Erratum: Cooperative Gas Adsorption without a Phase Transition in Metal-Organic Frameworks [Phys. Rev. Lett. 121, 015701 (2018)].

This corrects the article DOI: 10.1103/PhysRevLett.121.015701.

Cooperative Gas Adsorption without a Phase Transition in Metal-Organic Frameworks.

Cooperative adsorption of gases by porous frameworks, which permits more efficient uptake and removal than the more usual noncooperative (Langmuir-type) adsorption, usually results from a phase transition of the framework. Here we show how cooperativity emerges in the class of metal-organic frameworks mmen-M_{2}(dobpdc) in the absence of a phase transition. Our study provides a microscopic understanding of the emergent features of cooperative binding, including the position, slope, and height of the isotherm step, and indicates how to optimize gas storage and separation in these materials.

Entropy of polydisperse chains: solution on the Husimi lattice.

We consider the entropy of polydisperse chains placed on a lattice. In particular, we study a model for equilibrium polymerization, where the polydispersity is determined by two activities, for internal and endpoint monomers of a chain. We solve the problem exactly on a Husimi lattice built with squares and with arbitrary coordination number, obtaining an expression for the entropy as a function of the density of monomers and mean molecular weight of the chains. We compare this entropy with the one for the monodisperse case, and find that the excess of entropy due to polydispersity is identical to the one obtained for the one-dimensional case. Finally, we obtain a distribution of molecular weights with a rather complex behavior, but which becomes exponential for very large mean molecular weight of the chains, as required by scaling properties, which should apply in this limit.

Entropy of fully packed rigid rods on generalized Husimi trees: A route to the square-lattice limit.

Although hard rigid rods (k-mers) defined on the square lattice have been widely studied in the literature, their entropy per site, s(k), 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≤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}(k) for each type of cluster, whose extrapolations to L→∞ provide estimates of these entropies for the square lattice. For dimers, our estimates for s(2) differ from the exact result by only 0.03%, while that for s(3) differs from best available estimates by 3%. In this paper, we also obtain a new estimate for s(4). 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(k). 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≥4, in contrast to the results for the Bethe lattice wherein it is nematic.

Solution on the Bethe lattice of a hard core athermal gas with two kinds of particles.

Athermal lattice gases of particles with first neighbor exclusion have been studied for a long time as simple models exhibiting a fluid-solid transition. At low concentration the particles occupy randomly both sublattices, but as the concentration is increased one of the sublattices is occupied preferentially. Here, we study a mixed lattice gas with excluded volume interactions only in the grand-canonical formalism with two kinds of particles: small ones, which occupy a single lattice site and large ones, which, when placed on a site, do not allow other particles to occupy its first neighbors also. We solve the model on a Bethe lattice of arbitrary coordination number q. In the parameter space defined by the activities of both particles, at low values of the activity of small particles (z(1)) we find a continuous transition from the fluid to the solid phase as the activity of large particles (z(2)) is increased. At higher values of z(1) the transition becomes discontinuous, both regimes are separated by a tricritical point. The critical line has a negative slope at z(1) = 0 and displays a minimum before reaching the tricritical point, so that a re-entrant behavior is observed for constant values of z(2) in the region of low density of small particles. The isobaric curves of the total density of particles as a function of the density or the activity of small particles show a minimum in the fluid phase.

Order-disorder transition in a two-dimensional associating lattice gas.

We study an associating lattice gas (ALG) using Monte Carlo simulation on the triangular lattice and semianalytical solutions on Husimi lattices. In this model, the molecules have an orientational degree of freedom and the interactions depend on the relative orientations of nearest-neighbor molecules, mimicking the formation of hydrogen bonds. We focus on the transition between the high-density liquid (HDL) phase and the isotropic phase in the limit of full occupancy, corresponding to chemical potential µâ†’∞, which has not yet been studied systematically. Simulations yield a continuous phase transition at τ_{c}=k_{B}T_{c}/γ=0.4763(1) (where -γ is the bond energy) between the low-temperature HDL phase, with a nonvanishing mean orientation of the molecules, and the high-temperature isotropic phase. Results for critical exponents and the Binder cumulant indicate that the transition belongs to the three-state Potts model universality class, even though the ALG Hamiltonian does not have the full permutation symmetry of the Potts model. In contrast with simulation, the Husimi lattice analyses furnish a discontinuous phase transition, characterized by a discontinuity of the nematic order parameter. The transition temperatures (τ_{c}=0.51403 and 0.51207 for trees built with triangles and hexagons, respectively) are slightly higher than that found via simulation. Since the Husimi lattice studies show that the ALG phase diagram features a discontinuous isotropic-HDL line for finite µ, three possible scenarios arise for the triangular lattice. The first is that in the limit µâ†’∞ the first-order line ends in a critical point; the second is a change in the nature of the transition at some finite chemical potential; the third is that the entire line is one of continuous phase transitions. Results from other ALG models and the fact that mean-field approximations show a discontinuous phase transition for the three-state Potts model (known to possess a continuous transition) lends some weight to the third alternative.

Solution of a model of self-avoiding walks with multiple monomers per site on the Husimi lattice.

We solve a model of self-avoiding walks which allows for a site to be visited up to two times by the walk on the Husimi lattice. This model is inspired in the Domb-Joyce model and was proposed to describe the collapse transition of polymers with one-site interactions only. We consider the version in which immediate self-reversals of the walk are forbidden. The phase diagram we obtain for the grand-canonical version of the model is similar to the one found in the solution of the Bethe lattice, with two distinct polymerized phases: a tricritical point and a critical endpoint.

Solution of a model of self-avoiding walks with multiple monomers per site on the Bethe lattice.

We solve a model of self-avoiding walks with up to two monomers per site on the Bethe lattice. This model, inspired in the Domb-Joyce model, was recently proposed to describe the collapse transition observed in interacting polymers [J. Krawczyk, Phys. Rev. Lett. 96, 240603 (2006)]. When immediate self-reversals are allowed (reversion-allowed model), the solution displays a phase diagram with a polymerized phase and a nonpolymerized phase, separated by a phase transition which is of first order for a nonvanishing statistical weight of doubly occupied sites. If the configurations are restricted forbidding immediate self-reversals (reversion-forbidden model), a richer phase diagram with two distinct polymerized phases is found, displaying a tricritical point and a critical end point.

Grand-canonical solution of semiflexible self-avoiding trails on the Bethe lattice.

We consider a model of semiflexible interacting self-avoiding trails (sISATs) on a lattice, where the walks are constrained to visit each lattice edge at most once. Such models have been studied as an alternative to the self-attracting self-avoiding walks (SASAWs) to investigate the collapse transition of polymers, with the attractive interactions being on site as opposed to nearest-neighbor interactions in SASAWs. The grand-canonical version of the sISAT model is solved on a four-coordinated Bethe lattice, and four phases appear: non-polymerized (NP), regular polymerized (P), dense polymerized (DP), and anisotropic nematic (AN), the last one present in the phase diagram only for sufficiently stiff chains. The last two phases are dense, in the sense that all lattice sites are visited once in the AN phase and twice in the DP phase. In general, critical NP-P and DP-P transition surfaces meet with a NP-DP coexistence surface at a line of bicritical points. The region in which the AN phase is stable is limited by a discontinuous critical transition to the P phase, and we study this somewhat unusual transition in some detail. In the limit of rods, where the chains are totally rigid, the P phase is absent and the three coexistence lines (NP-AN, AN-DP, and NP-DP) meet at a triple point, which is the endpoint of the bicritical line.

Nature of the collapse transition in interacting self-avoiding trails.

We study the interacting self-avoiding trail (ISAT) model on a Bethe lattice of general coordination q and on a Husimi lattice built with squares and coordination q=4. The exact grand-canonical solutions of the model are obtained, considering that up to K monomers can be placed on a site and associating a weight ω_{i} with an i-fold visited site. Very rich phase diagrams are found with nonpolymerized, regular polymerized, and dense polymerized phases separated by lines (or surfaces) of continuous and discontinuous transitions. For a Bethe lattice with q=4 and K=2, the collapse transition is identified with a bicritical point and the collapsed phase is associated with the dense polymerized (solidlike) phase instead of the regular polymerized (liquidlike) phase. A similar result is found for the Husimi lattice, which may explain the difference between the collapse transition for ISATs and for interacting self-avoiding walks on the square lattice. For q=6 and K=3 (studied on the Bethe lattice only), a more complex phase diagram is found, with two critical planes and two coexistence surfaces, separated by two tricritical and two critical end-point lines meeting at a multicritical point. The mapping of the phase diagrams in the canonical ensemble is discussed and compared with simulational results for regular lattices.

Polydispersed rods on the square lattice.

We study the grand-canonical solution of a system of hard polydispersed rods placed on the square lattice using transfer matrix and finite-size scaling calculations. We determine the critical line separating an isotropic from a nematic phase. No second transition to a disordered phase is found at high density, contrary to what is observed in the monodispersed case. The estimates of critical exponents and the central charge on the critical line are consistent with the Ising universality class.

Transfer-matrix study of a hard-square lattice gas with two kinds of particles and density anomaly.

Using transfer matrix and finite-size scaling methods, we study the thermodynamic behavior of a lattice gas with two kinds of particles on the square lattice. Only excluded volume interactions are considered, so that the model is athermal. Large particles exclude the site they occupy and its four first neighbors, while small particles exclude only their site. Two thermodynamic phases are found: a disordered phase where large particles occupy both sublattices with the same probability and an ordered phase where one of the two sublattices is preferentially occupied by them. The transition between these phases is continuous at small concentrations of the small particles and discontinuous at larger concentrations, both transitions are separated by a tricritical point. Estimates of the central charge suggest that the critical line is in the Ising universality class, while the tricritical point has tricritical Ising (Blume-Emery-Griffiths) exponents. The isobaric curves of the total density as functions of the fugacity of small or large particles display a minimum in the disordered phase.

Entropy of chains placed on the square lattice.

We obtain the entropy of flexible linear chains composed of M monomers placed on the square lattice using a transfer matrix approach. An excluded volume interaction is included by considering the chains to be self-avoiding and mutually avoiding, and a fraction rho of the sites is occupied by monomers. We solve the problem exactly on stripes of increasing width m and then extrapolate our results to the two-dimensional limit m--> infinity using finite-size scaling. The extrapolated results for several finite values of M and in the polymer limit M--> infinity for the cases where all lattice sites are occupied (rho=1) and for the partially filled case rho<1 are compared with earlier results. These results are exact for dimers (M=2) and full occupation (rho=1) and derived from series expansions, mean-field-like approximations, and transfer matrix calculations for some other cases. For small values of M, as well as for the polymer limit M--> infinity, rather precise estimates of the entropy are obtained.

Walks with up to two monomers per site on the Bethe lattice: interpolation between models with immediate reversals allowed and immediate reversals forbidden.

We generalize earlier calculations of the RA (immediate reversals allowed) and RF (immediate reversals forbidden) model with multiple monomers per site, including a parameter in the model, so that both cases studied before are particular cases of the model considered here. Also, we calculate the bulk free energy of the model using Gujrati's prescription, which leads to corrections in the localization of coexistence lines and qualitative changes in the phase diagram of the model. Thus, this calculation provides an example in which the use of the method of iterating recursion relations starting with natural initial conditions to locate coexistence loci is not trustworthy and may lead to qualitatively different results. A continuous collapse transition appears in all cases as a tricritical point.

Hard rigid rods on a Bethe-like lattice.

We study a system of long rigid rods of fixed length k with only excluded volume interaction. We show that, contrary to the general expectation, the self-consistent field equations of the Bethe approximation do not give the exact solution of the problem on the Bethe lattice in this case. We construct a new lattice, called the random locally treelike layered lattice, which allows a dense packing of rods, and we show that the Bethe self-consistent equations are exact for this lattice. For a four-coordinated lattice, k-mers with k ≥ 4 undergo a continuous isotropic-nematic phase transition. For even coordination number q ≥ 6, the transition exists only for k ≥ k(min)(q), and is discontinuous.

Solution of an associating lattice-gas model with density anomaly on a Husimi lattice.

We study a model of a lattice gas with orientational degrees of freedom which resemble the formation of hydrogen bonds between the molecules. In this model, which is the simplified version of the Henriques-Barbosa model, no distinction is made between donors and acceptors in the bonding arms. We solve the model in the grand-canonical ensemble on a Husimi lattice built with hexagonal plaquettes with a central site. The ground state of the model, which was originally defined on the triangular lattice, is exactly reproduced by the solution on this Husimi lattice. In the phase diagram, one gas and two liquid [high density liquid (HDL) and low density liquid (LDL)] phases are present. All phase transitions (GAS-LDL, GAS-HDL, and LDL-HDL) are discontinuous, and the three phases coexist at a triple point. A line of temperatures of maximum density in the isobars is found in the metastable GAS phase, as well as another line of temperatures of minimum density appears in the LDL phase, part of it in the stable region and another in the metastable region of this phase. These findings are at variance with simulational results for the same model on the triangular lattice, which suggested a phase diagram with two critical points. However, our results show very good quantitative agreement with the simulations, both for the coexistence loci and the densities of particles and of hydrogen bonds. We discuss the comparison of the simulations with our results.

Grand-canonical and canonical solution of self-avoiding walks with up to three monomers per site on the Bethe lattice.

We solve a model of polymers represented by self-avoiding walks on a lattice, which may visit the same site up to three times in the grand-canonical formalism on the Bethe lattice. This may be a model for the collapse transition of polymers where only interactions between monomers at the same site are considered. The phase diagram of the model is very rich, displaying coexistence and critical surfaces, critical, critical end point, and tricritical lines, as well as a multicritical point. From the grand-canonical results, we present an argument to obtain the properties of the model in the canonical ensemble, and compare our results with simulations in the literature. We do actually find extended and collapsed phases, but the transition between them, composed by a line of critical end points and a line of tricritical points, separated by the multicritical point, is always continuous. This result is at variance with the simulations for the model, which suggest that part of the line should be a discontinuous transition. Finally, we discuss the connection of the present model with the standard model for the collapse of polymers (self-avoiding, self-attracting walks), where the transition between the extended and collapsed phases is a tricritical point.

Entropy of polydisperse chains: solution on the Bethe lattice.

We consider the entropy of polydisperse chains placed on a lattice. In particular, we study a model for equilibrium polymerization, where the polydispersivity is determined by two activities, for internal and endpoint monomers of a chain. We solve the problem exactly on a Bethe lattice with arbitrary coordination number, obtaining an expression for the entropy as a function of the density of monomers and mean molecular weight of the chains. We compare this entropy with the one for the monodisperse case and find that the excess of entropy due to polydispersivity is identical to the one obtained for the one-dimensional case. Finally, we obtain an exponential distribution of molecular weights.