Scaling laws for concentration-gradient-driven electrolyte transport through a 2D membrane

Electrolyte transport through porous membranes is central to many applications,Siwy, Bruening, and Howorka (2023); Marbach and Bocquet (2019) such as water desalination,DuChanois et al. (2021) energy harvesting from salinity gradients (osmotic power or "blue energy"), DuChanois et al. (2021); Zhang, Wen, and Jiang (2021); Siria, Bocquet, and Bocquet (2017); Rankin and Huang (2016) energy storage, Aluru et al. (2023); Pomerantseva et al. (2019) nanopore-based sensing,Siwy, Bruening, and Howorka (2023); Ying et al. (2022) and biomimetic ion-based information processing and storage (iontronics).Robin, Kavokine, and Bocquet (2021); Robin and Bocquet (2023); Kamsma et al. (2024); Gkoupidenis et al. (2024) Membranes made from two-dimensional (2D) nanomaterials, including graphene and molybdenum disulfide (MoS₂), exhibit unique properties compared with conventional membranes Sahu and Zwolak (2019) that are of practical interest for applications such as desalination,Zhang et al. (2022); Pakulski et al. (2020); Surwade et al. (2015) osmotic power, Pakulski et al. (2020); Macha et al. (2019); Feng et al. (2016) sensing,Venkatesan and Bashir (2011) and iontronics.Robin et al. (2023) For example, exceptionally high osmotic power densities observed in single-layer MoS₂ membranesFeng et al. (2016) and a 2D nanoporous polymer membraneCheng et al. (2023) have been partially attributed to the large concentration gradients that occur across these ultrathin membranes, which are up to several orders of magnitude larger than those for conventional membranes.

Despite the great promise of 2D membranes, significant knowledge gaps exist about the fluid-transport processes that govern applications. Most of the applications mentioned above, such as desalination and osmotic power, are underpinned by transport processes driven or modulated by electrolyte concentration gradients. Furthermore, in nanopore-based sensing, use of a salt gradient along with an applied electric field has been shown to improve measurement sensitivity significantly over an electric field alone.Wanunu et al. (2010) Nevertheless, although equations have been derived previously to quantify the pressure-driven flow of a pure liquid,Sampson (1891); Heiranian, Taqieddin, and Aluru (2020) the electric-field-driven solution flux Mao, Sherwood, and Ghosal (2014) and electric current of an electrolyte,Hall (1975); Lee et al. (2012) and the concentration-gradient-driven solution and solute fluxes of a solution containing a neutral solute through a 2D membrane,Rankin, Bocquet, and Huang (2019) no theory to date has quantified the relationships between the concentration-gradient-driven fluid fluxes of an electrolyte solution through a 2D membrane and relevant membrane and electrolyte properties. The development of such a theory would aid the predictive design of 2D membranes that are optimized for applications.

Of particular interest is understanding the parameters that control the process of diffusioosmosis, the net flow of a solution with respect to a surface due to a concentration gradient. Diffusioosmosis is responsible for the exceptionally high osmotic power densities measured in boron nitride nanotubes,Siria et al. (2013) and has been invoked to explain even higher osmotic power densities measured for 2D MoS₂ membranes,Feng et al. (2016) as well as hitherto unexplained discrepancies between the ionic conductance driven by concentration gradients and electric fields for graphene and MoS₂ membranes.Macha et al. (2019)

A better understanding of fluid transport through 2D membranes has significance that extends well beyond 2D membranes to thicker membranes, since fluid transport into (and out of) the pores of a thick membrane from (and to) the fluid outside strongly resembles transport through a 2D membrane (Fig. 1) and thus can be described by similar theories. For pressure-induced fluid flow Sisan and Lichter (2011) and electric-field-induced ionic currents Lee et al. (2012) through a pore in a finite-thickness membrane, the flow resistance due to the pore ends has been shown to be well described by the resistance of a 2D membrane. The influence of the pore ends to fluid transport through membranes, known as "entrance effects", becomes dominant as the membrane thickness approaches the pore size,Mao, Sherwood, and Ghosal (2014); Melnikov, Hulings, and Gracheva (2017) which occurs for nanoporous membranes at thicknesses significantly above the atomic scale, or when the flow resistance of the membrane is small, which is the case for carbon nanotube membranes due to their high interfacial slip.Sisan and Lichter (2011); Rankin and Huang (2016) To date, many theoretical models of fluid transport involving concentration gradients in porous media have only considered the flow inside the pores and have neglected entrance effects.Fair and Osterle (1971); Prieve (1982); Lavi and Green (2024); Bonthuis et al. (2011)

In this work, we derive general equations that quantify the variation of the concentration-gradient-driven flow rate, solute flux and electric current as functions of the pore radius, surface charge density and Debye screening length for transport of a dilute electrolyte solution through a circular aperture in an infinitesimally thin planar membrane, as well as the scaling laws of the electric-field-driven flow rate in the thin electric-double-layer limit in the same geometry. From these novel equations, whose accuracy we verify by comparison with finite-element numerical simulations in the Debye–Hückel regime, we determine the underlying scaling laws that govern the concentration-gradient-driven fluxes in the thick and thin electric-double-layer limits. We also determine the parameters that dictate the scaling of the fluxes with surface charge density and electrolyte concentration in simulations both inside and outside the Debye–Hückel regime when the width of the electric double layer in smaller than the pore radius. Using these relationships, we extend the analytical theory to outside the Debye–Hückel regime and determine general scaling laws for the fluxes as a function of surface charge density, pore size and electrolyte concentration in the thin electric-double-layer limit.

This derivation takes a similar approach to that of Ref. 29 for the concentration-gradient-driven transport of a solution containing a neutral solute through an ultra-thin membrane; however, a more rigorous consideration of a perturbation expansion is considered here to decouple the contributions due to the applied concentration gradient and induced electric field gradient in the case of an electrolyte solution. For low-Reynolds-number steady-state transport of a dilute electrolyte solution and assuming that the system can be described by continuum hydrodynamics, the governing equations areProbstein (1994)

	$\displaystyle\epsilon\epsilon_{0}\nabla^{2}\psi+\rho$	$\displaystyle=0,$		(1)
	$\displaystyle-\nabla p-\rho\nabla\psi+\eta\nabla^{2}\bm{u}$	$\displaystyle=0,$		(2)
	$\displaystyle\nabla\cdot\bm{j}_{i}=\nabla\cdot\left[c^{(i)}\bm{u}-D_{i}\nabla c^{(i)}-Z_{i}e\mu_{i}c^{(i)}\nabla\psi\right]$	$\displaystyle=0,$		(3)
	$\displaystyle\nabla\cdot\bm{u}$	$\displaystyle=0,$		(4)

where $\psi$ is the electric potential, $\rho=e\sum_{i}Z_{i}c^{(i)}$ is the total ionic charge density, $\bm{u}$ is the solution velocity, $p$ is the pressure, $\eta$ is the solution shear viscosity, and $\bm{j}_{i}$, $c^{(i)}$, $Z_{i}$, $D_{i}$ and $\mu_{i}$ are the flux density, concentration, valence, diffusivity and mobility, respectively, of species $i$. Moreover, $e$ is the elementary charge, $\epsilon$ is the dielectric constant of the medium (taken to be water here) and $\epsilon_{0}$ is the vacuum permittivity. At the membrane surface, we assume that the solution velocity and ion fluxes satisfy the no-slip ($\bm{u}=0$) and zero flux ($\bm{\hat{n}}\cdot\bm{j}_{i}=0$) boundary conditions, where $\bm{\hat{n}}$ is the unit vector normal to the surface.

We consider fluid flow through a circular aperture of radius $a$ in an infinitesimally thin planar wall, as illustrated in Fig. 1, induced by a concentration difference $\Delta c^{(i)}=c_{\mathrm{H}}^{(i)}-c_{\mathrm{L}}^{(i)}$ between the two sides of the membrane, where charge neutrality in the bulk solution in each reservoir requires that $\sum_{i}Z_{i}c^{(i)}_{\mathrm{H/L}}=0$. The pressure far from the pore is the same on both sides, i.e., $p_{\mathrm{H}}=p_{\mathrm{L}}$. Analogous to Ref. 29, we assume without loss of generality that the ion concentration in the presence of a concentration gradient for an oblate-spheroidal ($\zeta$, $\nu$, $\phi$) coordinate system has the form

$$c^{(i)}(\zeta,\nu)=c_{\mathrm{s}}^{(i)}(\zeta,\nu)\exp{\left(-\frac{Z_{i}e\psi(\zeta,\nu)}{k_{\mathrm{B}}T}\right)},$$

(5)

where $c_{\mathrm{s}}^{(i)}(\zeta,\nu)$ is to be determined. We can relate $\zeta$ and $\nu$ to a cylindrical coordinate system by $r=a\sqrt{(1+\nu^{2})(1-\zeta^{2})}$ in the radial direction and $z=a\nu\zeta$ in the axial direction, for which $0\leq\zeta\leq 1$, $-\infty<\nu<\infty$, and $0\leq\phi<2\pi$.Rankin, Bocquet, and Huang (2019); Morse and Feshbach (1953a)

We assume that $D_{i}$ and $\mu_{i}$ are related by the Einstein relation Probstein (1994) $\mu_{i}=\frac{D_{i}}{k_{\mathrm{B}}T}$, where $k_{\mathrm{B}}$ is the Boltzmann constant and $T$ is the temperature. Non-dimensionalizing the velocity, electric potential, concentrations, spatial gradients, ion diffusivities and ion flux densities according to $\bm{u}=\frac{\epsilon\epsilon_{0}}{\eta a}(\frac{k_{\mathrm{B}}T}{e})^{2}\hat{\bm{u}}$, $c^{(i)}=c_{\infty}\hat{c}^{(i)}$, $\psi=\frac{k_{\mathrm{B}}T}{e}\hat{\psi}$, $\nabla=\frac{\hat{\nabla}}{a}$, $\hat{D}_{i}=D_{0}\hat{D}_{i}$ and $\bm{j}_{i}=\frac{D_{0}c_{\infty}^{(i)}\hat{\bm{j}}_{i}}{a}$, respectively, we can write the ion flux density in Eq. (3) as

$$\hat{\bm{j}}_{i}=\mathrm{Pe}\ c^{(i)}\bm{u}-\hat{D}_{i}\left[\hat{\nabla}c^{(i)}+Z_{i}c^{(i)}\hat{\nabla}\hat{\psi}\right],$$

(6)

where $\mathrm{Pe}=\frac{\epsilon\epsilon_{0}}{\eta D_{0}}(\frac{k_{\mathrm{B}}T}{e})^{2}$ is the Péclet number, Mao, Sherwood, and Ghosal (2014) $D_{0}$ is a characteristic diffusivity, $c_{\infty}=\frac{1}{N}\sum_{i}c_{\infty}^{(i)}$ is the average bulk concentration of the electrolyte solution containing $N$ species, and $c_{\infty}^{(i)}=(c_{\mathrm{H}}^{(i)}+c_{\mathrm{L}}^{(i)})/2$ is the average bulk concentration of species $i$. Assuming that advection of the solute is negligible compared with diffusion ($\mathrm{Pe}\ll 1$), Eq. (6) reduces to

$$\hat{\bm{j}}_{i}=-\hat{D}_{i}\left[\hat{\nabla}\hat{c}^{(i)}+Z_{i}\hat{c}^{(i)}\hat{\nabla}\hat{\psi}\right].$$

(7)

From Eqs. (5) and (7),

$$\hat{\bm{j}}_{i}=-\hat{D}_{i}e^{-Z_{i}\hat{\psi}}\hat{\nabla}\hat{c}_{\mathrm{s}}^{(i)},$$

(8)

where $c_{\mathrm{s}}^{(i)}=c_{\infty}\hat{c}_{\mathrm{s}}^{(i)}$. Inserting Eq. (8) into Eq. (3) and using the Debye–Hückel approximation, $|Z_{i}\hat{\psi}|\ll 1$, gives

$$\hat{\nabla}\cdot\hat{\bm{j}}_{i}=-\hat{D}_{i}\hat{\nabla}\cdot\left[\left(1-Z_{i}\hat{\psi}\right)\hat{\nabla}\hat{c}_{\mathrm{s}}^{(i)}\right]=0.$$

(9)

We assume that each system variable can be represented by a perturbation expansion with respect to the equilibrium value, where $\beta\ll 1$ is a dimensionless quantity that characterizes the perturbation due to the applied concentration difference. Hence, to first order

	$\displaystyle\bm{u}$	$\displaystyle=\beta\bm{u}_{1},$		(10)
	$\displaystyle c_{\mathrm{s}}^{(i)}$	$\displaystyle=c_{\mathrm{s}_{0}}^{(i)}+\beta c_{\mathrm{s}_{1}}^{(i)},$		(11)
	$\displaystyle\psi$	$\displaystyle=\psi_{0}+\beta\psi_{1},$		(12)
	$\displaystyle p$	$\displaystyle=p_{0}+\beta p_{1},$		(13)

where $\bm{u}_{0}=\bm{0}$, since there is no net flow at equilibrium, and $c_{\mathrm{s}_{0}}^{(i)}=c_{\infty}^{(i)}$. Expanding to $O(\beta)$ and assuming negligible contributions due to $Z_{i}\hat{\psi}_{0}$ (i.e., a small equilibrium electric potential), Eq. (9) reduces to

$$\hat{\nabla}^{2}\hat{c}_{\mathrm{s}_{1}}^{(i)}=0,$$

(14)

where $\bm{\hat{n}}\cdot\hat{\nabla}\hat{c}_{\mathrm{s}_{1}}^{(i)}=0$ on the membrane surface. The boundary conditions far from the membrane are $c^{(i)}\rightarrow c_{\mathrm{H}}^{(i)}=c_{\infty}^{(i)}+\frac{\Delta c_{i}}{2}$ and $c^{(i)}\rightarrow c_{\mathrm{L}}^{(i)}=c_{\infty}^{(i)}-\frac{\Delta c_{i}}{2}$ in the upper and lower half planes, respectively, and $\psi\rightarrow 0$; hence, $\beta\hat{c}_{\mathrm{s}_{1}}^{(i)}\rightarrow\pm\frac{\Delta\hat{c}_{i}}{2}$ far from the membrane surface in the upper and lower half planes, respectively. Solving Eq. (14) subject to the boundary conditions on $\hat{c}_{\mathrm{s}_{1}}^{(i)}$ gives Morse and Feshbach (1953b)

$$\beta\hat{c}_{\mathrm{s}_{1}}^{(i)}(\zeta,\nu)=\frac{\Delta\hat{c}_{i}}{\pi}\tan^{-1}{\nu},$$

(15)

which is analogous to the result for a neutral solute.Rankin, Bocquet, and Huang (2019)

Summing Eq. (8) over $i$ for a binary $Z$:$Z$ electrolyte, where $Z_{+}=-Z_{-}=Z$, $\Delta c_{+}=\Delta c_{-}=\Delta c$, $c_{\infty}^{+}=c_{\infty}^{-}=c_{\infty}$ and $c^{+}_{\mathrm{s}}=c^{-}_{\mathrm{s}}=c_{\mathrm{s}}$, we can write the re-dimensionalized total ion flux density $\bm{j}=\bm{j}_{+}+\bm{j}_{-}$ and electric current density $\bm{j}_{\mathrm{e}}=e(Z_{+}\bm{j}_{+}+Z_{-}\bm{j}_{-})$, expanded up to $O(\beta)$ for $Ze|\psi|\ll k_{\mathrm{B}}T$, as

	$\displaystyle\bm{j}=-\beta\nabla c_{\mathrm{s}_{1}}$	$\displaystyle\left\{(D_{+}+D_{-})\left[1+\frac{1}{2}\left(\frac{Ze\psi_{0}}{k_{\mathrm{B}}T}\right)^{2}\right]\right.$
		$\displaystyle-\left.(D_{+}-D_{-})\left(\frac{Ze\psi_{0}}{k_{\mathrm{B}}T}\right)\right\}$		(16)

and

$$\bm{j}_{\mathrm{e}}=\beta Ze\nabla c_{\mathrm{s}_{1}}\left[(D_{+}+D_{-})\left(\frac{Ze\psi_{0}}{k_{\mathrm{B}}T}\right)-(D_{+}-D_{-})\right],$$

(17)

respectively. The total solute flux $J$ and electric current $I$ induced by the concentration difference across the membrane can be obtained from the surface integrals of the total ion flux density and electric current density, respectively, over the pore aperture;Rankin, Bocquet, and Huang (2019) i.e.,

	$\displaystyle J$	$\displaystyle=\iint_{S}\mathrm{d}S\,\bm{\hat{n}}\cdot\bm{j},$		(18)
	$\displaystyle I$	$\displaystyle=\iint_{S}\mathrm{d}S\,\bm{\hat{n}}\cdot\bm{j}_{\mathrm{e}},$		(19)

where $\bm{\hat{n}}=\bm{\hat{\nu}}$ is the unit vector normal to the pore mouth. Inserting Eq. (II) into Eq. (18) and evaluating the integral at the pore mouth $(\nu=0)$ gives

	$\displaystyle J=-a(D_{+}+D_{-})$	$\displaystyle\left\{2\Delta c+\frac{\epsilon\epsilon_{0}\Delta\ln c}{(\lambda_{\mathrm{D}}^{\infty})^{2}}\int^{1}_{0}\mathrm{d}\zeta\,\left[\frac{(\psi_{0}|_{\nu=0})^{2}}{2k_{\mathrm{B}}T}\right.\right.$
		$\displaystyle-\left.\left.\left(\frac{D_{+}-D_{-}}{D_{+}+D_{-}}\right)\frac{\psi_{0}|_{\nu=0}}{Ze}\right]\right\},$		(20)

where $\Delta c/c_{\infty}=(c_{\mathrm{H}}-c_{\mathrm{L}})/c_{\infty}\approx\Delta\ln c$ in the surface contributions to the fluxes at small $\Delta c$, and

$$\lambda_{\mathrm{D}}^{\infty}=\sqrt{\frac{\epsilon\epsilon_{0}k_{\mathrm{B}}T}{2(Ze)^{2}c_{\infty}}}$$

(21)

is the equilibrium (i.e., in the absence of a concentration difference $\Delta c$) Debye screening length of a $Z$:$Z$ electrolyte. Probstein (1994) The first term in Eq. (II) is the bulk contribution to the total solute flux, $J^{(0)}$. Inserting Eq. (17) into Eq. (19) and integrating over the pore mouth gives

	$\displaystyle I=$	$\displaystyle-2a(D_{+}-D_{-})ze\Delta c$
		$\displaystyle+a(D_{+}+D_{-})\frac{\epsilon\epsilon_{0}\Delta\ln c}{(\lambda_{\mathrm{D}}^{\infty})^{2}}\int^{1}_{0}\mathrm{d}\zeta\,\psi_{0}|_{\nu=0},$		(22)

where the first term is the bulk contribution to the electric current, $I^{(0)}$. The $O(\psi_{0}|_{\nu=0})$ term in Eq. (II) and the bulk contribution to Eq. (II) are negligible when the electric field in the pore mouth induced by the difference in ion diffusivities is small. Since the bulk and surface terms in the total solute flux and electric current have different dependencies on the concentration difference, it is not possible to define a single transport coefficient for the total flux; hence, we define the surface contributions $\delta J=J-J^{(0)}$ and $\delta I=I-I^{(0)}$.

For the related problem of electroosmosis through a circular aperture, it has previously been shown using the reciprocal theorem Happel and Brenner (1983); Mao, Sherwood, and Ghosal (2014) that

$$Q=-\frac{1}{\Delta p}\iiint_{V}\mathrm{d}V\,\bm{\bar{u}}\cdot\bm{F},$$

(23)

where $\bm{\bar{u}}$ is the fluid velocity induced by a pressure difference $\Delta p=p_{\mathrm{H}}-p_{\mathrm{L}}$ in the same system geometry for $\Delta c=0$, $\bm{F}$ is the electric body force in the Stokes equation (Eq. (2)), and the integral is over the volume occupied by the fluid. The analytical solution for the pressure-driven flow velocity in this geometry is Rankin, Bocquet, and Huang (2019)

$$\bm{\bar{u}}=-\frac{a\zeta^{2}\Delta p}{2\pi\eta\sqrt{(1+\nu^{2})(\nu^{2}+\zeta^{2})}}\bm{\hat{\nu}}.$$

(24)

For a binary electrolyte, the electric body force $\bm{F}$ is

$$\bm{F}=-\rho\nabla\psi=-e(Z_{+}c^{+}+Z_{-}c^{-})\nabla\psi.$$

(25)

Inserting Eq. (5) for $c^{\pm}$ into Eq. (25) and assuming $Ze|\psi|\ll k_{\mathrm{B}}T$ and a $Z$:$Z$ electrolyte gives

$$-\rho\nabla\psi=\frac{(Ze)^{2}}{k_{\mathrm{B}}T}c_{\mathrm{s}}\nabla\psi^{2},$$

(26)

where $Ze\sinh(\frac{Ze\psi}{k_{\mathrm{B}}T})\nabla\psi=k_{\mathrm{B}}T\nabla\cosh(\frac{Ze\psi}{k_{\mathrm{B}}T})\approx\frac{(Ze)^{2}}{2k_{\mathrm{B}}T}\nabla\psi^{2}$. Since there is zero fluid velocity at equilibrium, expanding Eq. (26) to $O(\beta)$ gives

	$\displaystyle-\rho_{1}\nabla\psi_{0}-\rho_{0}\nabla\psi_{1}$	$\displaystyle=\frac{(Ze)^{2}}{k_{\mathrm{B}}T}\left[c_{\mathrm{s}_{1}}\nabla\psi_{0}^{2}+2c_{\infty}\nabla(\psi_{0}\psi_{1})\right]$
		$\displaystyle=\nabla\left(\frac{\epsilon\epsilon_{0}}{(\lambda_{\mathrm{D}}^{\infty})^{2}}\psi_{0}\psi_{1}\right)+\frac{\epsilon\epsilon_{0}}{2(\lambda_{\mathrm{D}}^{\infty})^{2}}\frac{c_{\mathrm{s}_{1}}}{c_{\infty}}\nabla\psi_{0}^{2}.$		(27)

In the absence of an applied electric field, $\psi_{1}\rightarrow 0$ far from the membrane surface. Hence, there is no net contribution from $\nabla\left(\frac{\epsilon\epsilon_{0}}{(\lambda_{\mathrm{D}}^{\infty})^{2}}\psi_{0}\psi_{1}\right)$ to the total volumetric flow rate in Eq. (23). By inserting Eq. (15) for $c_{\mathrm{s}_{1}}$ into Eq. (II), we can write the electric body force as

$$\bm{F}=\frac{\epsilon\epsilon_{0}}{2(\lambda_{\mathrm{D}}^{\infty})^{2}}\frac{\Delta\ln c}{\pi}\tan^{-1}\nu\nabla\psi_{0}^{2}.$$

(28)

Inserting Eqs. (28) and (24) into Eq. (23) gives

	$\displaystyle Q$	$\displaystyle=\frac{a^{3}\epsilon\epsilon_{0}\Delta\ln c}{2\pi\eta(\lambda_{\mathrm{D}}^{\infty})^{2}}\int^{1}_{0}\mathrm{d}\zeta\,\zeta^{2}\int^{\mathrm{\infty}}_{-\mathrm{\infty}}\mathrm{d}\nu\,\tan^{-1}\nu\frac{\partial(\psi_{0}^{2})}{\partial\nu}$
		$\displaystyle=-\frac{a^{3}\epsilon\epsilon_{0}\Delta\ln c}{\pi\eta(\lambda_{\mathrm{D}}^{\infty})^{2}}\int^{1}_{0}\mathrm{d}\zeta\,\zeta^{2}\int^{\mathrm{\infty}}_{0}\mathrm{d}\nu\,\frac{\psi_{0}^{2}}{1+\nu^{2}}$		(29)

as the total flow rate, by integrating by parts in the second step.

To validate the derived scaling laws, finite-element method (FEM) simulations of concentration-gradient-driven transport of KCl solutions through $0.2$ nm thick membranes containing pores of various radii and surface charge densities, with various upper and lower reservoir concentrations, were carried out with COMSOL Multiphysics (version 4.3a).com Details of these simulations are given in Sec. LABEL:sec:FEM of the supplementary material, which we verified were all within the low Péclet-number regime. Simulations of electrolyte transport with zero electric potential were carried out to calculate $J^{(0)}$ and $I^{(0)}$ (see Sec LABEL:sec:bulk in the supplementary material), where $\delta J$ and $\delta I$ were recovered by calculating $J-J^{(0)}$ and $I-I^{(0)}$, respectively. The theory predicts that the flow rate $Q$ and the surface contributions to the total solute flux, $\delta J$, and electric current, $\delta I$, are linearly related to the logarithm of the concentration difference $\Delta\ln c$. We have verified this scaling for the range of concentration differences $1/100c_{\infty}\leq\Delta c\leq 18/11c_{\infty}$ studied in the numerical simulations (see Figs. LABEL:fig:Delc1–LABEL:fig:Delc3 in the supplementary material). The linear scaling of $Q$, $\delta J$ and $\delta I$ with $\Delta\ln c$ in the linear-response regime has been shown for concentration-gradient-driven electrolyte transport in cylindrical pores in this regime.Rankin and Huang (2016) These scaling relationships are expected for electrolyte transport and differ to the linear scaling of fluxes with $\Delta c$ for the neutral solute case in Ref. 29 as a result of the concentration dependence of the interaction range (Debye screening length) for electrolytes.