Ionic current and transport and transference numbersnumbers

Solute Transport

7.2 Ionic current and transport and transference numbersnumbers

Ionic current and transport and transference numbers 165 important, and a certain change inΔ𝛾important to make when𝛾is low, i.e., a small amount of extra stirring then helps a lot, at some point the incremental effect of stirring is small and the energy costs thereof will surpass the advantage of more stirring. In addition, at some point of increasing stirring intensity, another mass transport limitation in the process will become rate-limiting.

A very accurate approximation to the Airy equation (with the given boundary conditions) for the profile𝑐(𝑥˜)is

𝑐(𝑥˜) −𝑐_∞ 𝑐^∗−𝑐_∞

=1−tanh(𝛼 𝜉) (7.17)

where𝛼and𝜉are the same as above.

In conclusion, in this section we presented four models to describe steady-state diffusion and dispersion near a reactive or selective surface for neutral solutes. For the relationship between flux and concentration, these models all end up with the same result that flux is proportional to the concentration difference between bulk and surface. However, for the proportionality factor different expressions are obtained, with a dependence on diffusion coefficient that can be to the power¹/₂,²/₃, or 1.ⁱⁱⁱ

7.2 Ionic current and transport and transference

166 Solute Transport

Eq. (7.2), extended with convection, which describes the molar flux of an ion𝑖by 𝐽_𝑖=𝑐_𝑖𝑣

F−𝐷_𝑖 𝜕 𝑐_𝑖

𝜕 𝑥 +𝑧_𝑖𝑐_𝑖

𝜕 𝜙

𝜕 𝑥

where 𝐷_𝑖 is the diffusion coefficient of the ion in the electrolyte phase, and where the potential 𝜙 relates to a voltage 𝑉 by 𝜙 = 𝑉/𝑉

T. In §7.8 we explain how this extended NP-equation derives from a balance of forces acting on ions in solution.

A summation over all ions of their flux𝐽_𝑖times their valency𝑧_𝑖results for current density (unit A/m²) in

𝐼=𝐹

∑︁

𝑖

𝑧_𝑖𝐽_𝑖 =𝐹 𝑣

F

∑︁

𝑖

𝑧_𝑖𝑐_𝑖−𝐹 𝜀

∑︁

𝑖

𝐷_𝑖𝑧_𝑖

𝜕 𝑐_𝑖

𝜕 𝑥

−𝜀 𝜎_∞

𝜕𝑉

𝜕 𝑥 (7.18)

where we introduce the ionic conductivity of a solution (unit S/m, where S=A/V and A=C/m²)

𝜎_∞= 𝐹²/𝑅𝑇

∑︁

𝑖

𝑧²

𝑖𝐷_𝑖𝑐_𝑖 (7.19)

also called the electrical conductivity of a solution, 𝜅. In bulk electrolyte we have local electroneutrality,Í

𝑖𝑧_𝑖𝑐_𝑖 =0, and thus the convective term in Eq. (7.18) can be omitted, but that is not the case in a charged porous medium. We discuss ion conduction in a porous medium in the next box.

Conductivity in porous media, such as membranes. The conductivity of a porous medium where pores are filled with electrolyte, such as a flow channel that contains a mesh of material,𝜎

m, is lower than the conductivity of bulk solution by a factor𝜀, thus 𝜎m=𝜀 𝜎_∞, see p. 159. The factor𝜀is porosity, 𝑝, divided by the tortuosity factor,𝝉, i.e.,𝜀=𝑝/𝝉. For flow inside a nanoporous medium such as an ion-exchange membrane, ions also have friction with the structure, the matrix. In that case, the conductivity, 𝜎m is lower again, now by an additional factor𝐾

f,𝑖 that we discuss at p. 197. Thus 𝜎^∗

m=𝐾

f,𝑖𝜎

m =𝐾

f,𝑖𝜀 𝜎_∞.

An important topic is the electrolyte conductivity inside charged membranes, of special importance for electrodialysis (ED), see Ch. 12. Also here Eq. (7.19) can be directly applied, as conclusively proven by J.C. Díaz and J. Kamcev¹. We illustrate this fact by an extensive data set of Díaz and Kamcev for an AEM with|𝑋|=3.1 M and a CEM with|𝑋|=2.5 M for membranes with several thicknesses, tested with NaCl solutions with salt concentration from 1 mM to 1 M, see Fig. 7.1A for results obtained with an

Ionic current and transport and transference numbers 167

AEM membrane. From the slope of a trendline running through the origin, we can derive a conductivity of𝜎^∗

m∼16 mS/cm for salt concentrations 1–100 mM and𝜎^∗

m∼17 mS/m for𝑐_∞=1 M. Thus conductivity is independent of salt concentration, except for a slight increase around 1 M NaCl, see Fig. 7.1B. The constant conductance is in agreement with Eq. (7.19) given that at low enough𝑐_∞the concentration of counterions in the membrane is almost equal to|𝑋|and the concentration of coions is very low. The increase of𝜎^∗

mat 1 M NaCl is due to the increase in co- and counterion concentrations in the membrane.

To describe these data, we extend Eq. (2.11) in Ch. 2 to account for a non-unity partition coefficient,Φ𝑖(which we use to describe non-electrostatic contributions to the chemical potential of an ion), resulting in𝑐²

T,m=𝑋²+ (2Φ𝑖𝑐_∞)², which relates the total ion concentration in the membrane,𝑐_T,m, to salt concentration and membrane charge.

The counterion concentration in the membrane is then½ 𝑐

T,m+ |𝑋|

and the coion concentration½ 𝑐

T,m− |𝑋|

. These concentrations are then used in Eq. (7.19) which results in the theoretical lines in Fig. 7.1B (Φ𝑖=0.6). To fit to the data we introduce a membrane reduction factor, mrf, which describes by how much the rate of diffusion of ions is reduced in the membrane compared to solution, thus mrf=𝜎_∞/𝜎^∗

m=1/𝐾

f,𝑖𝜀. For the AEM considered in Fig. 7.1B, we then have mrf∼15 while for the CEM we derive mrf∼9.5. Note that measurement of𝜎^∗

mbased on conductivity (current-voltage relationship) is only correct when convection and diffusion can be neglected, see 1^stand 2^ndterm on the right in Eq. (7.18). For a charged membrane, convection is typically non-zero while diffusion is only zero if we can assume the same𝐷_𝑖 for all ions.

Interestingly, a similar analysis by Díaz and Kamcev of data for a commercial CEM membrane (Neosepta CMX) with|𝑋| ∼5.7 M, leads to a value of𝜎^∗

m∼5 mS/cm while other literature reports 7-10 mS/cm.These numbers lead to a reduction factor between mrf=30−60, in line with estimates for the mrf of Neosepta CMX membranes in Tedesco et al.(2018).²

We summarize these data of mrf vs. |𝑋| with the empirical function mrf = exp(𝛼

1|𝑋|^𝛼2) where|𝑋| has the unit M, with 𝛼

1=1.32 and𝛼

2 =0.6. (For 𝑋 =0 M, mrf=1, i.e., no reduction in ion mobility. In this analysis, we use mrf=√

30·60∼42 for the Neosepta membrane.) Data points and the fit line are presented in Fig. 7.2.

A similar relationship between conductivity and membrane charge density is given by Fanet al. (2022)³ where effectively mrf is expressed as an exponential function of the ionic valency squared, and with membrane charge density to the power 2/3, i.e., the empirical function above is modified to mrf∝exp 𝐴· |𝑋|^3/2

. The prefactor 𝐴is

168 Solute Transport

10 12 14 16 18

1 10 100 1000

0 0.5 1 1.5 2 2.5

0 100 200 300 400

membrane conductancem(mS/cm)

salt concentration (mM) CEM, X=2.5 M

AEM, X=3.1 M

membrane areal resistance(cm2)

membrane thickness (m) AEM, X=3.1 M

m~17 S/cm 1 M

1 mM 10 mM 100 mM

m~16 S/cm

Fig. 7.1:Resistance to ionic transport in an anion and cation exchange membrane (AEM and CEM).

A) Resistance as function of membrane thickness for an AEM. B) Ionic conductivity as function of external salt concentration. Data from Díaz and Kamcev (2021), and lines based on Eq. (7.19) and Donnan model, as explained in the nearby box.

inversely proportional to the dielectric constant in the membrane𝜀

rsquared, and the equation in Fanet al. matches that by us if𝜀

r=32 is used. Note that mrf as calculated by Fanet al. is extremely sensitive to the assumed value of𝜀_rin the membrane.

In Fanet al. (2022)³ also an expression is provided for the influence of membrane porosity, or water fraction,p, on the diffusion coefficient in the membrane, which is mrf = (2/𝑝−1)². This expression implies that when 𝑝=0.48, mrf=10, and when 𝑝=0.18, mrf=100, which are realistic predictions. However, it is not known whether this formula relates to an influence on diffusion itself (relating to porosity and tortuosity), or also to ion-membrane friction (resulting in𝐾

f,𝑖<1). In §7.8 we discuss approaches to include ion-membrane friction in a transport model.

We can now define the transference number of an ion,𝑡_𝑖, as 𝑡_𝑖 =

𝑧2 𝑖𝐷_𝑖𝑐_𝑖 Í

𝑖𝑧²

𝑖𝐷_𝑖𝑐_𝑖

. (7.20)

The transference numbers,𝑡_𝑖, are positive for all ions, and add up to unity, i.e.,Í

𝑖𝑡_𝑖=1. For a binary𝑧₊:𝑧₋ salt solution (only one cation, one anion), the transference number of each

Ionic current and transport and transference numbers 169

0 10 20 30 40 50

0 1 2 3 4 5 6

membrane reduction factor

membrane charge density (M) mrf=exp(₁|X| )

₁=1.32, ₂=0.6

2

Fig. 7.2:Data for ionic conductivity of an ion-exchange membrane (IEM) recalculated to a membrane reduction factor, mrf, which is the ratio of ion diffusion coefficient in an IEM relative to that in free solution, plotted as function of membrane charge density,|𝑋|. An empirical fit for mrf starts at mrf=1 at𝑋=0, and increases to above mrf=40 for the Neosepta CMX membrane (|𝑋| ∼5.7 M).

of the ions is

𝑡_𝑖 = |𝑧_𝑖|𝐷_𝑖

𝑧₊𝐷₊+ |𝑧₋|𝐷₋ (7.21)

and we have𝑡₊+𝑡₋=1.

In the absence of concentration gradients we only have electromigration as a driving force for ion transport, and then implementing the above equations in Eq. (7.2) results for the ionic flux in

𝐽_𝑖 =𝑐_𝑖𝑣

F− 𝑡_𝑖𝐼 𝑧_𝑖𝐹

. (7.22)

This is a very elegant equation, but it must be noted that it has a very small range of application: it is only valid in the complete absence of any concentration gradient, so in the boundary region near a selective surface it cannot be used.

Transference numbers in membranes. In this chapter we discuss transport in bulk electrolyte, where we have charge neutrality based on the ions. However, in a charged membrane, we also have the fixed charge groups, with concentration|𝑋|. In a description of transference numbers, these groups are mathematically treated as if they are ions, with a diffusion coefficient of zero. Thus, also in a membrane we can use Eq. (7.20) to calculate transference numbers of all (mobile) ions. Transference numbers will then strongly depend on the concentrations of counterions and of coions, leading to values

170 Solute Transport

of 𝑡_𝑖 close to zero or unity, even though the diffusion coefficients (mobilities) of the ions are similar. These numbers for𝑡_𝑖 in the membrane via their dependence on ion concentration in the membrane, will then depend on the concentrations of ions outside the membrane (because of Donnan equilibrium). But transference numbers have no direct relation to the rates of transport of ions, as we will discuss next.

—

Besides the transference number, 𝑡_𝑖, there is the transport number, 𝑇_𝑖, and this is a very different parameter. The transference number, 𝑡_𝑖, was defined by Eq. (7.20) and can be calculated when we know all ion valencies and diffusion coefficients. It describes the contribution of ions to the current only if the composition of the solution phase is uniform (all concentration gradients are zero). Instead, the transport number describes the actual or

‘real’ contribution of ion transport to the current, thus also when there are concentration gradients, which always is the case near a selective surface, such as near an electrode or membrane. Thus, the transport number𝑇_𝑖is the local contribution of a certain ion flux to the local current, whether or not there are ion concentration gradients. The differences between 𝑡_𝑖and𝑇_𝑖can be large, with the transference number,𝑡_𝑖, in a binary salt solution always at the same value, see Eq. (7.21), but the transport number,𝑇_𝑖, changing in time and with position.

Only with vigorous stirring, and away from selective interfaces, do the two parameters have the same value. But in general they have different values.

Transport numbers can be used to describe a process where bulk electrolyte is in contact with a selective interface, with current running through this interface. This interface can be an electrode, absorbent material, or membrane. As just mentioned, the transport numbers, 𝑇_𝑖, describe the contribution of each ion to the current density, and they are often position- dependent, and for a dynamic process also time-dependent. Only in a one-dimensional steady-state transport problem, without convection along the surface, i.e., without dispersion modelled as a ‘sideways’ refreshment, are they constant, i.e., independent of position. In other situations they vary with position. If the selective layer (for instance a membrane) operates in steady-state, with ion fluxes assumed to cross the membrane in one particular directionx, then inside the membrane the transport number of a certain ion will be the same at eachx-position, i.e.,𝑇_𝑖 will be invariant withx.^iv Note that this is only true for inert ions.

When we have reactive ions, transport numbers of individual ions will change with position, also for a membrane layer in steady state, see Ch. 10.

ivIn a full membrane module, with also a coordinatezdirected along the membrane, i.e., through the flow channel from entrance to exit, transport numbers will depend onz.

Ionic current and transport and transference numbers 171 So how to calculate transport numbers right next to a selective interface, such as a membrane? They are not material properties, or properties of the ionic solution, like the transference number is. They are not membrane properties in the same way that an ion mobility or a membrane charge density is. Instead, they follow from a combination of a transport model for the membrane, with a transport model for the flow channel, and they (i.e., their values) emerge in a calculation that considers the complete membrane-electrolyte system. In some cases we can assume the membrane (or other selective layer) to have a certain transport number for a certain ion, for instance we can assume that a certain membrane is perfectly selective and only allows access to a certain type of ion, so for this ion𝑇_𝑖=1 and for all other ions𝑇_𝑖=0. For a reactive electrode where only a certain species is consumed, for that species the transport number is unity, while it is zero for all other species.^v

The transport number is defined (at any position) as the fraction of the current density carried by a certain ion, i.e., it is the local ion flux times𝑧_𝑖over the local current density

𝑇_𝑖 =𝑧_𝑖𝐽_𝑖/𝐽

ch (7.23)

without any requirements such as uniformity of concentration profiles. Also, because 𝐽ch=Í

𝑖𝑧_𝑖𝐽_𝑖, it is the case thatÍ

𝑖𝑇_𝑖 =1, just like for transference numbers. However, what is different is that transport numbers can also be<0 or>1, while transference numbers are always between 0 and 1. For instance, in a process called ‘osmotic power generation’, or reverse electrodialysis (RED), both ions have fluxes in the same direction, thus for the coion the transport number is<0, while it is>1 for the counterion.

For solutions with all ions monovalent (a 1:1 salt, but more than one type of cation and anion possible), we can use the concept of the current efficiency,𝜆, which is the ratio of ions flux into a surface,𝐽

ions, over the current density,𝐽

ch. This current efficiency,𝜆, can be used to characterize ion adsorption in porous electrodes, see §15.3, and also describes ion selectivity in membrane transport, see §12.3. Current efficiency is defined as

𝜆= 𝐽

ions

𝐽_ch

= 𝐽₊+𝐽₋

𝐽_ch (7.24)

and for a 1:1 salt can be related to transport numbers according to

𝜆=𝑇₊−𝑇₋ (7.25)

where𝑇₊is calculated as a summation over the𝑇_𝑖’s of all cations, and the same for𝑇₋and all anions. Thus when all anions together contribute 50% to the current density, and likewise

vThis is then the case at the very surface, not necessarily at distances𝑥 >0, unless there is steady-state, no dispersion, etc.

172 Solute Transport

all cations contribute 50%, then𝑇₊=𝑇₋=½ and then𝜆=0, i.e., there is transport of current, with all cations going in one direction and all anions in the opposite direction, but there is no ‘net’ transport of ions as a whole.

As mentioned, when both the anions and cations go in the same direction (as is the case for the aforementioned RED-process), then the transport number will be<0 for the coion and>1 for the counterion, and consequently𝜆 >1. As a metric to define efficiency, numbers beyond unity are not very intuitive, and that is why in RED and similar processes, instead a different efficiency is used, which is the salt transport efficiency, 𝜗. This efficiency is defined as𝜗=1/𝜆and is a measure of how effectively the salt concentration difference is used to generate electrical current. For a well-designed RED process this ratio is close to unity. However, when membranes are used with large pores (a pore diameter of several nm’s or more), or membranes that are too thin (for instance a thickness less than 1𝜇m), then the ions flux (leakage of ions,𝐽₊+𝐽₋) is high relative to the generated current, and thus𝜗will be low. On p. 338 we discuss in more detail the low value of𝜗when very thin membranes are used, and the effect thereof on osmotic power production.