Although lattice gases composed of particles preventing up to their $k\mathrm{th}$ nearest neighbors from being occupied (the $k\mathrm{NN}$ models) have been widely investigated in the literature, the location and the universality class of the fluid-columnar transition in the 2NN model on the square lattice are still a topic of debate. Here, we present grand-canonical solutions of this model on Husimi lattices built with diagonal square lattices, with $2L(L+1)$ sites, for $L⩽7$. The systematic sequence of mean-field solutions confirms the existence of a continuous transition in this system, and extrapolations of the critical chemical potential ${\mu }_{2,c}\left(L\right)$ and particle density ${\rho }_{2,c}\left(L\right)$ to $L\to \infty$ yield estimates of these quantities in close agreement with previous results for the 2NN model on the square lattice. To confirm the reliability of this approach, we employ it also for the 1NN model, where very accurate estimates for the critical parameters ${\mu }_{1,c}$ and ${\rho }_{1,c}$—for the fluid-solid transition in this model on the square lattice—are found from extrapolations of data for $L⩽6$. The nonclassical critical exponents for these transitions are investigated through the coherent anomaly method (CAM), which in the 1NN case yields $\beta$ and $\nu$ differing by at most 6% from the expected Ising exponents. For the 2NN model, the CAM analysis is somewhat inconclusive, because the exponents sensibly depend on the value of ${\mu }_{2,c}$ used to calculate them. Notwithstanding, our results suggest that $\beta$ and $\nu$ are considerably larger than the Ashkin-Teller exponents reported in numerical studies of the 2NN system.

While the realistic modeling of the thermodynamic behavior of fluids usually demands elaborated atomistic models, much has been learned from simplified ones. Here, we investigate a model where pointlike particles (with activity z0) are mixed with molecules that exclude their first and second neighbors (i.e., cubes of lateral size λ=3a, with activity z2), both placed on the sites of a simple cubic lattice with parameter a. Only hard-core interactions exist among the particles so that the model is athermal. Despite its simplicity, the grand-canonical solution of this model on a Husimi lattice built with cubes revels a fluid-fluid demixing, yielding a phase diagram with two fluid phases (one of them dominated by small particles—F0) and a solidlike phase coexisting at a triple-point. Moreover, the fluid-fluid coexistence line ends at a critical point. An anomaly in the total density (ρT) of particles is also found, which is hallmarked by minima in the isobaric curves of ρT vs z0 (or z2). Interestingly, the line of minimum density crosses the phase diagram starting inside the region where both fluid phases are stable, passing through the F0 one and ending deep inside its metastable region, in a point where the spinodals of both fluid phases cross each other.

The apparent tortuosity due to adsorption of diffusing tracers in a porous material is determined by a scaling approach and is used to analyze recent data on LiCl and alkane diffusion in mesoporous silica. The slope of the adsorption isotherm at small loadings is written as β = q_A/q_G, where q_A is the adsorption-desorption ratio and q_G = ϵ/(as) - 1 is a geometrical factor depending on the range a of the tracer-wall interaction, the porosity ϵ, and the specific surface area s. The adsorption leads to a decrease of effective diffusion coefficient, which is quantified by multiplying the geometrical tortuosity factor τ_geom by an apparent tortuosity factor τ_app. In wide pores or when the adsorption barrier is high, τ_app = β + 1, as obtained in previous works, but in narrow pores there is an additional contribution from frequent adsorption-desorption transitions. These results are obtained in media with parallel pores of constant cross sections, where the ratio between the effective pore width ϵ/s and the actual width is ≈0.25. Applications to mesoporous silica samples are justified by the small deviations from this ideal ratio. In the analysis of alkane self-diffusion data, the fractions of adsorbed molecules predicted in a recent theoretical work are used to estimate τ_geom of the silica samples, which is ≫1 only in the sample with the narrowest pores (nominal 3 nm). The application of the model to Li⁺ ion diffusion leads to similar values of τ_geom and to a difference of energy barriers of desorption and adsorption for those ions of ∼0.06 eV. Comparatively, alkane self-diffusion provides the correct order of magnitude of τ_geom, with adsorption playing a less important role, whereas adsorption effects on Li⁺ diffusion are much more important.

Although hard rigid rods ($k$-mers) defined on the square lattice have been widely studied in the literature, their entropy per site, $s\left(k\right)$, in the full-packing limit is only known exactly for dimers ($k=2$) and numerically for trimers ($k=3$). Here, we investigate this entropy for rods with $k\le 7$, by defining and solving them on Husimi lattices built with diagonal and regular square-lattice clusters of effective lateral size $L$, where $L$ defines the level of approximation to the square lattice. Due to an $L$-parity effect, by increasing $L$ we obtain two systematic sequences of values for the entropies $s_L\left(k\right)$ for each type of cluster, whose extrapolations to $L\to \infty$ provide estimates of these entropies for the square lattice. For dimers, our estimates for $s\left(2\right)$ differ from the exact result by only $0.03\%$, while that for $s\left(3\right)$ differs from best available estimates by $3\%$. In this paper, we also obtain a new estimate for $s\left(4\right)$. For larger $k$, we find that the extrapolated results from the Husimi tree calculations do not lie between the lower and upper bounds established in the literature for $s\left(k\right)$. In fact, we observe that, to obtain reliable estimates for these entropies, we should deal with levels $L$ that increase with $k$. However, it is very challenging computationally to advance to solve the problem for large values of $L$ and for large rods. In addition, the exact calculations on the generalized Husimi trees provide strong evidence for the fully packed phase to be disordered for $k\ge 4$, in contrast to the results for the Bethe lattice wherein it is nematic.

We study binary mixtures of hard particles, which exclude up to their $k\mathrm{th}$ nearest neighbors $\left(k\mathrm{NN}\right)$ on the simple cubic lattice and have activities $z_k$. In the first model analyzed, point particles (0NN) are mixed with 1NN ones. The grand-canonical solution of this model on a Husimi lattice built with cubes unveils a phase diagram with a fluid and a solid phase separated by a continuous and a discontinuous transition line which meet at a tricritical point. A density anomaly, characterized by minima in isobaric curves of the total density of particles against $z_0$ (or $z_1)$, is also observed in this system. Overall, this scenario is identical to the one previously found for this model when defined on the square lattice. The second model investigated consists of the mixture of 1NN particles with 2NN ones. In this case, a very rich phase behavior is found in its Husimi lattice solution, with two solid phases—one associated with the ordering of 1NN particles $\left(S1\right)$ and the other with the ordering of 2NN ones $\left(S2\right)$—beyond the fluid $\left(F\right)$ phase. While the transitions between $F\text{−}S2$ and $S1\text{−}S2$ phases are always discontinuous, the $F\text{−}S1$ transition is continuous (discontinuous) for small (large) $z_2$. The critical and coexistence $F\text{−}S1$ lines meet at a tricritical point. Moreover, the coexistence $F\text{−}S1,F\text{−}S2$, and $S1\text{−}S2$ lines meet at a triple point. Density anomalies are absent in this case.