In all cases the kernels shown in Fig. 2 were used for computing the isotherms.

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9SCIentIFIC REPORtS | 7: 7216 | DOI:10.1038/s41598-017-07406-2

correspondingly leads to the fact that now larger portions of the pores can simultaneously be filled with the capillary-condensed phase by the axial domain growth. Hence, the transition becomes steeper. The boundary desorption transition is altered in a similar way. As shown in Fig. 6, with increasing ξ the corresponding interval Δλ of the gas activities determining the transition range for the given pore size interval decreases. This causes the desorption transition to occur at higher gas activities and in a narrower range Δλ.

It is interesting to note that the initial slope of the descending scans becomes flatter with increasing ξ . As revealed by Fig. 6, the increase of ξ by keeping σ constant leads to a strong increase of Λ0. As discussed earlier, this gives rise to a stronger pore blocking effect. Eventually, as shown for ξ = 16 nm in Fig. 5H, the adsorption tran- sition becomes so steep that no desorption scans can be obtained anymore. In this regime, condensation in all pores occurs in one step following the formation of a very first liquid bridge. This phenomenon may be observed irrespective of ξ . Crucial requirement is that PSD should be relatively narrow, see as an example Fig. 5L. This figure reveals also that, for narrow PSDs and for large L, one obtains H1 hysteresis loop with two sharp transitions (see, e.g., ref. 28) occurring at the gas pressures corresponding to the equilibrium (gas invasion) and metastable (liquid bridging) transitions in the smallest pores present in the system.

Increasing ξ by simultaneously keeping the relative disorder σ ξ/ constant does not preserves the shape of the boundary and scanning isotherms. Their evolution depends on which of the two quantities, Λ0 or σ, grow quicker with increasing ξ . In the particular case presented in Fig. 5J–L, the former one dominates and hence stronger pore blocking is observed for larger ξ .

Discussion The results obtained for a linear chain of pores with varying pore diameter along the channel axis are found to be in a remarkable qualitative agreement with the majority of experimental findings obtained for phase equilib- ria in materials with disordered pore structures. The model naturally leads to the complex cooperativity effects during the transitions by assuming the possibility of a simultaneous action of two transformation mechanisms, nucleation and phase growth. On a microscopic level, they are accounted for by the corresponding kernels, which can independently be obtained with a high accuracy. The general framework furnished with the appropriate kernels may be applied for a variety of systems and under very different thermodynamic conditions. Importantly, the resulting equations describing the transition pathways upon variation of the chemical potential are simple analytical expressions. Hence, they can easily be implemented into the pore structure determination algorithms. Essentially, the governing equations may be interpreted as the application of the BJH scheme, typically used in gas sorption, to the statistical arrays of serially connected individual pores. So far, BJH coupled with the independent pore model remained a pillar for structural characterization of porous solids. The results presented here provide now a closely related alternative allowing for a more reliable approach for structure determination of disordered porous materials. As a robust criterium for selecting between the two approaches one may consider the shapes of the scanning curves. As soon as one finds the scans crossing the boundary curves (the situations when the cavitation pressures are attained on desorption need to be considered separately), independent pore model can be applied, while the scans merging at the hysteresis closure points are the indications towards using the serial pore model.

The capability of the model to describe properly the scanning behavior opens further perspectives for both developing of the in-depth characterization algorithms, for addressing phenomena not yet fully covered, such as unveiling of the fluctuation and network effects, and for correlating the experimental data obtained for different