General force balances on ions and water in membranes

Solute Transport

Part 4. General force balances on ions and water in membranes

matrix of porous medium

ion ion ion

ion

water fills all space in the pores between the ions/solutes

ion ion ion

water water water

water

pores

feed membrane permeate

Donnan equilibrium Steric Partitioning

advection diffus ion diffus ion advection electrom igration electro-

migration

c₊

c

advection

acid-bas e equilibrium

NH₄⁺ NH₃+ H⁺ H₂CO₃ HCO₃+ H⁺

surface complexation

R R + H⁺

mem brane potential Passage =

feed conc.

permeate conc.

Donnan potential

pores

Fig. 7.7:Illustration of the terminology used to describe transport of solvent and solutes in a porous medium. Solutes, such as ions, flow through the pores in the porous medium (also called structure, or matrix), and all space around the solutes which is not occupied by the structure, is filled with solvent (fluid), often water. The porous medium can be a membrane, ion absorbent material, or a porous electrode. When water is the solvent, an ion will be hydrated. A bare ion together with its shell of tightly bound hydration water as a whole is called an ion. The solvent (for instance water) is ‘free’ and flows through all the space that is not occupied by the (hydrated) ions and other solutes, or occupied by the porous medium.

196 Solute Transport

of the ion with all other phases, and these depend on the velocity difference between the ion and that phase, resulting in

−𝜕 𝜇_𝑖

𝜕 𝑥^′ +∑︁

𝑘

𝑓_𝑖

-𝑘 𝑣^′

𝑘−𝑣^′

𝑖

=0 (7.64)

where the summation over 𝑘 is over all phases and species, including the solvent and the porous structure, and where 𝑓_𝑖

-𝑘 is a measure of the friction of a mole of ions𝑖with all of phase𝑘 present. In Eq. (7.64) we introduce the notation^′ for movement following a path through a tortuous pore. Velocities𝑣^′

𝑘 are ‘real’ interstitial velocities following this path.

When phase𝑘is a species of which the concentration can change, such as for any of the ion types, it is relevant to make the frictional term proportional to the amount (concentration) of that species. Thus for ion-ion friction, acting between ion type𝑖and𝑘, in the force balance for ion type𝑖, Eq. (7.64), 𝑓_𝑖-𝑘is made a linear function of the concentration of the other type of ion, 𝑘. Thus for ion-ion friction, in the force balance for species𝑖, 𝑓_𝑖

-𝑘 is replaced by 𝛽_𝑖

-𝑘𝑐_𝑘, where 𝛽_𝑖

-𝑘 is a constant, and𝑐_𝑘 is the local concentration of the other solute. (In doing so, there will ultimately be one ion-ion friction coefficient for each ion-ion pair, i.e., 𝛽_𝑖

-𝑘 =𝛽_𝑘

-𝑖.) However, when the other phase ‘is simply there’, the use of 𝑓_𝑖

-𝑘 as a friction factor between ionic species𝑖and that other phase, suffices. This applies for the friction with the porous medium, such as a membrane, and for friction with the fluid, because it surrounds all ions and solutes. There is no ‘concentration’ to speak of for these other phases. Indeed for friction of an ion with these two phases, there is no need to consider a proportionality with the ‘concentration of the porous medium’ or with ‘the concentration of water’. The porous medium is just there, while each ion is always completely enveloped by water molecules.

Of course, for friction of solutes with the porous medium, if we have an open structure with large pores, 𝑓_𝑖

-mwill be much lower than for a tight structure with small pores, but for a given porous structure, 𝑓_𝑖

-mcan be taken as a constant that describes the ion-matrix friction.

From this point onward, we first leave out ion-ion friction. Including that the porous medium has zero velocity, combination of Eqs. (7.63) and (7.64) results in

−𝜕ln𝑐_𝑖

𝜕 𝑥^′

−𝑧_𝑖

𝜕 𝜙

𝜕 𝑥^′

−𝜕 𝜇

exc,𝑖

𝜕 𝑥^′

−𝜐_𝑖

𝜕 𝑃^tot

𝜕 𝑥^′

=−𝑓_𝑖

-F 𝑣^′

F−𝑣^′

𝑖

+ 𝑓_𝑖

-m𝑣^′

𝑖 (7.65)

where we include the excess term that describes volumetric interactions, and the insertion pressure of a solute that has volume. These last two terms on the left would be zero when we assume the ions to be point charges.

From a force balance to the Nernst-Planck equation 197

Effect of temperature on transport. We continue here a discussion of temperature effects set up for equilibrium (EDL theory) on p. 39. As explained above, the driving force on (a mole of) ions is minus the gradient of the chemical potential, and in that analysis we should multiply equations given above with the factorRTto obtain a driving force with the correct unit of N/mol, for instance in Eq. (7.65). Then the right side is also multiplied byRT and the resulting friction factors𝑅𝑇 𝑓_𝑖

-Fand𝑅𝑇 𝑓_𝑖

-mdo not have a specific relation toRT. Below we replace 𝑓_𝑖

-Fwith 1/𝐷_𝑖 and thus 𝐷_𝑖 linearly increases with𝑅𝑇. Thus, in a typical NP equation, such as Eq. (7.67) below, diffusion coefficients linearly depend on temperature, T, and the origin of this effect is in a temperature dependence of driving forces, not in a an intrinsic change to its mobility. In addition, there will be such an intrinsic effect because the viscosity of solvent decreases with temperature and thus mobility will go up. So a T-dependence of the diffusion coefficient combines an intrinsic increase in solute mobility, with theRT-term.

In the next step we multiply each side by𝑐_𝑖and we make the replacement𝐽^′

𝑖 =𝑣^′

𝑖𝑐_𝑖, because the flux of an ion through a pore𝐽^′

𝑖 is equal to its own velocity𝑣^′

𝑖times its concentration𝑐_𝑖. We make the replacement 𝑓_𝑖

-F=1/𝐷_𝑖for the ion-fluid friction, resulting in

−𝜕 𝑐_𝑖

𝜕 𝑥^′

−𝑧_𝑖𝑐_𝑖

𝜕 𝜙

𝜕 𝑥^′

−𝑐_𝑖

𝜕 𝜇exc,𝑖

𝜕 𝑥^′

−𝑐_𝑖𝜐_𝑖

𝜕 𝑃^tot

𝜕 𝑥^′

=− 1 𝐷_𝑖

𝑐_𝑖𝑣^′

F−𝐽^′

𝑖

+ 𝑓_𝑖

-m𝐽^′

𝑖 (7.66)

which we rewrite to an expression explicit in flux𝐽^′

𝑖 that is 𝐽^′

𝑖 =𝐾

f,𝑖𝑐_𝑖𝑣^′

F−𝐾

f,𝑖𝐷_𝑖 𝜕 𝑐_𝑖

𝜕 𝑥^′ +𝑧_𝑖𝑐_𝑖

𝜕 𝜙

𝜕 𝑥^′ +𝑐_𝑖

𝜕 𝜇exc,𝑖

𝜕 𝑥^′ +𝑐_𝑖𝜐_𝑖

𝜕 𝑃^tot

𝜕 𝑥^′

(7.67) where (Spiegler and Kedem, 1966, Eq. 49)

𝐾f,𝑖=1/(1+ 𝑓_𝑖

-m/𝑓_𝑖

-F) . (7.68)

is a factor which describes the importance of ion-matrix friction relative to ion-fluid friction, with𝐾_f,𝑖 → 1 for very low ion-matrix friction, and𝐾_f,𝑖 → 0 when ion-matrix friction is very large.

In the next step, Eq. (7.67) –in which fluxes and velocities𝐽^′

𝑖 and𝑣^′

Fare defined inside a tortuous pore– is converted to an expression with fluxes and velocities defined on the macroscopic scale of the porous material,𝑣

Fand𝐽_𝑖, which is the common usage of fluxes and velocities in this book. To this end, we implement that𝐽_𝑖 =𝐽^′

𝑖·𝑝/𝜏wherepis porosity and𝜏is tortuosity. The latter describes that the path through a pore is longer than straight

198 Solute Transport

across a material, because it is constantly at an ever-changing angle to a direct line that goes from one to the other side of a material. If there is a certain flux𝐽^′

𝑖 along that curvy path, it is less by a factor𝜏(with𝜏larger than 1) if we project on the direction straight through the material. The factorpfor porosity enters because flux𝐽^′

𝑖 is per unit pore and with𝐽_𝑖 we aim to describe the superficial flux (which is per unit geometrical, i.e., ‘total’ area).

These conversions from𝐽^′

𝑖 to𝐽_𝑖 also apply to the conversion from𝑣

F

′to𝑣

F. But for the fluid, also a factor 1−𝜂enters in the conversion, because inside the pores, a volume fraction 𝜂 is blocked out for the fluid.^xii Thus the interstitial fluid velocity in the pore, along the pore direction,𝑣^′

F, relates to the superficial fluid velocity straight across the material,𝑣

F, by 𝑣F=𝑣^′

F· (1−𝜂) ·𝑝/𝜏. The final implementation is that the coordinate axis straight through the material is shorter than the curved path, and thus𝜕 𝑥=𝜕 𝑥^′/𝜏. If we include all of this in Eq. (7.67), we obtain the intermediate result that

𝐽_𝑖 𝜏 𝑝

=𝐾

f,𝑖𝑐_𝑖𝑣

F

𝜏

𝑝 (1−𝜂) −1 𝜏

𝐾f,𝑖𝐷_𝑖 𝜕 𝑐_𝑖

𝜕 𝑥^′ +𝑧_𝑖𝑐_𝑖

𝜕 𝜙

𝜕 𝑥^′ +𝑐_𝑖

𝜕 𝜇exc,𝑖

𝜕 𝑥^′ +𝑐_𝑖𝜐_𝑖

𝜕 𝑃^tot

𝜕 𝑥^′

. (7.69) Multiplying all sides by 𝑝/𝜏, and making use of the variable𝜀 =𝑝/𝜏², we obtain the most general NP equation^xiii,xiv

𝐽_𝑖 = 𝐾

f,𝑖

1−𝜂

𝑐_𝑖𝑣_F−𝜀 𝐾_f,𝑖𝐷_𝑖 𝜕 𝑐_𝑖

𝜕 𝑥 +𝑧_𝑖𝑐_𝑖

𝜕 𝜙

𝜕 𝑥 +𝑐_𝑖

𝜕 𝜇exc,𝑖

𝜕 𝑥

+𝑐_𝑖𝜐_𝑖

𝜕 𝑃^tot

𝜕 𝑥

(7.70) where the fraction of the pores occupied by solutes is𝜂and the porosity (fraction of total volume that is pores) isp. Fluxes are defined per cross-section of the total material (pores plus matrix), but concentrations are defined per volume of pore phase. Thus, a balance of forces acting on solutes inside a porous medium results in a general NP-equation that includes a group𝐾

f,𝑖/(1−𝜂) → 𝐾^′

c,𝑖 and𝜀 𝐾

f,𝑖 → 𝐾^′

d,𝑖, where𝐾^′

c,𝑖> 𝐾^′

d,𝑖. These latter 𝐾^′-values have similarities to the convective and diffusive hindrance functions derived on the basis of hydrodynamic theories for transport of spherical particles through cylindrical pores, where also the term for convection is larger than for diffusion, as used for instance in SEDE theory (see next box).

If we neglect all possible solute volume effects, Eq. (7.70) simplifies to the extended NP

xiiThe volume fraction𝜂used here is different from𝜂in Ch. 4 where it relates to the volume fraction of the immobile structure. Here it relates to the fraction of the pores (that themselves have a porosityp) that contains solutes.

xiiiOften this term𝜏2

is replaced by𝝉and then called the tortuosityfactor.

xivIn this book we generally present flux expressions as if there is only one direction,𝑥, to make the equations more insightful. The reader familiar with flow in multiple directions will understand that a gradient𝜕/𝜕 𝑥can be generalized, for instance by using the notation∇.

From a force balance to the Nernst-Planck equation 199

equation

𝐽_𝑖=𝐾

f,𝑖𝑐_𝑖𝑣

F−𝜀 𝐾

f,𝑖𝐷_𝑖 𝜕 𝑐_𝑖

𝜕 𝑥 +𝑧_𝑖𝑐_𝑖

𝜕 𝜙

𝜕 𝑥

(7.71)

which is an equation we will use extensively throughout this book. When solutes have no direct friction with a porous medium, then𝐾_f,𝑖=1 and Eq. (7.71) simplifies to Eq. (7.2).

The complete transport model based on Eq. (7.71), partition functions such as Eq. (2.1), and solvent transport equations given by Eq. (8.2), is called the two-fluid model (TFM), and in the context of membrane transport is called the solution-friction (SF) model.

The DSP model for ion transport in membranes. When the model explained in this section for multicomponent transport of charged and uncharged species through a membrane (which is typically also charged) is combined with a description for the Donnan equilibrium at the membrane-solution interfaces, and also includes contributions to the partitioning of ions at these interfaces such as related to ion volume, as discussed in §2.2, §2.8, and §4.2, we end up with a model called the Donnan Steric (Partitioning) Pore model, or DSP model. The state-of-the-art version is called the extended DSP-model. This ext-DSP model also includes how membrane charge is a function of local ion concentrations (especially pH and absorbing ions such as Ca²⁺) and also includes how ions react with one another in the pore space, as will be addressed in Ch. 10. For instance, the bicarbonate ion, HCO₃^–, can react with a proton to form carbonic acid, H₂CO₃, and this reaction is included in the ext-DSP model. And many similar reactions can be included at the same time as well. Also the formation of ion pairs, for instance because Mg²⁺ reacts with Cl^– to MgCl⁺, can be included in the ext-DSP model, see Kimaniet al.,J. Chem. Phys. 154124501 (2021), and references therein.

A different line of theory that was specifically developed for nanofiltration, is the SEDE model, which is a very detailed extension to the SF model, as explained in detail in Lanteriet al.,J. Colloid Interface Sci. 331, 148–155 (2009). For a comparison of this SEDE model with the SF model of this section, seehere.

Finally we explain the theory to describe ion fluxes in five different cases. The first case we consider is to include ion-ion friction in addition to the other frictional forces. Leaving out all effects related to solute volume, we then have an implicit relationship for ion flux,