Philippos J. Pomonis

Pore connectivity and pore entropy, that are crucial characteristics of porous solids controlling the mass transfer and diffusion of fluids through them, are linearly related.

Pore anisotropy ( B ), pore length ( L ), pore connectivity ( c ), pore entropy ( H ) and heteroscedasticity of pore size distributions are strongly related properties in oxidic materials with random porosity.

In this work we suggest that the common cause for the development of various power laws is the existence of a suitable exchangeable quantity between the agents of a set. Examples of such exchangeable quantities, leading to eponymous power laws, include money (Pareto’s Law), scientific knowledge (Lotka’s Law), people (Auerbach’s Law), and written or verbal information (Zipf’s Law), as well as less common cases like bullets during deadly conflicts, recognition in social networks, heat between the atmosphere and sea-ice floes, and, finally, mass of water vapors between pores in solids. This last case is examined closely in the present article based on extensive experimental data. It is shown that the transferred mass between pores, which eventually grow towards a power law distribution, may be expressed using different parameters, either transferred surface area, or transferred volume, or transferred pore length or transferred pore anisotropy. These distinctions lead to different power laws of variable strength as reflected by the corresponding exponent. The exponents depend quantitatively on the spread of frequency distribution of the examined parameter and tend to zero as the spread of distribution tends to a single order of magnitude. A comparison between the energy and the entropy of different kinds of pore distributions reveals that these two statistical parameters are linearly related, implying that the system poise at a critical state and the exchangeable quantities are the most convenient operations helping to keep this balance.

In this work it is shown that the range of Zipf's law for the ranking of random pores in solids upholds for ∼10 orders of magnitude, from nm pore sizes in laboratory-made materials, to cm pore sizes occurring in solidified volcanic magmas. Also it is demonstrated that for MCM-type materials with organized mesopores, the out-of-trend data in the Zipf plots, corresponding to the sudden pore filling during N₂ adsorption, can be used for the convenient estimation of local pore anisotropy.

The total mean pore anisotropy b_mean,total = (N·b) and the distribution of differential pore anisotropies b_mean,diff = (N_i·b_i) in porous solids were estimated from N₂ porosimetry data V = f(P/P₀). The pore numbers N and volumes V are ranked according to the Zipf's law log(N_i) = A − B log(V_i).

The Frenkel−Halsey−Hill (FHH) equation V/V_m ≈ [log(P₀/P)]^−1/s is revisited in relation to the meaning of its exponent in a specific intermediate range of pressure where capillary condensation occurs. It has been suggested in the past that plots of the form log V = constant − (D − 3)[loglog(P₀/P)], or its equivalent log S = const − (D − 2) log r, can be used for the estimation of the dimensionality D of the adsorbing surface from those parts of the slopes at low pressure corresponding to straight lines. In the present study it is shown that, for pores of cylindrical geometry and at a specific range of pressure where those pores are filled-up during the process of capillary condensation, the local slopes d log V/dloglog[(P₀/P)] or dlog S/d log r, of plots similar to the above, may be used to estimate the pore anisotropy b of the adsorbing space from the relationships log b = [[d log V/d loglog[(P₀/P)] − 3] log(0.5r) or log b = [[d log S/d log r] − 2] log(0.5r). These observations lead to the physicogeometrical conjunction that, during capillary condensation in cylindrical pores, usually assumed in nitrogen porosimetry, the scaling dimension of pore anisotropy b, scaled in units of radius r, is related to the dimensionality D of the process.