Sorption Hysteresis: A Statistical Thermodynamic Fluctuation Theory

Hysteresis is observed commonly in sorption isotherms of porous materials. Still, there has so far been no unified approach that can both model hysteresis and assess its underlying energetics. Standard approaches, such as capillary condensation and isotherms based on interfacial equations of state, have not proved to be up to the task. Here, we show that a statistical thermodynamic approach can achieve the following needs simultaneously: (i) showing why adsorption and desorption transitions may be sharp yet continuous; (ii) providing a simple (analytic) isotherm equation for hysteresis branches; (iii) clarifying the energetics underlying sorption hysteresis; and (iv) providing macroscopic and nanoscopic perspectives to understanding hysteresis. This approach identifies the two pairs of parameters (determinable by fitting experimental data) that are required to describe the hysteresis: the free energy per molecule within the pore clusters and the cluster size in the pores. The present paper focuses on providing mechanistic insights to IUPAC hysteresis types H1, H2(a), and H2(b) and can also be applied to the isotherm types IV and V.


A. Quasi-thermodynamic stability theory
Quasi-thermodynamic theory is advantageous over the Gibbsian ensemble approach for establishing a clear link between sorbate number fluctuation to thermodynamic stability.We consider a system surrounded by the reservoir.The key quantity is the minimum excess work done by an external medium on the system + reservoir, , that accompanies the exchange of sorbates (species 2)  =  +  (𝑟) (A1) where  is the change in the energy of the system and  (𝑟) is the change of energy of the reservoir.The superscript () represents the reservoir. (𝑟) accompanying the exchange of matter can be expressed as 1  () =  ()  () +  2 ()  2 () (A2) Since the system and the reservoir are in equilibrium,  2 () =  2 and  () =  hold.Hence, we use  2 and  without the superscript.The conservation of particles and energy between the system and the reservoir leads to  2 () = − 2 and  () = −.Taken all together, eqs A1 .Now we apply the stability theory to the interface.This can be achieved by repeating the same argument for the system (which contains an interface) and the reference systems on the solid and gas/vapor sides, denoted by  and .Subtracting  for the reference systems from that of the system, we obtain the following result for the interfacial minimum excess work, Δ ≡  −   −   : Neglecting the second and third terms of eq A6 following the postulates in the main text, we obtain which will be used as the foundation of thermodynamic stability theory for an interface.

B. Macroscopic versus nanoscopic stability conditions
Despite the same excess number ( 22 ) underlying the ln-ln gradients of macroscale and nanoscale isotherms (eq 7), there is a crucial difference between the two regarding how the stability condition is broken.To demonstrate this, we will carry out the order-of-magnitude analysis with and without nanoscopic subdivision.
Macroscopic System without Subdivision.For macroscopic systems, there are two types of thermodynamic quantities, intensive and extensive. 1,2When there is no subdivision into nanoscopic subsystems (pores), a macroscopic interface, with its characteristic length-scale , is two-tiered in terms of its thermodynamic quantities: 3 (1) (intensive) and ( 2 ) (extensive).As shown above, when the stability condition is satisfied, = (1).Since the system is two-tiered with (1) and ( 2 ) ,  22 + 1 = 〈 2  2 〉 〈 2 〉 , when the stability condition is broken, must reach ( 2 ).This takes place when the standard deviation reaches the system size order, √〈 2  2 〉 = ( 2 ).Consequently, the ln-ln gradient of the isotherm diverges, or precisely stated, becomes macroscopic ( ( 2 ) ) in its order of magnitude.However, as will be shown in the Results and Discussion, when a macroscopic interface is subdivided into nanoscopic subsystems, subdivision makes fluctuations in the order of the entire macroscopic system ( 2 ) impossible.

C. Relating the cooperative isotherm parameters with the functional shape
First, we show that   and  has a clear relationship to the functional shape of an isotherm, thereby eliminating ambiguity in the nonlinear fitting of experimental isotherm data.As shown in our recent paper,   is the activity at the steepest gradient of an isotherm, ℎ/4  , where ℎ is the step height (i.e., the difference between the lowest and highest) of the cooperative isotherm (see Appendix A of Ref [5]).Consequently, with a reasonable estimate of ℎ,  and   can either be obtained visually from an experimental isotherm or be used to check the validity of the parameters determined via nonlinear fitting.
Second, we establish what the "nearly parallel" 6 adsorption and desorption branches mean in the framework of the cooperative isotherm.Since nearly parallel means nearly equal gradient, because the isotherm step is common.Consequently, Since   ′ is lower than   , ′ is lower than .In fact, there is a slight reduction from  to  ′ as observed for the nearly parallel SBA-15 in Table 1.However, for a reasonably narrow isotherm loop,   ′ is not too different from   .In this case, ′ and  can be considered comparable.

D. The generalized Gibbs isotherm
Here we demonstrate that our approach can be applied naturally to interfaces of arbitrary geometry and porosity, unlike the conventional Gibbs isotherm.8][9] Instead of the conventional derivation of the Gibbs isotherm from the trio of Gibbs-Duhem equations plus the concentration profile, [7][8][9][10] we adopt a novel statistical thermodynamic approach based on Legendre transformation. 11For the conventional approach, a clearly defined coordinate system is indispensable, which introduces an unnecessary restriction to planar interfaces. 11Our approach is free from such a restriction, applicable to any surface geometry and porosity. 11,12 start with the definition of excess free energy,   , via   = Ω − Ω  − Ω  (D1) as the difference in the thermodynamic function ( Ω = − ) between the entire system (without superscript) and the two reference systems (superscripts  and  ) under the conservation of volume. 11,12The Gibbs dividing surface can be introduced via Legendre transformation, converting the thermodynamic function Ω (open to species 1 and 2) to the thermodynamic function  = Ω +  1  1 (open to species 2 but closed to 1), as   =  −   −   −  1 ( 1 −  1  −  1  ) (D2) using  1 , the chemical potential of species 1. 11 The Gibbs dividing surface, in this setup, is simply to impose the condition  1 −  1  −  1  = 0, which yields Using the partition function for the partially open ensembles, we can relate   to the Under the postulate that the effect of an interface is confined within a finite distance from the surface, 11 eq D4 can be rewritten while appreciating the finite-distanced nature of the interface, as Using (B) of the main text, eq D5 can be expressed in an approximate manner as Introducing the excess energy per nanoscale system via   =  ̃, we can rewrite eq D6 as which is the nanoscale version of eq D6.
Combining eq D6 with the interfacial free energy per unit sorbent,   , defined via eq 12a, we obtain which is the generalization of the Gibbs isotherm.The nanoscale counterpart of eq D8 can be derived by introducing the amount of sorbent per nanoscale system,  ̃1, defined via  1 =  ̃1, which converts eq D8 into to relate  1 to  22 .When the isotherm is a single-valued function, we can rewrite the macroscale relationship (eq D8) via the chain rule to yield .Under constant , expanding  in terms of  2 yields