Beyond the Tradeoff: Dynamic Selectivity in Ionic Transport and Current Selectivity
BBeyond the Trade-Off: Dynamic Selectivity inIonic Transport and Current Rectification
Anthony R. Poggioli, † , ‡ Alessandro Siria, † , ‡ and Lydéric Bocquet ∗ , † , ‡ † Laboratoire de Physique Statistique, Ecole Normale Supérieure, Paris ‡ Centre National de la Recherche Scientifique
E-mail: [email protected] a r X i v : . [ c ond - m a t . s o f t ] A ug bstract Traditionally, ion-selectivity in nanopores and nanoporous membranes is understoodto be a consequence of Debye overlap, in which the Debye screening length is compa-rable to the nanopore radius somewhere along the length of the nanopore(s). Thiscriterion sets a significant limitation on the size of ion-selective nanopores, as the De-bye length is on the order of − nm for typical ionic concentrations. However,the analytical results we present here demonstrate that surface conductance generatesa dynamical selectivity in ion transport, and this selectivity is controlled by so-calledDukhin, rather than Debye, overlap. The Dukhin length, defined as the ratio of surfaceto bulk conductance, reaches values of hundreds of nanometers for typical surface chargedensities and ionic concentrations, suggesting the possibility of designing large-nanopore( − nm), high-conductance membranes exhibiting significant ion-selectivity. Suchmembranes would have potentially dramatic implications for the efficiency of osmoticenergy conversion and separation techniques. Furthermore, we demonstrate that thismechanism of dynamic selectivity leads ultimately to the rectification of ionic current,rationalizing previous studies showing that Debye overlap is not a necessary conditionfor the occurrence of rectifying behavior in nanopores. Introduction
Ionic and water transport in nanometric confinement has been an active topic of research fortwo decades , with practical applications to macro-/bio-molecular analysis , energy gener-ation , and desalination . Of particular interest is understanding via what mechanismsand under what conditions one may obtain nonlinear responses of fluxes (charge, solute mass,solvent mass) to external forcings (voltage, pressure, concentration difference). A detailedtheoretical understanding of such nonlinear transport processes is a necessary first step indesigning exotic tailored nanopore and membrane functionalities . The most well-knownexample of such a nonlinear response is ionic current rectification (ICR), in which the ionic2urrents driven through nanopores by applied voltages of equal magnitude and opposite signare found to be of unequal magnitude, in analogy toff solid-state semiconductor diodes .ICR has been extensively studied experimentally and via continuum simulations, e.g. ,though comparatively few studies have examined the phenomenon analytically . Ex-perimentally, it is found that ICR may be induced by unequal reservoir concentrations ,asymmetric geometries , or asymmetric surface charge distributions .ICR is generally understood to be a consequence of the accumulation or depletion ofionic concentration induced by a gradient in ion selectivity along the length of a nanopore .The key parameter controlling the local selectivity is accordingly the ratio of the local Debyelength λ D to the local nanopore radius R . The Debye length is defined as λ D ≡ (cid:115) (cid:15) r (cid:15) k B Te (cid:80) j z j c j , (1)where (cid:15) r and (cid:15) are, respectively, the relative dielectric permittivity of the solvent and thevacuum permittivity, k B is the Boltzmann constant, T is the thermodynamic temperature, e is the elementary charge, z j is the valence of the j th ionic species, and c j is the concentrationof the j th ionic species. This length scale characterizes the thickness of the diffuse layer ofnet ionic charge that forms in the vicinity of charged surfaces . As the diffuse layer mustcounterbalance the surface charge, a region of strong Debye layer overlap, λ D /R (cid:38) O (1) ,should be highly selective to counterions, while a region of weak Debye overlap, λ D /R (cid:28) ,should be essentially non-selective.Debye overlap has significant consequences for ionic transport through the pore. In orderto illustrate the underlying physical mechanisms, consider a conical nanopore with a uniformsurface charge density, as discussed in . When λ D /R (cid:38) O (1) in the vicinity of the nanoporetip, a Donnan equilibrium between the tip and the connected reservoir forms, and this regionis highly selective to (positive) counterions. The tip is then a region of increased transferenceof cations and suppressed transference of anions. Thus, when an external electric field is3irected from the (non-selective) base to the (highly selective) tip, both cations and anionspass from regions of relatively lower to relatively higher transference, and the result at steadystate is a depletion of ionic concentration in the nanopore interior. The depletion of chargecarriers results in a decrease of the local electrophoretic conductivity and corresponds tothe low conductance (reverse-bias) diode state. On the other hand, if the direction of theapplied field is inverted, both ionic species pass from regions of relatively high to relativelylow transference, resulting in an accumulation of ionic concentration. This accumulation ofcharge carriers likewise results in a high conductance (forward-bias) state. Altogether, thecriterion λ D /R (cid:38) is expected to be a prerequisite to observe non-linear ionic transport,and in particular ionic current rectification. The mechanism of concentration accumulationand depletion in ICR has since been extensively corroborated numerically andexperimentally .However, recent studies have indicated not only that the parameter λ D /R fails to predictthe occurrence or strength of rectification, but also that significant rectification may be ob-tained even when the Debye length is one-to-three orders of magnitude smaller than the mini-mum nanopore radius . To our knowledge, no consistent alternative criterion hasbeen proposed to predict the occurrence of ICR. Furthermore, while there have been manynumerical simulations of ICR within the Poisson-Nernst-Planck (PNP) or Poisson-Nernst-Planck-Stokes (PNPS) framework, few studies have offered a detailed theoretical analysisof the phenomenon, and these studies are typically confined to quite specialized scenarios,e.g. .In this paper, we demonstrate that the nanopore selectivity is not determined by therelative value of the Debye length λ D compared to the pore radius, but rather by a dynamiccriterion related to the relative magnitudes of the surface and bulk ionic conductions. Thisintroduces the so-called Dukhin length (cid:96) Du , defined as the ratio of the surface conductivityto the bulk conductivity in the nanopore . The Dukhin length can be adequatly rewrittenin terms of the charge density on the pore surface, σ , as (cid:96) Du = ( | σ | /e ) /c , with c the bulk salt4oncentration . A dimensionless Dukhin number is accordingly introduced as Du ≡ (cid:96) Du R ≡ | σ | ecR . (2)The Dukhin length approaches values of hundreds of nanometers for typical surface chargesin the range of − mC m − and concentrations in the range of . − mM, one-to-two orders of magnitude larger than the corresponding Debye lengths. Written in termsof the surface charge density (Eq. 2), we note that the criterion Du > for substantialselectivity/ICR is equivalent to the criterion | σ | > ecR noted by some authors .As we demonstrate below, substantial ionic selectivity may be obtained when the nanoporeradius is comparable to the Dukhin length. This is consistent with numerical results indi-cating substantial ion-selectivity may be obtained in highly charged pores with radii muchlarger than the Debye length , and it is in stark contrast to traditional ion-selective mem-branes, which typically have subnanometric pore sizes . We term this mechanism dynamicselectivity, in contrast to the ’thermodynamic’ picture of ionic selectivity based on Debyeoverlap and the formation of a Donnan equilibrium at the ends of the nanopore, e.g. .The possibility of obtaining significant selectivity for large ( − nm) pores may havesignificant implications for, e.g., osmotic energy generation; we will discuss this point in moredetail below.Furthermore, we will show below that a gradient in the local Dukhin number along thelength of the nanopore results in a repartitioning of the fraction of the ionic transport carriedin the non-selective bulk and in the highly selective Debye layer. It is this repartitioning,rather than Debye overlap and the formation of a local Donnan equilibrium at one end ofthe nanopore, that results in ICR.Our starting point will be the derivation of one-dimensional transport equations from aradial integration of the axisymmetric PNP equations in the limit that λ D /R (cid:28) but Du ∼ ; we focus on this limit as we are particularly interested in rationalizing those experimen-5al and numerical results indicating that substantial rectification andselectivity may occur even when λ D /R (cid:28) . We note that a reanalysis of the data reportedin the literature corroborates the assertion that significant rectification occurs when Du ∼ ,irrespective of the value of λ D /R ; this is discussed below.From these transport equations, we are able to derive a simple implicit expression forthe current-voltage (IV) response in a concentration diode. To our knowledge, this is thefirst time such a solution has been presented. Additionally, we will give numerical solutionsof these transport equations for a geometric diode (rectification induced by an asymmetric,continuously varying radius) and a charge diode (rectification induced by an asymmetric,continuously varying surface charge density distribution). These solutions will directly il-lustrate the key role of the local Dukhin number and the mechanism of dynamic selectivityin ICR. Finally, we derive analytical expressions for general limiting conductances directlyfrom the transport equations that will be useful in estimating, for example, surface chargedensities from rectified IV curves. Theory
1D Transport Equations in the Absence of Debye Overlap
In this section, we derive one-dimensional transport equations for the electrostatic potentialand total ionic concentration at the nanopore centerline from the axisymmetric Poisson-Nernst-Planck (PNP) equations. Our derivation relies on the geometric constraint R /(cid:96) (cid:28) ,where R is a scale of the radial extent of the nanopore, and (cid:96) is a characteristic scale ofvariation of the nanopore geometry. Such a slowly varying approximation implies that aPoisson-Boltzmann (PB) equilibrium holds locally on each cross-section . As we intendto demonstrate that the Dukhin number is the principal parameter controlling selectivityand ICR, and that ICR may occur even in the absence of Debye overlap anywhere along thelength of the nanopore, we focus on the regime that λ D /R (cid:28) . There is no Debye overlap6n the center of the nanopore so that the electrolyte there is electroneutral, and we maypartition the ionic concentrations and electrostatic potential as follows : c ± ( x, r ) = c ( x )2 + δc ± ( x, r ) , and (3) φ ( x, r ) = φ ( x ) + δφ ( x, r ) , (4)where c ( x ) ≡ c ( x, is the value of the total ionic concentration c ≡ c + + c − at thenanopore centerline ( r = 0 ), φ ( x ) ≡ φ ( x, is the electrostatic potential at the nanoporecenterline, and δc ± ( x, r ) and δφ ( x, r ) are the radial deviations in the ionic concentrations andelectrostatic potential induced by the formation of a screening Debye layer in the vicinity ofthe nanopore wall. In Eq. 3, we have assumed a symmetric (z:z) salt. In what follows, wewill assume a monovalent (1:1) salt in which the cation and anion have identical mobilityand diffusion coefficients.With this notation and these assumptions, the steady-state PNP equations reduce to j ± = − D (cid:18) ∂c ± ∂x ± ek B T c ± ∂φ∂x (cid:19) , (5) ddx (cid:18) π (cid:90) R dr rj ± (cid:19) ≡ dJ ± dx = 0 , (6) (cid:15) r (cid:15) r ∂∂r (cid:18) r ∂δφ∂r (cid:19) + n c ≈ (cid:15) r (cid:15) ∂ δφ∂Z + n c = 0 , (7) c = c cosh (cid:18) eδφk B T (cid:19) , and (8) n c = − ec sinh (cid:18) eδφk B T (cid:19) , (9)where D is the diffusion coefficient, e is the elementary charge, k B is the Boltzmann constant, T is the thermodynamic temperature, R is the local nanopore radius, (cid:15) r ( ≈ for water atroom temperature) is the relative permittivity of the solvent and (cid:15) is the vacuum permittiv-ity, and n c ≡ e ( c + − c − ) is the ionic charge density. In Eq. 7 we have neglected the portion7f the radial gradient induced by the curvature of the nanopore wall as it is suppressedby a factor λ D /R (cid:28) relative to ∂ r , and we have introduced the coordinate Z ≡ R − r .Eq. 5 is the ionic flux density in the along-flow ( x ) direction; Eq. 6 is the cross-sectionallyintegrated continuity equation at steady state; Eq. 7 is the Poisson equation, retaining onlythe radial component of the electric field divergence in accordance with the slowly varyingapproximation; and Eqs. 8 and 9 are the distributions of the total ionic concentration andionic charge density obtained from the Boltzmann distribution, applied on the assumptionof a slowly varying geometry. Finally, we have neglected advection as it is not expected toaffect our conclusions , and such an approach allows for tractable analytical derivations.Before continuing, we introduce dimensionless rescaled variables, as listed in Table 1;we have rescaled the x -coordinate by the total nanopore length L and the surface charge σ by a reference magnitude | σ ref | , and we have introduced c , the average of the reservoirconcentrations, as a scale of ionic concentration in the nanopore. In what follows, we willtake the reference surface charge magnitude | σ ref | to be either the magnitude of the surfacecharge density when it is uniform, or the maximum surface charge magnitude when thesurface charge density is nonuniform. We further recast Eqs. 5 and 6 in terms of therescaled solute flux J ≡ J + + J − ≡ (cid:82) dA j and ionic current I ≡ J + − J − ≡ (cid:82) dA i . Withthese modifications, the governing equations become j = − (cid:18) ∂c∂x + n c ∂φ∂x (cid:19) , (10) i = − (cid:18) ∂n c ∂x + c ∂φ∂x (cid:19) , (11) dJdx = dIdx = 0 , (12) (cid:18) λ ref D R (cid:19) ∂ δφ∂Z + n c = 0 , (13) c = c cosh ( δφ ) , and (14)8 c = − c sinh ( δφ ) , (15)where we have introduced a reference Debye length λ ref D ≡ (cid:112) k B T (cid:15) r (cid:15) /e c , defined in termsof the mean reservoir concentration c . Table 1: Independent and dependent variables and their rescaled dimensionlesscounterparts. quantity variable rescaledposition x x → Lx radius R R → R min R concentration c c → cc ionic charge density n c n c → ecn c electrostatic potential φ φ → ( k B T /e ) φ electrochem. potential µ ± µ ± → k B T µ ± flux density j ± j ± → ( Dc/L ) j ± surface charge σ σ → | σ ref | σ conductance G G → ( R /L )( e D/k B T ) c We differentiate Eq. 14 (15) with respect to x , insert the result into Eq. 10 (11), andintegrate on the cross-section to obtain JπR = − dc dx − (cid:104) δc (cid:105) c dc dx − (cid:104) n c (cid:105) c c dφ dx , and (16) IπR = − c dφ dx − (cid:104) n c (cid:105) c dc dx − (cid:104) δc (cid:105) c c dφ dx , (17)where (cid:104)(cid:105) ≡ A − (cid:82) dA denotes a cross-sectional average. The integral of the charge densityis set by the condition of local electroneutrality, a necessary consequence of a local PBequilibrium; the integral of δc may be evaluated using PB equilibrium theory (Eqs. 13, 14,and 15) in the limit λ D /R (cid:28) . The condition of local electroneutrality requires that (cid:104) n c (cid:105) c = − Du ref σRc ≡ − Du ( x ) , (18)where Du ref ≡ | σ ref | /ecR is a reference Dukhin number, and Du ( x ) is the local Dukhinnumber. We note that Du ref is defined to be always positive, but Du ( x ) carries the sign of9he local surface charge density. The integral of δc can be evaluated using Eqs. 13 through15. The result is (cid:104) δc (cid:105) c = 4 λ D R R (cid:115)(cid:20) Du λ D /R R ) (cid:21) + 1 − , (19)where Du is the local value of the Dukhin number, and λ D ( x ) ≡ λ ref D / (cid:112) c ( x ) is the localDebye length. In the limit ( λ D /R R ) / Du → , this reduces to (cid:104) δc (cid:105) c = 2 | Du ( x ) | . (20)We will consider this limit in developing an analytical solution for the concentration diode be-low, as we are most interested in the scenario that there is no Debye overlap ( λ D ( x ) /R ( x ) R (cid:28) everywhere) but the Dukhin number is of order one (Du(x) ∼ somewhere). In this case,the transport equations, Eqs. 16 and 17, become JπR = − (cid:20) dc dx + 2 | Du ( x ) | dc dx − Du ( x ) c dφ dx (cid:21) , and (21) IπR = − (cid:20) c dφ dx − Du ( x ) dc dx + 2 | Du ( x ) | c dφ dx (cid:21) , (22)where I and J are of course constant along the length of the nanopore (Eq. 12). It is usefulto distinguish between those terms in Eqs. 21 and 22 that arise from transport outside ofthe Debye layer (bulk transport) and those arising from transport within the Debye layer(surface transport). In the equation for the solute flux (Eq. 21), the terms represent, fromleft to right, bulk diffusion, surface diffusion, and surface electrophoretic mass transport. Inthe equation for the ionic current (Eq. 22), the terms represent bulk electrophoretic current,surface charge diffusion, and surface electrophoretic current. We see that the local Dukhinnumber, which sets the cross-sectionally averaged ionic charge (Eq. 18), as well as the cross-sectionally averaged excess concentration when λ D /R R (cid:28) Du (Eq. 20), determines theratio of surface to bulk transport. Motivated by this observation, we quantify the ratio of10he surface transport to the total transport by the following surface transport ratio (STR):STR ( x ) ≡ | Du ( x ) | | Du ( x ) | . (23)We see immediately that the partitioning of the transport into surface and bulk componentsadjusts along the length of the nanopore and is controlled locally by the Dukhin number. Inthe case that there is a large asymmetry in Dukhin number on either end of the nanopore,this can result in a substantial repartitioning of the transport in the nanopore interior, theconsequences of which will be explored in the following sections.An illustrative form of Eqs. 21 and 22 is obtained by introducing the definition of the localDukhin number, Eq. 18, and defining the coion and counterion fluxes J co / count ≡ ( J ± S I ) / and electrochemical potentials µ co / count ≡ ln( c / ± S φ . In the preceding definition, S ≡ sign( σ ) is the sign of the surface charge. Inserting these definitions into Eqs. 21 and 22, weobtain J co πR = − c dµ co dx , and (24) J count πR = − (cid:18) c ref | σ | R (cid:19) dµ count dx . (25)Note that we have implicitly assumed in deriving these equations from Eqs. 21 and 22 thatthe sign of the surface charge S does not change along the length of the nanopore. If the signof the surface charge does change at one (or several) points along the length of the nanopore,Eqs. 24 and 25 may be applied in each region delineated by a discontinuity in S ( x ) , matching c and φ at each discontinuity. (More about the proper boundary conditions for Eqs. 21/24and 22/25 will be said below.)In Eqs. 24 and 25, we recognize c / as the concentration of both coions and counterionsat the nanopore centerline. Further, in Eq. 25, we recognize 2Du ref | σ | /R as the additionalconcentration accumulated in the Debye layer (Eq. 20). The term 2Du ref | σ | /R × dµ count /dx in Eq. 25 represents the entirety of the surface transport in the nanopore; this indicates that11oions are perfectly excluded from the Debye layer in the limit λ D /R (cid:28) Du ( R dimensioned).This will be shown to be the case from PB equilibrium theory and the condition of localequilibrium below.We will employ Eqs. 21 and 22 below to develop an implicit analytical solution for thecurrent-voltage (IV) relationship in a concentration diode. Dynamic Selectivity
As noted above, in the slowly varying limit ( R /(cid:96) (cid:28) ) considered here, the ionic concentra-tion profiles on the cross-section deviate negligibly from those predicted by PB equilibriumtheory. Thus, we may apply this equilibrium theory to determine how the ionic selectivity onthe cross-section is influenced by Debye overlap ( λ D /R ) and the Dukhin number. Using Eqs.18 and 19, we determine the cross-sectionally averaged total concentration, (cid:104) c (cid:105) = (cid:104) δc (cid:105) + c ,and the cross-sectionally averaged counterion concentration, (cid:104) c count (cid:105) = ( (cid:104) c (cid:105) + |(cid:104) n c (cid:105)| ) / . Wetake the ratio (cid:104) c count (cid:105) / (cid:104) c (cid:105) as a metric of selectivity; this parameter ranges between / , indi-cating the total average concentration is equally partitioned between coions and counterionsand the nanopore is locally non-selective, and unity, indicating that the entirety of the av-erage concentration is due to counterions and the nanopore is perfectly selective. We findthat, in the range < λ D /R < − ) , the nanopore selectivity is strongly influenced by thelocal Dukhin number in the range − < Du < (Fig. 1). In comparison, the influenceof λ D /R on selectivity for this range of λ D /R is substantially less pronounced. From equilibrium to dynamical selectivity
As noted above, Eqs. 24 and 25 suggest that the Debye layer is perfectly selective in thelimit ( λ D /R ) / Du → . This can be confirmed as follows: For a monovalent ionic species,the deviations in counterion (coion) concentration from the centerline potential in the Debyelayer are related to δc and n c by δc count = ( δc + | n c | ) / ( δc co = ( δc − | n c | ) / ). With theseexpressions and Eqs. 18 and 20, we find (cid:104) δc count (cid:105) = 2 | Du | c and (cid:104) δc co (cid:105) = 0 , confirming that12 u -2 -1 h c c oun t i / h c i λ D /R >> 12(10 -1 )10 -1 -2 )<< Du Figure 1: The ratio of the cross-sectionally averaged counterion concentration to the cross-sectionally averaged total concentration, a metric of the nanopore selectivity, as a functionof Dukhin number. The lines are colored according to λ D /R , as indicated in the legend. Thedashed (dot-dashed) line indicates the curve obtained in the limit λ D /R → ∞ ( → ).the Debye layer is indeed perfectly selective in this limit.An upper limit on the selectivity induced by Debye overlap may be obtained by consid-ering the selectivity of the nanopore entrance/exit in the limit λ D /R → ∞ . In this case,we may impose electrochemical equilibrium across the junction between the interior of thenanopore and the reservoir and local electroneutrality on either side . The ratio of counterionto total concentration is then found to be (cid:104) c count (cid:105)(cid:104) c (cid:105) = (cid:113)(cid:0) (cid:1) + | Du | + | Du | (cid:113) | Du | ) . (26)This upper limit exceeds the selectivity obtained above for λ D /R = 2(10 − ) only slightly(Fig. 1), indicating that the nanopore selectivity rapidly saturates for values of λ D /R (cid:38) / .A lower limit, valid when λ D /R (cid:28) Du, is obtained via Eqns. 18 and 20. In this case, wefind (cid:104) c count (cid:105)(cid:104) c (cid:105) = 12 (cid:18) | Du | | Du | (cid:19) ≡
12 (1 +
STR ) , (27)13here in the second equality we have made use of Eq. 23. Eq. 27 shows the deep connectionbetween transport and selectivity and can be made more intuitive as follows. As shownabove, the Debye layer in this limit is perfectly selective, so that the ratio of counterionconcentration to total concentration is unity. As always, the bulk is perfectly non-selective,so that the ratio there is / . As the STR is the ratio of surface transport to total transport,the ratio of bulk-to-total transport is − STR, and we may estimate the total selectivity onthe cross-section based on the partitioning of the ionic transport as × (1 − STR) (cid:124) (cid:123)(cid:122) (cid:125) bulk + 1 × STR (cid:124) (cid:123)(cid:122) (cid:125) surface = 12 (1 + STR) . We thus recover the result given in Eq. 27. This result illustrates that it is the selectivityin the bulk and surface weighted by the dynamic partitioning of the ionic transport thatcontrols local selectivity.Together, the upper and lower limits on selectivity (Eqs. 26 and 27, respectively), definean envelope of selectivity variation with λ D /R (Fig. 1). The conclusion of these results isapparent: the principal parameter controlling nanopore selectivity is not λ D /R but the localDukhin number. This result may be understood as follows: when λ D /R is small, the localDukhin number controls both the fraction of the transport in the Debye layer (Eq. 23) andthe selectivity of the Debye layer (Eq. 27). A large value of Du means that the majorityof the ionic flux is carried within the Debye layer, and that this region is highly selective.Thus, even though the unselective bulk region takes up the majority of the cross-section, themajority of the transport must pass through the highly selective but relatively small Debyelayer. As this process is controlled by the local Dukhin number and the local partitioning ofionic currents, both of which adjust dynamically, we refer to it as dynamic selectivity.On the other hand, when λ D /R is large, a significant surface charge (as quantified by theDukhin number) must still be present to draw counterions into (and exclude coions from)the nanopore and thus render the nanopore highly selective. This is indicated in the Donnan14esult for the selectivity (Eq. 26). All together, the result is the dominance of the localDukhin number in determining the local nanopore selectivity (Fig. 1).These results suggest that a nanopore may exhibit significant selectivity when the poresize is comparable to the Dukhin length. As noted above, the Dukhin length reaches hundredsof nanometers for typical ionic concentrations ( . − mM) and surface charge densities( − mC m − ). This is in strong contrast to traditional ion-selective membranes, whichhave typically subnanometric pore sizes , and indeed to the typical picture of ion-selectivityas occurring only in the presence of strong Debye overlap ( λ D /R (cid:29) ) .We explore this idea by examining the ion selectivity under an applied concentrationdifference and voltage in a uniform nanopore, and under an applied voltage in a conicalnanopore. Transport and dynamic selectivity
We first consider a nanopore of uniform negative surface charge density σ and constant radius R connecting a left reservoir of concentration c L and applied voltage ∆ V to a grounded rightreservoir of concentration c R ≤ c L (Fig. 2, inset). We anticipate here some of the resultsderived later in order to illustrate the concept of dynamic selectivity.As detailed below, we may solve the transport equations, Eqs. 21 and 22, to obtain animplicit algebraic solution for the ionic current and solute flux (Eqs. 35 and 36), from whichwe calculate the ionic flux J ± ≡ ( J ± I ) / and the ion selectivities S ± ≡ | J ± || J + | + | J − | . (28)Note from the above definition that the counterion selectivity will fall between / and ,while the coion selectivity will fall between and / , such that S + + S − = 1 .We show the results of a calculation of the cation (counterion) selectivity in a uniformnanopore under an applied concentration difference in Fig. 2. The results are colored15ccording to concentration ratio c L /c R > and are plotted against the maximum Dukhinnumber Du R ≡ | σ | /ec R R imposed at the right end of the nanotube and corresponding to thesmaller reservoir concentration c R . We see that for moderate concentration ratios ( c L /c R ≤ ), a maximum Dukhin number Du R of order one results in selectivities of ∼ − . Du R -2 -1 c a t i on s e l e c t i v i t y ∆ V = 0 c L /c R = 31010 ∆ V Du R = 1 Figure 2: Cation selectivity for the diffusive flux in the absence of an applied voltage ( ∆ V =0 ) as a function of the larger imposed Dukhin number Du R corresponding to the smallerreservoir concentration c R . The curves are colored according to the corresponding valueof c L /c R , as indicated in the legend. The fluxes are calculated for a nanopore of uniformnegative surface charge density and constant radius, as indicated in the schematic in theupper left. The inset shows the selectivity as a function of voltage applied to the reservoircontaining the larger concentration, c L , and for a fixed value of Du R = 1 . The curves arecolored according to c L /c R , as in the main panel, and the voltage is rescaled according tothe thermal voltage k B T /e ≈ mV.In the inset of Fig. 2 we show the cation selectivity as a function of applied voltage ∆ V , again colored according to concentration ratio c L /c R , and at a fixed Du R = 1 . Werecall that the voltage is applied in the high concentration reservoir; this corresponds tothe reverse-bias (low conductance) state of the concentration diode. As we will see below,generically ion selectivity is maximized in a diode when voltage is applied in the reverse-biasdirection. Under an applied voltage, the anion flux is reduced and eventually shut downas the anion chemical potential differential ∆ µ − = ln( c L /c R ) − ∆ V decreases and vanishes.16his results in a rapid increase in the cation selectivity as small voltages are applied in thehigh concentration reservoir and a peak cation selectivity of when ∆ V = ln( c L /c R ) and J − = 0 . We see in the inset of Fig. 2 that the cation selectivity saturates at a valuethat is independent of c L /c R as ∆ V is increased above ln( c L /c R ) and anions begin to flowfrom the right to the left reservoir. This saturation value is given by S + (∆ V → + ∞ ) = 1 + 4Du max max , (29)where Du max is the larger of the two Dukhin numbers imposed on either end of the nanopore;in this case, Du max = Du R . For the present case, Du R = 1 , and the saturation value of thecation selectivity is ∼ . These results suggest that the nanopore selectivity may betuned with the application of small external applied voltages. (Note that ∆ V is rescaled by k B T /e ≈ mV such that the maximum plotted voltage in the inset of Fig. 2 ∆ V = 20 corresponds to ∼ mV.)While the zero-voltage selectivity is not as optimal as traditional ion-selective membranes,which have counterion selectivity ratios ∼ , this tradeoff is more than made up for bypore diameters that are one-to-two orders-of-magnitude larger.Let us consider two prototypical situations in which such effects could be usefully har-nessed. Reverse electrodialysis (RED) is one of a few proposed techniques for the conversionof the osmotic energy associated with the salinity contrast between fresh and saltwater tomechanical energy. This technique depends on ion-selective diffusive fluxes of the type dis-cussed above and shown in Fig. 2 across stacks of alternating cation- and anion-selectivemembranes. The principal limiting factor in commercialization of this process is the lowconversion efficiency engendered by the high membrane resistance due to the subnanometricpores in typical ion-selective membranes . Our results suggest that this problem may becircumvented by using large-pore ( − nm) membranes with pore diameters and surfacecharges tailored to the operating concentrations such that a maximum Dukhin number of17rder one is achieved.Another phenomenon of interest is traditional electrodialysis (ED), in which an electricfield is applied across stacks of cation- and anion-selective membranes in order to separateions from brackish source water. In this case, what is of interest is the selectivity of the ionicflux induced by an applied voltage in the absence of a concentration differential. To thisend, we first consider as a benchmark the performance (both selectivity and conductance)of a uniform nanopore–i.e., a nanopore with constant (negative) surface charge density andradius; we then compare this to the performance of a conical nanopore having a fixed length,surface charge density, and tip radius R tip . That is, we hold the tip Dukhin number Du tip ≡| σ | /ecR tip fixed while increasing the ratio of base and tip radii α ≡ R base /R tip above unity.The scenario under consideration is sketched in an inset in Fig. 3. Our goal in increasingthe opening angle is to improve the conductance compared to the uniform nanopore withouta great cost to the nanopore selectivity.In the case of a uniform nanopore, the transport equations, Eqs. 21 and 22, are triviallysolved for the conductance (given by Eqs. 37 through 41) and the cation selectivity (givenby Eq. 29 with Du max = Du tip ). Note that in the case of a uniform nanopore the cationselectivity is voltage-independent, as the Dukhin number is equal to Du tip everywhere. Thisis indicated by the blue curves in Figs. 3a, showing the cation selectivity as a function ofvoltage, and 3b, showing the apparent conductance G app ≡ I/ ∆ V normalized by the uniformnanopore conductance.The influence of introducing a conical structure (i.e., increasing α ≡ R base /R tip aboveone) on the cation selectivity is shown in Fig. 3a. In the vicinity of ∆ V = 0 , the selectivitydrops rapidly as α is increased, decreasing from the theoretical limit of ∼ (Eq. 29)to ∼ as α is increased from to ; however, as ∆ V is increased, the selectivityagain rapidly approaches the theoretical limit for large positive voltages. We note that thevoltage is applied to the larger end of the conical nanopore; this again corresponds to thelow conductance and high selectivity configuration of the diode. We see in Fig. 3a that a18 V c a t i on s e l e c t i v i t y α = 1 ∆ V appa r en t c ond . a)b) Du tip = 1 Figure 3: a) Cation selectivity in a conical nanopore as a function of applied voltage andcolored according to the ratio of base and tip radii α ≡ R base /R tip ≥ , as indicated inthe legend. b) Apparent conductance G app ≡ I/ ∆ V normalized by the conductance of auniform ( α = 1 ) nanopore (Eqs. 37 through 39) as a function of ∆ V and colored according to α ≡ R base /R tip , as in a. The inset in panel a shows a schematic representation of the geometryconsidered here. All curves are calculated with a Dukhin number at the tip Du tip = 1 . Thevoltage is rescaled by the thermal voltage k B T /e ≈ mV, and the variables indicated inthe schematic are rescaled according to Table 1.voltage as small as ∼ V ( ∆ V = 40 ) is enough to achieve a selectivity very nearly identicalto the uniform nanopore selectivity.We plot in Fig. 3b the influence of increasing α ≡ R base /R tip on the apparent conduc-tance. We see that the limiting apparent conductance for large positive applied voltagesis substantially increased as α is increased. Increasing α to is enough to approximatelydouble the conductance, while α = 10 results in a conductance that is more than four timeslarger than the uniform pore conductance. We will show below that the limiting conductanceis related to the uniform nanopore conductance G uni (Eqs. 37 through 41) by G (∆ V → + ∞ ) = α − α G uni . (30)19he results for a conical nanopore shown in Fig. 3 and given in Eqs. 29 and 30 indicatethat 1) substantial selectivity may be achieved in large (i.e., high conductance) uniform radiusnanopores if the surface charge and pore radius is tailored to the operating concentrationssuch that Du tip (cid:38) and 2) the conductance may be even further enhanced by introducinga conical shape to the nanopore while holding the tip radius fixed. Together, these resultssuggest that, e.g., desalination processes based on ED may be made substantially moreefficient by using high surface charge, large, conical nanopores. Results
The Role of Dynamic Selectivity in the Concentration Diode
In this section, we illustrate the principal role of the Dukhin number in ICR by develop-ing an implicit expression for the IV relationship in a concentration diode using Eqs. 21and 22. For a nanopore of uniform cross-section, these equations are exactly valid cross-sectional integrations of the governing PNP equations in the limits
R/L → , λ D /R → ,and ( λ D /R ) / Du → . Therefore, the fact that they produce rectified IV curves is strongevidence that λ D /R (cid:38) O (1) somewhere in the nanopore is not a necessary condition for theexistence of ICR. As the rectification is fundamentally a consequence of the mechanism ofdynamic selectivity outlined above, there is no fundamental mechanistic difference betweenionic diodes induced by asymmetric geometry, differences in reservoir concentration, or (con-tinuous) asymmetric surface charge profiles induced by, e.g., differences in reservoir pH. Ineach case, the asymmetry induces an asymmetry in the Dukhin number | Du | = | σ | /ecR across the nanopore, resulting in an asymmetry in the selectivity of the Debye layer andin the partitioning of the ionic transport between surface and bulk. We focus here on theconcentration diode because we are able to derive an illustrative algebraic solution.20 luxes across a concentration diode We consider here the same uniform nanopore configuration as described above and shownschematically in the inset of Fig. 2. The local Dukhin number along the length of thenanopore is given by Du ( x ) = − Du ref c , (31)where the reference Dukhin number Du ref ≡ | σ | /ecR is defined in terms of the magnitude ofthe uniform surface charge density, the average of the two reservoir concentrations, and theuniform nanotube radius. We have omitted the subscript zero on the concentration, and wewill continue to omit the subscript in what follows, recalling that the indicated total ionicconcentrations and electrostatic potentials are centerline values.We note that, because the surface charge density and nanotube radius do not vary alongthe length of the nanopore, the variation in the local Dukhin number is determined entirelyby the variation in the local concentration. In addition to the reference Dukhin number, weintroduce the local Dukhin numbers on the left and right of the nanotube, Du L ≡ Du ref /c L and Du R ≡ Du ref /c R , respectively. We note that the ratio of Dukhin numbers Du R / Du L ≡ c L /c R ≥ is simply the ratio of reservoir concentrations. With these definitions, we canexpress the reference Dukhin number asDu ref = 2 Du − L + Du − R . (32)We will formulate our results below in terms of the maximum Dukhin number in the system, Du R , corresponding to the minimum reservoir concentration, and the concentration ratio Du R / Du L ≡ c L /c R . Note that, while we do not impose particular values for the reservoirconcentrations in our rescaled, dimensionless model, the concentration ratios considered hereare consistent with the range of concentrations considered experimentally, typically between0.1 mM and 1 M. 21ith the local Dukhin number given in Eq. 31, Eqs. 21 and 22 become Jπ = − (cid:20) dcdx + 2 Du ref (cid:18) d ln cdx + dφdx (cid:19)(cid:21) , and (33) Iπ = − (cid:20) c dφdx + 2 Du ref (cid:18) d ln cdx + dφdx (cid:19)(cid:21) . (34)Solving Eqs. 33 and 34 is a straightforward procedure, but before integrating we musttake care to ensure that we are imposing appropriate boundary conditions at the nanotubeends. This is a nontrivial question because the rapid variation in local Dukhin number (froma nonzero value in the nanotube interior to zero in the reservoir) that occurs on either endof the nanotube means that the slowly varying approximation we have used to impose alocal PB equilibrium breaks down . In one-dimensional PNP-based models of nanoporeionic transport, this is typically taken into account by imposing continuity of the electro-chemical potential µ ± = ln c ± ± φ across the junction and local electroneutrality on eitherside . This is justified as follows: The rapid variation in local geometry and con-sequently in the local Dukhin number results in localized deviations from equilibrium andelectroneutrality and rapid variations in electrostatic potential and ionic concentrations. Thescale of this adjustment region is given by the Debye length. (See the paper of Shockley for an extensive discussion of this point in the equivalent context of semiconductor physics.)Thus, in the limit λ D /L → , the adjustment region may be treated as a point discontinuityin the ionic concentrations and electrostatic potential. However, as the ionic flux densities j ± = − c ± ∂ x µ ± are proportional to the gradient of the electrochemical potential, the discon-tinuities in ionic concentration and electrostatic potential must be such that continuity ofelectrochemical potential is maintained, ensuring finite ionic fluxes. Outside of this adjust-ment region, the ions again locally equilibrate, and thus local electroneutrality is imposedon either side of the junction.As we have already assumed a radially uniform electrochemical potential in locally ap-plying a PB equilibrium, imposing continuity of the electrochemical potential between the22niform reservoir values and the values over the entire cross-section in the nanotube interiorsreduces to imposing electrochemical continuity between the reservoir and nanopore center-line. Furthermore, as the ionic charge vanishes at the nanopore centerline, imposition ofelectroneutrality there amounts to imposing c int+ = c int − = c int / , where ’int’ indicates thevalue on the interior of the nanopore. Electrochemical continuity across the junction thusrequires ln( c res / ± φ res = ln( c int / ± φ int , where ’res’ indicates the value in the reservoir.This condition is satisfied by imposing continuity of the ionic concentration and potential, c res = c int and φ res = φ int , respectively .We note that 1) this still corresponds to finite discontinuities in the ionic concentrationsand electrostatic potential within the Debye layer, and 2) there would be a discontinuity incenterline concentrations and electrostatic potential in the case of Debye overlap, as the ioniccharge density would no longer vanish at the nanopore centerline. In the limit of completeDebye overlap, this corresponds to the formation of a local Donnan equilibrium between theends of the nanotube and the adjacent reservoirs .With the above boundary conditions, we can directly integrate Eq. 33 for the solute flux: (cid:18) Du R Du L + 1 (cid:19) J π = (cid:18) Du R Du L − (cid:19) + 2Du R (cid:20) ln (cid:18) Du R Du L (cid:19) + ∆ V (cid:21) , (35)where we have used Eq. 32 and the definitions of Du L and Du R to rewrite the result in termsof the maximum Dukhin number Du R and the ratio Du R /Du L ≡ c L /c R .Using Eq. 33, we solve for cdφ/dx in terms of J and dc/dx , insert the result into Eq. 34,and integrate. The result can be combined with Eq. 35 to obtainln (cid:18) Du R Du L (cid:19) + ∆ V = (cid:18) IJ (cid:19) × ln (cid:34) Du R Du L + 2Du R (cid:0) − IJ (cid:1) R (cid:0) − IJ (cid:1) (cid:35) . (36)23q. 36 may be used to determine the applied voltage as a function of the ratio I/J for givenvalues of Du R and Du R / Du L , and the result can be combined with Eq. 35 to determinethe solute flux and ionic current as a function of applied voltage. IV curves obtained usingEqs. 35 and 36 are plotted in Fig. 4 for a fixed value of Du R = 1 and several values ofDu R / Du L ≡ c L /c R ≥ . ∆ V -40 -20 0 20 40 I -200-150-100-50050100150200 Du R = 1 I → I + I - ← I Du R /Du L = 131010 Figure 4: IV curves obtained from Eqs. 35 and 36 for a fixed value of Du R = 1 . The curvesare colored according to Du R / Du L ≡ c L /c R as indicated in the legend. The dashed blacklines indicate I + and I − , the currents obtained in the limit ∆ V → + ∞ and −∞ , respectively,for Du R / Du L = 10 . (See Eqs. 42 and 44.) The voltage is rescaled by the thermal voltage k B T /e ≈ mV, and the current is rescaled according to Table 1. Limiting Conductances
As anticipated, as Du R / Du L → , representing equal reservoir concentrations, the IV curvebecomes progressively more linear, and the conductance approaches a limiting value repre-senting the sum of the bulk and surface electrophoretic contributions : G = G bulk + G surf , with (37) G bulk ≡ π, and (38)24 surf ≡ π Du ref . (39)This result is obtained by solving Eqs. 33 and 34 with dc/dx = 0 . In dimensioned terms,these conductances are given by G bulk = πR L e Dk B T c res , and (40) G surf = 2 πRL eDk B T | σ | , (41)where c res is the concentration in both reservoirs. This limiting conductance is indicated bythe blue curve in Fig. 4.As a concentration difference is applied (Du R / Du L > ) and increased, the IV curvesbecome progressively more rectified. This is due to the asymmetry in selectivity betweenthe left and right end of the nanotube. The Dukhin number at the right end is held fixedat one, resulting in substantial selectivity for positive coions at that end (Fig. 1). On theother hand, as the Dukhin number on the left end is decreased via an increasing reservoirconcentration c L , the counterion selectivity at this end rapidly decreases, approaching thenon-selective limit for values of Du R / Du L > (Fig. 1).In order to fully understand under what conditions ICR occurs, we examine limitingvoltage-independent conductances obtained under various conditions. We have already noted(Eqs. 37 through 41) the limiting conductance when the reservoir concentrations are equal(Du R / Du L = 1 ). We next examine the limiting currents and differential conductances G ≡ ∂I/∂ ∆ V when Du R and Du R /Du L are held fixed and ∆ V → ±∞ , denoted I ± and G ± ,respectively. From Eq. 36, we see that the logarithm on the right-hand side must diverge asthe voltage diverges for a fixed concentration ratio Du R /Du L . In the limit | ∆ V | → ∞ , theionic current and solute flux will be in the same direction, such that the coefficient of thelogarithm I/J > . Thus, when ∆ V → + ∞ , the argument of the logarithm must diverge.This requires that I/J → Du R ) − ≡ STR − R , where STR R is the surface transport ratio25t the right end of the nanopore (Eq. 23). Combined with Eq. 35, this gives for the currentand conductance when ∆ V → + ∞ (cid:18) π Du ref STR R (cid:19) − I + = 12Du R (cid:18) Du R Du L − (cid:19) + ln (cid:18) Du R Du L (cid:19) + ∆ V, and (42) G + = G surf STR R ≡ π Du ref R R . (43)Likewise, as ∆ V → −∞ , Eq. 36 indicates that the argument of the logarithm must vanish,and thus I/J → STR − L . By the same procedure we find (cid:18) π Du ref STR L (cid:19) − I − = 12Du R (cid:18) Du R Du L − (cid:19) + ln (cid:18) Du R Du L (cid:19) + ∆ V, and (44) G − = G surf STR L ≡ π Du ref L L . (45)These limiting currents are indicated in Fig. 4 for Du R = 1 and Du R / Du L = 10 .As Du R becomes smaller for a fixed value of the ratio Du R / Du L , the voltage magnitudethat leads to significant concentration accumulation or depletion becomes larger. This isbecause the asymmetry in nanopore selectivity between either end of the nanopore becomesweaker (Fig. 1). This means that, for an experimentally feasible range of applied voltages,the IV curve linearizes as Du R is decreased. From Eq. 36, we see that as Du R → for afixed value of Du R / Du L , I/J → ∆ V / ln ( Du R / Du L ) . This is true so long as the numeratorand denominator of the logarithm are (cid:29) , which, from our discussion above, requires thatthe voltage magnitude not be too large. We find from this limit and Eq. 35 a limitingconductance G → (cid:16) Du R Du L − (cid:17)(cid:16) Du R Du L + 1 (cid:17) ln (cid:16) Du R Du L (cid:17) G bulk , (46)26alid for fixed Du R / Du L when Du R (cid:28) . We note that the prefactor in Eq. 46 approachesunity and G → G bulk in the limit Du R / Du L → , as it must.Finally, we examine the limit Du R → ∞ for fixed Du R / Du L . In this case, the entirenanopore becomes perfectly selective for counterions (Fig. 1), and the IV curve again lin-earizes. Unlike in the case that Du R (cid:28) , however, we find that the IV curve is linearirrespective of the magnitude of the applied voltage for large Du R . From Eq. 23, we notethat, as Du → ∞ , STR → . Thus, as Du R → ∞ for fixed Du R / Du L , STR L → STR R → ,and, from Eqns. 43 and 45, we find G + → G − → G surf . In this case, the conductance isdominated by the (concentration-independent) surface contribution.Each of the limiting conductances discussed above, and the conditions under which theyobtain, are listed in Table 2. Table 2: Limiting conductances for the concentration diode and the conditionsunder which they obtain. The results are written generically in terms of themaximum and minimum reservoir concentrations, c max and c min , respectively, andthe corresponding maximum and minimum Dukhin numbers, Du max ≡ | σ | /ec min R and Du min ≡ | σ | /ec max R , imposed on either end of the nanopore. G max and G min are,respectively, the maximum and minimum conductances obtained as | ∆ V | → ∞ .(See Eqs. 43, 45 and 53, 54.) The bulk ( G bulk ) and surface ( G surf ) electrophoreticcondutances are given in Eqs. 38 and 39, respectively. condition G c max /c min = 1 G bulk + G surf Du max (cid:28) c max /c min − c max /c min +1) ln ( c max /c min ) × G bulk Du max → ∞ G surf | ∆ V | → ∞ G min = max max × G surf G max = min min × G surf Rectification Ratio
We are now in a position to discuss the rectification ratio, defined asrectification ratio ≡ | I ( −| ∆ V | ) − I (∆ V = 0) || I (+ | ∆ V | ) − I (∆ V = 0) | . (47)27e plot this ratio in Fig. 5 as a function of Du R for fixed concentration ratios Du R /Du L = c L /c R . In general, the rectification ratio is a function of the voltage magnitude | ∆ V | ; fordefiniteness, we take | ∆ V | = 40 , which corresponds to a dimensioned applied voltage ofapproximately V.The rectification ratios display a peak for a finite value of Du R (Fig. 5). This peakgrows and is shifted to higher values of Du R as Du R /Du L is increased. The value of Du R corresponding to the peak rectification ratio Du peak R is of order one over much of the parameterspace ( . < Du peak R < for ≤ Du R / Du L ≤ ). Du R -2 -1 r e c t i f i c a t i on r a t i o R /Du L = 31010 Figure 5: Rectification ratio obtained from Eqs. 35 and 36 and evaluated at | ∆ V | = 40 ( ≈ V), as a function of Du R . The curves are colored according to Du R / Du L = c L /c R asindicated in the legend. The dashed line indicates the theoretical maximum rectificationratio, STR R /STR L , valid in the limit | ∆ V | → ∞ , for Du R /Du L = 10 .The fact that the rectification ratio shows a peak at a finite value of Du R can be under-stood as follows: By taking the ratio of the limiting conductances valid when | ∆ V | → ∞ (Eqs. 43 and 45), we see that there is an upper limit on the maximum rectification, given by G − /G + = STR R / STR L and valid in the limit | ∆ V | → ∞ . As Du R → for a fixed value ofDu R /Du L , this ratio reaches a maximum value equal to the concentration ratio Du R /Du L .However, in this case, the local Dukhin number is much smaller than one everywhere and thenanopore is therefore only weakly selective for counterions (Fig. 1). It thus takes very large28oltages to engender significant ion accumulation or depletion, voltages much larger thanthe practical upper limit in nanofluidic experiments ( ∼ V), and the IV curve is effectivelylinearized.On the other hand, when Du R → ∞ (with Du R /Du L fixed), STR R → STR L → , indi-cating that the IV curve is strictly linear in this limit, irrespective of the voltage magnitude.This is because the local Dukhin number is everywhere much larger than one and the entiretyof the ionic transport is carried within the Debye layer. There is therefore no repartitioningof the transport between bulk and surface and no gradient in ion selectivity along the lengthof the nanopore. In this case, the conductance is given by the (concentration-independent)surface conductance (Eq. 41). Thus, the location of Du max R represents a compromise betweenthe non-selective (Du R → ) and perfectly selective (Du R → ∞ ) limits.The occurrence of a maximum rectification ratio for a finite value of Du R and a fixedvalue of the ratio Du R /Du L is exactly analogous to the common observation of a maximumrectification ratio for a finite concentration or surface charge density and a fixed geometry inconical diodes, e.g. . In that case, the ratio of Dukhin numbers is fixed by the ratio ofbase and tip radii, while the variation of concentration or surface charge results in a variationof the maximum Dukhin number occuring at the tip of the conical nanopore. As in theconcentration diode, the location of the maximum is determined by a compromise betweenthe non-selective (high concentration or low surface charge) and perfectly selective (lowconcentration or high surface charge) limits. We will discuss the role of dynamic selectivityin diodes induced by asymmetric geometry (and surface charge density distributions) below. Dynamic Selectivity and Limiting Conductances in Generic Diodes
Rectification in Geometric and Concentration Diodes
From our understanding of the role of the Dukhin number in controlling local selectivity,and of the mechanism of dynamic selectivity in controlling rectification, we conclude thatICR is generically a consequence of inequality of the Dukhin numbers imposed on either29nd of a nanopore, irrespective of whether that asymmetry is induced by a difference inreservoir concentrations, an asymmetric geometry, or an asymmetric surface charge densitydistribution (or any combination thereof). This is corroborated by the results shown in Fig.6, where we compare an IV curve obtained from the above solution for a concentration diode(Eqs. 35 and 36, Figs. 6a and d) to numerical solutions of the transport equations (Eqs. 21and 22) for the IV curve in a geometric (Figs. 6b and e) and a charge diode (Figs. 6c and f).In order to illustrate rectification induced by an asymmetric geometry, we assume a linearvariation in the nanopore radius from the maximum radius (minimum Dukhin number) onthe left to the minimum radius (maximum Dukhin number) on the right. The surface chargedensity is taken to be fixed and negative, and the reservoir concentrations are taken to beequal. This configuration is shown schematically in Fig. 6b. ∆ V -40 -20 0 20 40 I × - -2-101 ∆ V -40 -20 0 20 40 I × - -4-202 IVG min G max G ∆ V -40 -20 0 20 40 I × - -4-202 a) conc. diode b) geo. diode c) charge dioded) e) f) Figure 6: a-c) Schematics of diodes induced by a) unequal reservoir concentrations, b) asym-metric geometry, and c) asymmetric surface charge distribution. d-f) IV curves obtained forDu R = 1 and Du L = 0 . for d) the concentration diode shown in a, e) the geometric diodeshown in b, and f) the charge diode shown in c. In panels d-f, the dashed yellow and red linesshow the maximum and minimum conductances obtained when | ∆ V | → ∞ and calculatedaccording to Eqs. 53 and 54, respectively. In panels e and f, the dashed purple line indicatesthe linear response conductance valid in the vicinity of ∆ V = 0 for the charge and geometricdiodes and calculated according to Eq. 63. The voltage is rescaled by the thermal voltage k B T /e ≈ mV, and the current as well as the variables indicated in the schematics arerescaled according to Table 1.Likewise, we illustrate rectification induced by a continuous, asymmetric surface chargeprofile by imposing a negative surface charge density whose magnitude varies linearly from aminimum density (minimum Dukhin number) on the left to a maximum density (maximum30ukhin number) on the right. In this case, we impose a constant nanopore radius and equalreservoir concentrations. This configuration is shown schematically in Fig. 6c. In all threecases, we take Du L = 0 . and Du R = 1 . We immediately see from Figs. 6d through f thatthe qualitative structure of the rectified IV curve is essentially the same over a given range ofvoltage ( − ≤ ∆ V ≤ +40 here). Indeed, in each case the rectification ratio (Eq. 47) is ∼ for | ∆ V | = 40 . The qualitative similarity of the IV curves obtained in these three differentconfigurations illustrates the equivalence of the mechanism resulting in rectification. General Expressions for the Limiting Conductances and Selectivities When | ∆ V | →∞ In this section, we show that the above expressions obtained for the limiting conductanceswhen | ∆ V | → ∞ (Eqs. 43 and 45) obtained for the concentration diode are particularexamples of general expressions relating the minimum (maximum) Dukhin number imposedat one end of the nanopore to the maximum (minimum) conductance obtained for largeimposed voltages. We also derive expressions for the limiting ion-selectivities when | ∆ V | →∞ . In the case of the geometric and charge diodes, these correspond to the minimum andmaximum selectivities obtainable by varying the applied voltage. These expressions are validfor concentration, geometric, and charge diodes (or any combination thereof), as illustratedin Fig. 6 for the limiting conductances. In the course of this discussion, we illustrategeneral features of the evolution of the Dukhin number profile in the nanopore interior as afunction of applied voltage, further illustrating the principal role of dynamic selectivity inthe accumulation or depletion of concentration in the nanopore interior and hence in ICR.The general expressions for limiting conductances in generic diodes, along with thosederived below for the linear response near ∆ V = 0 in geometric and charge diodes, will allowobservations of rectified IV curves to be related to, for example, the surface charge density.The surface charge density is difficult to estimate directly and is often estimated by observ-ing the saturation of the conductance at the surface-dominated value for low concentrations,31.g. . However, the inference of surface charge from conductance measurements typi-cally relies on a linear response, in which case the relation between surface charge and thesaturating conductance at low concentration is known analytically, e.g. . It is not clear a priori how this framework may be extended to, e.g., conical nanopores, where ICR isinherent to the IV response below a certain concentration. To our knowledge, general ana-lytical results for the relationship between surface charge and conductance do not exist inthe literature for rectified IV curves, except in certain specialized scenarios, e.g. .Starting from Eqs. 21 and 22, we may write the transport equations generally as JπR = − (cid:20) dcdx + 2 | Du | (cid:18) dcdx − S c dφdx (cid:19)(cid:21) , and (48) IπR = − (cid:20) c dφdx − S2 | Du | (cid:18) dcdx − S c dφdx (cid:19)(cid:21) . (49)Note that we have assumed nothing about the sign of the surface charge S (except that itdoes not change along the length of the nanopore) or the nature of the variation of the localDukhin number | Du | .As | ∆ V | → ∞ , the solute flux will be dominated by the surface electrophoretic masstransport, J/πR ∼ S2 | Du | cdφ/dx = S2Du ref ( | σ | /R ) dφ/dx . Integrating in x , we find J = − S (cid:18)(cid:90) L dx2 πR | σ | (cid:19) − Du ref ∆ V. (50)In order to obtain a condition on the flux ratio I/J that holds in the limit | ∆ V | → ∞ , wesolve Eqs. 48 and 49 for the concentration gradient dc/dx in terms of I/J , the local Dukhinnumber, and the (divergent) solute flux: dcdx = − S IJ + S (cid:16) | Du | (cid:17) | Du | JπR . (51)On physical grounds, the concentration gradient cannot diverge everywhere in the nanopore32nterior, even in the limit that | ∆ V | → ∞ . Accordingly, the prefactor in Eq. 51 mustvanish as the solute flux diverges. This requires that I/J → −
S(1 + 1 / | Du | ) ≡ − S / STR ,where in the second equality we have made use of Eq. 23. As the ratio
I/J is spatiallyuniform at steady state, this condition requires the Dukhin number in the nanopore interiorto approach a uniform value, which we will designate Du u . Likewise, we designate thecorresponding surface transport ratio STR u ≡ u / (1 + 2Du u ) . With this result for theflux ratio I/J and Eq. 50 we find for the current I = (cid:18)(cid:90) L dx2 πR | σ | (cid:19) − Du ref STR u ∆ V. (52)The mechanism of concentration accumulation/depletion is driven by the gradient inDukhin number induced by the asymmetry between the maximum (Du max ) and minimum(Du min ) Dukhin numbers imposed on either end of the nanopore. Thus, for very strongapplied voltages, the accumulation (depletion) will cease when the concentration everywhereis such that the uniform Dukhin number in the interior is equal to Du u = Du min (Du max ).At one end of the nanopore, the concentration gradient then must diverge to match thedivergence in the solute flux (Eq. 50) while allowing the Dukhin number to deviate from itsuniform interior value and adjust to the appropriate boundary condition.This mechanism is illustrated in Fig. 7: In Figs. 7a through c, we show the profilesof centerline concentration as a function of the applied voltage for the concentration (Fig.7a), geometric (Fig. 7b), and charge (Fig. 7c) diodes shown schematically in Figs. 6athrough c, respectively. In the case of the concentration diode, Eq. 51 may be integratedto obtain an implicit expression for the concentration profile, while for the geometric andcharge diodes it is necessary to solve the transport equations (Eqs. 21 and 22) numericallyto obtain the concentration profiles. As for the IV curves shown in Figs. 6d through f,the profiles are calculated for Du L = 0 . and Du R = 1 . The qualitative structure of theconcentration profiles is quite different in the three configurations considered; however, in33ach case, there is increasing depletion (accumulation) of concentration in the nanoporeinterior for increasing magnitude positive (negative) voltage. Note that the sign of thevoltage resulting in accumulation/depletion would be inverted for a positive surface charge,rather than the negative surface charge considered here. x -1 -0.5 0 c a) conc. diode x -1 -0.5 0 c b) geom. diode x -1 -0.5 0 c c) charge diode -40-200 +20+40 ∆ V x -1 -0.5 0 | D u | d) x -1 -0.5 0 | D u | e) x -1 -0.5 0 | D u | f) Figure 7: a-c) Profiles of total ionic concentration along the length of the a) concentrationdiode shown schematically in Fig. 6a, b) the geometric diode (Fig. 6b), and c) the chargediode (Fig. 6c). d-f) The corresponding profiles of local Dukhin number for the concentra-tion (d), geometric (e), and charge (f) diodes. The dashed black lines in d-f indicate theminimum and maximum imposed Dukhin numbers at either end of the nanopore, Du L = 0 . and Du R = 1 , respectively. In all panels, the curves are colored according to the appliedvoltage, as indicated in the colorbar on the right. The voltages are rescaled by the thermalvoltage k B T /e ≈ mV, and the concentrations are rescaled by the mean of the reservoirconcentrations.However, in Figs. 7d through f, we see that the evolution of the local Dukhin numberprofiles | Du( x ) | with applied voltage are strikingly similar in the three configurations, eventhough the concentration profiles are quite different. In each case, an increasing magnitudepositive (negative) voltage results in a growing region in the nanopore interior where | Du | ≈ Du max = Du R ( | Du | ≈ Du min = Du L ). Fig. 7 illustrates the key role of the local Dukhinnumber in controlling the accumulation/depletion of concentration in the nanopore and hencein ICR. It also illustrates that, in the extreme limit that | ∆ V | → ∞ , the Dukhin numberwill approach a uniform value equal to the maximum or minimum Dukhin number imposedat one end of the nanopore. (Whether it approaches Du max or Du min depends on the signof the applied voltage and the surface charge density.) Thus, denoting the maximum and34inimum limiting conductances as G max and G min , respectively, we find G max = G surf STR min ≡ G surf min min , and (53) G min = G surf STR max ≡ G surf max max . (54)In these equations, G surf is the surface conductance obtained when Du ref → ∞ and only thesurface terms in Eq. 49 are relevant : G surf = (cid:18)(cid:90) L dx2 πR | σ | (cid:19) − Du ref dim. −−→ eDk B T (cid:18)(cid:90) L dx2 πR | σ | (cid:19) − , (55)where we have redimensionalized in the second line. The limiting conductances predictedusing Eqs. 53 and 54 are shown in Figs. 6d through f (dashed yellow and red lines).We note that, by setting | σ | ≡ R ≡ and identifying Du min = Du L and Du max = Du R ,we recover Eqs. 43 and 45 from Eqs. 54 and 53, respectively.Figs. 7d through f show that the Dukhin number has not fully approached a uniformvalue everywhere in the nanopore interior even for | ∆ V | = +40 ; however, Figs. 6d throughf indicate that the differential conductance is roughly equal to its limiting value for | ∆ V | (cid:38) − ( ∼ − mV). We note, however, that the voltage necessary to reach the limitingconductance depends on both the maximum Dukhin number in the system Du max and theasymmetry in Dukhin numbers, quantified by the ratio Du max /Du min . (See Fig. 5 andrelated discussion.) Thus, we note that care must be taken in applying Eqs. 53 and 54 toexperimental IV curves. This difficulty can be avoided by instead fitting the surface chargeto the linear response conductance G obtained in the vicinity of ∆ V = 0 ; we will obtain ananalytical expression for G in the following section.Using the asymptotic expressions for the solute flux (Eq. 50) and ionic current (Eqs.52 through 55) we can calculate the co-/counterion fluxes J co / count = ( J ± S I ) / and derive35xpressions for the counterion selectivities S count ≡ | J count | / ( | J co | + | J count | ) in the limits ∆ V → ±∞ . The results are S maxcount = 1 + 4Du max max (reverse − bias); (56) S mincount = 1 + 4Du min min (forward − bias) . (57)Eq. 56 confirms Eq. 29. Furthermore, we see that the ion-selectivity is maximized in thereverse-bias (low conductance) configuration, as previously noted. The coion selectivity is ofcourse given by the relation S co + S count = 1 .Finally, before leaving this section, we derive Eq. 30 relating the conical conductance inthe reverse-bias configuration to the conductance of a uniform nanopore having the samesurface charge density and a constant radius equal to the tip radius. Eq. 54 holds in thereverse-bias configuration. We perform the integration in Eq. 55 with | σ | = 1 and imposinga linear variation in the radius to find G surf = 2 π Du tip ( α − / ln α . We combine this resultwith Eq. 54, recognizing that π Du tip × (1 + 2Du tip ) / tip = π + 2 π Du tip ≡ G uni , theuniform reference nanopore conductance (Eqs. 37 through 39), to find G min = α − α G uni . (58)This confirms Eq. 30. Conductance in the Vicinity of ∆ V = 0 In the case of the concentration diode, the imposed concentration difference means thatthere is a difference in electrochemical potentials between the reservoirs for at least one ofthe ionic species irrespective of the applied voltage. However, for the geometric and chargediodes equilibrium obtains when ∆ V = 0 , and we can linearize about this equilibrium toobtain an expression for the differential conductance G ≡ ∂I/∂ ∆ V | ∆ V =0 in the vicinity36f ∆ V = 0 . The equilibrium state is characterized by ∆ V = 0 = ⇒ J co = J count = 0 and the concentration, electrostatic potential, and electrochemical potential profiles c ≡ , φ ≡ , and µ co ≡ µ count ≡ − ln2 , respectively. We introduce a perturbative forcing δV (cid:28) ,which induces fluxes δJ co and δJ count . The applied voltage perturbs the concentration andelectrostatic potential profiles such that c → c (cid:48) and φ → φ (cid:48) , with c (cid:48) = 0 on either endof the nanopore and φ (cid:48) varying between δV on the left and on the right end of the nanopore.The electrochemical potentials become µ co → − ln2 + c (cid:48) + S φ (cid:48) and µ count → − ln2 + c (cid:48) − S φ (cid:48) ,from which we define µ (cid:48) co / count ≡ c (cid:48) ± S φ (cid:48) .We linearize Eqs. 24 and 25 to find δJ co πR = − dµ (cid:48) co dx , and (59) δJ count πR = − (cid:18)
12 + 2Du ref | σ | R (cid:19) dµ (cid:48) count dx . (60)Integration of Eqs. 59 and 60 along the length of the nanopore gives δJ co = +S 12 (cid:18)(cid:90) L dx πR (cid:19) − δV, and (61) δJ count = − S 12 (cid:20)(cid:90) L dx πR (1 + 4Du ref | σ | /R ) (cid:21) − δV. (62)From these results we may calculate the conductance at ∆ V = 0 as G ≡ δI/δV ≡ S( δJ co − δJ count ) /δV . We find G = G bulk0 (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) (cid:90) L dx πR (cid:19) − + (cid:20) (cid:90) L dx πR (1 + 4Du ref | σ | /R ) (cid:21) − (cid:124) (cid:123)(cid:122) (cid:125) G surf0 , (63)where we have partitioned the result into bulk and surface contributions. The conductances37redicted from Eq. 63 are shown in Fig. 6 for a diode induced by a linear variation in i)nanopore radius (Fig. 6e) and ii) surface charge density (Fig. 6f).For the sake of illustration, we derive an explicit expression for G in a conical nanoporewith a linearly varying radius and uniform surface charge density. Many studies have lookedat ICR in such nanopores, e.g. , and our result will be useful in relatingrectified IV curves to surface charge densities in conical nanopores.We take the radius to vary linearly between a maximum at the base of the conicalnanopore, R base , and a minimum at the tip, R tip . This gives for the magnitude of the radialslope | dR/dx | = α − (in rescaled variables), where α ≡ R base /R tip , as introduced above. Weinsert this into Eq. 63, along with the condition that the surface charge density is uniform | σ | = 1 , and evaluate the integrals to find G bulk0 = π α , and (64) G surf0 = α − (cid:16) tip tip /α (cid:17) π Du tip . (65)In the above, we have recognized that Du ref = Du tip ≡ | σ | /ec res R tip , the Dukhin numberdefined in terms of the uniform surface charge density magnitude, reservoir concentration,and tip radius. Redimensioning Eqs. 64 and 65, we find G bulk0 = πR base R tip L e Dk B T c res , and (66) G surf0 = α − (cid:16) tip tip /α (cid:17) πR tip L eDk B T | σ | . (67)Note that, in the limit of a uniform nanopore α → , the sum of Eqs. 64 (66) and 65 (67)agrees with the sum of Eqs. 38 (40) and 39 (41), as it must.We find that the dependence of the surface conductance on the surface charge (throughthe terms proportional to Du tip appearing in the logarithm) is more complicated than a linear38roportionality, indicating that the results for the conductance in a conical nanopore reportedin , for example, cannot be naïvely applied to the linear response conductance obtained for | ∆ V | (cid:28) k B T /e (in dimensioned terms). The conductance derived in indicates a linearproportionality between the surface conductance and the magnitude of the surface chargedensity, and we obtain the expression for the surface conductance given therein from Eq. 67only in the limit that 4Du tip (cid:29) α . Discussion
A Reanalysis of ICR Data in the Literature
Our results suggest that inequality of the Dukhin numbers imposed on either end of ananopore is the only criterion for the occurrence of ICR. Further, they suggest that Du ∼ is a (rough) criterion for the maximization of rectification. To this end, we have reinterpreteddata for conical nanopores in the literature in terms of the (maximal) tip Dukhin number(Table 3). We see that substantial rectification may be obtained even when the minimumradius is two-to-three orders-of-magnitude larger than the Debye length, but that peak rec-tification consistently corresponds to Du ∼ , consistent with our theoretical description ofdynamic selectivity and its role in ICR.In selecting experimental and numerical data from the literature, we searched for anyrectification data that was obtained by imposing a continuous variation in concentration,geometry, and/or surface charge. Immediately, this excludes data obtained in charge diodescontaining a discontinuity in the magnitude and/or sign of the surface charge, e.g. . Theimportant distinction between diodes containing discontinuities in the local Dukhin numberor the sign of the local surface charge and those considered here will be discussed in thefollowing section. We additionally searched for rectification ratios (either directly reportedor inferred from reported IV curves) that displayed a local maximum as the maximum Dukhinnumber was varied (via variations either in reservoir concentration or surface charge density)39 able 3: Maximum rectification ratios in conical nanopores for fixed Dukhinratios (Du max /Du min = R base /R tip ) along with corresponding values of λ D /R tip andDu tip , as estimated from the literature. (The superscripts on the peak values ofthe rectification ratio indicate the corresponding reference.) Note that the peakrectification ratios cannot be directly compared as they are not all calculated atthe same reference voltage magnitude. (The reference voltage magnitudes rangebetween mV and V.) peak rect. ratio λ D /R tip Du tip − ) 1 . .
014 0 . . .
046 0 . . .
082 1 . . .
17 4 . . .
33 2 . . .
61 1 . while the Dukhin ratio was held fixed. In the end, all of the data that fit our criteria werefound to come from conical nanopores. A Note on the Distinction Between Intrinsic and Extrinsic Diodes
Briefly, we note that we have been concerned here with the ICR induced by continuousvariations in the local Dukhin number and in the presence of surface charge of a single sign.This is in contrast to both classical bipolar diodes, containing regions of both positive andnegative surface charge, e.g. , and unipolar diodes, e.g. , containing regions ofzero and nonzero surface charge. We term the latter intrinsic diodes, as in this case thezone of depletion or accumulation is localized to the intrinsic discontinuity in either the localDukhin number or the sign of the surface charge. We term the type of diodes consideredhere extrinsic diodes, in contrast to the previous terminology and in recognition of the factthat, in this case, the rectification is due to an imposed inequality in the Dukhin numberson either end of the nanopore, rather than an intrinsic discontinuity.Intrinsic diodes are typically found to exhibit much stronger rectification , due tothe presence of a localized intrinsic accumulation/depletion zone. Picallo et al. showed40nalytically that, in the limit of high surface charge (Du → ∞ ), bipolar diodes exhibit idealShockley behavior, typical of classical p-n junction semiconductor diodes . In this case,the current is described by I = I sat [1 − exp ( − e ∆ V /k B T )] , where I sat is the finite saturationcurrent obtained for large positive (reverse-biased) voltages. This is in strong contrast to thebehavior of extrinsic diodes as detailed above, in which two finite limiting conductances areobserved for large positive or negative voltages, and it is further notable because it illustratesthat rectification in intrinsic diodes is maximized as Du → ∞ , rather than being washedout. Conclusions and Perspectives
The principal conclusion of this work is that the key parameter controlling nanopore ion-selectivity, through the mechanism of dynamic selectivity, is the Dukhin number Du ≡| σ | /ecR ; the ratio of Debye length to nanopore radius λ D /R is of secondary importance. Asthe Dukhin length (cid:96) Du ≡ | σ | /ec can reach values of hundreds of nanometers for typical surfacecharge densities and ionic concentrations, this suggests that significant ionic selectivity canbe obtained for large ( − nm) nanopores. This result has allowed us to rationalize theobservation that significant ionic current rectification is routinely observed even when thenanopore radius is one-to-three orders of magnitude larger than the Debye length. In doingso, we have obtained several general formulae for limiting and linear response conductancesin ionic diodes, which will allow researchers to relate IV measurements in asymmetric systemsto unknown parameters such as the surface charge density.Crucially, this mechanism suggests the possibility of designing large, conical pore ion-selective membranes. The tip radius and surface charge density of such membrane nanoporescan be tailored to the operating concentrations in order to obtain significant ion-selectivity( − ) while achieving orders-of-magnitude larger conductances than those obtainedin traditional (subnanometric) ion-selective membranes. The implications for the design41f much more efficient osmotic energy conversion (RED) and desalination/filtration (ED)devices is profound, as the key limiting factor in commercialization of such technologies isthe poor efficiency due to low membrane conductance. Future work will focus on developmentof prototypical devices to demonstrate high-efficiency energy conversion and desalination inlarge nanopores. Acknowledgement
A.R.P. acknowledges funding from the European Union’s Horizon 2020 Framework Pro-gram/European Training Program 674979 - NanoTRANS. A.S. acknowledges funding fromthe European Union’s Horizon 2020 Framework Programme/ERC Starting Grant agreementnumber 637748 - NanoSOFT. L.B. acknowledges funding from ANR project Neptune andfrom the European Union’s Horizon 2020 Framework Programme/ERC Advanced Grantagreement number 785911 - Shadoks.
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