Ion volume effects in electrochemical processes
4.3 Ion-ion Coulombic interactions in electrolytes
100 Ion volume effects in electrochemical processes
According to Eq. (4.24), a difference in size between the ions of 20%, with the smaller ion having a size equal toℎ
p, thus𝛼′=1.0, leads to a selectivity of𝑆
1−2∼1.11, which is not very impressive. Using Eq. (4.18) instead of Eq. (4.15) leads to the more correct result that for these parameters,𝑆
1−2∼1.22, which is twice larger but still not very large. But when we analyze the situation beyond the tracer limit, selectivity becomes significantly higher, as we show next. We will now use the full BMCSL approach with two ions modelled as spheres of different sizes, and a third type of particle that represents the porous medium and is modelled as spheres that are all connected to one another. This is Eq. (7.2) in Spruijt and Biesheuvel (2014) where we use𝑁=∞for this third type of particle (𝜂
3=50%). We make a calculation of ion selectivity at a 20% total volume fraction of the two ions in the pores (thus the two ions together occupy 10% of the total volume of the porous medium) including that the two types of ions have volumetric interactions with one another. Like in the previous example, the larger ion is only 20% larger than the smaller one.iii Results are that we now have a selectivity factor of𝑆
1−2=4.44! Thus ion volume effect can be very significant in a realistic porous medium, significantly beyond what the simple Eq. (4.24) might suggest.
Ion-ion Coulombic interactions in electrolytes 101
+ +
+
- -
-
+ + +
+ + +
- -
- - - -
A B C
‘sea’ of neutral outside solution
+
-
ion and its countercharge within a certain volume
Fig. 4.2:In a dilute electrolyte solution, ions are randomly distributed as depicted in panel A, but the electrostatic energy is lowered when ions of opposite charge stay nearer to one another as illustrated in panel B for a 1:1 salt. There is a distribution of distances between an ion and its nearest countercharge.
is another contributing factor. Another energy term is of electrostatic origin, because ions of like sign repel and of opposite charge sign attract. For dilute solutions the attraction of an ion with its most nearby countercharge is the most relevant, and it is the Coulombic attraction between them which leads to a decrease of the energy of the electrolyte, and thus to a reduction of the activity of the ions. When very dilute, these Coulombic interactions are effectively zero, because the ions remain far apart and the likelihood of an ion starting to orbit its counterpart is small. But if concentrations go up, ions are ‘pushed together’, in a statistical sense, and trajectories of a certain ion more frequently get close to ions of opposite sign. Then it will happen more frequently that these trajectories are further deflected such that ions stay closer for a longer time, i.e., their paths become correlated. A snapshot for a dilute solution would not show ion-ion positional correlations, but a snapshot at higher concentrations shows correlations of cations with anions, see Fig. 4.2. The Coulombic interactions result in a negative contribution to an ion’s chemical potential, i.e., to a value of 𝛾less than unity.
In a calculation of the chemical potential, we can focus on the Coulombic interactions between an ion and its most nearby countercharge, and we do not consider the trajectories of other cations and anions. Thus we can focus on an ion and the most nearby countercharge, and analyze the Coulombic energy of the interaction between them. In this calculation we must consider all possible distances between the two ions, and take into account the likelihood of each separation. First of all, being very close is increasingly unlikely because around any point ‘space’ expands with distance squared, making two ions being further apart more likely then being close. On the other hand, further apart is less likely because the Coulombic energy is less, i.e., because of the higher attraction short distances are favoured by Boltzmann’s distribution law. The minimum distance between the ions in a pair is determined by the sum of their radii, which is then also part of the theory. There is not a sharp maximum for the range of possible distances where the first countercharge is to
102 Ion volume effects in electrochemical processes
found relative to a certain ion, but very large distances for the first countercharge, are very unlikely. A simplification assumes a distinct maximum separation calculated from equally distributing all ions over space, away from one another to a maximum, and then taking this ion-ion distance as the maximum. In the dilute limit this entire calculation can be done analytically for a 1:1 salt and leads to the result that the natural logarithm of𝛾is given by
ln𝛾±=−𝛼 𝜆
B
√︁3
𝑁av𝑐∞ (4.26)
where±refers to the mean activity coefficient which is equally attributed to the cation as to the anion, with𝛼a numerical prefactor that we empirically determine to be𝛼=1, and 𝜆B is the Bjerrum length, at room temperature about𝜆B=0.716 nm. This expression for the dilute limit, Eq. (4.26), is independent of ion radii, but ion radii do play a role in a full numerical calculation that for non-dilute conditions is more exact. These complete numerical calculations are provided as continuous lines in Fig. 4.3 and accurately describe data of the activity coefficients of several 1:1 salts up to quite high salt concentrations. For a low average ion radius of⟨𝑎⟩=0.18 nm, the analytical solution, Eq. (4.26) is close to the numerical calculation, certainly in the dilute limit. In Biesheuvel (2020) also an excellent fit of this theory to data is provided for symmetric 2:2 and 3:3 salts, as well as for asymmetric 2:1 and 3:1 salts. In the latter cases, instead of considering only one countercharge, an ‘ion ensemble’ with 3 and 4 ions, respectively, must be numerically evaluated.
—
For a 1:1 fully dissociated salt, the contribution to the osmotic pressure because of this ion-ion Coulombic interaction (i.i.c.i.) is as follows. We can make use of Eq. (1.3) with the right hand side now a summation over two ions, and we can also use the expression for free energy densityf that we discuss further on, Eq. (III-14), noting thatΠ =𝑐2𝜕(𝑓/𝑐) /𝜕 𝑐. For a symmetric salt this relation can be directly used withctwice the salt concentration,𝑐∞.iv We now obtain for the contribution to the osmotic pressure
Πi.i.c.i.=−½𝛼 𝜆
B𝑁1/3
av 𝑐4∞/3. (4.27)
At room temperature, this contribution isΠi.i.c.i∼ −0.03𝑅𝑇 𝑐4∞/3, with𝑐∞expressed in mM.
Thus at𝑐∞=1000 mM this contribution reduces the osmotic pressure by 7.7 bar.
The same result can also be expressed as an osmotic coefficient,𝜙, which describes the osmotic pressure relative to the ideal situation thatΠid=2𝑅𝑇 𝑐∞, thus𝜙= Π/Πid. For a 1:1 salt, at room temperature,𝜙is given by𝜙∼1−0.015 3
√
𝑐∞(Biesheuvel, 2020). Thus at 1 M, the osmotic pressure is reduced because of electrostatic interactions by∼15%.
ivFor the ideal part, we have 𝑓=2𝑐∞(ln(𝑐∞/𝑐ref) −1), and we then correctly end up withΠ =2𝑐∞.
Ion-ion Coulombic interactions in electrolytes 103
—
It is possible to include the effect of this ion-ion Coulombic interaction in a flux expression for ions where it will be combined with other driving forces acting on an ion, such as diffusion and electromigration. If we combine ion-ion Coulomibic interaction with regular diffusion by Fick’s law, we arrive for a 1:1 salt atv
𝐽=−𝐷
1+ 𝜕ln𝛾
𝜕ln𝑐∞ 𝜕 𝑐∞
𝜕 𝑥
. (4.28)
The term within brackets becomes 1−1/3𝛼𝜆B3
√
𝑁av𝑐∞, and that term multiplied with𝐷𝑖 can be called an apparent diffusion coefficient,𝐷
app,𝑖. A large range of data for the diffusion of KCl, NaCl and LiCl up to concentrations of𝑐∞∼200 mM is excellently described by Eq. (4.28), see Biesheuvel (2020). Due to the activity correction, at∼200 mM, the apparent diffusion coefficient, 𝐷
app,𝑖, is about 10% lower than its value at infinite dilution, where 𝐷app,𝑖=𝐷𝑖.
—
A general conclusion of a study of ion activity coefficients is also that certainly for 1:1 salts the correction to an ion’s activity due to ion-ion Coulombic interactions is in many cases not very large, and can be neglected. A drop in ln𝛾by 0.3 points, as shown in Fig. 4.3, relates to an contribution to an ion’s chemical potential that is equivalent to the effect of an electrical field with a voltage change of 8 mV, not a very large number. Thus, electromigration, and regular diffusion, are in most cases more important than ion-ion Coulombic interactions.
But for 2:2 and 3:3 salts, these energies are much stronger, see Biesheuvel (2020).
—
Thus, Eq. (4.26) is a first contribution to the activity correction of an ion, ln𝛾, with the effect of ion volume a second contribution, which develops at moderate to high salt concentrations. We describe this volume effect with the Carnahan-Starling equation of state, Eq. (4.4), and take for the anion and cation the same size. The volume fraction𝜂is calculated from multiplying the ion’s volume𝑣, with the total ions concentration, which is two times the salt concentration. We present in Fig. 4.4 curves for ln𝛾(the summation of the two 𝜇-terms discussed in this section) as function of salt concentration and ion size (the ion
vThis derivation is based on the solute friction balances that will be extensively discussed in Ch. 7 where the extra contribution to an ion’s chemical potential is included in the force acting on an ion which is minus the gradient of the chemical potential.
104 Ion volume effects in electrochemical processes
-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0
0 0.1 0.2 0.3 0.4 0.5
-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0
0 0.1 c0.2⅓(M⅓)0.3 0.4 0.5
KF NaBr KCl CsCl
ln g
-0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02
0 0.1 0.2 0.3 0.4 0.5
<a>=0.18 nm
<a>=0.35 nm
Fig. 4.3:Data and theory for (the natural logarithm of) the mean ion activity coefficient of four 1:1 salts as function of the cube root of salt concentration. Lines are based on a calculation of the Coulombic energy between an ion and its most nearby countercharge, for different values of the average ion radius,
⟨𝑎⟩, which for low𝑐∞and/or low average ion radius⟨𝑎⟩has the analytical solution given by Eq. (4.26).
size is only required in the volumetric excess contribution). For ions with a non-zero size, Fig. 4.4 shows that with increasing salt concentration the curves first go down and then up again, and do so much quicker when the ion size is larger.
The effects just discussed, relating to Coulombic interactions and ion volume, have an impact on many aspects of electrochemical processes, including ion transport and thermodynamics, such as the calculation of the minimum energy required for water desalination. The impact of these two non-idealities on this thermodynamic energy for desalination is discussed on p. 288.
Salting in/salting out. The solubility of a salt (ion pair) depends on the chemical potential of the ions, and thus depends on the effects discussed above, the ion pair energy, and volume exclusion. The more that ions of a salt A are ‘stabilized’ (reduced chemical potential), the more they remain dissolved and do not condense into neutral aggregates containing the two ions. If now another salt,X, is added, at low concentration ofXthe solubility ofAincreases. This is because extraXreduces the Coulombic energy of the ions ofA, and thus the ions ofAare stabilized. This is a well-known effect, called
‘salting in’: adding an additional saltXleads to a higher solubility of a saltA. However,
Ion-ion Coulombic interactions in electrolytes 105
-1 -0.75 -0.5 -0.25 0 0.25
0 0.5 1 1.5 2
c
⅓(M
⅓)
ln g
ion size = 0.3 nm ion size = 0.4 nm
Fig. 4.4:Theory for ln𝛾±in a 1:1 salt solution as function of the cube root of salt concentration including the analytical solution for ion-ion Coulombic interaction, Eq. (4.26), and a volumetric excess effect described by the Carnahan-Starling equation of state.
beyond a certain concentration ofX, the effect is reversed: with moreXthe solubility ofAgoes down, and this is called ‘salting out’.
One common explanation of salting out is that so few water molecules remain to hydrate the ions, that a significant competition develops, and clustering into neutral salt aggregates becomes more favourable. The other route to explain this reduction in solubility is via an effect of ion volume, because now higher concentrations of salts (of whichever type) increase the energy of the ions, see Fig. 4.4, and clustering into aggregates of salt A is enhanced. Thus in the same way that a reduction of the chemical potential of an ion (i.e., reduction of ln𝛾) leads to salting in, its increase leads to salting out.