Finally, equations 6 and 10 compose the so-called boundary hysteresis loop, reminiscent of the adsorption/ desorption isotherms in gas sorption, freezing/melting curves in thermoporometry, and intrusion/extrusion curves in mercury intrusion. For monolithic materials, i.e. materials with large L, the pair of equations 7 and 11 can be used instead. It is worth mentioning that equations 7 and 11 describe quite generally the conditions when nucleation is effective. The relevant examples here are capillary condensation and cavitation-driven evaporation. If the transition have purely percolation-like character, such as evaporation via gas invasion or mercury intrusion, pn and ′pn in equations 7 and 11 can be shown to be equal to 1/L, hence these equations become, respectively,

The results of the simulations performed with the disordered pores are shown in Fig. 3. In these simulations the long pores were composed of statistically distributed pore sections whose pore sizes were given by PSDs shown in Fig. 3A. Figure 3B shows schematically a part of the typical pore configuration. The relative amounts of the liquid phase obtained for different gas activities (sorption isotherms) and different temperatures (melting and freezing curves) are shown in Fig. 3C,D, respectively. Importantly, the simulation results are found to deviate notably from those predicted by IPM as shown by the broken lines. With the kernels available (Fig. 2C,D), the mean phase compositions obtained upon monotonous changes of the thermodynamic parameters can readily be obtained using Eqs 6 and 10. They are shown in Fig. 3C,D by the solid lines and are found to be in excellent agreement with the simulation data. This agreement proves the robustness of SPM in describing the boundary transitions in disordered pore systems. In turn, the theory would allow a reliable pore size analysis. It may be be noted that for one-dimensional channels with geometric disorder the application of IPM with the BJH analysis scheme yields PSDs significantly underestimating the real pore sizes (see Fig. 3A). The latter finding supports some observations reported in the literature showing that PSDs experimentally obtained using gas sorption or thermoporometry methods for materials possessing geometric disorder delivered pore sizes smaller than those obtained by electron microscopy methods50, 51.

Scanning behavior. In the previous section only the full cycles of the χ variation were considered. Let us address now how would θ vary if χ is varied non-monotonically, i.e. if the variation direction is inverted at some intermediate stages. The family of the thus collected ascending and descending curves are commonly referred to as scanning behavior15. It is often argued in the literature that the scanning curves contain more detailed informa- tion on both the pore space properties and on the phase transition mechanisms13.

Let us first consider an ascending scan started at χ0 attained upon a monotonous decrease of χ to χ0 along the descending boundary transition line. As the first step, the mean number ′N0 of the continuous domains of the

Figure 2. Phase transition in ideal pores and transition kernels. (A) Adsorption and desorption isotherms obtained using GCMC simulations of a lattice-gas in a slit pore of 5 nm width as a function of relative activity at T/Tc = 2/3. Different transitions are obtained using different pore geometries. A slit pore open at both ends exhibited capillary-condensation (denoted as ‘liquid bridging’) and evaporation (denoted as ‘desorption’) upon increasing and decreasing gas activity, respectively. A capped slit pore exhibited the reversible capillary-filling (‘advanced adsorption’) and evaporation (‘desorption’) transitions. A slit pore with the length of 10 nm and closed at both ends was used to locate the cavitation pressure (‘cavitation’). (B) The fraction of the liquid phase in a cylindrical pore with a diameter of 6 nm obtained upon temperature variation for the Kossel crystal. As in (A), different pore geometries were used to address different transitions. The channel open at both ends exhibited the irreversible melting (‘liquid nucleation’) and freezing (‘ice invasion’) transitions, while in the capped cylinder two reversible transitions (‘ice crystal shrinkage’ and ‘ice invasion’, respectively) were obtained. The infinitely long cylinder, modeled using periodic boundary conditions, was used to locate the homogeneous nucleation temperature (‘homogeneous ice nucleation’). (C,D) The gas activities and temperatures at which the different transitions occur in ideal pores (kernels). The solid lines are the best fits of the appropriate analytical expressions used further to compute phase compositions in disordered pores.

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