Confinement effects and acid strength in zeolites

Chemical reactivity and sorption in zeolites are coupled to confinement and—to a lesser extent—to the acid strength of Brønsted acid sites (BAS). In presence of water the zeolite Brønsted acid sites eventually convert into hydronium ions. The gradual transition from zeolite Brønsted acid sites to hydronium ions in zeolites of varying pore size is examined by ab initio molecular dynamics combined with enhanced sampling based on Well-Tempered Metadynamics and a recently developed set of collective variables. While at low water content (1–2 water/BAS) the acidic protons prefer to be shared between zeolites and water, higher water contents (n > 2) invariably lead to solvation of the protons within a localized water cluster adjacent to the BAS. At low water loadings the standard free energy of the formed complexes is dominated by enthalpy and is associated with the acid strength of the BAS and the space around the site. Conversely, the entropy increases linearly with the concentration of waters in the pores, favors proton solvation and is independent of the pore size/shape.


Supplementary Note 1: Collective Variables.
In Voronoi diagrams, volume is partitioned in polyhedra around points called seeds or generators. Every polyhedron is defined as the portion of volume whose points are closer to a specific seed than to any other. Similarly, hydrogen atoms assigned to the site are taken as the fraction of protons that are much closer to this site than from all the other ones. Softmax functions allow to reproduce this behaviour without any discontinuity or singularity as shown in the Eqs. (1) and (2), where is a vector of the atomic positions, the indexes and run over the atoms able to bond or release hydrogen atoms, runs over the hydrogen atoms and is a parameter that controls steepness and selectivity of this function. ! ( ) approaches 1 when the atom is the closest to and 0 otherwise, see Finally, the summation over the hydrogen atoms returns ! , the total number of protons within the th voronoi polyhedron.

CV1: .
Once we have obtained the instantaneous number of hydrogen atoms around each site, the difference between these values, ! , and the respective references, ! + , returns their deviation from neutrality, see

Eq.
(3). Then, the overall excess or defect of protons for every moiety is taken as the summation of their site contributions, see Eq. (4).
In this specific case, + describes the overall excess or defect of protons of the BAS and is obtained by summing all the ! with that runs over each BAS oxygen atom. Similarly, -describes the overall excess or defect of protons of the water clusters and it is obtained by summing all the ! with that runs over each water oxygen atoms.
The result of this operation is a vector 9 9⃗ = ( + , -) that describes with its components the protonation state of our system. In order to further reduce the dimensionality of the problem, we project this vector into a single a scalar value. This is done by linearly combining + andas in Eq. (5).
In systems such as ours characterized by two different inequivalent moieties able to exchange protons (BAS and water cluster), in principle three theoretical protonation states are possible, see

Supplementary
As we said, + andhave been chosen in order to represent the overall deviation from neutrality of BAS and water clusters respectively. Since the BAS cannot act like a base and therefore cannot gain a proton from the water cluster, the state described by 9 9⃗ = (+1, −1) is totally unphysical and . will never reach negative values. Then, . = 0 ( 9999⃗ = (0,0)) describes the undissociated states, while . = 1 (9 9⃗ = (−1, +1)) describes the states in which a proton is transferred from the BAS to the water cluster.
This CV ensures the possibility to explore every protonation states starting from the most energetically accessible up to the highest one in energy without imposing any restriction.

CV2: .
With this CV we measure the distance between the sites that have exchanged a proton and, thus, not lying in their reference states.
Here the reference value of protons in each Voronoi polyhedron is taken as the total number of protons assigned to the entire group *∈, divided by the the total number of sites belonging to the -th group, , . Starting from the Eq. (2), the instantaneous deviation from its reference is computed as follow: In the example of a BAS and 4 water molecules, let us take the protolysis of the BAS and consequent

Restraint: .
In order to prevent simultaneous dissociation events, we restraint a third CV used only to monitor how many sites out of their reference states are present. The functional form of this CV is: where runs over the site indexes and is a positive number much less than 1. With a proper value of the square root term is a good approximation of the absolute value that allows to avoid the singularity for ! = 0 (see Supplementary Figure 2). This CV returns the summation of the partial charge moduli and, by restraining it, we can limit at the given time the number of reacted pairs simultaneously present.

Supplementary Note 2: Ab-initio MD setup.
All the simulations have been performed with CP2K package 1 and set up as reported in Supplementary

Samples preparation.
Each system has a composition of 96 units of SiO 3 with an aluminum atom replacing a silicon. All of them have been thermalized with a 5 ps NVT MD simulation.

Simulation lengths.
Supplementary

4.3: Water cluster sphericities.
An indication that the framework does not affect the water cluster behavior comes from the analysis of their shapes. A volume and a surface can be computed for polyhedra whose vertices are defined by 4 or more water molecules. From these we can get an estimation of their sphericity according to the Wadell .
This formula defines the sphericity, . , of a generic polyhedron as the ratio between its volume . and its surface . . . can assume values between 0 and 1, where, by definition, the sphericity of the cluster is unity for a sphere and decreases in solids with lower symmetry. The sphericities of these clusters inside the zeolites were calculated and compared with the values extracted from the equivalent MD simulations in gas phase, see Supplementary

Supplementary Note 5: Validation of interatomic potential.
Errore. L'origine riferimento non è stata trovata. shows the Ow-Ow radial distribution functions of gas p hase water clusters simulated with PBE+D2 and B3LYP+D2 exchange and correlation functionals. This analysis proved that our structures are in agreement with the ones described by a higher level of theory, in that they give identical peak maxima for both first and second shells, and that the over-binding errors due to the PBE are well compensated by the Grimme dispersion corrections.

Supplementary Figure 5 O-O radial distribution functions and their integrals over distances.
Blue lines report water clusters behavior simulated with PBE functional, while orange lines report the same results performed with B3LYP functional.