierke, M. A. Functionalization of Porous Carbon Materials with Designed Pore Architecture. Adv. Mater.

2009, 21, 265293.

(68) Jani, A. M. M.; Anglin, E. J.; McInnes, S. J. P.; Losic, D.; Shapter, J. G.; Voelcker, N. H. Nanoporous anodic aluminium oxide membranes with layered surface chemistry. Chem.

Commun. 2009, 30623064. (69) Sisan, T.; Lichter, S. The end of nanochannels. Microuid. Nanouid.

2011, 11, 787

791. (70) Castro, P.; Huber, M. Marine Biology, 9th ed.; McGraw-Hill Education: New York, 2012. (71) Kim, H.-K.; Lee, M.-S.; Lee, S.-Y.; Choi, Y.-W.; Jeong, N.-J.; Kim, C.-S. High power density of reverse electrodialysis with pore-lling ion exchange membranes and a highopen-area spacer. J. Mater. Chem. A

2015, 3, 1630216306.

39


Langmuir

1 2 3 4 5 6 7 8 9 10 11 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 51 52 53 54 55 56 57 58 59 60

Graphical TOC Entry

40


Page 40 of 40

The Effect of Hydrodynamic Slip on Membrane-Based Salinity

Recommend Documents