transitions in statistically disordered porous solids.

Another important aspect of the theory presented is the option provided to model more accurately complex physical processes accompanying phase equilibria. Many theoretical approaches aimed at describing physical properties of composite systems, including different phases coexisting in the pore spaces of mesoporous solids, are based on the effective-medium approximations56. However, a number of experimentally observed measures or processes can crucially depend not only on the average phase composition, but also on the finer details of the mutual spatial distribution of the coexisting phases. As to the most prominent examples, one may refer to the light or ultrasonic scattering and to diffusion studies in mesoporous materials at different stages of adsorption and desorption or freezing and melting57–62. In these experiments it has been unequivocally demonstrated that the light and sound transmissions and the diffusivities measured at one and the same loading, but attained during adsorption and desorption, can be different. In particular, light scattering is controlled by the inhomogeneity of the liquid distribution along the pore space and their number densities58, 62. In disordered pore systems, geometric disorder controls the distribution of the effective radii of the scatterers, namely of the domains with the liquid phase. In a close analogy, the diffusivities are the function of how liquid domains are distributed spatially and how they are interconnected via the wetting films59, 63, 64. Importantly, the number densities of the different domains (N0 and ′N0) as well as their spatial extensions (Λ0 and Λ′0), which are different for different preparation histories of one and the same average phase composition, are readily obtained within the model outlined here. Hence, the model provides an effective approach for predicting scattering and transport properties along different transition pathways.

With all potentials highlighted, there are still several open questions which need to be addressed in future studies. In the first line, it is critical to establish how accurately can the single pore model describe real porous solids with the pore spaces of higher dimensions, rather than one-dimensional pore systems considered here. In this respect, no distinctive conclusions may be inferred by referring to percolation theories because the nucle- ation processes may have substantial impact on the phase equilibria. At the same time, some features inherent in percolation problems are of relevance, in particular in determining the phase growth processes. The most essential difference between the serial and independent pore models is the fact that the former one takes account of the cooperativity effects in phase transitions. Most notably, they control growth of the phase domains at given thermodynamic conditions. Hence, for discussing the dimensionality effects upon phase equilibria it is absolutely essential to consider in all details what happens at the junctions between different pores. On the one hand, the junctions may provide an effective mechanism for transferring the phase state to the neighbouring pores. For example, a junction with a relatively large size can promote gas invasion. In this case, this will render the overall behavior even further from that predicted by the independent pore model. On the other hand, the junctions may hinder phase growth, like the same junction of a large size can prohibit advanced adsorption. Hence, the overall behavior may become to be shifted towards the independent pore model. With the two examples mentioned, it is evident that one and the same junction may have opposite impacts upon different transition mechanisms. While one of the transition branches may develop more features inherent in the independent pore model, the other one may become affected in the opposite way. Hence, in order to gain deeper insight into the dimensionality effects, more close attention needs to be paid first to assess geometric features of the pore junctions and to understand their role in promoting or hindering the transfer of phase state to adjacent pores. These studies may also shed light on the question whether the network effects can be reliably determined from the experimental data. Another interesting problems which have become apparent during this work are how accurate the conventionally obtained kernels are for describing phase transitions in pores with complex pore geometries and how to relate the length parameter L of the present work to real extensions of porous materials.

Methods GCMC simulations of gas sorption. A simple cubic single occupancy lattice-gas model in an external field and with nearest neighbor interactions was considered. The external field serves to model the pore geometry confining the lattice gas. The Hamiltonian for this model is given by

∑ ∑ φ= − +n n n , (22)i j

i j i

i i ,

H ε

where ε denotes the nearest neighbor interaction strength, ⟨i, j⟩ indicates the sum over all nearest neighbor bonds, ni ∈ {0, 1} is the occupation number, and φi the external field at the lattice node i. For lattice nodes adjacent to pore walls, the external field is chosen to be φi = 3ε. This value is based roughly on the energy minimum experienced by a Lennard-Jones-(12, 6) atom close to a solid constituted by the same atoms31. For all other lattice nodes φi equals zero. The lattice constant was chosen to be 0.5 nm. The Grand canonical Monte Carlo (GCMC) simulations were performed using the Metropolis algorithm65, which, in order to change the state of the lattice gas towards the equilibrium state, at each step attempts to reverse the occupancy of every node with the probabilities

= β µ∆ −p emin[1, ], (23) E

add/remove ( )

where β = 1/(kT), ΔE is the energy change between the new and the old system state and μ denotes the chemical potential. In order to obtain the sorption isotherms, the relative activity λ λ = β µ µ−e/ 0

( )0 is used as thermody-

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