Cooperativity in Sorption Isotherms

We present a general theory of cooperativity in sorption isotherms that can be applied to sorbent/gas and sorbent/solution isotherms and is valid even when sorbates dissolve into or penetrate the sorbent. Our universal foundation, based on the principles of statistical thermodynamics, is the excess number of sorbates (around a probe sorbate), which can capture the cooperativities of sigmoidal and divergent isotherms alike via the ln–ln gradient of an isotherm (the excess number relationship). The excess number relationship plays a central role in deriving isotherm equations. Its combination with the characteristic relationship (i.e., a succinct summary of the sorption mechanism via the dependence of excess number on interfacial coverage or sorbate activity) yields a differential equation whose solution is an isotherm equation. The cooperative isotherm equations for convergent and divergent cooperativities derived from this novel method can be applied to fit experimental data traditionally fitted via various isotherm models, with a clear statistical thermodynamic interpretation of their parameters..

Consequently, the value for  in the cooperative characteristic relationship (eq 5) is not the limiting value at  2 → 0 but is an extrapolated value based on the finite  2 behavior.
We can prove a similar result for the sorbent/solution interface, based on the requirement that the KBIs should not diverge at the  2 → 0 limit.To do so, let us use the following expression derived in B. Derivation of the cooperative isotherm for the sorbent/solution interface: which is valid both for * and .Since KBIs converge at the  2 → 0 limit and  2 → 0 at this limit, we can prove that  22 + 1 → 1.
The limiting behavior of  22 + 1, in classical adsorption thermodynamics, has been known as the consequence of Henry's Law for sorbent/gas interface, which can be expressed at the  2 → 0 limit as where   is Henry's law constant.Substituting eq A3 into eq 2 leads to  22 = 1 at this limit.
Note that Henry's law-based discussion is limited to sorbent/gas interfaces.
To be strictly compliant with this limiting behavior at  2 → 0, we have added, in our previous papers, the linear term ( 1  2 ) to the cooperative isotherm, such that Such a modification could also be introduced for the sorbent/solution isotherm.However, we shall neglect this linear term in this paper for the following reasons.Firstly, the addition of the linear term complicates the equation away from the simple, tractable form of eq 7. Secondly, the practical analysis of sorption isotherms often focuses on low  2 behavior away from  2 → 0. This is true, particularly for the sorbent/solution interfaces for which sorption isotherm becomes increasingly difficult to measure as the sorbate concentration goes down.Note also that the amount of sorption is very small at  2 → 0 and  2 → 0, making the error introduced by omitting the linear term negligible.For these reasons, we shall focus on the low and finite  2 , instead of the limiting behavior, in this paper.
The cooperative isotherm for solid/gas and solid/vapor interfaces (eq A4) has a track record of applications to experimental isotherms.The examples include (1) water vapor adsorption isotherms on hydrophobic activated carbon fibers with pore widths of 0.5 and 0.6 nm, (2) water vapor adsorption isotherms on the pitch resin-based activated carbon materials with slit sizes of 0.5, 0.6, 1.0, and 1.1 nm. 1 Extension to multiple types of microscopic patches leads to the linear sum of eq A4 with different  and   . 2 This has enable to apply our theory to heterogeneous materials with multiple pore sizes, including (3) NH3 adsorption on a hydrogenbonded organic framework, Kuf-1a, 2 (4) water vapor adsorption on an aluminophosphate molecular sieve, 2 and (5) CO2 adsorption on a metal-organic framework, PCN-53. 2 The S3 determination of sorbate cluster number and the free energy of sorption from these isotherms has led to mechanistic insights. 2" B. Derivation of the cooperative isotherm for the sorbent/solution interface.
Here we derive a solution-phase analogue to eq 2. The derivation can be facilitated significantly via statistical variable transformation. 3,4As a preparation, here we show how the number variance in the {, ,   ,  1 ,  2 } ensemble can be converted to that in the {, ,   ,  1 ,  2 } ensemble under the invariance of the mole ratio and its fluctuation, via where { 1 } and { 1 } have been used for the shorthand notations for {, ,   ,  1 ,  2 } and {, ,   ,  1 ,  2 }, respectively.Carrying out the Maclaurin expansion of eq B1a yields where  2 = 〈 2 〉/〈 1 〉.This ensemble transformation (eq B1b) applies both to the interface ( * ) and solution ().
With the preparation above, let us carry out a differentiation of Γ 2 (1) with respect to ln  2 , as required for a sorbent/solution generalization of eq 2. This differentiation can be carried out straightforwardly under constant  1 , which yields ln  2 where we have expressed the constant  1 ensembles adopted therein emphatically by { 1 * } and { 1  }.Analogous to eq 1, we can introduce the excess numbers for the interface and the solution phases,  22 * and  22  , defined in the constant  1 ensemble, via and using it to rewrite eq B2 as ln  2 Dividing both sides with Γ 2 (1) transforms eq B4 into the following form: where   =  2 * / 2  is the sorbate-solvent swapping constant.

S4
Here we carry out the statistical variable transformation 3,4 to express  22 * + 1 and  22  + 1 in eq B3 in terms of the fluctuations in the {, ,   ,  1 ,  2 } ensemble.This can be carried out using eq B1b as where the right-hand side can be expanded as which represents the difference between self-and mutual interactions.

C. Penetration of solvent and sorbate into sorbent.
Let us consider a three-component system consisting of sorbent (species ), solvent (1) and sorbate (2).The system forms two phases: sorbent and solution phases.The only constraint here is that the sorbent molecules are absent in phase .According to the Gibbs phase rule, this there-component system forming two phases has  = 3 − 2 + 2 = 3 degrees of freedom.
Under constant temperature and pressure, the Gibbs-Duhem equations for the entire system ( * ) and the reference solution phase (), defined in the same way as in the main text, 5 are Our condition, that the sorbent molecules are absent in the solution phase, can be expressed as which serves as the alternative for the Gibbs dividing surface condition.Under this condition (eq C3), subtracting eq C2 from eq C1 yields Under the same condition (eq C3), we obtain from eq C2 Combining eqs C4 and C5, we obtain To summarize, we have derived eq C6 for the sorbent-solution phase equilibrium under the only condition that the sorbent molecules do not dissolve into the solution phase.This means that eq C6 (eq 8a) is valid even when solvent and sorbate molecules dissolve into or penetrate the sorbent.Consequently, starting from Γ 2 (1) introduced in eq C6, we can follow the discussion as presented in B. Derivation of the cooperative isotherm for the sorbent/solution interface to derive our fundamental equations, eq 9a for sorbent/solution and eq 2 for sorbent/gas sorption where the latter can be derived as the special case of eq 9a.Note that the connection to the discussions in B. Derivation of the cooperative isotherm for the sorbent/solution interface, carried out in a constant   ensemble, can be made seamlessly by the fact that Γ 2 is an intensive quantity independent of the choice of the ensemble.
We have previously discussed the similarity and differences between (a) the preferential solvation (involving a solute in dilution) and (b) the Gibbs adsorption isotherm (of an interface in a system of a two-component mixture). 6The difference between the two within this setup was rationalized by the Gibbs phase rule.The solute, in the case of the preferential solvation (a), does not contribute to the degrees of freedom because it is at infinite dilution.However, the presence of the interface in (b) reduces the degrees of freedom in the context of the surface excess.In contrast to the above comparison, the discussion in this section shows the parallel between (c) the preferential solvation of a solute at infinite dilution in an -component solution and (d) sorption from an -component solution mixture species onto a sorbent that is absent in the solution.In comparison to (c), (d) has one more degree of freedom in the Gibbs phase rule by the introduction of the sorbent, which is yet is reduced by one due to the presence of an interface.The absence of the sorbent in the solution in (d) corresponds to the absence of the infinitely dilute solute in the bulk in (c).

D. Isotherms via differential equations.
The ABC Isotherm.8][9] Its derivation follows the same logical steps as the cooperative isotherm summarized in the main text.
• the fundamental equation, which has been rewritten using the sorbate-sorbate Kirkwood-Buff integral, Integrating eq D1a in combination with eq D1b yields 8][9] This isotherm has also been generalized to sorbent/solution interfaces.
The Cubic Isotherm.Here we start with a different expression for the fundamental equation, which can be derived straightforwardly from eq D1a, in combination with a different characteristic relationship, Integrating eq D2a with eq D2b yields the following isotherm: where  ′ was introduced upon integration.This is the polynomial isotherm founded upon sorbate number correlation and the meaning of its parameters  ′ ,  ′ , and ′, will be discussed below.The parameters  ′ ,  ′ , and  ′ can be interpreted by comparing eq D2c with the Maclaurin expansion of the ABC isotherm (eq D1c), which yields through which the parameters can be expressed using KBIs.
The Virial Isotherm.Here we adopt eq D2a again for the fundamental equation yet with a characteristic relationship different from eq D2a, as Integrating eq D2a along with eq D3a, we obtain where the constant  ′′ was introduced upon integration.This isotherm has been known as the virial isotherm. 10,11The parameters of the exponential isotherm can be related to those of the ABC isotherm  ′′ ,  ′′ and  ′′ have a clear link to , , and  , hence to the sorbate-surface, sorbate pair, and triplet interactions. 9 E. Divergent cooperativity via the differential equation approach.
Here we show that the AB isotherm for divergent cooperativity can be derived from the fluctuation equation (eq 2) in combination with the characteristic relationship (eq 14c) that signifies a linearly increasing sorbate cluster number.Combining eqs 2 and 14c under constant  yields the following differential equation:

S6•
the characteristic relationship for sorbate fluctuation, stating how  22 depends on sorbate activity, via is an integration constant.Solving eq E3 for 〈 2 〉 yields leads to the AB isotherm.