Boundary equations for binary salts

Solute Transport

7.4 Boundary equations for binary salts

176 Solute Transport

Convection in a model for concentration polarization (CP) in membrane transport.

When we make use of the full extended NP equation also including convection, Eq. (7.2), then a DBL model solved for a binary𝑧₊:|𝑧₋|salt for zero current leads to the result that the ‘monovalent equivalent’ (m.e.) molar flux (𝐽

m.e.=𝑧₊𝐽₊=|𝑧₋|𝐽₋) is given by

𝐽m.e.=𝑐 𝑣

F−𝐷

hm

𝜕 𝑐

𝜕 𝑥 (7.34)

where the harmonic mean diffusion coefficient 𝐷

hm is given by Eq. (7.30). For a 1:1 salt,𝐽

m.e.simply corresponds to the salt flux,𝐽_𝑖(the same molar flux for cations as for anions), andcis simply the salt concentration, at any point in the DBL. In Ch. 11 this expression is integrated to obtain an analytical expression for the DBL in front of a membrane used for water desalination.

Boundary equations for binary salts 177 with ‘dispersion parallel to diffusion’ in this section, because it is not exactly described by Eq. (7.35).

We will explain how to make use of results from §7.1. There several results were obtained for the relationship between the boundary flux𝐽in steady state, and the difference in concentration between bulk and surface based on solving Eq. (7.11) with the boundary condition at the surface𝐽=−𝐷·𝜕 𝑐/𝜕 𝑥|^∗(and𝜕 𝑐/𝜕 𝑥 =0 far away). First of all, termsJ in the expressions in §7.1 must be replaced by−𝐷 𝜕 𝑐/𝜕 𝑥|^∗, and we must interpretDnow as 𝐷_hm. Then we have an expression for𝑐^∗−𝑐_∞ versus𝜕 𝑐/𝜕 𝑥|^∗based on Eq. (7.11) with Dreplaced by𝐷

hm. Next we need to know what is𝜕 𝑐/𝜕 𝑥|^∗in the new model for a binary salt, as function of current,𝐽

ch, transport numbers at the surface,𝑇_𝑖, ion valencies, and ion diffusion coefficients. This is the step we make next.

We can evaluate the expression for current density, Eq. (7.18), for a binary salt, use 𝐽ch=𝐼/𝐹, assume electroneutrality, thusÍ

𝑖𝑧_𝑖𝑐_𝑖 =0, make use of𝑧²₋=|𝑧₋|², definecagain as the ‘monovalent equivalent’ salt concentration, and thus obtain

𝐽ch=− (𝐷₊−𝐷₋) 𝜕 𝑐

𝜕 𝑥

− (𝑧₊𝐷₊+ |𝑧₋|𝐷₋)𝑐

𝜕 𝜙

𝜕 𝑥

. (7.36)

Free diffusional potential. When current density 𝐽

ch is zero, we can derive what is the potential gradient that develops when the asymmetric salt diffuses through ‘free’

solution, or through a porous medium without fixed charges. This ‘free diffusional potential’ (Sasidhar and Ruckenstein, 1982; p. 351)⁴can be derived from Eq. (7.36) by setting𝐽_chto zero, leading to

𝜕 𝜙

𝜕 𝑥

=− 𝐷₊−𝐷₋ 𝑧₊𝐷₊+ |𝑧₋|𝐷₋

𝜕ln𝑐

𝜕 𝑥 (7.37)

which can be integrated from a position I to II, leading to 𝜙II−𝜙

I=− 𝐷₊−𝐷₋ 𝑧₊𝐷₊+ |𝑧₋|𝐷₋ln

𝑐II

𝑐I

(7.38) which we can also rewrite to

𝑐=𝑐_∞·exp(−𝛼 𝜙) (7.39)

where we replaced 𝑐

II by justc,𝑐

I by𝑐_∞, set𝜙

I=0, replace𝜙

II by 𝜙, and use𝛼 = (𝑧₊𝐷₊+ |𝑧₋|𝐷₋) /(𝐷₊−𝐷₋). Eq. (7.39) seems similar to the Boltzmann equation, Eq. (2.2), but this similarity is only superficial and is not real. Because the Boltzmann

178 Solute Transport

distribution exists independent of diffusion coefficients, while here the sign of𝐷₊−𝐷₋ determines the slope of the potential change versus the concentration change, setting this slope to negative, zero, or positive. This different dependence on diffusion coefficients, shows that there is no relation between Eq. (7.39) and Boltzmann’s law.

Second, we make use of the definition of the transport number, 𝑇_𝑖 = 𝑧_𝑖𝐽_𝑖/𝐽

ch, which we combine with Eq. (7.2) evaluated for one of the ions (we choose the cation), and then combine with Eq. (7.36), resulting finally in

𝜕 𝑐

𝜕 𝑥

· (𝑇₊𝐷₋+𝑇₋𝐷₊)=𝑐

𝜕 𝜙

𝜕 𝑥

· (|𝑧₋|𝐷₋𝑇₊−𝑧₊𝐷₊𝑇₋) . (7.40) This we combine again with Eq. (7.36), resulting in

𝐽_ch=− 𝐷₊𝐷₋ (𝑧₊+ |𝑧₋|)

|𝑧₋|𝐷₋𝑇₊−𝑧₊𝐷₊𝑇₋

· 𝜕 𝑐

𝜕 𝑥

∗

=−𝑐

𝐷₊𝐷₋ (𝑧₊+ |𝑧₋|) 𝐷₋𝑇₊+𝐷₊𝑇₋

· 𝜕 𝜙

𝜕 𝑥

∗

(7.41) reiterating that index * refers to a position at the surface. This general relationship provides for any binary 𝑧₊:|𝑧₋| salt the required boundary condition for 𝜕 𝑐/𝜕 𝑥|^∗ at the surface as function of current density 𝐽

ch, transport numbers, and ion diffusion coefficients and valencies.

Thus when in a certain experiment with a certain binary salt solution we know𝐽

chand the 𝑇_𝑖’s, we can use Eq. (7.41) to calculate𝜕 𝑐/𝜕 𝑥|^∗, which is then equal to the related expression for𝐽_𝑖/𝐷

hmfrom §7.1. Below we provide examples of how this works out exactly, but first let us discuss certain simplifications of Eq. (7.41).

As a first simplified case of Eq. (7.41), we can assume that the surface is only accessible to cations (i.e., only cations can cross the surface), and thus we have𝑇₊=1 and𝑇₋=0 there, and then the general expression, Eq. (7.41), simplifies to

𝐽ch=−𝑧₊+ |𝑧₋|

|𝑧₋| 𝐷₊

𝜕 𝑐

𝜕 𝑥

∗

(7.42) which shows that for a certain current density the concentration gradient in solution next to a selective interface (for instance a membrane or electrode) only depends on the diffusion coefficient, D, of the ion that goes through the interface, and not onDof the ion that is blocked.^viii For a given current density, the resulting concentration gradient also depends on the valencies of both ions. Note that this result that𝐽_chis independent of𝐷_coion, as Eq. (7.42)

viiiEq. (7.42) is the same as Eq. (72-11) in Newman (1983) when we use𝑢_𝑖∝𝐷_𝑖. Note that in Newman,𝜐₊𝑧₊=1.

Boundary equations for binary salts 179 shows, is only correct for a membrane or electrode that completely blocks the coions, and is only valid at the very interface. Eq. (7.42) is also valid away from the interface but then only in the simple DBL model of §7.1.1 for steady state. It is not valid away from the surface for the models that include refreshment if only because transport numbers𝑇_𝑖gradually change from their values at the surface (which relate to the selectivity of the interface) to values in bulk where they are equal to the transference numbers,𝑡_𝑖.

For a 1:1 salt, Eq. (7.42) simplifies to the classical result 𝐽ch=−2𝐷₊

𝜕 𝑐

𝜕 𝑥

∗

=−2𝑐 𝐷₊

𝜕 𝜙

𝜕 𝑥

∗

(7.43) which illustrates that the cation flux (which is equal to the current density,𝐽_ch, in this case of a perfectly selective interface) is twice the flux by molecular diffusion, and twice the Ohmic transport of cations. Thus the cation flux is now for 50% due to diffusion, and for 50% due to electromigration. The electrical field that drags the cations (counterions) to and through the surface (which leads to the Ohmic contribution to cation transport), also pushes the coions away from the surface. This leads to the salt concentration near the surface to go down until the concentration gradient becomes steep enough that diffusion attracts the coions with the same force as the force of the electric field pushes them away. That concentration profile –with the concentration now decreasing towards the interface– now acts as a diffusional driving force for the counterions towards the surface. The end result is that the counterion has a twice larger flux than one would guess based on Ohm’s law only and assuming only cations to be mobile charge carriers. Instead, the coions, even if they don’t go through the interface, play a role, are coupled to the counterions, and for a given current they reduce the voltage drop over a film or channel by a factor of 2 compared to the case that they would be fixed in space, homogeneously distributed. For a 2:1 salt and again a membrane which does not allow anions through, the proportionality between𝐽

chand−𝐷₊𝜕 𝑐₊/𝜕 𝑥(with𝑐₊=𝑐/𝑧₊) is not 2 but 6. For a 1:2 salt it is a factor 1.5 (𝑐₊=𝑐).

The examples here for an interface that only allows cations through, can be easily modified to represent the opposite case of an interface only allowing access to the anions, by replacing each𝑧₊by|𝑧₋|, and vice-versa. To derive these equations, it is best to start again at Eq. (7.41).

A different simplification is when we have a symmetric z:z salt, with equal diffusion coefficients,𝐷=𝐷_hm, but the surface is no longer perfectly selective to one of the ions. We then simplify Eq. (7.41) to

𝐽ch=− 2𝐷 𝑇₊−𝑇₋

𝜕 𝑐

𝜕 𝑥

∗

(7.44) and, interestingly, this relationship is independent of the ion valency, i.e., Eq. (7.44) is

180 Solute Transport

equally valid for a 1:1 and a 2:2 salt. Note that for a 2:2 salt, concentrationcis twice higher than the salt concentration.

As explained, an important application of the above expressions for 𝐽

ch as function of 𝜕 𝑐/𝜕 𝑥|^∗ is that it allows us to generalize results from §7.1. Thus, for instance, we can use Eq. (7.13) for the convection-along-the-surface-with-constant-𝜏-model, implement 𝐽=−𝐷

hm·𝜕 𝑐/𝜕 𝑥|^∗, and have 𝜕 𝑐/𝜕 𝑥|^∗replaced by an expression involving𝐽

chand the𝑇_𝑖’s based on any of the expressions just discussed, such as Eq. (7.41) in the general case.

Here let us analyze the example of a solution of a 1:1 salt, with equal diffusion coefficients, 𝐷=𝐷_hm, based on Eq. (7.44). We use Eq. (7.13) and then arrive at

𝐽ch=

√︂

𝐷 𝜏

· 2

𝑇₊−𝑇₋

· (𝑐_∞−𝑐^∗) . (7.45)

In a different example, again for a 1:1 salt, but now with unequal diffusion coefficients, for a perfectly selective interface (only allowing cations through, thus𝑇₊=1), combination of Eqs. (7.13) and (7.42) leads to

𝐽_ch =2

√︂

𝐷₊ 𝜏

√︂

𝐷₊ 𝐷hm

· (𝑐_∞−𝑐^∗) (7.46)

which interestingly still includes a dependence on the coion (anion) diffusion coefficient (via 𝐷hm). There will not be such an influence of the coion when we use the classical film model without a refreshment effect from §7.1.1. The two last expressions lead to the same result when we set𝑇₊=1 and𝑇₋=0 in the first, and𝐷

hm=𝐷₊=𝐷in the second expression.

Note that when expressions similar to Eqs. (7.45) and (7.46) are derived for 2:1, 2:2, etc., salts, that𝑐_∞and𝑐^∗again refer to bulk and surface, but they are not the salt concentration, in the way that throughout this book𝑐_∞is defined as a salt concentration of a 1:1 salt. Instead they are monovalent equivalent salt concentrations.