Exploring the thermodynamic criteria for responsive adsorption processes

A general thermodynamic model to investigate responsive adsorption processes in flexible porous materials.


Detailed simulation methodology
In our model the mean field density functional theory describing the adsorption of a Lennard-Jones fluid into an ideal slit pore follows a previously reported approach. S1 . The equation for describing the grand potential is displayed in Eq. S1, where w is the pore width, µ is the chemical potential, A is pore area, V is the external potential and f is intrinsic free energy density.
We partition the intristic free energy density, f , into repulsive and attractive contributions according to the Weeks-Chandler-Anderson (WCA) division of the Lennard-Jones potential f = f rep + f att , where the repulsive contribution is approximated by hard spheres. For convenience, we set the effective hard sphere diameter, d, equal to σ, the Lennard-Jones length parameter. Rosenfeld's fundamental measure functional S2 was used for the hard sphere term and standard mean field treatment of the attractive contribution, Eqn. S2.
where Φ(r) has the form described in Eqn. S3.
The Lennard-Jones potential (Φ lj ) for fluid interacting with the walls of the pore is described by Eq. S4 S3 and the fluid-fluid interactions are described by a standard Lennard-Jones potential, Eq. S5. We ignore wall-wall interactions.
The deviation of the pore width from some equilibrium width is described by athermal Helmholtz free energy function of two general forms, Eq. S6 and Eq. S7.
The mixture of quadratic and parabolic terms in Eq. S7 produce a double well function with coefficients (c i ) defining the location and depth of the minima. We consider the wall of the solid to be constructed from same particles as the adsorbate thus σ sf = σ ff and ε sf = ε ff . However, the strength of the solid-fluid interactions can be controlled by the density of interaction sites ρ sf . For a carbon wall, this density is equal to ≈38.2 nm -2 or ≈4.41 σ -2 cc , where σ cc =0.34 nm. Numaguchi et al argued in their studies of flexible MOFs that for stacked-layer porous coordination polymers the ρ sf should be equal to ≈2.2 σ -2 cc . S4 As such MOFs are considered to have half the interaction site density compared to carbon walls, leading to weaker solid-fluid interaction, if ε sf = ε ff . We use this weaker interaction by setting ρ sf = 2.0 in the present study.
Using the above parameters are library of adsorption isotherms were generated for pore widths ranging between 2.0 ≤ w * ≤ 20 , with a resolution of 0.175.
The osmotic potential, Eq. S8 is then straightforwardly constructed by combining the free energy profiles of host system and the library of adsorption isotherms, as the classical density functional theory method explicitly gives the grand potential Ω(T, P, w).
For a given host free energy potential (F host ) we can generate an adsorption isotherm by looping over configurational activity (λ = exp µ kT ), where µ is the configurational contribution to the chemical potential only, and computing the minima of the osmotic potential (Ω os ) for the range of pore widths (w). Thus allowing the system to change pore width for a lower energy pore according to the osmotic potential. We impose an additional kinetic criteria that the system can only change pore width if the maximum height of the osmotic potential between the current pore width and the new pore width is less than a threshold energy (Ω crit ) to be 6T * .
Gas adsorption isotherm generation is computed using a python script using the numpy, scipy and pandas packages. An example of the code used to generate the responsive isotherms can be found at the data repository of J. D. Evans at https://github.com/jackevansadl/supp-data. We encourage researchers to explore this methodology by combining alternative isotherm kernels and complex host free energy potentials to produce entirely new adsorption profiles.
To calculate the relative frequency for three temperatures (main manuscript Fig. 4) similar conditions were scanned for the responsive adsorption maps (30 points between 0 ≤ E * thin ≤ 20 and 30 points between 0 ≤ E * barrier ≤ 20)but in addition w wide was also scanned for 30 values between 2.6 ≤ w * wide ≤ 15 this produced a total of 12180 different combinations of E * thin , E * barrier and w * wide for each temperature. From these combinations isotherms were generated and the number of isotherms which resulted in NGA and an sdw value > 0.5 (denoted flex.) were counted to give the results in Table S2 and Fig. 4 (main manuscript).

Metastability in Adsorption Processes
To understand the relationship between the free energy barriers in the system with structural transitions, critical energy fluctuations and metastability, consider the narrow pore to open pore transition as shown in the Fig. S1.

Fig. S1
Representative gas adsorption isotherm for a material which shows gate-opening, where N * is the adsorbed density and λ * is configurational activity. The dashed line corresponds to an equilibrium process and the solid line a processes that includes metastability. (b) Osmotic potentials at five points during gas adsorption including labeling of the critical energy barrier for a w thin → w wide transition.
This figure shows the behavior of volume and the amount adsorbed as a function of pressure and the osmotic potential profiles at different points on the isotherm as a function of pore widths. At the beginning (P1), there are two minima in the osmotic potential associated with the pore state at width w thin and w wide . The pore width w thin , at this point, is lower in energy than w wide , which persists at pressure P2. At P3 the w wide has a lower value of Ω os * so the system can expand to this lower energy state, this corresponds to the equilibrium transition. However, this transition from w thin to w wide is associated with an energy barrier, E barrier . If this barrier is too great, the transition will not occur while the system remains in a metastable w thin state. This metastable state extends until the energy barrier falls below some characteristic thermal fluctuation value (designated here as Ω crit ); at this point (P4) the transition can occur spontaneously. Finally at P5 the system remains in this stable state, associated with an open pore, w wide . The plausible values of the energy fluctuations have been explored by Numaguchi et al. S4 Their argument is energy fluctuation of the system should depend on the observation time and the size of a local domain of a crystal. They employed transition state theory to formulate the rate constant for structural transition of a crystal domain (in their case, cp → op pore transition). They set the observation time to one hour and assumed that spontaneous transition occurs at the pressure for which the time constant of the spontaneous transition (=1/rate constant) is equal to the observation time. The larger is the crystal domain under consideration the deeper into S4 metastable region of pressure one needs to go to observe spontaneous transition within one hour. According to this analysis, at T * = 0.8, the energy fluctuation of 6kT per unit cell corresponds to the transition domain size of 8.3 × 8.3 nm 2 of their model of ELM-11. It is not straightforward to compare the numbers used in the current study to that of Numaguchi et al. however we set Ω crit ) to be 6kT and also consider the effect of this parameter.

Comparison of reduced units to a previous atomistic study
There are several studies of atomistic simulations of methane adsorption in DUT-49 at 111K. S5,S6 Let us consider how the atomistic system approximately maps onto our reduced units, considering the critical parameters of TraPPE methane (σ ff = 3.73 Åand ε ff = 1.23 kJ mol −1 ). S7 The structure of DUT-49 consists of one large pore ≈24 Å (6.43σ ff ) in diameter, 8 smaller cages of about ≈20 Å (5.36σ ff ) and 14 ≈10 Å cages (2.68σ ff ). During the collapse of the structure, under NGA conditions, the largest pore shrinks to a pore of about ≈8 Å (2.14σ ff ), ≈20 Å pore reduces to ≈7 Å (1.85σ ff ), while the smallest pore remains unchanged.
Mapping the experimental DUT-49 system on to the slit pore DFT model employed here is not a trivial task due to the difference in geometry of the systems and lack of direct correspondence between certain properties (i.e. wall density required in the slit pore model). To model the free energy profile of the host material, obtained by previous simulations shown Fig. S2, we adopt the following simplified approach. We approximate that the profile is a result of structural changes of 9 pore cages, and therefore energy change per pore cage can be easily obtained (for example, the barrier becomes 127 kJ mol −1 ). In order to translate this energy profile into reduced units we further need to consider this energy per unit area, which is the most problematic transition. Firstly, we assume that all 9 cages are the same with pore diameter of d = 20 Å (w * wide = 5.4) that deform to 8 Å (w * thin = 2.1). The surface are of this pore can be treated as a fraction of a surface of a sphere S = f πd 2 , where f = 0.2 is a fitting parameter.
Subsequently, the critical parameters of the atomistic free energy landscape (per cage E thin = 112 kJ mol −1 and E barrier = 126 kJ mol −1 ) are reduced to E * thin = 5.0 and E * barrier = 5.6. The resulting isotherm for this model, Fig. S3 for a slightly reduced threshold energy (Ω crit = 5.5T * ) displays a negative step. It would be naive to expect some quantitative accuracy here as the model does not accurately reproduce isotherms of spherical pore structures.  Table S1 A description of the reduced units displayed in this study.