Modelling nanofiltration of electrolyte solutions

Advances in Colloid and Interface Science 268 (2019) 39–63

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Historical Perspective

Modelling nanofiltration of electrolyte solutions Andriy Yaroshchuk a,b,⁎, Merlin L. Bruening c, Emiliy Zholkovskiy d a

ICREA, Barcelona, Spain Department of Chemical Engineering, Polytechnic University of Catalonia, Barcelona Tech, Spain c Department of Chemical & Biomolecular Engineering, University of Notre Dame, Notre Dame, IN, USA d F.D.Ovcharenko Institute of Bio-Colloid Chemistry, National Academy of Science of Ukraine, Kyiv, Ukraine b

a r t i c l e

i n f o

a b s t r a c t

Article history: 8 March 2019 Available online 13 March 2019

This review critically examines current models for nanofiltration (NF) of electrolyte solutions. We start from linear irreversible thermodynamics, we derive a basic equation set for ion transfer in terms of gradients of ion electrochemical potentials and transmembrane volume flux. These equations are extended to the case of significant differences of thermodynamic forces across the membrane (continuous version of irreversible thermodynamics) and solved in quadratures for single salts and trace ions added to single salts in the case of macroscopically-homogeneous membranes. These solutions reduce to (quasi)analytical expressions in the popular Spiegler-Kedem approximation (composition-independent phenomenological coefficients), which we extend to the case of trace ions. This enables us to identify membrane properties (e.g. ion permeances, ion reflection coefficients, electrokinetic charge density) that control its performance in NF of multi-ion solutions. Further, we specify the phenomenological coefficients of irreversible thermodynamics in terms of ion partitioning, hindrance and diffusion coefficients for the model of straight cylindrical capillaries. The corresponding expressions enable assessment of the applicability of the popular nanopore model of NF. This model (based on the use of macroscopic approaches at nanoscale) leads to a number of trends that have never been observed experimentally. We also show that the use of the Born formula (frequently employed for the description of dielectric exclusion) hardly leads to meaningful values of solvent dielectric constant in membrane pores because this formula disregards the very solvent structure whose changes are supposed to bring about the reduction of dielectric permittivity in nanopores. We conclude that the effect should better be quantified in terms of ion excess solvation energies in the membrane phase. As an alternative to the nanopore description of NF, we review recent work on the development of an advanced engineering model for NF of multi-ion solutions in terms of a solution-diffusion-electromigration mechanism. This model (taking into account spontaneously arising transmembrane electric fields) captures several trends observed experimentally, and the use of trace ions can provide model parameters (ion permeances in the membrane) from experiment. We also consider a recent model (ultrathin barrier layers with deviations from local electroneutrality) that may reproduce observed feed-salt concentration dependences of membrane performance in terms of concentration-independent properties like excess ion solvation energies. Due to its complexity, practical modelling of nanofiltration will probably be performed with advanced engineering models for the foreseeable future. Although mechanistic studies are vital for understanding transport and developing membranes, future simulations in this area will likely need to depart from typical continuum models to provide physical insight. For enhancing the quality of modelling input, it is essential to improve the control of concentration polarization in membrane test cells. © 2019 Elsevier B.V. All rights reserved.

Contents 1. 2.

Introduction . . . . . . . . . Basic equations and concepts . 2.1. Derivation of equations . 2.1.1. Single salts . .

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⁎ Corresponding author. E-mail address: [email protected] (A. Yaroshchuk).

https://doi.org/10.1016/j.cis.2019.03.004 0001-8686/© 2019 Elsevier B.V. All rights reserved.

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A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

2.1.2. Trace ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution of transport equations for macroscopically homogeneous membranes: single salts and trace ions. 2.2.1. Spiegler-Kedem approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Trace ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Specification of phenomenological coefficients within the scope of a model of straight, narrow capillaries . 3. Nanopore models of NF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Steric exclusion and hindrance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Local-equilibrium partitioning mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Donnan exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Superposition of Donnan exclusion and steric hindrance/exclusion . . . . . . . . . . . . . . 3.2.3. Dielectric exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Solution-diffusion-electromigration models of nanofiltration . . . . . . . . . . . . . . . . . . . . . . . . 4.1. An analytical solution to transport of three ions with different charges . . . . . . . . . . . . . . . . . 4.2. Determining single-ion permeances using NF with trace ions . . . . . . . . . . . . . . . . . . . . . 4.3. “Under-osmotic” operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Deviations from local electrical neutrality in ultrathin barrier layers . . . . . . . . . . . . . . . . . . 5. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.

1. Introduction This critical review explores models of ion and water transport through nanofiltration (NF) membranes and examines some of the implications of these models. In general, transport models fall into two classes, mechanistic models and irreversible thermodynamics descriptions that do not specify a molecular mechanism, although in principle one can relate mechanisms to coefficients in irreversible thermodynamics models. With specific assumptions, the irreversible thermodynamics treatment leads to simpler models such as solution-diffusion, whereas the mechanistic approaches frequently invoke nanoporous materials in which ion exclusion and hindered diffusion and convection occur. This critical review aims to show the relationships between mechanistic and irreversible thermodynamics descriptions and examines some of the challenges in the widely used model of NF as a nanoporous membrane with steric and electrostatic exclusion mechanisms. For interested readers, the supporting information of this critical review considers the basics of irreversible thermodynamics and introduces the fundamental equations for describing fluxes in terms of gradients of electrochemical potentials. Section 2 begins with these fundamental equations and develops expressions for ion and solvent fluxes based on locally linear relationships between fluxes and driving forces in irreversible thermodynamics. This yields expression for fluxes in terms of a minimal number of membrane properties. For the specific cases of single-salt transport and transport of a dominant salt along with trace ions, transport equations greatly simplify. Using one simplified approach, the Spiegler-Kedem model, Section 2 examines volume flows as a function of transmembrane pressure drop, even at pressures below the osmotic pressure of the feed. An extension of this model to the case of one dominant salt and trace ions also affords a convenient (quasi)analytical solution. The additional assumption of no convective coupling leads to the solution-diffusion treatment, which for the case of a dominant salt and trace ions gives straightforward equations for ion passages as a function of ion permeances in the membrane. Although some might argue that irreversible thermodynamics treatments needlessly complicates modelling of membrane transport [1], we think it is often important to understand the assumptions behind common transport models (e.g. Spiegler-Kedem, solution-diffusion, and solution-diffusion-electromigration). Readers who are less interested in the origin of transport equations may wish to skip section 2. Section 3 investigates nanopore models that provide mechanistic approaches for obtaining values for the coefficients in the irreversible thermodynamics treatment. In nm-sized pores steric, Donnan and dielectric exclusion as well as hindrance of diffusion will decrease ion

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42 43 44 44 45 46 46 47 47 48 51 54 54 55 56 57 59 60 61 61

transport. However, Section 3 suggests that the treatment of exclusion and hindrance in nanopores relies on inconsistent assumptions. For example, the steric exclusion treatment takes into account solute sizes but neglects the finite size of molecules in the pore-filling solvent, which is important in sub-nm pores. Additionally Donnan exclusion treats ions as point charges although steric exclusion accounts for their size. The steric/Donnan exclusion treatment predicts some unusual trends in NF that do not agree with experimental observations, although this mechanism does account for trends such as increasing ion rejection with decreasing pore size or increasing ion hydrated radius. Due to experimental challenges in characterizing nanopores (if they exist) in NF membranes, and the questionable assumptions in models of transport through such pores, the irreversible thermodynamics model with constant ion permeances (at a given feed composition) likely provides the most practical engineering model of NF. Section 4 of this critical review presents an analytical solution of the solution-diffusionelectromigration model with constant ion permeances for transport of solutions containing 3 different ions. The model effectively predicts ion rejections, including negative rejections, as a function of volume flow and gives insight into the electric fields that arise in the membrane. Moreover, the use of the solution-diffusion-electromigration treatment for dominant salts and trace ions gives single-ion permeances that describe negative ion rejections. The end of Section 4 considers the possibility that deviations from local electrical neutrality may control ion transport in ultrathin barrier layers. In such thin films, space charge regions that form due to selective exclusion of cations or anions will control transport to increase rejection. Finally, Section 5 summarizes conclusions from the critical review and considers future challenges such as determining ion reflection coefficients and accounting for inhomogeneous concentration polarization, which occurs to some extent in all NF modules.

2. Basic equations and concepts 2.1. Derivation of equations The irreversible thermodynamic analysis starts from these basic equations for solute flux, Ji, and volume flux, Jv, ðeÞ

Ji ¼ −

dμ Pi ∙c ∙ i þ J v ∙τ i ∙ci RT i dx

ð1Þ

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

J v ¼ −χ∙

dp − dx

X i

ci ∙ð1−τi Þ∙

ðeÞ dμ i

! ð2Þ

dx

where Pi are the ion permeabilities in the membrane, ci are the virtual ion concentrations (see below), μ(e) i are the ion electrochemical potentials, x is the membrane coordinate, τi are the ion transmission coefficients (indicative of the extent of convective coupling in the membrane), p is the hydrostatic pressure and χ is the membrane hydraulic permeability defined at zero gradients of electrochemical potentials of all the ions. The absence of such gradients is largely hypothetical in NF (especially under typically non-linear conditions). Below this section derives an expression using a more practical version of hydraulic permeability defined at zero current. Eq. (1) is a system of equations, one for each ion. Moreover, the form of Eqs. (1, 2) implies neglect of the flux of a given ion arising due to the electrochemical potential gradients of other ions at zero transmembrane volume flow. As discussed below in the section devoted to capillary models, this is justified in narrow nanopores. Eqs. (1, 2) can apply to either porous or dense membranes, but in dense membranes the values of τi are often negligible, which leads to the solutiondiffusion-electromigration model. In NF the variation of thermodynamic potentials at the membrane scale is often too large for globally linear irreversible thermodynamics relations to apply. However, one can often conceptually divide the actual membrane barrier into “sub-membranes” (Fig. 1) whose thickness is small enough to ensure local linearity of relationships between fluxes and forces. This procedure is appropriate if the microscopic membrane structure is sufficiently homogeneous to permit averaging over crosssections of those “sub-membranes”. By writing linear equations for each “sub-membrane” and making their thicknesses infinitesimal, one obtains the system of differential equations represented in Eqs. (1, 2). This leads to the so-called continuous version of irreversible thermodynamics where local fluxes vary linearly with forces but global linearity is not required. Notably, the concentrations and hydrostatic pressure in Eqs. (1, 2) are not those in the membrane phase but those in a virtual solution that is defined as a solution that could be in thermodynamic equilibrium with a given “sub-membrane” (see Fig. 1). This follows from the fact that in the globally linear approximation the hydrostatic and osmotic pressures as well as the ion concentrations and electrostatic potential are not those in the membrane phase but those in the adjacent solutions. In the case of the locally linear approach of Eqs. (1, 2) such adjacent solutions do not exist. However their properties can serve as a change of variables in the transport equations following physically from the local thermodynamic equilibrium. Yaroshchuk discusses in detail the applicability of local thermodynamic equilibrium [2].

41

Introducing the virtual solution is the only correct way to proceed within the scope of the continuous version of irreversible thermodynamics where one should not specify any membrane properties other than phenomenological coefficients. The solution of Eqs. (1, 2) with appropriate boundary conditions yields the ion concentrations and electrochemical potentials in the virtual solutions with the phenomenological coefficients as the only specified membrane properties. Calculation of the ion concentrations, the electrostatic potential, and the hydrostatic pressure in the membrane requires more detailed (and, correspondingly, less phenomenological) information about the membrane. Spiegler and Kedem first proposed the virtual (or reference as they termed it) solution concept, [3] and a number of researchers used virtual solutions to model transport in a capillary space-charge model, [4–8] though sometimes without clear understanding of its physical background. In fact, within the scope of the capillary space-charge model the usefulness of the virtual solution concept becomes especially clear. Indeed, the real ionic concentrations inside the capillary depend on the radial coordinate whereas the virtual ones do not. Thus, the equations of transport in the axial direction are more straightforward in terms of virtual quantities than with real quantities. In the absence of chemical reactions, the one-dimensional ion and volume fluxes should be constant in any membrane cross-section perpendicular to the transport axis. Eq. (3) gives the standard expression for μ(e) i ðeÞ

ð cÞ

μ i ≡ μ i þ FZ i φ

ð3Þ

where μ(c) i is the chemical potential of the ion in the virtual solution, F is Faraday's constant, Zi is the ion charge and φ is the electrical potential in the virtual solution. Eq. (4) describes the electric-current density, Ie , X

Ie ≡ F∙

ð4Þ

Z i Ji

i

Substituting Eqs. (1, 3) into Eq. (4) yields ð cÞ

dφ 1 X t i dμ i þ ∙ ∙ dx F i Z i dx

Ie ¼ −g∙

! þ ρek ∙ J v

ð5Þ

where g≡

F2 X 2 ∙ Z ∙ci ∙P i RT i i

ð6Þ

is the electrical conductivity at zero volume flow, Z 2 ∙c ∙P t i ≡ X i 2i i Z j ∙c j ∙P j

ð7Þ

j

Feed Permeate Membrane

are the ion transport numbers at zero volume flow, and ρek ≡ F∙

X

Z i ∙ci ∙τi

ð8Þ

i

Virtual Solutions Membrane Sections

dx

is the electrokinetic charge density. Eq. (5) shows that the total electriccurrent density is a superposition of electromigration, diffusion and streaming currents. In pressure-driven membrane processes like NF, Ie is usually zero. However, this does not mean that the electric field is zero. For the virtual electric field, E, at zero-current conditions one obtains ð cÞ

E≡− Fig. 1. Illustration of a model of membrane transport with virtual solutions that are in local equilibrium with membrane sections (sub-membranes).

dφ 1 X t i dμ i ρ ¼ ∙ − ek ∙J v ∙ dx F i Z i dx g

ð9Þ

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A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

Thus, the electric field contains components related to both diffusion and streaming potentials. Combining Eqs. (9, 3, 1) removes the virtual electric field from the expression for ion flux to give 0 1 ð cÞ X t j dμ ðjcÞ dμ Pi i A þ J v ∙Τi ∙ci −Z i J i ¼ − ∙ci ∙@ ∙ RT dx Z j dx j

ð10Þ

ð cÞ

dμ dp dΠ X − þ ci ∙Τ i ∙ i J v ¼ −χ  ∙ dx dx dx i

χ

χ ≡ t i ρek FZ i ci

is the ion transmission coefficient at zero electric current. (This transmission coefficient includes ion movement due to the streamingpotential gradient.) Eq. (10) clearly demonstrates the coupling between the flows of various ions via spontaneously-arising electric fields. This will be important below for the analysis within the scope of the Solution-Diffusion-Electromigration model. Eq. (10) is similar to modelling transport using the extended NernstPlanck equation, which many groups have applied [9–16]. The extended Nernst-Planck approach assumes that transport through the membrane occurs by diffusion, electromigration, and convection. Alternatively, one can derive the Nernst-Planck equation using Stefan-Maxwell frictional treatment [17]. However, Eq. (10) employs virtual solution concentrations and the Nernst-Planck equation uses real concentrations. Thus, the boundary conditions for Eq. (10) are concentrations and electrical potentials in the solutions just outside the membrane, whereas the Nernst-Planck boundary conditions are the corresponding values just inside the membrane, which requires a value for the partition coefficient. The extended Nernst-Planck approach uses both diffusion and partition coefficients, while Eq. (10) contains only phenomenological coefficients, namely, the ion permeability and transmission coefficient in the membrane. For engineering purposes, fitting data with only phenomenological coefficients (obtainable from membrane-transport experiments only) is advantageous, but the extended Nernst-Planck equation can provide more insight into the transport mechanisms. This is particularly valuable if one can determine partition coefficients experimentally. When taking the microscopic heterogeneity of ion distributions inside pores into account, this method requires a procedure for averaging ion flows over the pore cross-section. Early studies on the space-charge model [4,5,18,19] developed such a procedure for the specific case of straight capillaries with circular and/or slit-like cross sections. The method splits the real electrostatic potential into two parts: one that varies slowly and depends only on the macroscopic trans-membrane coordinate and another that depends on both macroscopic and cross-sectional coordinates inside the pore. The latter included a calculation procedure that effectively relied on the assumption of local thermodynamic equilibrium across the pore cross-section. Although correct, this method may give the wrong impression that it applies only to the specific case of straight capillaries of particular cross-sections. Actually, it implements the general approach of local thermodynamic equilibrium. One can also transform Eq. (2) using the zero electric-current condition. Combining Eqs. (2, 8, 9) gives ð cÞ

!

ð14Þ

ρ2ek g

ð15Þ

The second term in the denominator of Eq. (15) accounts for electroosmosis due to the streaming potential. If all the ion transmission coefficients equal zero (limiting case of solution-diffusion mechanism), Eq. (14) expectedly reduces to the well-known case of an ideally semipermeable membrane where the driving force for the volume flow is the difference between the gradients of hydrostatic and osmotic pressure in the virtual solution (not in the membrane phase). Eq. (10) is convenient for numerical solution because it excludes the electric field, but the equations for different ions are still strongly coupled so obtaining an analytical solution for salt mixtures is difficult (see however Section 4). Nonetheless, useful analytical solutions are possible in the particular cases of single salts or one dominant salt and (any number of) trace ions. 2.1.1. Single salts If there are only two ions (single binary salt), Eq. (10) reduces to Eq. (16) (accounting for the electroneutrality of virtual solutions) Js ¼ −

Ps dμ ∙cs ∙ s þ J v ∙Τs ∙cs RTν dx

ð16Þ

where Js ≡

Jþ J ¼ − νþ ν−

ð17Þ

is the salt flux (note that under zero electric-current conditions the ion fluxes must be stoichiometric), ν+ and ν− are the stoichiometric coefficients of the cation and anion, respectively, in the salt, and ν = ν+ + ν−. The variables J+ and J− are the cation and anion fluxes, respectively. ð cÞ

μ s ≡ ν þ μ þ þ ν − μ ð−cÞ

ð18Þ

is the salt chemical potential, Ps ≡

ðZ þ −Z − Þ∙P þ P − Z þ P þ −Z − P −

ð19Þ

is the salt diffusion permeability of the membrane, and Τs ≡ τþ −t þ ∙ðτþ −τ− Þ ≡ τ þ ∙t − þ τ− ∙t þ

ð20Þ

is the salt transmission coefficient. Eq. (14) reduces to J v ¼ −χ  ∙

  dp dμ −σ s ∙cs ∙ s dx dx

ð21Þ

where ð12Þ

σ s ≡ 1−Τþ ≡ 1−Τ − ≡ 1−Τ s

ð22Þ

is the zero-current salt reflection coefficient.

where ð cÞ dΠ X dμ i ≡ ci ∙ dx dx i

1 þ χ∙

ð11Þ

X dμ dp dΠ ρ2ek − þ ∙J þ ci ∙Τi ∙ i J v ¼ −χ∙ dx dx g v dx i

!

where χ∗, the hydraulic permeability at zero current, is

where Τi ≡ τi −

is the gradient of osmotic pressure, Π, in the virtual solution. Solving Eq. (12) for Jv yields

ð13Þ

2.1.2. Trace ions Assuming that trace-ion concentrations are low enough that they do not influence the dominant salt transport, Eq. (16) still applies to the

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

dominant salt. In deriving equations for the fluxes of the trace ions, their transport numbers are negligible so one can neglect terms containing trace-ion transport numbers on the right-hand side of Eq. (10). However, the first term in the parentheses of Eq. (10) remains because a chemical potential gradient can be large even if the concentration of the corresponding species is low [20]. Taking this into account (and neglecting the contribution of traces to electrokinetic-charge density and electrical conductivity) yields ð cÞ

Jt ¼ −

P t ðcs Þ dμ t 1 dμ ∙ct ∙ −Z t ∙θs ðcs Þ∙ ∙ s RT ν dx dx

where csp is the salt permeate concentration. Using Eqs. (27, 29, 30), one can obtain −

csp −ð1−σ s ðcs ÞÞ∙cs dcs ¼ Jv ∙ dx P s ðcs Þ∙Φs ðcs Þ

ð23Þ

Zcsf Jv L ¼ csp

where t þ ðcs Þ t − ðcs Þ þ θs ðcs Þ ≡ Zþ Z−

ð24Þ

P t ðcs Þ∙ðτ þ ðcs Þ−τ− ðcs ÞÞ Τt ðcs Þ ≡ τt ðcs Þ−Z t ∙ Z þ P þ ðcs Þ−Z − P − ðcs Þ

ð25Þ

   ctp dct Τt ðcs Þ Τs ðcs Þ−csp =cs 1 dlnðγt ðcs ÞÞ ¼ J v ∙ ct ∙ þ ∙ Z t ∙θs ðcs Þ− ∙ − dx P s ðcs Þ P t ðc s Þ P t ðcs Þ Φs ðcs Þ dlnðcs Þ



ð26Þ where γt(cs) is the activity coefficient of the trace and Φs(cs) is the differential salt osmotic coefficient in the virtual solution defined as sÞ . Eq. (26) is a first-order ordinary differential equation Φs ðcs Þ ≡ 1 þ dlnðγ dlnðcs Þ

with variable coefficients that can always be solved in quadratures (see below). 2.2. Solution of transport equations for macroscopically homogeneous membranes: single salts and trace ions This section assumes that the phenomenological coefficients do not explicitly depend on the macroscopic trans-membrane coordinate, x. This implies that the membrane is macroscopically homogeneous. However, the coefficients can vary arbitrarily with the virtual salt concentration to give Js ¼ −

P s ðcs Þ dμ ∙cs ∙ s þ J v ∙ð1−σ s ðcs ÞÞ∙cs RTν dx

  dp dμ −σ s ðcs Þ∙cs ∙ s J v ¼ −χ  ðcs Þ∙ dx dx cs ∙

  dμ s dlnðγ s Þ dcs dcs ≡ RTν∙ 1 þ ∙ ≡ RTν∙Φs ðcs Þ∙ dlnðcs Þ dx dx dx

J s ¼ J v ∙csp

P s ðcs Þ∙Φs ðcs Þ dcs csp −ð1−σ s ðcs ÞÞ∙cs

ð32Þ

where csf is the salt concentration in the solution at the feed membrane surface. Eq. (32) provides a relationship between the solute concentrations in the feed and permeate and the trans-membrane volume flow. Employing Eqs. (27, 29) and taking into account that dp ≡ dx

(Note that the plus and minus subscripts always refer to the cation and anion of the dominant salt, respectively, and t denotes the trace ion.) Τt(cs) is a zero-current transmission coefficient of the trace ion. Eqs. (23–25) explicitly denote that all the phenomenological coefficients (including those related to the trace) are functions of the virtual dominant-salt concentration only (due to the low concentrations of trace ions). Using the definitions of chemical potentials in terms of activity coefficients and concentrations, the expression for the gradient of dominant-salt concentration (obtainable from Eq. (16)), and the NF boundary condition for the dominant salt, Js = csp ∙ Jv, where csp is the dominant-salt concentration in the permeate, and the NF boundary condition for the trace, Jt = ctp ∙ Jv, where ctp is the trace concentration in the permeate, Eq. (14) becomes 

ð31Þ

Separation of variables and integration of this equation over the membrane (active layer) thickness yields

! þ J v ∙Τt ðcs Þ∙ct

43

ð27Þ ð28Þ

dp P s ðcs Þ∙Φs ðcs Þ 1 ∙ ¼ þ RTν∙σ s ðcs Þ∙Φs ðcs Þ dcs χ  ðcs Þ csp −ð1−σ s ðcs ÞÞ∙cs

ð30Þ

gives ð33Þ

One can also separate variables and integrate this expression over the membrane thickness to obtain an equation for the transmembrane hydrostatic-pressure difference, Δp. Zcsf σ s ðcs Þ∙Φs ðcs Þdcs

Δp ≡ pð0Þ−pðLÞ ¼ RTν∙ csp

Zcsf þ csp

P s ðcs Þ∙Φs ðcs Þ dcs ∙ χ  ðcs Þ csp −ð1−σ s ðcs ÞÞ∙cs

ð34Þ

Eqs. (32, 34) are a parametric relationship between the transmembrane volume flow and the hydrostatic-pressure difference. These equations allow the salt diffusion permeability, salt reflection coefficient and membrane hydraulic permeability at zero current to vary with the virtual salt concentration. They also account for the solution non-ideality through the differential osmotic coefficient of the virtual solution. The differential osmotic coefficient dependence on concentration can be obtained from independent experiments on bulk solutions, and the literature contains corresponding information for a large number of solutes [21,22]. Remarkably, as long as the membrane active layer can be considered macroscopically homogeneous, Eqs. (32, 34) apply to any mechanistic model of membrane transport. Eq. (34) simplifies considerably with the assumption that the membrane mechanical permeability at zero current is concentrationindependent. As an example of such a case, within the scope of the Donnan-exclusion model a concentration dependence of mechanical permeability stems from the so-called electro-viscosity effect, whose impact is generally b10–20% and is especially weak for the narrow pores typical for NF. On the contrary, the same model predicts a much larger dependence of salt diffusion permeability and salt reflection coefficient on virtual concentration. With constant mechanical permeability, using Eqs. (32) and (34) becomes Zcsf Δp ¼ RTν∙

σ s ðcs Þ∙Φs ðcs Þdc þ csp

ð29Þ

dp dcs ∙ dcs dx

Jv L χ

ð35Þ

Eq. (35) highlights the physical meaning of the terms: the first relates to osmotic-pressure while the second stems from dissipation. With the additional assumption of a concentration-independent salt

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

1.0

200

reflection coefficient, Eq. (35) further reduces to this very simple relationship

0.8

150

Jv L χ

ð36Þ

Δ P (bar)

Δp ¼ σ s ∙ΔΠ þ where

0.6

100 0.4 50

0.2

Zcsf ΔΠ ≡ RTν

Φs ðcs Þdc

ð37Þ

0

is the thermodynamic definition of the osmotic-pressure difference between the feed and permeate solutions. Remarkably, Eq. (36) still accounts for the solution non-ideality and allows an arbitrary concentration dependence of the membrane salt diffusion permeability. 2.2.1. Spiegler-Kedem approximation This subsection considers the Spiegler-Kedem approximation, which assumes that all the phenomenological coefficients in Eqs. (32, 34) are concentration-independent [3]. Additionally, it neglects the nonideality of the virtual solution (Φs(cs) ≡ 1). Yaroshchuk investigated the error associated with the use of this approximation by considering Donnan exclusion (with constant fixed-charge density) as an example of a mechanism for the dependence of phenomenological coefficients on the virtual salt concentration [23]. These dependences did not change the general picture of the rejection-flow relationship, and even semi-quantitatively the Spiegler-Kedem approximation worked well. With constant coefficients, the integral in the right-hand side of Eq. (32) has an analytical expression, which gives this well-known result for the salt rejection, Res Res ≡ 1−

0.0 0

csp

csp 1− expð−Pes ∙ð1−σ s ÞÞ ¼ σ s∙ 1−σ s ∙ expð−Pes ∙ð1−σ s ÞÞ csf

ð38Þ

Res

44

50

100

150 200 Jv (μm/sec)

250

300

Fig. 2. Plot of transmembrane pressure drop (ΔP, red line, calculated with Eq. (41)) and salt rejection (Res, dashed blue line, calculated with Eq. (38)) versus volume flux for a membrane with a hydraulic permeance of 1.7 μm/(bar sec), a salt permeance of 40 μm/s, and σs=0.99. The feed osmotic pressure was 28 bar. Membrane parameters are similar to those for a DL4040F1020 GE NF membrane, as determined in reference [24]. The black line depicts the asymptotic linear dependence.

transmembrane pressures than at “over-osmotic” pressures. This is clear in the limiting case of small Péclet numbers where Eq. (41) reduces to

1 Π f ∙σ 2s Δp ≈ J v ∙ þ A B

! ð42Þ

In Eq. (42), A ≡ χ  =L

ð43Þ

where Pes ≡

is the hydraulic permeance of the membrane and

Jv L Ps

ð39Þ

According to Eq. (38), the rejection tends to the salt reflection coefficient at sufficiently large Péclet numbers, Pes. At small Pes rejection increases linearly according to Eq. (40) Res ≈ σ s ∙Pes

ð40Þ

Thus, at low trans-membrane volume flows, salt rejection is inversely proportional to the diffusion permeability of barrier layer. In the same Spiegler-Kedem approximation, from Eq. (34) one obtains Δp ¼ Π f ∙σ 2s ∙

1− expð−Pes ∙ð1−σ s ÞÞ J L þ v 1−σ s ∙ expð−Pes ∙ð1−σ s ÞÞ χ 

ð41Þ

where Πf is the osmotic pressure in the feed solution. At larger transmembrane volume flows (high Pes), this relationship predicts the well-known asymptotically-linear dependence of pressure drop on volume flow with a non-zero intersection of its extension with the hydrostatic-pressure axis (see Fig. 2). This intersection occurs at the product of the feed osmotic pressure and the square of the salt reflection coefficient and will only occur at the feed osmotic pressure if the reflection coefficient is unity. Importantly, Eq. (41) and Fig. 2 show that the volume flow is N0 even at transmembrane pressure drops below the osmotic pressure, Πf, of the feed solution (28 bar in the figure). This occurs because the rejection is low enough that the osmotic pressure difference between the feed and permeate solution is less than the transmembrane pressure drop. However, the slope of the pressure-vs-flow dependence is larger at these low

B ≡ P s =L

ð44Þ

is the solute (diffusion) permeance of the membrane. In contrast, at “over-osmotic” pressures where the Péclet number is not small, the slope in a plot of Δp versus Jv approaches 1/A. Thus, Eq. (42) shows that the osmotic-pressure phenomena give rise to an additional apparent mechanical resistance at low transmembrane pressures. The extent of flux reduction at “under-osmotic” pressures depends on the combination of membrane properties represented by A ∙ Πf ∙ σ2s /B. If this parameter is large the flux reduction is strong so the flux at “under-osmotic” pressures is almost negligible (RO-like behavior). When the parameter is small the flow-pressure relationship is practically linear everywhere (UF-like behavior). One of the characteristic features of NF is that it can exhibit both kinds of behavior depending on the solute. This becomes especially interesting in the case of mixed solutions (see Section 4 below).

2.2.2. Trace ions Using the expression for the gradient of virtual dominant-salt concentration (Eq. (31)), the solution to Eq. (26) can become 0 ct ðxÞ ¼ expð−F ðxÞÞ∙@ctf −ctp ∙ J v ∙

Zx 0

1 expð F ðx0 ÞÞ 0 A dx P t ðcs ðx0 ÞÞ

ð45Þ

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

where Zx F ðxÞ ≡ −J v ∙ 0

Τt ðx0 Þ 0 dx P t ðx0 Þ ð46Þ

  γt csf θs ðcs Þ∙Φs ðcs Þdlnðcs Þ− ln γ t ðcs ðxÞÞ 

Zcsf þ Zt ∙ cs ðxÞ

At x = L, ct(L) = ctp (the trace permeate concentration), the reciprocal transmission, ft, is ft ≡

ctf 1 ≡ ¼ expð F ðLÞÞ þ J v ∙ ctp 1− Ret

ZL 0

expð F ðxÞÞ dx P t ðcðxÞÞ

2 f t ¼ ð1−T s ∙ f s Þ

b 6 ∙f s ∙4

1 þ K∙ ð1−T s Þa

Zf s

ρek ≡ FZ þ ν þ cs ∙ðτþ −τ− Þ

1

3 7 5 yb ∙ð1−T s ∙yÞaþ1 dy

ð48Þ

where a≡

T t Ps T s Pt

ð49Þ

  tþ t− b ≡ Zt ∙ þ Zþ Z− K≡

ð50Þ

Ps Pt

ð51Þ

Yaroshchuk and Ribitsch [20] provide the derivation of an equivalent expression. That study also investigated the applicability of the approximation of constant coefficients when using the Donnan-exclusion model without steric hindrance for specification of the concentration dependence of coefficients in Eqs. (46, 47). The approximation of constant coefficients reproduced all the qualitative features of the Donnan-exclusion model, and reciprocal transmission coefficients agreed to within 20%. For the reciprocal transmission of the dominant salt, the SpieglerKedem approximation yields  fs ¼ 1 þ

 1 −1 ∙ð1− expð−Pes ∙T s ÞÞ Ts

ð52Þ

At Ts, Tt = 0 (solution-diffusion-electromigration model), Eqs. (48, 52) reduce to this very simple result [25]. b

f t ¼ f s þ K∙ f s ¼ 1 þ Pes

b

f s− f s 1−b

• Dominant (single) salt reflection coefficient, σs, or the related salt transmission coefficient, Τs ≡ 1 − σs. Eq. (20) describes this coefficient. • Membrane diffusion permeability to the dominant (single) salt, Ps. This parameter controls the rate of increase of the dominant (single)-salt rejection as a function of transmembrane volume flow (see Eq. (40)). Eq. (19) defines Ps. • Membrane electrochemical activity with respect to the dominant salt, θs, defined in Eq. (24). Although this does not explicitly manifest itself in the transport of the dominant salt, it can greatly impact the transport of trace ions. • Electrokinetic charge density, ρek, defined by Eq. (8). This general definition can be specified for the case of one dominant salt

ð47Þ

where ctf is the trace concentration in the feed. This relationship accounts for the non-ideality of virtual solution for both dominant salt and the trace ion. However, explicit accounting for the non-ideality of the trace ion requires values for its activity coefficient in a range of virtual solutions containing the dominant salt. This information is not readily available. In the approximation of composition-independent phenomenological coefficients and ideal solution (Spiegler-Kedem model extended to include trace ions) most of the integrals in Eqs. (46, 47) have analytical expressions that lead to Eq. (48)

a

45

ð53Þ ð54Þ

Eqs. (48, 52) show that in NF of single salts and trace ions, ion transmission (1-rejection) depends on the following membrane phenomenological coefficients:

ð55Þ

• Membrane permeability to the trace ion, Pt.

The next sub-section explores how these coefficients may relate to such properties as ion partition and hindrance coefficients within the scope of capillary models. Section 3 subsequently investigates these coefficients within the scope of nanopore equilibrium and kinetic mechanisms suggested in the literature. 2.3. Specification of phenomenological coefficients within the scope of a model of straight, narrow capillaries This section develops a procedure to relate phenomenological coefficients to membrane structure and physico-chemical properties. The procedure starts with this relationship between the local flux, ji(r), of the i-th ion, the gradient of its electrochemical potential and the local volume-flow velocity, v(r). ðeÞ

ji ðr Þ ¼ −

βi ðr Þ∙Di ∙Γ i ðr Þ dμ i ∙ci ∙ þ ci ∙α i ðr Þ∙Γi ðr Þ∙vðrÞ RT dx

ð56Þ

In this equation Di is the ion diffusion coefficient in solution, Γi is the coefficient for ion partitioning between a point inside the capillary (pore) and the virtual solution, βi is the coefficient of diffusion (steric) hindrance (generally may include also other hindrances, such as electrostatic mechanisms), and αi is the coefficient of local convective steric hindrance, which quantifies the extent of coupling between the convective flow of volume and ion (αi(r) = 1 means complete coupling). These equations postulate that the local gradients of (electro)chemical potentials drive local solute flows relative to the movement of the center of mass (which is identified with the movement of volume assuming the solutions are dilute although not necessarily ideal). This assumption is based on the combination of Fick's and Ohm's laws and the StokesEinstein relationship between diffusion coefficient and electrochemical mobility. In general, the coefficients in Eq. (56) are functions of the radial coordinate, r, inside the capillary. This equation demonstrates one of the advantages of the concept of virtual solution and virtual quantities: although the coefficients generally depend on the micro-coordinate (for example, the radial coordinate inside a nanopore), the virtual concentration and gradient of electrochemical potential do not. This will be useful below for averaging the local fluxes over the capillary crosssection. In real porous systems the flows and gradients are not onedimensional. Capillary models consider all the flows and gradients 1D, which is a considerable simplification. The fluid velocity is a superposition of pressure-driven and (electro) osmotic components. Generally, the velocity profiles (at the capillary cross-section scale) of these two components are different. The

46

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

(electro)osmotic flow depends on the gradients of electrochemical potentials of all the ions. Thus, even at zero net volume flow (fluid velocity averaged over the capillary cross-section), generally there are contributions to the flux of a given ion that are proportional to the gradients of electrochemical potentials of other ions. Yaroshchuk [2] developed an approach to the calculation of phenomenological coefficients taking into account such direct coupling of ion flows. However, the formulation only included the limiting case of complete local convective coupling (αi(r) = 1). Meanwhile, for modelling of NF membranes with their very narrow pores, this approximation is clearly insufficient. The same study demonstrated that in very narrow pores the difference between electro(osmotic) and pressure-driven velocity profiles becomes insignificant and one can consider both of them parabolic as in pressure-driven flows. The next step is averaging of local flows over the capillary crosssection. For capillaries of circular cross-section, Eq. (57) give the average flux Ji over the capillary cross-section. ðeÞ

Ji ¼ −

ci Di dμ i 2 ∙ ∙ ∙ RT dx r2p

þ ci ∙J v ∙

4 ∙ r 4p

Zrp

Zrp rβi ðrÞ∙Γ i ðr Þdr 0

 rα i ðr Þ∙Γi ðrÞ∙ r 2p −r 2 dr

ð57Þ

0

where rp is the pore radius. This equation assumes a parabolic flowvelocity profile, and Jv is the average volume flux (in m/s) through the capillary. Using Eq. (57) and Eq. (1), one can obtain these expressions for the ion permeabilities and transmission coefficients.

Pi ≡

2Di γ ∙ r 2p

4 τi ≡ 4 ∙ rp

Zrp

Zrp rβi ðr Þ∙Γ i ðr Þdr

ð58Þ

0

 rα i ðr Þ∙Γi ðr Þ∙ r 2p −r 2 dr

ð59Þ

0

In this equation γ is the porosity of the membrane with cylindrical pores. Eqs. (57–59) take into account that virtual concentrations and gradients of electrochemical potentials can move outside of the integrals because they are properties of the virtual solution and do not depend on r. These equations apply to capillaries of circular crosssections, but one can consider capillaries with other cross-sectional geometries similarly. Further details on various aspects of capillary models related to charged nano-pores can be found in [26]. 3. Nanopore models of NF 3.1. Steric exclusion and hindrance The original models of steric exclusion/hindrance aimed to describe ultrafiltration of macromolecules [27–29] based on “geometric” exclusion of the centers of solute molecules from near-pore-wall layers and hydrodynamic interactions of solutes with the solid pore surfaces. Later, many research groups applied these macroscopic models to NF, where the pore radii approach molecular dimensions [9–12,30–33] and the continuum assumptions of the model likely do not apply (see below). Additionally, most expressions for hindrance factors were developed with neutral solutes and neutral pores, although they are widely used with ions in charged pores. Perhaps the correlations with neutral solutes are a reasonable first-order approximation for ion hindrances, but this subject needs further investigation. At the high-end of NF pore sizes of several nanometers, the continuum assumption should be more accurate and rejected solutes will be significantly larger

Excluded Area

Solute

Pore Fig. 3. Representation of the steric exclusion model for partitioning of a solute in a pore. The center of the dashed circle (solute) can only reside within the white area of the pore due to steric exclusion. This creates the shaded region where the center of the solute is excluded [27–29].

than the solvent. However, such membranes will show relatively low ion rejections. Fig. 3 illustrates the steric exclusion model, where the distance between the center of the solute and the pore wall must be greater than the solute radius, i.e. no part of the solute can penetrate the pore wall so the center of the solute cannot reside outside of the white circle in Fig. 3. This leads to the ion partition coefficient, Γi, in Eq. (60), where λi is the ratio of the solute radius to the pore radius. Γi ¼ ð1−λi Þ2 ; λi b1

ð60Þ

(for λi N 1, Γi = 0). This partition coefficient, the average concentration inside the pore divided by the concentration outside the pore, is simply the cross-sectional area represented by the white circle in Fig. 3 divided by the cross-sectional pore area (or the cross-sectional area in which the center of the solute can reside divided by the total area). In addition to exclusion from the pore, hindered convection and diffusion also limit transport. Dechadilok and Deen reviewed diffusion and convection hindrance factors obtained through fitting of macroscopic hydrodynamic calculations (which may not apply at nanopore dimensions) [34]. These hindrance factors represent the theoretical ratio of fluxes obtained with and without accounting for steric constraints and particle-wall interactions in diffusion or convection. The hindrance factors occur in addition to steric effects on partitioning. Fig. 4a shows the product of the partition coefficients calculated using Eq. (60) and hindrance factors obtained from literature correlations (eq. (16) of reference [34] for the diffusion-reduction hindrance factor, λi b 0.95, and eq. (18) of this reference for the solute-convection reduction hindrance factor). Notably, the hindrance to solute diffusion is much stronger than the hindrance to its convection. In the products of steric exclusion (partition coefficient) and hindrance factors, the partition coefficient plays a prominent role, especially for convective hindrance. In fact, the convective hindrance factor is larger than one over the whole range of λi, as Fig. 4b shows. This factor is large primarily because the solute concentration is low at the pore walls where the convective velocity is lowest. However, as the particle radius approaches the pore radius, the particle essentially flows with the solvent and the convection hindrance factor approaches 1. In addition to applying continuum hydrodynamic models at the nm scale, another possible challenge in the steric exclusion model for NF is that it accounts for the finite size of the solute using the exclusion zone, but it disregards the finite size of solvent molecules. In ultrafiltration or microfiltration of macromolecules or nanoparticles, neglecting the solvent size is reasonable because the solute radius is much larger than that of the solvent. However, for ions and small-molecule solutes, this assumption may not hold. Moreover, as Eq. (60) and Fig. 4a show, steric exclusion becomes especially strong only when the pore size approaches the solute size. Interestingly, the self-diffusion coefficient of water at 20 °C (2 × 10−9 m2/s) [35] is essentially the same as the diffusion coefficient of chloride ions in water. Accordingly, the Stokes' radii estimated from the (self)-diffusion coefficients of chloride and water are the same, in contrast to the large difference required for the

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

47

Fig. 4. (a) Plots of the product of the partition coefficient (Γi, Eq. (60)) and the hindrance factors for diffusion and convection in a cylindrical pore as a function of the ratio, λi, of the solute radius to pore radius [34]. (b) Convective and diffusive hindrance factors as a function of λi, as calculated using correlations in [34].

continuum treatment of the solvent. If one accounted for exclusion of both chloride and water from the near-pore-wall layer, there would be no steric exclusion. Even for larger hydrated solutes, an excluded layer of water would still decrease steric exclusion. One might argue that the water molecules may specifically interact with the surface of the membranes and can approach closely to the surface, whereas ions do not. However, specifically adsorbed water and ions will both likely be immobile and simply reduce the membrane pore size. Related challenges may apply to the hydrodynamic calculations of diffusion and convective hindrance. Recent studies from the Freger group suggest that the pore sizes obtained from steric exclusion/hindrance models and filtration data are inconsistent with diffusion coefficients obtained with attenuated total reflectance spectroscopy [36–38]. Moreover, other work suggests minimal convective coupling in NF [39,40]. These data along with the challenges mentioned above for applying the steric exclusion/hindrance model to nm-sized pores call into question the applicability of this approach for modelling NF. Despite these inconsistencies, many research groups employed the steric exclusion/hindrance mechanism to model NF [9–12,30–33,41]. The standard approach uses hydraulic permeabilities, non-charged solute rejections, and the steric hindrance/exclusion model to fit experimental data and obtain the pore size [9]. Different approaches assume either uniform cylindrical pores or a distribution of pore sizes in the membrane barrier layer [42]. Of course, the pore size distribution introduces additional fitting parameter(s). Moreover, for NF of ions, additional factors including Donnan and dielectric exclusion may decrease the partition coefficient. The section below discusses these mechanisms.

mechanisms taking into account the inhomogeneity of the ion distribution inside nanopores (finite extent of overlap of electric double layers) [43]. This analysis used the standard space-charge model [4] and concluded that in the narrow (b1 nm) pores typical of NF membranes one can neglect the inhomogeneity. Therefore, this subsection uses the fine-pore (or Teorell-MeyerSievers) model [44–46] to explore Donnan exclusion. In this case, the ion partition coefficients are independent of micro-coordinate inside the pores and constrained by local electroneutrality. Additionally, due to the local thermodynamic equilibrium, all the ion partition coefficients can be expressed through the (local) dimensionless Donnan potential, ψ ≡ FΨ RT where Ψ is the dimensional Donnan potential. Eq. (62) gives the local electrical neutrality condition in the membrane X

Z i ci ∙ expð−Z i ψÞ þ Z X cX ¼ 0

ð62Þ

i

where ZX and cX are the valence and concentration of fixed charges, respectively. (In principle, at this stage one could even account for the non-ideality of the virtual solution.) Due to the electroneutrality of the virtual solution X

Z i ci ¼ 0

ð63Þ

i

At given virtual concentrations of ions and fixed charges, Eq. (62) contains only one unknown, the Donnan potential. For single salts, one can transform Eq. (62) to

3.2. Local-equilibrium partitioning mechanisms 3.2.1. Donnan exclusion Donnan exclusion results from electrostatic interactions of ions with fixed electric charges on the pore walls. The fixed charge may result from adsorbed ions or ionized functional groups in the membrane, e.g. deprotonated carboxylic acids. Such charges give rise to electrical double layers next to the wall surface, and the thickness of these layers varies with the Debye screening length, κ−1, which depends on the concentration, c, of a 1:1 electrolyte with monovalent ions according to Eq. (61)

κ

−1

≡

sffiffiffiffiffiffiffiffiffiffiffiffiffi RTεε0 2

2F c

ð61Þ

where ε is the medium relative dielectric constant, and ε0 is the dielectric permittivity of free space. The screening length ranges from ca. 1 nm in 100 mM solutions up to ca. 10 nm in 1 mM solutions of 1:1 monovalent salts. Wang and coworkers examined Donnan exclusion

expð−Z þ ψÞ− expð−Z − ψÞ þ

Z X cX ¼0 Z þ ν þ cs

ð64Þ

For symmetrical salts, the solution of Eq. (64) is ψ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ∙ ln X þ X 2 þ 1 Zþ

ð65Þ

where X≡

Z X cX 2Z þ ν þ cs

ð66Þ

is the dimensionless fixed-charge concentration. Eqs. (65, 66) show that the Donnan potential increases with increasing fixed-charge concentration and decreases with increasing virtual salt concentration. Within the scope of the fine-pore model, the salt transmission coefficient in Eq. (20) (which controls the salt rejection at sufficiently large

48

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

trans-membrane volume flows) becomes Τs ¼

1



Z þ βþ Dþ α − −Z − β− D− α þ ∙ expð−ðZ þ þ Z − ÞψÞ Z þ βþ Dþ expð−Z þ ψÞ−Z − β− D− expð−Z − ψÞ

ð67Þ

where α  ; β are cross-section-averaged convection- and diffusionreduction factors (see below). In the simplest case of a (1:1) salt with ions of equal size (i.e. α+ = α−, β+ = β−), Eq. (67) reduces to Τs ¼

α α ≡ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi coshψ X2 þ 1

ð68Þ

where α ≡ α þ ¼ α − . Eq. (68) shows that when the virtual salt concentrations becomes noticeably lower than the fixed-charge concentrations (large X2) the salt limiting transmission (one minus limiting rejection) should become a linear function of feed concentration (if the fixed-charge density is concentration-independent). Such a strong dependence does not occur experimentally, rather an approximate square-root dependence often appears. This made researchers think about possible mechanisms for reduction of the fixed-charge concentration with decreasing salt concentration. One of the suggested mechanisms includes the adsorption of anions according to a Freundlich isotherm, cX = kf ∙ cns [47,48]. If one assumes that the Freundlich constant, n, is around ½ (that is the fixed charge density increases proportionally to the square root of salt concentration) Eqs. (66, 68) show that, indeed, the limiting salt transmission increases sub-linearly with concentration in agreement with experiment. However, the hypothesis of anion adsorption lacks physico-chemical grounds and so does the Freundlich isotherm. (A recent paper also suggests that such behavior could result from a very high affinity for protons relative to other cations [49].) Even more complex charge-formation mechanisms involving counterion binding and the existence of amphoteric groups have been invoked in the literature [50–53]. Although they could explain some of the trends observed experimentally the number of fitting parameters in such models was too large for those parameters to be determinable in a unique way. A simple physical mechanism, variation in the dissociation of weakacid groups with changing salt concentration, can also give rise to weaker dependences of salt rejection on feed concentration. Combining the adsorption of hydrogen ions according to the Langmuir-Stern isotherm [54] and Eq. (64) gives Eq. (69), expð−ψÞ− expðψÞ þ

X0 ¼0 1 þ K a ∙ expð−2:3∙pH−ψÞ

ð69Þ

which for simplicity considers (1:1) salts where X0 is the maximum dimensionless fixed-charge concentration (occurring at full dissociation), and Ka is the dimensionless association constant of hydrogen ions and weak-acid groups. Fig. 5 shows the salt transmission coefficient (one minus reflection coefficient) vs. feed concentration. Indeed, the concentration dependence is noticeably less at low pH values, and the transmission coefficient is approximately proportional to the square root of concentration because decreasing the salt concentration makes the concentration of hydrogen ions in the pores (electrostatic adsorption) higher. This reduces the degree of dissociation. At higher pH values, the dependences in Fig. 6 become stronger and expectedly tend to the case of constant charge at pH values where all the groups dissociate fully irrespective of the salt concentration. Of course, one can select the parameters of the model such that the slower concentration dependence occurs at higher pH. 3.2.2. Superposition of Donnan exclusion and steric hindrance/exclusion Although the physico-chemical foundations of the steric exclusion/ hindrance model for NF are questionable, the model nonetheless semi-quantitatively reflects these important features.

Salt Transmission Coefficient



0.1

pH3 pH4 pH5

0.01 pH6

0.001 0.001

0.01 0.1 Concentration (mM)

1

Fig. 5. Salt transmission coefficient (1-Reflection coefficient) as a function of pH and salt concentration, as calculated using Eqs. (69 and 68). This calculation assumes no steric exclusion/hindrance for simplicity and Ka = 1000, cX0 = 2M. The small dashed red lines are guides that show a slope of 1 for the lower dashed line and 0.5 for the upper dashed line.

• Hindrance gets stronger with smaller pores (or, more generally, denser membranes) • Larger solutes are hindered more than smaller ones. • Diffusion is probably hindered more strongly than convection because the convective flow is dominated by the central parts of the pores (or by larger pores if there is a pore-size distribution) where the steric hindrance is weaker.1 The physico-chemical foundations of the simple approach to combining steric and Donnan exclusion as published in the literature (and reproduced below) may be even less certain than the foundations of steric hindrance. The Donnan-exclusion model considers ions as point charges whereas for significant steric exclusion/hindrance the solute radius should approach that of the pore. Several studies considered electrostatic interactions of charged nanoparticles with charged capillaries, taking into account the finite size of the nanoparticles, which one can view as a kind of macro-ion [55–57]. However, those studies considered a distinctly different range of parameters (the “macro-ion” was always much larger than all the other ions, which were considered point-like) than NF, so they are not of direct use here. The simplest way to superimpose the above versions of Donnan and steric exclusion models is to suppose that they are additive. With the fine-pore model of Donnan exclusion, the ion partition coefficients do not depend on radial position and can be taken out of the integration signs in Eqs. (58, 59). The remaining local steric hindrance/exclusion factors are functions of coordinate, but the more sophisticated versions of the corresponding integral factors were obtained via integration of local hindrance coefficients over the capillary cross-section. With such steric exclusion/hindrance coefficients, they should just be multiplied by the corresponding Donnan-partitioning factors (appropriate exponents of the Donnan potential) to obtain the total exclusion/hindrance. This yields P i ≡ Di ∙βi ∙γ∙ expð−Z i ψÞ

ð70Þ

τi ≡ α i ∙ expð−Z i ψÞ

ð71Þ

1 Nonetheless, the quantitative difference between these two kinds of coupling factors appears to be overestimated by the model (see below).

a Salt Reflection Coefficient

1

rp = 0.45 nm

b

MgCl2 MgCl2

Salt Reflection Coefficient

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

0.8

0.6 0.4

0.2

NaCl

0

KCl

-0.2 -0.4

0.1

1

X

10

rp = 0.70 nm

1 0.8 0.6 0.4

0.2 MgCl2 MgCl2 NaCl KCl Na2SO4 Na2SO4

0

-0.2

Na2SO4 Na2SO4

-0.4

100

49

0.1

1

X

10

100

Fig. 6. Reflection coefficients for several salts as a function of the ratio, X, of fixed negative-charge concentration to feed concentration (calculated by using Eqs. (64, 67), Stokes ion radii and hindrance factors from ref. [34]). The figures show simulations from the Donnan-steric exclusion/hindrance model with pore radii of (a) 0.45 nm and (b) 0.7 nm.

These expressions also introduced active membrane porosity, γ, taking into account the pore tortuosity and denoted α i, βi the cross-sectionaveraged steric exclusion/hindrance factors. For co-ionic solutes (those with the same charge sign as the fixed charges), Donnan exclusion reduces the ion partition coefficient (for partitioning between the pores and the virtual equilibrium solution). Therefore, both Donnan and steric exclusion/hindrance factors act in the same direction to enhance the preferential transfer of solvent relative to these solutes. Nonetheless, this does not necessarily mean that solute rejection is always positive. As discussed below, spontaneously arising electric fields may strongly influence the transfer of ionic solutes. For counter-ionic solutes (those whose charge sign is opposite to the fixed charges), Donnan exclusion enhances their concentration in the membrane phase, but steric hindrance/exclusion counteracts this enhancement. Therefore, for this kind of solute, the transmission coefficients can be larger or smaller than one, depending on the relative contributions of the two rejection mechanisms. Transmission coefficients larger than unity may mean preferential (relative to the solvent) transfer of the solute and negative rejections. (At high volume flows, transmission coefficients larger than one imply negative rejection, as the rejection approaches the reflection coefficient, which is 1-transmission coefficient.) However, in zero-current pressure-driven membrane processes, the electromigration counterflow generated by spontaneously-arising electric fields largely compensates (sometimes even overcompensates) the enhanced partitioning of counter-ions. Nonetheless, with salts whose counterions have lower mobility than coions, the purely Donnan exclusion mechanism can give rise to slightly negative rejections at small to moderate dimensionless fixed-charge concentrations [58,59]. The absolute value of these negative rejections initially increases with the dimensionless fixedcharge concentration, passes through a maximum, decreases, and eventually the rejection turns positive. Introduction of even weak steric exclusion/hindrance can easily make these negative rejections positive. However, as Fig. 6 shows there is still a range of dimensionless fixedcharge concentrations where the salt reflection coefficient decreases with increasing fixed charge concentration (or decreasing salt concentration). This decrease can turn into negative rejections. Moreover, in membranes with smaller pores this phenomenon is even more pronounced (see Fig. 6 for MgCl2.) This counterintuitive behavior results because steric exclusion/ hindrance amplifies the difference in the aqueous diffusion coefficients of ions. Additionally, the convective coupling factor, α i , decreases with decreasing pore size more slowly than the diffusion-reduction factor, βi , especially for solutes with radii near that of the pore (see Fig. 4). Therefore, the convective flow of relatively large counterions is not, yet, greatly reduced due to steric hindrance so their transmission coefficient can be larger than one (due to the Donnan enhancement).

At the same time, their (electro)diffusion is hindered more strongly. Under zero-current conditions, the convective current due to the preferential transfer of counterions is compensated by the electromigration current generated by the electric field of the streaming potential. Nevertheless, because of the strongly reduced diffusion permeability of the counterions, this electric field primarily enhances the electromigration flow of relatively small coions toward the permeate. This enhanced flow can mean even negative rejections. Fig. 7 schematically illustrates this picture. Thus, paradoxically, smaller pores may mean lower (and even negative) rejections for certain kinds of salts, but only at high ratios of fixed charge to ion concentration (compare the solid black curves in Fig. 6a and b). The decrease in the reflection coefficient with increasing dimensionless negative fixed-charge concentration is especially pronounced for MgCl2 because the Stokes radius (calculated from the diffusion coefficient) of hydrated Mg2+ ions (0.349 nm) is almost 3 times larger than that of Cl− (0.121 nm). For the smaller pores, this gives rise to a decrease of reflection coefficient with the dimensionless fixed-charge concentration over the whole studied range. The NaCl calculations show that this decrease is not due to the cation charge but to the ion-size asymmetry. Even though the size-asymmetry is relatively small for NaCl, the reflection coefficient decreases with the dimensionless fixed-charge concentration over a given range (|X| b ca. 1 or 2 in Fig. 7a). Moreover, the KCl reflection coefficient always increases with increasing dimensionless

Counterion Convection Counterion Electromigration Counterion Diffusion Counterion Net

Co-ion Convection

Donnan enhancement Convection enhancement Steric reduction

High mobility reduction High diffusion reduction Overall enhancement

Donnan reduction Convection enhancement Small steric reduction

Co-ion Electromigration Low mobility reduction Co-ion Diffusion

Co-ion Net

Low diffusion reduction Overall enhancement

Fig. 7. Scheme showing the factors that could give rise to negative salt rejection at a high ratio of fixed charged density to feed salt concentration in the steric/Donnan exclusion model. The lengths of arrows qualitatively represent the magnitudes of different fluxes. In this scheme the counter-ion is larger than the co-ion.

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fixed-charge concentration, in agreement with the equal sizes of potassium cations and chloride anions. Notably, the increase in the dimensionless fixed-charge concentration can occur not only due to an increase in the fixed-charge concentration but also because of a decrease in the salt concentration. Therefore, Fig. 6 predicts a considerable increase in the MgCl2 reflection coefficient with increasing salt concentration. Several studies observed this trend of increasing NF salt rejection with increasing concentrations of salts containing doubly charged cations (primarily CaCl2) [52,60,61]. However, in general, the trends in Fig. 7 disagree with experimental data. For instance, the differences in the reflection coefficients of NaCl and Na2SO4 are not very large in Fig. 7, but experimentally one often observes sulfate rejections of 0.99 accompanied by chloride rejections of only 0.5–0.6. Of course, one can reach sulfate rejections of 0.99 by increasing the dimensionless fixed-charge concentration. However, this will also make the NaCl rejection much higher (N0.9) than experimental values. In the fine-pore approximation, the salt transmission coefficient and salt diffusion permeability of the membrane are proportional, and τs 1 α− αþ ¼ ∙ Zþ∙ −Z − ∙ P s Z þ −Z − β − D− β þ Dþ

! ð72Þ

This ratio does not depend on the Donnan potential because changes in partition coefficients due to fixed charge affect both transmission and diffusion permeability, but the ratio in Eq. (72) does depend on the pore size. According to Eq. (40), the reciprocal salt diffusion permeability controls the initial slope of the dependence of salt rejection on the trans-membrane volume flow. This proportionality means that smaller transmission coefficients also imply more rapid increases of rejection with the flow and, thus, higher rejections at a given flow. Therefore, everything said above about the behavior of the salt-reflection coefficient transfers to the salt rejections occurring at small trans-membrane volume flows and most probably also at intermediate ones. 3.2.2.1. Rejection of trace ions in the Donnan/steric exclusion model. Many practical applications of NF involve multi-ion solutions. Therefore, NF models must apply to this type of feed solution. The (quasi)analytical solution obtained above in the limiting case of one dominant salt and trace ions (Eq. (48)) enables one to explore to what extent the Donnan-Steric Exclusion/hindrance model reproduces experimentally observed trends in NF of multi-ion solutions. This section considers a practically-relevant case of dominant NaCl with traces (b1%) of MgSO4 (this could be an example of some hardness in tap water). Fig. 8 shows results of sample calculations of rejections of dominant salt (NaCl) and trace ions. (Note that the different traces permeate through the membrane independently from each other, i.e. their concentrations are low enough that they do not significantly affect the electrical potential established by the dominant salt, so one can consider any number of traces simultaneously. Additionally, the trace ions have no significant

effect on rejections of the dominant salt.). Fig. 8a shows that the rejection of traces of sulfate increases monotonically with the dimensionless concentration of fixed charges and reaches the typically observed rejections of N0.99 at X = − 9. The rejection of dominant NaCl initially decreases with the dimensionless fixed-charge concentration and then increases to reach the typically observed values of 0.5–0.6 at X = -9. As discussed above, the initial decrease of NaCl rejection with the dimensionless fixed-charge concentration occurs because hydrated Na+ is larger than hydrated chloride. This decrease can also correspond to increasing rejection of NaCl with its increasing feed concentration within a concentration range at a fixed charge density. This is not a known experimental trend, but the effect is relatively small and could have been overlooked. The predicted trends in rejection of traces of Mg2+ (Fig. 8b) have never been observed. Although the rejection has a correct “order of magnitude” of 0.85 for the slightly negatively-charged membrane (X = − 1), for the fixed-charge concentration required to reproduce the observed rejections of NaCl and traces of sulfate (X = -9), traces of Mg2+ experience negative rejections of −3.5-4! The physics of negative rejections of “large” counterions like Mg2+ was discussed above. For such closely fitting solutes, hindrance of diffusion solute transport is much stronger than hindrance of convective transport. As a result, “large” counterions undergo extensive convective transport, and electromigration does not counteract convection because of the strongly reduced diffusion coefficient. This effect is more pronounced for traces of multivalent “large” counterions than for dominant ones because the preferential partitioning of such traces is stronger (the monovalent Na+ counterions control the Donnan potential, which is thus relatively large at high absolute values of X). Thus, the predicted behavior makes physical sense within the scope of this model. However, it clearly disagrees with experimental data. Calculations show that increasing the supposed pore radius (within the typical NF range that is up to ca. 1 nm) does not help and, in fact, results in even more pronounced Mg2+ negative rejections. Literature studies often explain mismatches of this kind through a possible dependence of membrane fixed-charge density on the salt [10,11,61,62]. However, the charge density must be the same for all the ions in the case of mixed solutions. In other words, the Steric-Donnan-Exclusion model predicts that if traces of MgSO4 are added to dominant NaCl (or a similar monovalent salt) the rejection of sulfates is highly positive (in agreement with numerous reports), but the rejection of magnesium ions is simultaneously strongly negative (which has never been observed). Thus, the model is probably inadequate. There are two possible reasons for these dramatic deviations from experiment: the model does not take into account other exclusion mechanisms that reduce the partitioning of Mg2+ (e.g. dielectric exclusion, see below), or the model unrealistically exaggerates the steric hindrance of diffusion vs. the steric hindrance of convective solute transfer. The section below shows that the dielectric-exclusion mechanism is much stronger for doubly charged ions and thus, indeed, would increase Mg2+ rejection. The second reason for the high negative Mg2+ trace

Fig. 8. Salt and trace-ion rejections calculated using the Steric/Donnan exclusion model assuming a pore radius of 0.45 nm and different values for the Péclet number, Pes, and for the ratio, 2+ X, of fixed-charge to feed salt concentrations. Plot (a) shows rejections of dominant NaCl (solid lines) and trace SO2− . 4 (dashed lines), whereas plot (b) shows rejections of trace Mg Rejections were calculated using Eqs. (48, 52, 64, 67, 70, 71), Stokes ion radii, and hindrance factors from ref. [34].

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

0.8

X=-9 (KCl) X=0 (Li+) X=-3 (KCl) X=-1 (Li+) X=-1 (KCl) X=0 (KCl)

Rejection

0.6 0.4 0.2

X=-3 (Li+)

0 -0.2

X=-9 (Li+)

-0.4 0

1

2

Pes 3

4

5

Fig. 9. Salt and trace-ion rejections calculated using the Steric/Donnan exclusion model assuming a pore radius of 0.45 nm and different values for the Péclet number, Pes, and for the ratio, X, of fixed-charge to feed salt concentrations. Solid lines represent rejections of dominant KCl and dashed lines indicate rejections of trace Li+. Calculated by using Eqs. (48, 52, 64, 67, 70, 71), Stokes ion radii and hindrance factors from ref. [34].

rejections mostly relates to the ion size. Accordingly, one can calculate the rejections of traces of “large” hydrated monovalent cations such as Li+. Fig. 9 shows calculated rejections of traces of Li+ added to dominant KCl. The negative rejections at larger dimensionless fixed-charge concentrations still occur at larger Péclet numbers (as for Mg2+), but they are less pronounced. Even if these relatively large Péclet numbers are unattainable in experiment, the strong difference in the (positive) rejections of K+ and Li+ at smaller Pes should be easily detectable. The authors are unaware of corresponding experiments with commercial NF membranes but such experiments could support this concept. Negative rejections of lithium from a LiCl/KCl mixture containing 10% lithium and 90% potassium were observed experimentally with nanoporous negatively-charged track-etched membranes [2]. However, the structure, in particular the pore size, of such membranes is very different from commercial NF membranes. From Fig. 8, one can also see that for NaCl rejections N0.2–0.3, the Péclet number should approach or exceed 1. Within the scope of the Steric-Donnan-Exclusion model, obtaining sufficiently large Péclet numbers requires assuming that either the barrier layer thickness is unrealistically large or its porosity is very low (below b1% in most cases). Besides disagreeing with experimental data (like direct observation of the membrane thickness by electron microscopy) such fitted parameters do not match information on the observed membrane hydraulic permeability, which is much larger than the hydraulic permeability based on calculated pore sizes and porosities fitted to the solute rejections. Thus for instance, by using Eqs. (60, 70, 71), an expression for the steric exclusion factors found in the literature (eqs. (16) and (18) of reference [34]), and Eq. (65) for the Donnan potential, one can show that in NaCl solutions the salt diffusion permeability of a membrane with a pore radius of 0.45 nm and a dimensionless concentration of fixed charges X = − 9 is about Ps ≈ 10−10m2/s. Given that the NaCl rejection of 0.2–0.3 is experimentally reached (NF270 membrane) at transmembrane volume fluxes of about 10 μm/s [63] one can conclude that for Pes ≈ 1 to occur at such fluxes the thickness-over-porosity ratio has to be about 10 μm. By substituting this thickness to porosity ratio, this pore radius and the bulk water viscosity to Hagen-Poiseuille equation, one obtains the hydraulic permeance of about 3 ∙ 10−12m/s ∙ Pa whereas the measured permeance is about one order of magnitude larger. The large experimental hydraulic permeances can hardly be explained by a change in the viscosity of water in the nanopores (relative to the bulk) because one would need to assume this viscosity to be strongly reduced (and not increased as one would surmise for confined water). Assuming somewhat larger pore radii helps to reduce this discrepancy to some extent but within the typical NF pore-size range the discrepancy remains significant.

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3.2.3. Dielectric exclusion Dielectric exclusion (DE) results from differences (between the membrane “interior” and the equilibrium bulk solution) in the interactions between charged solutes and bound charges induced due to dielectric polarization. The literature distinguishes dielectric exclusion that results from image forces and dielectric exclusion described by the Born formula, but the physical mechanism of both is basically the same. The dielectric properties of polar solvents like water are almost certainly different in b1 nm nanopores compared to the bulk solvent. Nonetheless, this is still a hypothesis, albeit a very plausible one, because unambiguous determination of the dielectric properties of water in nanopores (especially those of composite/asymmetric NF membranes that have relatively thick support layers) is a daunting task. Therefore, in a first approximation this section treats DE assuming the water properties in nanopores are the same as in the bulk. This assumption that the water structure does not change in nano-confinement does not necessarily disregard this structure altogether. As discussed below, nanopores with dimensions commensurate with the bulk water correlation length may considerably change the picture of dielectric exclusion (as compared to the purely macroscopic electrostatic analysis). Yaroshchuk reviewed the macroscopic picture of DE, including some manifestations of water structure in nano-confinement [64]. DE explicitly relates to the electrostatic component of ion solvation energy. Due to the local thermodynamic equilibrium, the ion partition coefficient depends on the excess solvation energy, or the difference in the solvation energy inside, Wi, pore, and outside, Wi, bulk, the pore.   Γi  exp − W i;pore −W i;bulk

ð73Þ

3.2.3.1. Single ions. For a single point-like charge in a water-filled pore surrounded by a membrane matrix of a low dielectric constant, the ion charge induces bound charges at the pore surface due to the mismatch of the dielectric constants of water and matrix. If the matrix dielectric constant is lower than that in the surrounding water, the sign of the induced charge is the same as that of the ion. Moreover, in this limiting case the excess solvation energy is proportional to this dimensionless parameter Z2 F2 Z 2 λB ≡ 8πRTεs ε0 N A h 2h

ð74Þ

where h is the characteristic pore dimension (the radius in the case of cylindrical pores) and λBis the Bjerrum length, the distance at which two unit charges interact with energy equal to kT. In bulk water λB ≈ 0.7 nm. The most prominent feature of Eq. (74) is the proportionality to the square of ion charge which implies that both cations and anions are excluded, and multiply charged ions experience especially strong exclusion (remember that the excess solvation energy goes into the exponent of the expression for the partition coefficient). Thus, in this limiting case, the excess solvation energy becomes ΔW ≡ W pore −W bulk ¼

Z 2 λB ∙f ðε m =εs Þ 2h

ð75Þ

where f(εm/εs) is a dimensionless function of the ratio of dielectric constants of membrane matrix and solvent. The form of this function depends on the pore geometry, and the corresponding expressions can be found in [64]. An interesting situation arises in non-symmetrical salts. Cations and anions are excluded to different extents, but the membrane “interior” has to be electrically neutral (here the membrane is considered without fixed charges for illustration). As a result, an interphase electrostatic potential difference arises between the membrane and the equilibrium solution to make the partition coefficient of the more excluded (larger-

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Excess Solvation Energy (kT)

charge) ions larger and that of the less excluded ions smaller. This leads to overall electrical neutrality, and for both ions the logarithm of the partition coefficient is controlled by the negative product of their charges, −Z+Z− [64]. Incidentally, the situation becomes more complex in ultrathin barrier layers with deviations from local electroneutrality (see Section 4 below). Another interesting feature is the dependence of excess solvation energy on the pore geometry. Fig. 10 shows that this dependence is very strong, especially if the dielectric constants of membrane matrix and solvent are very different. This shape-dependence can be explained in two ways. First, the electrostatic interactions of a charge with an infinite flat interface between media with different dielectric constants can be formally represented as interactions with an image charge located at the same distance on the other side of the interface (whence, the term “image forces”). With two interfaces (slit-like pores), the interaction is not just a simple sum of interactions with two interfaces because there is also a mutual polarization of the two interfaces, which one can consider as multiple images. This effect is significant. For instance, for a membrane matrix dielectric constant of 3, the excess solvation energy for the slit is almost 3 times larger than the simple sum of repulsion energies by two planes. The excess solvation energy in the cylinder is further ca. 4 times larger than in the slit, and that from the spherical pore is 2.5 times more than from the cylinder (note the log scale in Fig. 10). In the case of cylindrical and spherical pores one cannot use the concept of multiple images, but one can still argue that various parts of the pore surface get polarized not only by the ion but also by other (equally polarized) parts of the surface. The strong dependence of DE on the pore geometry is a problem because the pore geometries in NF are complex and not easy to determine. In the case of pores with more “closed” geometries (cylinders or spheres), the energy of the electric field helps to explain this strong dependence on the geometry even better. The energy of an electric field is a volume integral of the product of dielectric constant and the square of the electric-field strength. Because the field strength is roughly inversely proportional to the dielectric constant, for the product one obtains an approximate inverse proportionality to the dielectric permittivity. This qualitatively explains why the electrostatic energy increases when an ion is transferred close to (or even into) media with low dielectric constants. Of course, this picture is too vague because the field lines tend to redistribute and favor media with higher dielectric constant. However, in pores with “closed” geometries there is not much room for field lines to “escape” the low-dielectric-constant domains, so electrostatic energy increases relatively more than in more “open” pores.

100

sphere cylinder

10

3.2.3.2. Ionic screening of dielectric exclusion. Screening of electrostatic interactions in electrolyte solutions occurs due to correlation of ion positions in space, which increases the probability of finding an oppositely charged ion close to an ion of a given charge. As a result, on average the strength of ion-ion interactions decays exponentially with interion distance and is not inversely proportional with this distance as one would expect for Coulombic forces. This has important implications for dielectric exclusion. An ion interacts not only with its “own” polarization charges but also with those induced by other ions. Those polarization charges with higher probability tend to have the opposite sign of the polarization charge of the given ion. Thus, screening can reduce the DE energy considerably. In this context, the mutual polarization of different parts of pore surfaces can play a prominent role. The polarization charges tend to extend along the pore, and the more distant charges get screened more easily. In terms of multiple images (in slit-like pores for example), the more distant images are screened first. As a result, the contribution of mutual surface polarization is lost quite early and what remains are basically the interactions with the closest images due to the additive polarization of two pore surfaces. As shown previously [64], this leads to the screening of DE starting at quite small ratios of pore size to the screening length. The introduction of fixed charges (with the associated counterions whose concentration cannot decrease because of the electroneutrality requirement) also gives rise to considerable screening of interactions with polarized interfaces. However, as discussed below this strong screening is essentially due to the use of purely macroscopic electrostatics. Accounting for the effects of water structure in terms of non-local electrostatics makes the effects of mutual polarization (and the “early” screening with them) essentially weaker. Finally, in Yaroshchuk's analysis of screening of DE, a major part of the equations had to be linearized in the so-called binary potential (otherwise no analytical solutions could be obtained for the more realistic pore geometries like cylinders or slits) [64]. This linearization, however, neglected the effect of intensification of ion-ion interactions in narrow nanopores and led to the appearance of unphysical multiple values of salt partitioning coefficients as functions of equilibrium concentration. Although the physical reason for this non-physical behavior was pointed out in [64] it was (and still is) difficult to suggest effective ways of taking the non-linearity into account. Meanwhile, the use of linearized equations beyond the scope of their applicability (for example, in quite narrow nanopores often evoked in the context of NF) gave rise to technical problems in the modelling of experimental data [65]. 3.2.3.3. Born formula, its physical meaning and applicability; effects of water structure. In view of the complexity of this picture and the uncertainty concerning essential properties like the pore geometry, the “image forces” approach outlined above was considered too complex for practical modelling of membranes. As a result, most studies that modelled DE took the alternative approach of using the Born formula while assuming the bulk dielectric constant of water in nanopores decreased due to confinement effects [60,66–75]. In this case, Eq. (76) describes the excess solvation energy

slit

plane

ΔW Born ¼

1 0.1

0.01 0

0.2

0.4 0.6 0.8 εmembrane/εsolvent

1

Fig. 10. Dielectric exclusion excess solvation energies for different pore geometries as a function of the membrane matrix dielectric constant. The characteristic pore dimension (cylinder and sphere diameter, slit height, double distance to the plane) is 0.7 nm (Bjerrum length of water). Calculated by using formulae available in ref. [64].

  Z2 F 2 1 1 ∙ − 8πRTε 0 NA ai εp εs

ð76Þ

where εp is the dielectric constant of “pore” water, εs is the dielectric constant of bulk water, and ai is the ion radius, typically assumed to be the bare (not hydrated) radius. The varying radii and charges of different ions lead to a wide range of excess solvation energies. Simplicity is the incontestable advantage of Eq. (76). However, such an equation must also at least generally describe the underlying physics. Therefore, one must take a closer look at the assumptions made when deriving the Born formula. First, the equation results from macroscopic electrostatics where the ion is considered a non-polarizable conducting sphere whose electric charge resides in an infinitely thin layer on the ion surface. The solvent

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

is considered a structureless dielectric continuum. Thus, the model assumes that the induced (bound) charges due to dielectric polarization are localized within an infinitely thin layer at the ion/solvent “interface”. The main reason that the dielectric properties of water would change in nanoconfinement is a confined supramolecular structure. However, the Born formula largely neglects just this structure. Meanwhile, due to short-range correlations in the orientation of water molecules, in strongly inhomogeneous electric fields (like those around ions) the bound charges induced due to the dielectric polarization of solvent do not reside exclusively within geometrically thin layers but extend over a certain space around the ion [76–79]. The size of this space varies with the solvent correlation length. In bulk water, it is about 0.7 nm, which is around the typical pore radii reported for NF membranes. The water correlation length in nanoconfinement is unknown, but presuming the water in nanopores is more “structured” than in the bulk, the correlation length should be at least the same as in the bulk water if not larger. A relatively simple way to account for the water structure is the use of non-local electrostatics [76–79]. This treatment replaces the classical local constituent relationship between the electric field and electric displacement with a more general (still linear) but nonlocal relationship that postulates that the electric displacement at a given point is controlled by the electric-field strength not only at the same point but also in its vicinity. This results from strong particleparticle correlations between the solvent molecules that make the prevailing dipole orientations depend not only on the external electric field and the mean field created by all other molecules but also on the orientation of the closest neighbors. Accounting for the strong correlations between the orientations of closest water molecules has rather dramatic consequences for the electrostatic component of ion-solvation energy because the latter is controlled by the closest vicinity of the ion. In particular, this energy becomes much less sensitive to the bulk (long-wavelength in terms of non-local electrostatics) dielectric permittivity than within the scope of a macroscopic approach. This energy is largely controlled by the socalled short-wavelength component of the dielectric response function. This component primarily depends on the infrared polarizability of water molecules and not on their dipole orientation. Meanwhile only this dipole-orientation water structure is likely to change in nanoconfinement, and it will reduce only the long-wavelength dielectric permittivity. Due to the non-locality of the dielectric response, the solvation energy is much less sensitive to the property (bulk dielectric permittivity) that is most probable to change in nanoconfinement. Semi-quantitatively, the effect of non-locality of solvent dielectric response can be viewed as an effective reduction of the dielectric constant in the close vicinity of the ion. In nanopores whose size is comparable with the solvent correlation length, the ion interactions with polarized solvent/matrix interfaces are effectively controlled by this reduced dielectric constant. The short-wave-length dielectric permittivity of water is likely around 5, which is quite close to the probable dielectric constant of polymer membrane matrices (about 3). On the other hand, pronounced effects of mutual polarization, extension of induced charges along the pores or multiple images are especially strong when the dielectric constant of solvent and matrix are very different. As discussed above, these effects alone make the ionic screening of DE start at quite small ratios of pore size to the screening length. When the (effective) dielectric constant of solvent and matrix are close to each other, the ionic screening becomes much more “local” and starts at smaller screening lengths (higher salt concentrations). The screening of DE due to the fixed charges also gets weaker. As a result, in sufficiently dilute solutions in a crude approximation one can probably disregard the effects of ionic screening altogether and consider the DE-related excess solvation energies independent of concentration. This approximation is even more appropriate if one assumes that the solvent correlation length increases in nanopores, in agreement with the usually postulated more “structured” water in nano-confinement. Some dependence of rejection on the feed concentration observed experimentally at the points of zero charge of

53

several NF membranes [75] can still be explained (at constant excess solvation energies), for example, by the deviations from local electric neutrality occurring in ultrathin barrier layers due to unequal ion partition (see Section 4 below). Due to the linearity of non-local electrostatics, the quadratic scaling of excess solvation energy with the ion charge magnitude remains in place. An important difference with the classical Born formula is that the exclusion solvation energy will scale with the reciprocal pore size instead of the ion radius. However, some dependence on the ion size may occur (in a qualitative agreement with experiments, which appear to indicate such a dependence). Actually, this is a subtle matter related to the possible redistribution (in the nanoconfinement as compared to the bulk solution) of the dielectric-polarization charges induced in the ion vicinity. At present, it is not quite clear how this redistribution can be described within the scope of non-local electrostatics. However, from physical considerations it seems plausible that this redistribution may be somewhat influenced by the ion size. As a result, the excess solvation energies may be ion-specific even for ions of the same charge magnitude. To summarize, it probably does not make physical sense to parameterize the effects of excess solvation energies in terms of altered macroscopic dielectric constant of water in nanopores by using the classical Born formula. Fitting experimental data directly in terms of excess ion solvation energies is preferable. At the same time, these energies may well have a predominantly “dielectric-exclusion” origin and, thus, be proportional to the squares of ion charges. They may also depend on the ion size. These observations are important for rational parameter selection in the section below dealing with the effects of unequal ion partitioning into ultrathin barrier layers. In most published studies that model DE in membranes [60,66–75], the dielectric permittivity featured in the Born formula serves as an adjustable parameter. However, some studies attempted to relate this property to the membrane pore size. Eq. (77) gives a relatively simple relation [80–82] for calculation of the dielectric constant in nanopores, εpore.  2 dw εpore ¼ εbound þ ðεwater −εbound Þ∙ 1− rp

ð77Þ

In this equation, εbound is the dielectric constant of water bound near the pore surface, εwater is the dielectric constant of “free” water in the pore, and dw is the diameter of a water molecule. This equation assumes a monolayer of “bound” water whose dielectric constant may be around 5. The remaining part of the water is bulk-like water with εwater ≈ 80. The next step is calculating the “average” dielectric constant via multiplying “bound” and bulk dielectric constants by the volume (actually surface for this cylindrical pore geometry) fractions of the corresponding kinds of water. As a result, one obtains Eq. (77). Typical procedures for calculating the dielectric constants of mixtures of components

Fig. 11. Scheme of a model sometimes used to estimate “average” dielectric constants in nanopores. The water consists of a layer of bound water near the pore surface and free water in the interior.

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having different dielectric properties are more sophisticated, especially if the permittivities of constituent media are very different [83,84]. However, the larger problem with Eq. (3–18) is that the “bound” and bulk water do not have equal effects on ion partitioning. If one imagines an ion transferred to the pore sketched in Fig. 11, this ion will not “enter” the layer of bound water due to both steric and dielectric exclusion. As a result, the ion will remain in the bulk-water part of the pore where it will experience DE due to the “image-force”-like interactions with the polarized interface between the bound water and bulk-like water. The interface between the bound water and membrane matrix may also make a contribution. This has little resemblance to the mechanism obtained via substituting the “average” dielectric constant of Eq. (77) into the Born formula. 4. Solution-diffusion-electromigration models of nanofiltration 4.1. An analytical solution to transport of three ions with different charges As mentioned above, nanopore NF models rely on macroscopic approaches that do not quantitatively apply at the nanoscale. Moreover these models require membrane parameters such as pore size distributions and geometries, surface charge densities, and dielectric constants that are not readily accessible experimentally. Thus, although the nanopore model may provide qualitative insight into NF, phenomenological descriptions based on irreversible thermodynamics will likely prove more useful in describing NF for engineering purposes. Eq. (1) provides the starting point for irreversible thermodynamics models. To simplify the treatment of this series of equations, the solutiondiffusion-electromigration model (SDEM) assumes that the transmission coefficient, τi(c), is zero so the last term in Eq. (1) is also zero. Some recent studies argue that this assumption agrees well with experimental data obtained with several commercial NF membranes [25,39,40,63]. Moreover, using differences in ion permeabilities the SDEM easily models the very high chloride/sulfate selectivity usually observed in NF [85]. With the solution-diffusion assumption, the definition of electrochemical potential, and a constant activity coefficient, Eq. (1) becomes J i ¼ −P i

dci F dφ −ci Pi Z i RT dx dx

ð78Þ

Note that mechanistically, Pi is the product of the local diffusion and partition coefficients, where the partition coefficient is the ratio of concentrations in the real and virtual solutions and ci is the concentration in the virtual solution (see Section 2). In principle, the value of Pi is a function of position and/or concentration, but practical solutions of the system of equations represented in Eq. (78) typically require the assumption of constant permeabilities throughout the membrane. Pi may depend on the feed composition, but its value throughout the membrane should be constant for the solution procedure below to be feasible. Presuming the diffusion coefficient varies little with position, the constant permeance assumption implies that the local partition coefficient is essentially constant across the membrane. Considering all species, Eq. (78) is a system of n equations, where n is the number of ions in solution. The equations contain n + 1 variables, the position-dependent electrical potential and ion concentrations, but due to the electrical neutrality of virtual solutions and the zero-current condition, only n − 1 variables are independent. Commercial software such as MatLab readily affords solutions to systems of ordinary differential equations such as that in Eq. (78). Nevertheless, even with fixed values of ion permeabilities, finding the appropriate solution includes iterative matching of multiple ion fluxes and concentrations. This is challenging because the solutions to the equations depend on the initial guess values, and local minima may exist. To overcome this challenge and create a computationally simple method, Yaroshchuk and Bruening developed an analytical solution to

Eq. (78) for a system with three ions of different charges [86]. (Note that this method does not apply to cases where two of the charges are equal; a special method for this is also described in reference [86]). The specific solution enables spreadsheet calculations of ion fluxes, concentrations, and electric fields, and an example spreadsheet is available in the supporting information of reference [86]. The analytical solution employs the relative double ionic strength, u, defined in Eq. (79) u¼

Z 21 c1 þ Z 22 c2 þ Z 23 c3 2I ≡ C c1 þ c2 þ c3

ð79Þ

for the virtual solution where the subscripts refer to the specific 3 ions, I is the ionic strength, and C is the sum of all the ion concentrations in the virtual solution. One can define the permeate concentrations as a function of up according to Eq. (80) where the subscript p indicates the value in the permeate solution [86]. c1p ¼ C p ∙

up þ Z 2 Z 3 ðZ 1 −Z 2 Þ∙ðZ 1 −Z 3 Þ

c2p ¼ C p ∙

up þ Z 1 Z 3 ðZ 2 −Z 1 Þ∙ðZ 2 −Z 3 Þ

c3p ¼ C p ∙

up þ Z 1 Z 2 ðZ 3 −Z 1 Þ∙ðZ 3 −Z 2 Þ

ð80Þ

With these equations, the expression for Cp as a function of up in Eq. (81) (obtained as a result of integration of the system of differential equations of Eq. (78)) defines the sum of the permeate concentrations of all ions. 

C p ≡ C up



"

# Fm ðup Þ "  # F p ðup Þ  F m up þ u0 F p ðup Þþ F m ðup Þ F p up −u0 F p ðup Þþ F m ðup Þ   ∙ ð81Þ ¼ C0∙ F m up þ up F p up −up

Eqs. (82–85) describe the functions Fm(up) and Fp(up), which depend only on up and ion permeances (the permeability scaled on the barrier layer thickness) and charges [86]. 

F p up



qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2    Ζ∙ρ02 þ ρ12 ∙up −4Ζ∙ Ζ∙ρ01 þ ρ11 ∙up ∙ Ζ∙ρ−11 þ ρ01 ∙up − Ζ∙ρ02 þ ρ12 ∙up  ≡ 2∙ Ζ∙ρ−11 þ ρ01 ∙up

ð82Þ  F m up ≡

ρij ≡

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2    Ζ∙ρ02 þ ρ12 ∙up −4Ζ∙ Ζρ01 þ ρ11 ∙up ∙ Ζ∙ρ−11 þ ρ01 ∙up þ Ζ∙ρ02 þ ρ12 ∙up  2∙ Ζ∙ρ−11 þ ρ01 ∙up

 Z i1 ∙ Z 2j −Z 3j P1

−

 Z i2 ∙ Z 1j −Z 3j P2

þ

 Z i3 ∙ Z 1j −Z 2j P3

ð83Þ

Ζ ≡ Z1Z2Z3

ð84Þ

Pi L

ð85Þ

Pi ≡

(In Eq. (83) i = − 1, 0, or 1, and j = 1 or 2 to give the values of ρ−11, ρ01, ρ02, or ρ12.) Thus, in Eq. (81) Cp is a function of the total ion concentration, C0, and the relative double ionic strength, u0, in the feed solutions as well as up and ion permeances and charges. Finally, Eq. (86) gives the volume flux, Jv, also as a function of up, ion charges and permeances, and the total ion concentrations in the permeate and feed solutions, permeances, and the total ion concentrations in

Dimensionless Electric Field

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

the permeate and feed solutions.  J v up ¼

! C ðZ −Z 2 ÞðZ 2 −Z 3 ÞðZ 1 −Z 3 Þ  0 −1 ∙ 1 Ζ∙ρ−11 þ ρ01 ∙up C up

ð86Þ

For a given value of up, Eqs. (80, 81, and 86) allow calculation of the permeate ion concentrations and the volume flux. The value of up defines the permeate concentrations that in turn can only occur at a specific volume flux. Section 4.3 describes how one may estimate the pressure drop using this method of solution. Although these equations appear complicated, one can easily program them into a spreadsheet to calculate permeate concentrations and ion rejections as a function of volume flux [86]. Fig. 12 shows an example of using the spreadsheet solutions of Eqs. (80, 81, and 86) to model literature NF data. The simulation matches the high experimental rejection of the divalent ion as well as the negative rejection of Na+ at low volumetric flow rates. Overall, the fitting procedure includes obtaining P Ca2þ from separate measurements of single-salt CaCl2 rejection and varying the value of P Cl− , which was assumed equal to P Naþ . Thus the fit in Fig. 12 employed only one adjustable parameter, P Cl− . Despite the agreement between experimental and calculated rejections in Fig. 12 (differences are always b0.1), these data do not enable fitting of separate values for the permeances to Cl− and Na+. In reality, the permeances of these two monovalent ions will differ somewhat as even their aqueous diffusion coefficients are 35% different. Determination of the individual ion permeances of” rapid” ions such as Cl− and Na+ will likely require experiments with trace ions (see below). Nevertheless, single-ion permeances are important for fully understanding the effects of spontaneously arising electric fields on ion transport. Accounting for these fields is vital to explain the negative rejections that frequently occur in NF as well as trends in ion rejections as a function of composition and flow rate. Fig. 13 shows some calculated spontaneously arising electric fields for different NF conditions during the filtration of MA and M2B salts, where the anion B2− has the lowest permeance. The first point to note is that the electric field is highly nonlinear, and such information is available only through computation. Secondly, the sign of the electric field depends on the relative permeances of M+ and A−, as well as the relative fractions of MA and M2B in solution. With a higher permeance to A− than M+ and a large fraction of MA (relative to the total salt) in the solution, the electric field is positive to retard A− (primary effect) and enhance M+ (smaller effect) transport to achieve zero current. Electromigration of B2− is small due to its very low permeance. In contrast, when the permeance to M+ is much larger than that to A−, and MA is the dominant salt in the solution, the electric field takes the

55

5 0 -5 -10

E(u) chi 0.7 PM=10 P =1, PM+PA=10, χMA=0.7 A-

-15

E(u) =10, P P =1, χMA=0.7 AM+

-20

E(u) chi 0.05 PA=10 PM=1, χ P =10, PM+ MA=0.05 A-

-25

0

0.2 0.4 0.6 0.8 Scaled x Coordinate

1

Fig. 13. Calculated dimensionless electric fields (in RT/FL units, where L is the barrier layer thickness) as a function of scaled distance into the membrane selective layer during NF of solutions containing MA and M2B salts. The calculation employs Jv = 10 μm/s and permeances (in μm/s) as indicated in the legend. The permeance of B2− was 0.01 μm/s, and the figure indicates the salt mole fraction of MA, χMA, defined as [MA]/([MA] + [M2B]). Adapted from reference [86] with permission from Elsevier.

opposite sign to maintain zero current. Finally, the electric field is negative when MA is present at trace levels, even if the permeance to M+ is smaller than the permeance to A−. The large excess of M+ over A− leads to a negative electric field that retards M+ (primary effect) and enhances A− transport (smaller effect) to maintain zero current. In this case, the large electric field is sufficient to give higher concentrations of A− in the permeate than in the feed, which is equivalent to negative rejection. An additional challenge in using this approach for modelling NF is that ion permeances may depend on the feed-solution composition [39,87–92]. Adsorption of ions, particularly divalent ions, in membranes will change partition coefficients, and for charged membranes partition coefficients will vary with ionic strength due to screening of the surface charge. Thus, when possible one should determine ion permeances in conditions similar to those in which NF occurs. Freger and coworkers used numerical solution of Eq. (78) to model permeate concentrations of Na+, Mg2+, Ca2+, Cl−, and SO42− in seawater NF [24]. They accounted for concentration polarization using a thin-film model and varied the single-ion permeances to fit the ion concentrations in the permeate solution. Although they could not accurately determine separate permeances of Na+ and Cl−, they fitted the permeances of the less permeable divalent ions, and model and experimental permeate concentrations differed by 15% or less. Adding a concentration dependence to the divalent ion permeances decreased the deviations between experiment and calculations by up to a factor of two, but given the extra fitting parameters, this is not surprising. Overall, the fit to experiment is nearly within the experimental uncertainty, showing that the single-ion permeance model (even with constant ion permeances) effectively describes NF performance.

1.0 Ca2+

Ion Rejection

0.8

4.2. Determining single-ion permeances using NF with trace ions

Cl-

0.6

0.4 Na+

0.2 0.0 -0.2 -0.4 0

10

20

30

40

Jv (μm/sec)

NF with feed solutions containing a dominant salt along with trace concentrations of other ions can readily yield permeance values for the trace and dominant ions in the framework of the SDEM. To obtain membrane permeances from permeate ion concentrations, one must first account for concentration polarization to obtain intrinsic ion rejections, R(int) , defined in Eq. (87), where cim is the ion concentration in the i solution immediately adjacent to the membrane (on the feed side) and cip is the ion concentration in the permeate. ð int Þ

Fig. 12. Experimental (squares) and simulated (lines) real ion rejections (corrected for concentration polarization) during nanofiltration of a solution containing 0.0286 M CaCl2 and 0.0429 M NaCl (0.667 ratio of Ca2+ to Na2+) through a Desal DK membrane. The literature data come from fig. 8b in reference [71], and the simulation employs ion permeances of 0.57 μm/s for Ca2+ and 24 μm/s for Cl− and Na+. Reproduced from reference [86] with permission from Elsevier.

Rei

¼ 1−

cip cim

ð87Þ

Yaroshchuk employed a standard thin-film model to first determine the solution boundary layer thickness [25,63]. Eq. (88) allows calculaðδÞ

tion of the dominant-salt permeance of the boundary layer, P s , and

56

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

of the membrane, P s , from observed salt rejection data, Re(obs) , at differs ent volume flows.

ReðsobsÞ ¼

Jv J exp − ðvδÞ Ps Ps

!

J J 1 þ v ∙ exp − ðvδÞ Ps Ps ðδÞ

!

ð88Þ

4.3. “Under-osmotic” operation

ðδÞ

Because P s ¼ Dδs , where δ is the boundary layer thickness and D(δ) s is the diffusion coefficient in the boundary layer solution, with a literature ðδÞ

value of D(δ) s , one can calculate δ from P s . However, calculating the concentration of a trace ion in the solution immediately adjacent to the membrane requires not only the boundary layer thickness and the trace ion diffusion coefficient, but also must account for the electric field if the diffusion coefficients of the dominant salt cation and anion in solution are different. Yaroshchuk and coworkers developed an expression that takes this into account, but the equation is lengthy and not included here [25,63]. With the calculated value for cim and a measured value for cip, one can easily calculate the intrinsic trace ion rejection. Additionally based on the volume flow and the value of P s, Eq. (89) gives the intrinsic rejection of the dominant salt.

Reðs int Þ

¼

Jv Ps 1þ

Jv Ps

ð89Þ

Finally, as described earlier in the context of the extended SpieglerKedem model, Eqs. (50, 51, and 53) describe the reciprocal intrinsic transmission of trace ions, ft, in terms of the reciprocal intrinsic dominant-salt transmission, fs, and the parameters b and K. Obtaining rejections at a series of transmembrane flow rates will give ft as a function of fs, and Eq. (53) enables fitting of these data to give values of K and b. The salt permeance is accessible from the dominant-salt rejection and Eq. (89), so the fitted K affords values of P t . One can also obtain values of the single-ion permeances of the dominant salt using Eq. (90). P ¼

Ps Zþ 1þ b Zt

appear in a solution containing MgCl2 as the dominant salt because the membranes are much less permeable to Mg2+ than to Cl−, and the spontaneously arising electric field accelerates cations and decelerates anions. In contrast, for Cl− the most negative rejections occur when Na2SO4 is the dominant salt because membranes are much more permeable to Na+ than SO42−.

ð90Þ

Pagès and coworkers used the permeances of trace ions to fit Na+ and Cl− rejections in NF of solutions containing different dominant salts (Fig. 14) [63]. Although in most cases the ion permeances of the NF270 membrane do not vary more than a factor of 2 with the choice of dominant salt, the ion rejections show wide ranges due to the spontaneously arising electric fields. For Na+, the most negative rejections

This section employs the SDEM to explore NF in the “under-osmotic” operation mode where the hydrostatic pressure is smaller than the feed-solution osmotic pressure (versus deionized water). Eq. (41) gives an expression for the trans-membrane hydrostatic pressure drop as a function of volume flow for NF of single salts or non-electrolytes. In the limit that the reflection coefficient approaches unity, this equation becomes Δp ¼

Jv þ Π f Res A

ð91Þ

For membranes with high salt permeability, despite the assumed ideal salt reflection, salt rejection is low and significant volume flow can occur at pressures below the feed osmotic pressure, Πf. In mixtures containing dominant salts with high permeances in the membrane, NF at pressures below the feed osmotic pressure is possible and sometimes desirable. The analytical solution for NF permeate ion concentrations in ternary mixtures allows rapid exploration of pressure drop as a function of volume flow and salt permeances. Eq. (86) enables calculation of Jv as a function of up, Cp, and ion charges and permeances, and Eq. (81) gives Cp as a function of the feed composition and the same ion properties. Keeping in mind that (in the approximation of ideal solutions) osmotic pressure varies linearly with the total concentration of ions and that the SDEM assumes all the ion reflection coefficients are unity, one can substitute Eq. (86) for Jv and total ion rejection for Res in Eq. (91) to obtain this expression for the trans-membrane hydrostatic pressure drop as a function of up.      1 Cf ðZ 1 −Z 2 ÞðZ 2 −Z 3 ÞðZ 1 −Z 3 Þ Cp Δp up ¼ ∙ −1 ∙ þ Π f ∙ 1− Ζ∙ρ−11 þ ρ01 ∙up A Cp Cf

ð92Þ

Eqs. (81, 86, and 92) set a parametric relationship between the trans-membrane volume flux and hydro-static pressure difference. The relationship between Δp and Jv is particularly important in applications such as the removal of sulfates from solutions containing high concentrations of NaCl, where filtration will occur below the feed-solution osmotic pressure. For example, in NF removal of sulfates from brine in the chlor-alkali industry [93] and vacuum salt production [94], the feed NaCl concentrations range from 2 to 6 M whereas the corresponding sulfate concentrations are 5–10% of these values. From such feeds, NF membranes show sulfate rejections N0.97 at trans-membrane

Fig. 14. Rejections of (a) Na+ and (b) Cl− in NF through Dow NF270 membranes (negatively charged) using solutions containing different dominant salts. When Na+ and Cl− are not part of the dominant salt, they are present in trace concentrations. Lines are fits to the data using the SDEM model. Taken from reference [63] with permission from Elsevier.

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

pressures between 20 and 40 bar. The chloride rejection is zero or even slightly negative, and the applied pressures are only a fraction of the feed osmotic pressure. The contribution of strongly-rejected species (e.g. sulfates) to the permeate osmolality is small so the transmembrane osmotic-pressure difference stems primarily from the feed partial osmolality of these highly rejected species. Thus, NF can purify only brine-like feeds with relatively low concentrations of sulfates. As the concentration of sulfates increases, for a given hydrostatic pressure both volume flux and sulfate rejection will decrease, despite decreases in chloride rejection. Fig. 15 illustrates the calculation of these trends for specific values of the Na+, Cl−, and SO42− permeances. Additionally, if one can create a membrane with higher Cl− permeances, enhanced negative Cl− rejections will lead to lower transmembrane osmotic pressure differences, higher fluxes, and higher SO42− rejections. These higher SO42− rejections will stem from both the increased volume flow and a decrease in spontaneously arising electrical fields. 4.4. Deviations from local electrical neutrality in ultrathin barrier layers The sub-section on dielectric exclusion demonstrated that accounting for the effects of water structure gives ion partition coefficients that are approximately concentration-independent. However, these coefficients vary greatly with the magnitude of the ion charge and can depend on the ion size. Indeed NF rejections depend greatly on ion charge [95]. This subsection demonstrates how a model using such concentration-independent (but ion-specific) partition coefficients can reproduce some trends in the concentration dependences of experimental NF salt rejections. Such models may potentially quantify the membrane performance (including its dependence on the feed composition) in terms of a limited number of parameters such as excess ion solvation energies and ion diffusion coefficients in the membrane barrier layer. Most models of NF assume electrical neutrality (including fixed charges) throughout the membrane. However, with the development of barrier layers as thin as 20 nm [96–99], this assumption may not apply. Deviations from electrical neutrality can stem from either unequal excess solvation energies for.

57

cations and anions in the membrane or charged membrane surfaces. As Fig. 16 shows, differences in intrinsic ion partition coefficients (partition coefficients in the absence of an electrical potential, Γ(int) ≡ exp i (−Wi)) of anions and cations will lead to space-charge regions in the ultrathin barrier layer, and for sufficiently thin membranes, these regions may extend throughout the barrier. (In Fig. 16, the potential drop eventually gives rise to equal cation and anion partition coefficients in the center of the barrier layer.) Space-charge effects will become more important with smaller ion partition coefficients, larger differences between cation and anion intrinsic partition coefficients, lower ion concentrations in solution, increases in membrane surface charge, and thinner barrier layers. Partition coefficients for cations and anions should show especially large differences for salts such as MgCl2 and Na2SO4 due to the strong exclusion of the divalent ion. As Fig. 16 shows, for sufficiently thick membranes, in the interior of the membrane the magnitude of the electrical potential is sufficient to give equal cation and anion concentrations, despite the higher intrinsic partition coefficient for the anion. Nevertheless, the space-charge regions may still dominate transport resistances. With deviations from electrical neutrality, full modelling of transport in barrier layers requires simultaneous solution of the Nernst-Planck and Poisson equations. However, Yaroshchuk and coworkers developed a simplified treatment that assumes equilibrium Poisson-Boltzmann ion concentration profiles in membranes with small differences between feed and permeate ion concentrations [100]. This approximate treatment gives salt transport resistances (see Eq. (96) below) that are often within 10% of those calculated with the Nernst-Planck-Poisson approach, and at most a factor of 2 different for the conditions that were investigated [100]. Moreover, the quasi-equilibrium approach demonstrates how partition coefficients of one ion can control salt transport. The simplified approach starts with the SDEM, where Eq. (93) describes transport. ðeÞ

Ji ¼ −

Di Γi ðxÞci dμ i ∙ RT dx

ð93Þ

Fig. 15. Plots of calculated (a) volume fluxes, (b) chloride rejections, and (c) sulfate rejections as a function of the Na2SO4 salt mole fraction (indicated as a percentage on each plot) and the ratio of applied transmembrane pressure to the feed osmotic pressure. The assumed feed solutions contained NaCl and Na2SO4. Calculations employed the SDEM model with P Naþ ¼ 10 0μm=s; P Cl− ¼ 10μm=s; P SO2− ¼ 0:05μm=s, A = 10−11m/s ∙ Pa, and the percentages are the salt mole fractions of sulfate. 4

58

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

Membrane Skin

Concentration or Electrical Potential

Feed

Permeate

Potential .

Anion Cation

Non-neutral regions, High salt transport resistance Fig. 16. Scheme of cation and anion concentration profiles as well as electrical potential in a membrane barrier layer that exhibits a much higher partition coefficient for the anion than the cation. Negatively charged regions near the faces of this membrane create large potential gradients. Potential drops in solution are smaller than those in the membrane due to higher ion concentrations in solution. Adapted from reference [100] with permission of the American Chemical Society.

energies of the two ions are not the same or because there is fixed charge on the external membrane surface that creates a potential difference between the membrane and solution. In either case, the space charge region may control transport, even when the interior of the membrane is electrically neutral [100]. Fig. 17 shows the calculated resistances to salt transport as a function of barrier-layer thickness (with 3 different surface charge densities on the membrane) for NF of a salt MA2, where the cation M2+ has a much lower intrinsic partition coefficient in the membrane than the anion A−. Particularly for the positively charged membrane but to some extent for all membranes, the resistance to salt transport initially increases rapidly, then decreases and finally increases at the same slope for all three membranes. Eq. (98) describes salt rejection, Res, in terms of salt diffusion resistance, Rs, and volume flux Jv. Res ≡ 1−

In this equation, Di is the ion diffusion coefficient that includes all the effects of short-range interactions of the ion with the membrane matrix, and the ion partition coefficient, Γi, may vary with position across the membrane. This partition coefficient is simply the ratio of the real and virtual ion concentrations, where at quasi-equilibrium the virtual ion concentration is that in the external solutions. Separating variables and integrating this equation across the membrane while noting that the value of the virtual concentration is nearly constant due to the quasi-equilibrium assumption, one obtains ðeÞ

Ji ¼

Di ci Δμ i RT R L dx 0 Γ ðxÞ i

ð94Þ

Substituting for the electrochemical potential difference across the membrane leads to Ji ¼

  Di F Z i ci Δφ Δci þ R L dx RT 0 ð Þ Γi x

ð95Þ

Noting the stoichiometry of the salt and invoking the zero-current condition yields an expression for Δφ. Finally, substituting the expression for Δφ into Eq. (95) leads to Eq. (96) (see reference [100] for more details on the derivation of these equations).

Rs ≡

Δcs Zþ ∙ ¼ Js D− ∙ðZ þ −Z − Þ

ZL 0

dx Z− − ∙ Γ− ðxÞ Dþ ∙ðZ þ −Z − Þ

ZL 0

dx Γþ ðxÞ

ð96Þ

In Eq. (96), Rs is the diffusion resistance to salt transport and Δcs is the salt concentration difference across the barrier layer. Importantly, this equation shows that if the partition coefficient for one of the ions is small, that ion controls transport. Moreover, space-charge regions where the partition coefficient of one of the ions is very low will also dominate the resistance to salt transport. Importantly, ion partition coefficients depend on both electrical potential differences and excess solvation energies as Eq. (97) shows. Γ ðxÞ ¼ expð−Z  ψðxÞ−W  Þ

csp Rs J v ¼ csf 1 þ Rs J v

ð98Þ

High rejections require large values of the product RsJv. For a given pressure drop, Jv is inversely proportional to the membrane thickness. Moreover, because of the initial rapid rise in Rs with membrane thickness, the largest rejections at a given pressure drop will occur for membranes with thicknesses b10 nm (half thicknesses b5 nm). Although such thin barrier layers will likely contain defects that reduce rejection, the calculations show the value of creating ultrathin membranes to achieve high rejections. Interestingly, lipid bilayer membranes are highly impermeable and yet have thicknesses around 4 nm [101,102]. Unequal cation and anion partitioning or surface charge could greatly affect the permeability of those membranes. The shapes of the plots in Fig. 17 reflect interplay between the double-layer thickness in the membrane and the surface potential. (Note that with the assumed dielectric constants and intrinsic partition coefficients in this simulation, the thickness of the electrical double layer in the membrane is much larger than in solution due to low ion concentrations in the membrane [100].) At large barrier-layer thicknesses (N25 nm), the electrical double-layer thickness is constant, and the interior of the membrane is electrically neutral. Making the membrane even thicker simply increases the thickness of the neutral region of the membrane and increases the diffusion length. Thus, the resistance increases linearly with the same slope for all three membranes at large barrier-layer thicknesses. Initially in Fig. 17, Rs rapidly rises with a nearly constant slope because the barrier layer is thin enough that the electrical potential does not noticeably alter the ion partitioning. The low concentration of excluded ion throughout these barrier layers leads to a rapid rise in the resistance with thickness. Subsequently, when the barrier-layer thickness increases the screening effect increases as well, so the partitioning of the excluded ion grows even near the surface of

ð97Þ

In this equation, ψ(x) is the dimensionless equilibrium electrical potential difference between the solution and the point in the membrane and W± is the dimensionless excess ion solvation energy. In the center of a thick membrane the equilibrium potential takes a value that yields equal partition coefficients for the cation and anion to give electrical neutrality. Space charge regions where the cation and anion partition coefficients are not equal can occur either because the solvation

Fig. 17. Resistances to salt transport, Rs, as a function of barrier-layer half-thickness and fixed surface-charge density for a membrane equilibrated on both sides with a solution containing 10 mM MA2. The simulation assumes Γ(int) = 0.14 for A− and 3.4 × 10−4 for i M2+, as well as barrier-layer diffusion coefficients of 5 × 10−12 m2/s for M2+ and 1 × 10−11 m2/s for A−. Dashed lines show the linear relationship between thickness and resistance when the barrier layer is significantly thicker than the electrical double layer. Used with permission of the American Chemical Society [100].

Rs or Rejection

A. Yaroshchuk et al. / Advances in Colloid and Interface Science 268 (2019) 39–63

Dominant Ca2+ exclusion increases with surface-charge screening

Dominant Ca2+ Dominant Cl- exclusion decreases with surfacecharge screening

space-charge layer thickness decreases

Concentration (log scale) Fig. 18. Qualitative plot of CaCl2 NF rejection or salt resistance as a function of the CaCl2 concentration in the feed solution. The NF membrane contains a negative surface charge.

the membrane to reduce the overall resistance to salt transfer. Thus, the resistance shows a local maximum as a function of thickness. Deviations from electroneutrality could explain some unusual trends in the rejection of ions as a function of solution concentration. In NF of solutions containing CaCl2, rejection shows a maximum as the concentration of CaCl2 increases [52,53,60,103]. Moreover, in some cases rejection may first decrease with concentration, then increase and finally decrease again as shown schematically in Fig. 18. Calculations of Rs as a function of CaCl2 concentration show such a trend without resorting to changing ion excess solvation energies or membrane surface charge densities as a function of salt concentration. The calculation assumes a negatively charged membrane. At low CaCl2 concentrations, Cl− is likely the more highly excluded ion due to the negatively charged membrane surface. Increasing the salt concentration screens the surface charge to decrease Cl− exclusion and decrease rejection. However, with sufficient screening of the surface charge, Ca2+ becomes the more excluded ion because it probably has a higher excess solvation energy. This Ca2+ exclusion leads to a negatively charged region in the ultrathin barrier layer, and exclusion increases with more surface-charge screening. Finally, as the CaCl2 concentration continues to increase, the increased concentration of ions in the membrane leads to a smaller space-charge layer and less salt rejection. Other phenomena such as Ca2+ adsorption might partially explain such trends, but deviation from electroneutrality naturally yields these trends without variation of any parameters. Although unequal partitioning of cations and anions to create spacecharge regions may explain some trends in NF, the extent to which this occurs in membranes under real conditions is unknown because there are no data on single-ion partition coefficients for the ultra-thin barriers in thin-film composite membranes. Values of partition coefficients for NaCl in cellulose acetate are as low as 0.014, [104] and one can expect much lower values for divalent ions based on their relatively high rejections in NF (and stronger dielectric exclusion, for example). Thus, deviations from electroneutrality may be significant. Some recent experiments [100] clearly show that surface charge affects the rejections of divalent ions. Coating a Dow NF270 membrane with a polycation to give a positively charged surface decreases Na2SO4 rejection but increases MgCl2 rejection. Because the polyelectrolyte layer is very thin, this layer primarily affects ion partitioning, and the opposite effects on Na2SO4 and MgCl2 strongly suggest changes in partitioning and deviations from electroneutrality in the membrane barrier layer. 5. Conclusions This critical review examined thermodynamic and mechanistic approaches to modelling NF. These approaches are complementary in that mechanistic models provide insight into the thermodynamic coefficients, and the thermodynamic methods set appropriate limits on mechanistic approaches. The irreversible thermodynamics approach employed the concepts of local (at the pore-radius scale) thermodynamic equilibrium and

59

virtual solutions to derive (in terms of simple integrals) relationships between solute rejections, transmembrane volume flows and hydrostatic pressure differences for macroscopically homogeneous membranes in single-salt solutions. These relationships apply to arbitrary dependences of membrane transport properties on virtual salt concentration, account for solution non-ideality and can serve as the basis for analyzing various rejection mechanisms. Irreversible thermodynamic treatments of multi-ion feed solutions are more complicated, but relationships between solute rejections, transmembrane flows, and transmembrane pressure differences are accessible for the limiting case of one dominant single salt and any number of trace ions. These relationships explain many of the peculiarities, e.g. negative rejections or strong dependences of trace rejections on the kind of dominant salt, in NF of electrolyte mixtures. The approximation of composition-independent membrane transport properties yields analytical expressions for the integrals in the thermodynamic treatment and provides further insight. Besides the well-known Spiegler-Kedem relationship between the rejection of a single salt and transmembrane volume flow, this critical review also derived expressions for the dependence of volume flux on the transmembrane pressure difference and for the rejections of trace ions. The derivations identify the parameters that control the NF membrane performance in single-salt and mixed electrolyte solutions (trace approximation). These parameters include: salt reflection coefficients, salt permeabilities, electrokinetic-charge densities, membrane electrochemical perm-selectivities, and permeance to trace ions. A model of straight cylindrical capillaries can relate these parameters to the membrane pore size, ion partition and diffusion coefficients and steric hindrance factors for diffusive and convective transfer of solutes. Using the capillary model, this critical review explored the NF mechanisms usually invoked in the literature: steric exclusion/hindrance, Donnan exclusion and dielectric exclusion. The principal problem with the models based on steric exclusion/hindrance phenomena is that they use macroscopic hydrodynamics to account for the finite size of the solute but neglect the commensurate size of solvent molecules. Moreover, macroscopic hydrodynamics may not apply in the sub-nm pores invoked to explain NF. Perhaps for this reason the ubiquitous model of combined Donnan-Steric Exclusion/Hindrance predicts several trends that have never been observed experimentally. Thus, even if this model can satisfactorily fit a limited set of experimental data, the resulting adjustable parameters probably do not correspond to true physical characteristics of the membranes. Mechanistic studies are clearly very important for designing new membranes, but future simulations in this area may need to depart from typical continuum models. In the case of dielectric exclusion, a brief recapitulation of the macroscopic approach shows that strong ionic screening of interactions of ions with polarized surfaces of pores of dielectric membranes is largely due to the high bulk dielectric constant of water. Accounting for the supramolecular structure of water (within the scope of non-local electrostatics) can make this screening much weaker and the ion excess solvation energies concentration-independent in a crude approximation. This also lends justification of sorts to some features of the popular Born formula such as the scaling of excess solvation energy with the square of ion charge. At the same time, this analysis shows that the parameterization in terms of a macroscopic dielectric constant of “pore” water is not physically meaningful, and one might more profitably use ion excess solvation energies as model parameters. Overall, the nanopore models suffer from both questionable application of macroscopic principles to nanopores and challenges in verifying the membrane properties that they use as inputs. Thus, until experimental characterization of the complex chemical and pore structure of NF membranes is possible, alternative advanced engineering models may be preferable for describing NF with a few thermodynamic coefficients that one can determine from experiment.

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One such model relies on a minimum set of compositionindependent membrane parameters, namely ion permeances, and assumes a solution-diffusion-electromigration ion-transport mechanism in the membrane. Despite the limited number of fitting parameters, the model reproduces rather complex experimental trends in NF of electrolyte mixtures containing various dominant single salts and trace ions. It also reproduces reasonably well the patterns observed in NF of ternary ion mixtures of arbitrary composition. An obvious weakness of this engineering model is that its use, in principle, requires separate experimentation for each feed composition. In some instances, the ion permeances may not depend greatly on the solutions composition, but this will not always be the case. Another (somewhat more mechanistic) model postulates composition-independent ion excess solvation energies and takes into account the small, finite thickness of barrier layers of NF membranes. Despite constant intrinsic ion-partition coefficients the model predicts concentration-dependent ion rejections and even qualitatively reproduces some experimentally observed trends. Along these lines, one could try and develop an approach to the parameterization of NF performance in terms of a limited number of constant (but ion-specific) parameters such as ion excess solvation energies, ion diffusion coefficients in the membrane and surface charge density. The scope of applicability of the engineering model could also be increased by relaxing the approximation of zero ion transmission coefficients (adding convection to the SDEM ion transport mechanism). Technically, this is not a problem, especially in the case of one dominant salt and trace ions where a (quasi)analytical solution of Eq. (48) is available. However, this would mean increasing the number of adjustable parameters by a factor of two (a transmission coefficient for each ion in addition to the ion permeances). Unambiguous determination of this increased number of model parameters from experiment crucially depends on the quality of experimental data, primarily on quantitative control of concentration polarization. As demonstrated in [105] an inhomogeneous distribution of concentration polarization along the membrane surface, which is typical for most membrane test cells, can have especially dramatic consequences for interpreting very different simultaneous rejections of different ions in NF of electrolyte mixtures. Very recently, a novel test-cell design with a rotating disk-like membrane was developed, and the concentration polarization was homogeneous over the membrane surface in this case [106]. The use of this kind of test cell may provide the quality input required for advanced engineering models. Given the challenges in characterizing the properties and structure of ultrathin barrier layers on practical membranes, we think that engineering models with composition-independent ion excess solvation energies will likely provide the most insight into NF transport for the foreseeable future. Glossary and Symbols

a ai A b B ci cim cs csf csp ct ctf ctp

constant defined in Eq. (49) to help describe trace-ion reciprocal transmission ion radius hydraulic permeance of the membrane (Eq. (43)) constant defined in Eq. (50) to help describe trace-ion reciprocal transmission solute diffusion permeance of the membrane (Eq. (44)) concentration of species i in a virtual solution concentration of species i in the feed solution immediately adjacent to the membrane concentration of salt in virtual solution concentration of salt in the feed solution concentration of salt in the permeate solution concentration of a trace ion in virtual solution concentration of a trace ion in the feed solution concentration of a trace ion in the permeate solution

cX C Cp C0 dw Di D(δ) s fn fs ft F F(x) Fm Fp i g h h Ie I Ji Jk Jt Jv Jw kf K Ka lkn L Lnk n NA Nk p Pi Pi Pk Ps Ps ðδÞ Ps Pt P+ P P− Pes r rp R Rs Re(int) i Res Re(obs) s Re(int) s Ret s s S t ti t+

concentration of fixed charge in the membrane sum of the concentrations of all ions in solution sum of the concentration of all ions in the permeate solution sum of the concentrations of all ions in the feed solution diameter of a water molecule diffusion coefficient for ion i in the membrane salt diffusion coefficient in the feed boundary layer next to the membrane volume fraction of solute n reciprocal transmission of salt reciprocal transmission of a trace ion (Eq. (47)) Faraday constant function defined for describing trace ion concentrations (Eq. (46)) function in Eq. (82) defined for concise presentation of Eq. (81) function in Eq. (82) defined for concise presentation of Eq. (81) index number representing a specific ion electrical conductivity at zero volume flow (Eq. (6)) membrane thickness (Section 1 only) characteristic pore dimension electric current density ionic strength in virtual solution flux of ion i transmembrane flux of component k flux of a trace ion volume flux solvent flux Freundlich isotherm equilibrium constant ratio of salt permeability to trace ion permeability dimensionless association constant of weak-acidic groups symmetric kinetic coefficients membrane thickness kinetic coefficients in irreversible thermodynamics Freundlich isotherm exponent Avogadro's number moles of component k hydrostatic pressure permeability of ion i in the membrane permeance of ion i (Eq. (85)) in the membrane diffusion permeance of species k permeability of salt in the membrane salt permeance in the membrane salt permeance in the boundary layer adjacent to the membrane permeability of a trace ion in the membrane permeability of the cation of a salt in the membrane permeance of the cation or anion of a salt permeability of the anion of a salt salt Péclet number radial coordinate pore radius gas constant membrane resistance to salt transport (see Eq. (96)) intrinsic rejection of ion i (Eq. (97)) salt rejection observed salt rejection intrinsic salt rejection rejection of a trace ion entropy subscript denoting a salt property membrane area time transport number of ion i transport number for the cation of a salt

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t− T Τi Τs Τt Τ+ Τ− u up u0 v Vk Vw W ΔWBorn Wi, bulk Wi, pore W± x X X0 Xn Yn Zi Zt ZX Z+ Z− αi αi α+ αþ α− α− βi βi β+ βþ β− β− γ γt γs Γi Γ− Γ+ δ δkn ΔΠ ε

transport number for the anion of a salt temperature in K ion i transmission coefficient at zero electric current (Eq. (11)) salt transmission coefficient at zero current (Eq. (20)) transmission coefficient at zero current for a trace ion transmission coefficient at zero electric current for the cation of a salt transmission coefficient at zero electric current for the anion of a salt relative double ionic strength (Eq. (89)) relative double ionic strength in the permeate solution relative double ionic strength in the feed solution solution velocity at a specific point molar volume of component k molar volume of solvent entropy production function (per unit membrane area) Born excess solvation energy. solvation energy for ion i in a bulk solution. solvation energy for ion i in a pore. dimensionless excess ion solvation energy for the cation or anion of a salt membrane coordinate dimensionless fixed charge concentration (Eq. 67) maximum dimensionless fixed charge concentration at full dissociation thermodynamic driving forces for transport thermodynamic fluxes such as volume flow and individual component fluxes charge number of ion i charge number of a trace ion valence of the fixed charge species charge number on the cation of a salt charge number on the anions of a salt hindrance factor for the convection of ion i cross-sectional averaged hindrance factor for the convection of ion i hindrance factor for convection of the cation of a salt cross-sectional averaged hindrance factor for convection of the cation of a salt hindrance factor for convection of the anion of a salt cross-sectional averaged hindrance factor for convection of the anion of a salt diffusion hindrance factor for ion i cross-sectional averaged diffusion hindrance factor for ion i diffusion hindrance factor for the cation of a salt cross-sectional averaged hindrance factor for diffusion of the cation of a salt diffusion hindrance factor for the anion of a salt cross-sectional averaged hindrance factor for diffusion of the anion of a salt membrane porosity divided by tortuosity activity coefficient of a trace ion in the virtual solution activity coefficient of salt in the virtual solution coefficient for partitioning of ion i between the real and virtual solution coefficient for partitioning of a salt anion between real and virtual solutions coefficient for partitioning of a salt cation between real and virtual solutions thickness of the solution boundary layer adjacent to the membrane Kronecker delta osmotic pressure difference between the feed and permeate solutions dielectric constant of a medium

εbound εm εp εs εwater ε0 Ζ θs κ λB λi μ(c) i μ(e) i μs μ + (c) μ − (c) ν ν+ ν− Π Πf ρek ρij σs τi τ+ τ− φ Φs χ χ∗ ψ Ψ

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dielectric constant of water bound on a surface. dielectric constant of membrane matrix dielectric constant of solvent in a pore dielectric constant of a solution dielectric constant of free water. permittivity of free space product of the ion charges in a solution with three ions membrane electrochemical activity in dominant salt (defined by Eq. (24)) reciprocal of the Debye screening length Bjerrum length ratio of the solute radius to the pore radius chemical potential of species i electrochemical potential of species i salt electrochemical potential chemical potential of the cation of a salt chemical potential of the anion of a salt sum of the anion and cation stoichiometric coefficients in a salt cation stoichiometric coefficient in a salt anion stoichiometric coefficient in a salt osmotic pressure in the virtual solution osmotic pressure of the feed solution (versus pure solvent) electrokinetic charge density (defined in Eq. (8)) function defined in Eq. (83) for concise expression of Eq. (81) salt reflection coefficient transmission coefficient of ion i transmission coefficient of the cation of a salt transmission coefficient of the anion of a salt electrical potential in the virtual solution salt differential osmotic coefficient in the virtual solution membrane hydraulic permeability at zero gradients of ion electrochemical potentials hydraulic permeability at zero current (Eq. (15)) dimensionless Donnan potential dimensional Donnan potential

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