Effect of ion volume on the partitioning into a porous mediumporous medium

Ion volume effects in electrochemical processes

4.2 Effect of ion volume on the partitioning into a porous mediumporous medium

A simple way to introduce the effect of ion size is how for a spherical particle (ion) that enters a perfectly cylindrical pore, not all positions are available. In this case, the partitioning function,Φ_exc,𝑖, is the ratio between the area of the pore cross-section that is accessible to

Effect of ion volume on the partitioning into a porous medium 93 the centre of the ion (with ion size𝜎_𝑖), over the total cross-sectional area, and is given by

Φ_exc,𝑖=

𝜋/4 𝐷

p−𝜎_𝑖2 𝜋/₄𝐷²

p

=(1−𝜆)² (4.1)

where𝜆is the ratio of ion size𝜎_𝑖over pore size𝐷_p. In Eq. (4.1) it is assumed that the ions are unrestricted in their motion outside the pores, in bulk solution. Eq. (4.1) is based on the idea that an ion’s center cannot get closer to the wall of a pore than a distance equal to its own radius,𝜎_𝑖/2, or in other words, only a center region with diameter𝐷

p−𝜎_𝑖 is available for the center of an ion. This is based on the concept of the ion as a perfect sphere and the pore of a perfect cylindrical shape.

This very elegant equation has only one problem. Which is that most porous materials do not consist of (one type of) perfect cylindrical pores. But fortunately, there are other, more appropriate, approximations of the structure of a porous network that we can make use of. The approach we will use is to approximate the porous structure as a random assembly of more-or-less spherical particles. They can be very close-packed, or form a very dilute assembly. In the theory they are fixed in position, i.e., they form a rigid structure. We can make a structure with mixtures of different fractions of spheres of different sizes, but we will assume one type of sphere in this chapter. We base our derivation on a very accurate equation-of-state (EOS) that was originally developed for hard sphere mixtures, and modify it to make it apply to the case of a porous medium consisting of a network of spheres, where all spherical particles are connected and held rigidly in space.

The starting point is the Carnahan-Starling equation of state (CS EOS), which very accurately describes volumetric interactions in a ‘hard-sphere mixture’, i.e., for a solution of particles that all have the same size. The osmotic pressure according to the CS EOS is given by

Π 𝑐

= 1+𝜂+𝜂²−𝜂³

(1−𝜂)³ =1+2𝜂(2−𝜂)

(1−𝜂)³ (4.2)

where𝜂is the volume fraction occupied by the spherical particles. Eq. (4.2) includes both an ideal term, which is the factor 1 on the right, and a volumetric excess term, which is the term 2𝜂(2−...there.ⁱ The expansion of Eq. (4.2) around𝜂=0 leads to

Π 𝑐

=1+4𝜂+10𝜂²+18𝜂³... (4.3)

iHere, as throughout this book, pressures and chemical potentials can be multiplied by𝑅𝑇to obtain a dimensional pressure in J/m³and potential in J/mol. For the dimensionless potential,𝜇, it is common to then write that the chemical potential is so many ‘kT’s, or ‘𝑘

B𝑇’s.

94 Ion volume effects in electrochemical processes which shows that the CS virial coefficients are 𝐵

2=4, 𝐵

3=10, 𝐵

4=18, etc., where the 2^nd and 3^rd virial coefficients are exact for hard sphere mixtures, but higher order virial coefficients are slightly off (18 must be 18.36. . . , etc.).

Based on Eq. (4.2), and the Gibbs-Duhem equation, Eq. (1.4), we can derive the excess contribution to the chemical potential of a species

𝜇exc,i= 3−𝜂

(1−𝜂)³ −3 (4.4)

which has the expansion around𝜂=0 of

𝜇exc,i=8𝜂+15𝜂²+ ... (4.5)

where the factor 8 can be interpreted as due to a single sphere of volume𝑣excluding 8×its own volume for (the placement of the center of) another sphere of the same size. Note that this 8×larger volume does not imply that (CS predicts that) there is a maximum packing density of 1/8, or 12.5%. This limit is not there because when two spheres are so close that theirexcludedvolumes overlap, then their joint excluded volume is less than the sum of their individual excluded volumes. (It may be good to reiterate that our interest in𝜇_𝑖-values for volume or other effect is because a difference in𝜇_𝑖 of a solute between two phases leads to a contribution to the partitioning coefficient.)

Besides its exact nature, the elegance of the CS EOS is that it can be extended to multicomponent mixtures of spheres of different sizes. And that equation can be further modified to consider mixtures where some of the particles are connected in pairs, triplets, etc., and even into long strings of particles, to describe polymer molecules, or to represent random porous media. The generalization of the CS EOS to mixtures of spheres of unequal size. is the Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state. The complete equation is rather unwieldy, but it is an explicit expression forΠ and 𝜇_exc,i as function of the volume fractions and sizes of all the species involved. When the sizes of all particles are the same, the BMCSL EOS simplifies to the CS EOS.

Some useful limits of BMCSL are as follows. When particles of size𝜎_𝑖 are dispersed in a solution containing other particles 𝑗 that are infinitely small (point charges), then the CS-equations above can be used with𝜂based on particle𝑖, with an additional term based on the osmotic pressure exerted by the small particles 𝑗, which results in an additional contribution to the total osmotic pressure of the system of

Πsea of small particles=𝑐_𝑗 (4.6)

where the concentration𝑐_𝑗of the small particles in Eq. (4.6) is based on the volume available

Effect of ion volume on the partitioning into a porous medium 95 to them, which is the volume not occupied by the large particles𝑖, thus𝑐_𝑗 = 𝑐^∗

𝑗/(1−𝜂), where𝑐^∗

𝑗 is the concentration per total volume.

The small particles 𝑗 also modify the chemical potential of species𝑖by a term that is equal to the volume of particle𝑖times the pressure exerted by all the small particles,𝑗

𝜇additional,𝑖=𝑣_𝑖 ·Πsea of small particles (4.7) which can be understood as an insertion pressure: it is the energy that must be invested by the large particle to open up a space with the volume𝑣_𝑖 against the pressure exerted by the small particles 𝑗.

The excess contribution to the chemical potential for the small particles,j, because of the presence of larger particles,i, is

𝜇exc, 𝑗 =−ln(1−𝜂). (4.8)

A different limit is when we haveNtypes of particles, i.e., a polidisperse system, and all species can have non-zero sizes, but all particles, indexed as type𝑖in the remainder of section are present at a very low concentration, except for one majority species that is indexedj. All equations below until Eq. (4.18) are valid for such a polidisperse system. In this case we can take the elegant ‘test particle limit’ or ‘tracer limit’ of the BMCSL EOS. For all particles typei, present in low quantities, and with size𝜎_𝑖 in a mixture that predominantly consists of spherical particlesj, of size𝜎_𝑗, with𝛼=𝜎_𝑖/𝜎_𝑗, the excess term for moleculesiin the tracer limit of the BMCSL equation is given by

𝜇exc,𝑖=−

1−3𝛼²+2𝛼³

ln(1−𝜂) +𝜂𝛼 (

𝛼(3+2𝛼−3𝜂)

(1−𝜂)³ +3 1+𝛼−𝛼² 1−𝜂

) (4.9) where𝜂 is the volume fraction of total space filled up with particles type 𝑗. Eq. (4.9) correctly simplifies to all of the earlier mentioned limits when𝛼is either 0, 1, or∞.ⁱⁱ

When now the other species 𝑗 is to some extent restricted in its movement, by being connected to other spheres of species 𝑗, such as when they are combined into doublets, triplets, or into long (polymer) chains, Eq. (4.9) is modified to account for the lower exclusion effect. This is because the entropic contribution to the excess function is reduced or altogether switched off when these species 𝑗 cannot move as freely as before. We here extend this approach to describe volumetric effects in a porous medium that is described as a dense packing of connected spheres (the 𝑗-particles). To account for this connectedness, for the

iiNote that in this section we are leaving out an indexiin our notation of𝛼, even though𝛼is a factor that for each soluteican be different.

96 Ion volume effects in electrochemical processes

particles typeithat move through the porous medium consisting of particles typejthat have fixed positions in space, we add to Eq. (4.9) a contribution given by

𝜇additional,i=− 𝜂 1−𝜂

𝛼³+ln(1−𝜂) (4.10)

which consists of two terms, the first one being a correction because all spheres of type 𝑗 are connected to one another, and the second term is a constant factor that now allows for concentrations to be defined per unit pore volume. Thus, for all equations that follow in this section, until Eq. (4.24), note that all concentrations in the porous medium are defined per volume of the open fraction (the pores).

Combining these two terms, we can easily group all terms according to their dependency on𝛼, resulting in

𝜇_exc,i= ^3𝜂

1−𝜂𝛼+3

ln(1−𝜂) +^{𝜂(2−𝜂)}

(1−𝜂)²

𝛼²−2

ln(1−𝜂) +^𝜂(2𝜂2−4𝜂+1)

(1−𝜂)³

𝛼³. _(4.11) We can analyze this new excess function for a particle of size𝜎_𝑖 in a porous material that has a volume fraction𝜂and is made of spheres of size𝜎_𝑗. Interestingly, the resulting curves as function of size ratio𝛼can be made to almost overlap when we plot the excess functions 𝜇exc,inot versus the size ratio as such, but versus the ratio of ion size over an effective pore size. This effective pore size,ℎ

p, or ‘characteristic pore dimension’ (also discussed in §2.2), is equal to the pore volume over the area, and thusℎ_p is the inverse of the specific ‘liquid phase’ surface area,𝑎

L). For the type of porous medium just discussed,ℎ

pis given by ℎp= 𝜎_𝑗

6 1−𝜂

𝜂

. (4.12)

Thus we define a modified size ratio,𝛼^′,

𝛼^′= 𝜎_𝑖

ℎ_p (4.13)

and combine Eqs. (4.12) and (4.13) to arrive at 𝛼^′= 6𝜂

1−𝜂

𝛼 . (4.14)

Combination of Eqs. (4.11) and (4.14) leads to 𝜇exc,i=½𝛼^′+

(1−𝜂)²

12𝜂2 ln(1−𝜂) + 2−𝜂 12𝜂

𝛼^′2−

(1−𝜂)³

108𝜂3 ln(1−𝜂) + 2𝜂2−4𝜂+1 108𝜂2

𝛼^′3. (4.15)

Effect of ion volume on the partitioning into a porous medium 97

0 1 2 3 4 5 6

0 1 2 3 4

ˈ = 

_i

/ h

_p



exc,i

(kT )

porous medium h=0.4 0.5 0.6

Fig. 4.1:The excess contribution to the chemical potential,𝜇

exc,i, of a spherical particle of size𝜎_𝑖, inside a dense porous medium with various packing degrees𝜂, as well as comparison with𝜇

exc,ifor the ideal case of a sphere in a cylindrical pore.

For this porous medium approach, we can now plot the excess term𝜇_exc,𝑖of Eq. (4.11) versus 𝛼^′in Fig. 4.1 as function of packing degree𝜂. Interestingly, all curves for𝜇

exc,ivs.𝛼^′now start off with the same slope of𝜕 𝜇

exc,i/𝜕 𝛼^′=1/2, irrespective of the chosen packing degree of the porous medium,𝜂, just as described by the first term in Eq. (4.15).

And we can now analyze this slope for the classical partitioning function based on the cylindrical pore, Eq. (4.1). Also here we must recalculate, now from the ratio of ion size over pore size,𝜆, to𝛼^′. We must also convert the partitioning function for volume effects, Φ_exc,𝑖, from Eq. (4.1), to an excess contribution to the chemical potential,𝜇

exc,i, resulting in

𝜇exc,cyl,𝑖 =−2 ln(1−𝜆) (4.16)

in which we can implement that for a cylindrical pore𝛼^′ = 4𝜆. Taylor expansion around 𝛼^′=0 leads to

𝜇exc,cyl,i=¹/₂𝛼^′+1/₁₆𝛼^′2+... (4.17) which has the same first term, linear in𝛼^′, as for the porous medium approach, Eq. (4.15).

Thus, for small enough ions the two approaches overlap, which is a comforting result.

It is advantageous that the new approach based on a porous medium does not depend on a virtual ‘diameter of an ideal cylindrical pore’ but uses the much more accessible and insightful property of the specific surface area, 𝑎

L, or its inverse, the characteristic pore

98 Ion volume effects in electrochemical processes dimension,ℎ

p.

As Fig. 4.1 shows, until around𝛼^′∼3, the new expression for𝜇

exc,i changes faster than the prior function based on an ideal cylindrical pore, i.e., there is now a larger impact of a size change on 𝜇

exc,i. This trend only reverses at larger values of 𝛼^′ when for the cylinder approach the excess function more rapidly increases and diverges when the ion size approaches the diameter of the cylindrical pore. This rapid increase and divergence is not seen with the new approach. Importantly, both approaches give results that are not too different, with very similar 𝜇_exc,i up to𝛼^′∼1, and with a maximum deviation of∼1 kT around𝛼^′∼3 (2.5 vs. 3.5 kT).

The new approach has a small dependency of the function 𝛼^′−𝜇

exc,i on the packing density of the porous medium,𝜂, as Fig. 4.1 shows, but the effect is relatively small. For further analysis, let us use the curve for 50% packing as an appropriate choice, for which the resulting expression forΦ_exc,𝑖becomes

Φ_exc,𝑖=𝑒⁻(^𝜇_exc,𝑖−𝜇_exc,∞,𝑖)=exp n

−

1/2𝛼^′+5/26𝛼^′2+1/40𝛼^′3 o

(4.18) where the first numerical value is exact, the other two are approximations. Note that the predicted partitioning effect is still a function of the density of the porous medium, via the value ofℎ

p, which is a strong function of𝜂, see Eq. (4.12). (In Eq. (4.18) we assumed that 𝜇exc,∞,𝑖=0.)

The expressions discussed in this section can also be compared to another empirical method that was used to develop an expression for𝜇_exc,ifor ions inside a slit-shaped pore, results of which compared favorably with full density functional theory (DFT) calculations.

Here the standard CS EOS was used, with a dependence on𝜂_𝑖, which is the volume fraction of the spheresthemselvesin the slit. To correct for the presence of the slit-shaped pore walls (to get a good fit to the DFT calculation results), this volume fraction𝜂_𝑖 was corrected by adding an extra term𝛾 𝛼^′that depended on the ratio of ion size over slit size (𝛾=0.0725).

Thus to each factor𝜂_𝑖 in the CS expression, Eq. (4.5), this extra factor is added to account for pore constriction, which results in

𝜇_exc,𝑖= 3− (𝜂_𝑖+𝛾 𝛼^′)

(1− (𝜂_𝑖+𝛾 𝛼^′))³ −3. (4.19) If we take the tracer limit, so a dependence on the ion concentration itself is neglected, i.e., we set𝜂_𝑖=0, and do a Taylor expansion around𝛼^′=0, we obtain

𝜇exc,i=8𝛾 𝛼^′+ O 𝛼^′2

(4.20) and thus we have around𝛼^′ = 0 a linear dependence on 𝛼^′ with a prefactor for𝜇

exc,i of 8𝛾=0.58, which is close to the value of½derived for the approaches discussed earlier on.

Effect of ion volume on the partitioning into a porous medium 99 Eq. (4.19) can also be used to describe the interactions between the ions in the pore, i.e., a dependence of the excess function on𝜂_𝑖, but we use a different function for that later.

Thus, in conclusion, all of these approaches for the effect of ion volume on the excess function in porous media, give similar results, though the original ‘sphere-in-cylinder’

approach has the weakest dependence on pore size below𝛼^′∼3 and after that the effect of pore size increases very steeply. The more gradual dependencies predicted by the two other approaches seem better. Note that all expressions until now (except Eq. (4.19)) assume that the absorbing species is present only in trace quantities, and thus do not interfere with one another. This effect we discuss further on.

—

We can use these expressions to estimate the effect of ion size on ion selectivity, for equilibrium conditions, thus neglecting for now how different transport rates influence selectivity. At equilibrium, ion selectivity describes the difference in ion adsorption in a microporous material (porous electrodes, gel, membrane) between two different ions (Gamaethiralalageet al., 2021). We can define the selectivity between ion 1 and ion 2 as

𝑆1−2 = 𝑐₁ 𝑐2

·𝑐_∞,

2

𝑐_∞,

1

(4.21) where index∞refers to outside the porous material, whilecwithout an extra index refers to inside the micropores. To find an expression for𝑆

1−2as function of the two ion sizes, we use the partitioning coefficient related to volume, given by

Φ_exc,𝑖=exp − 𝜇

exc,𝑖−𝜇

exc,∞,𝑖 . (4.22)

If we now combine with Eqs. (2.34) and (4.21), we arrive at 𝑆1−2 =exp 𝜇

exc,2−𝜇

exc,1

(4.23)

where we assumed that for both ions outside the pore𝜇

exc,∞,𝑖is zero, which for a sufficiently dilute solution is a valid approximation.

We can now insert any of the above-discussed expressions for𝜇

exc,𝑖 in Eq. (4.23). The expression we will analyze is Eq. (4.15) and thus we obtain

𝑆1−2=exp −½(𝜎

1−𝜎

2) /ℎ

p

(4.24)

which illustrates that when ion 1 is smaller than ion 2, 𝑆

1−2 is larger than unity, i.e., the smaller ion is preferentially adsorbed. As Eq. (4.24) also shows, the effect of ion size increases with decreasing pore size,ℎ

p.

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.