Ising Paradigm in Isobaric Ensembles.

We review recent work on Ising-like models with "compressible cells" of fluctuating volume that, as such, are naturally treated in NpT and µpT ensembles. Besides volumetric phenomena, local entropic effects crucially underlie the models. We focus on "compressible cell gases" (CCG), namely, lattice gases with fluctuating cell volumes, and "compressible cell liquids" (CCL) with singly occupied cells and fluctuating cell volumes. CCGs contemplate singular diameters and "Yang-Yang features" predicted by the "complete scaling" formulation of asymmetric fluid criticality, with a specific version incorporating "ice-like" hydrogen bonding further describing the "singularity-free scenario" for the low-temperature unusual thermodynamics of supercooled water. In turn, suitable CCL variants constitute adequate prototypes of water-like liquid-liquid criticality and the freezing transition of a system of hard spheres. On incorporating vacant cells to such two-state CCL variants, one obtains three-state, BEG-like models providing a satisfactory description of water's "second-critical-point scenario" and the whole phase behavior of a simple substance like argon. Future challenges comprise water's crystal-fluid phase behavior and metastable states.

The temperature of maximum density for aqueous solutions.

Experimental and theoretical advances for understanding the temperature of maximum density (TMD) of aqueous solutions are outlined. The main equations that relate the TMD behavior to key thermodynamic properties are stated. The experimental TMD data are classified as a function of the nature of the solute (inorganic electrolytes, non-electrolytes, organic salts and ionic liquids, and amino acids and proteins). In addition, the experimental results that explore the effect of pressure are detailed. These experimental data are rationalized by making use of qualitative and semi-quantitative arguments based on the thermodynamics of aqueous systems. The main theoretical and simulation advances in TMD for aqueous solutions are also shown-including new calculations in the context of the scaled particle theory-and their ability to reproduce the experimental data is evaluated. Finally, new experiments and theoretical and simulation developments, which could give important insights into the problem of TMD for aqueous solutions, are proposed.

Ising model for the freezing transition.

We introduce a three-state Ising model with entropy-volume coupling suitably incorporating a packing mechanism into a lattice gas with no attractive interactions. On working in a great grand canonical ensemble in which the energy, volume, and number of particles are all allowed to fluctuate simultaneously, the model's mean-field solutions illuminate a strictly first-order transition akin to hard-sphere freezing while describing the thermodynamics of solid and fluid phases. Further implementation of attractive interactions in a natural way allows every aspect of the phase diagram of a simple substance to be reproduced, thereby accomplishing the van der Waals picture of the states of matter from first principles of statistical mechanics. This fairly accurate qualitative description plausibly renders mean-field theory a reasonable approach for freezing in three dimensions. At the same time, our mean-field treatment itself suggests freezing to persist in infinitely many dimensions, as advanced from recent simulations [Charbonneau et al., Eur. Phys. J. E 44, 101 (2021)10.1140/epje/s10189-021-00104-y].

Liquid-Liquid Criticality in TIP4P/2005 and Three-State Models of Water.

Molecular dynamics simulations leading to the isothermal compressibility, the isobaric thermal expansivity, and the isobaric heat capacity of TIP4P/2005 water are found to be consistent with the coordinates of its second, liquid-liquid critical point reported recently by Debenedetti et al. [ Science 2020, 369, 289-292]. In accord with the theory of critical phenomena, we encounter that the rise in the magnitude of these response functions as temperature is lowered is especially marked along the critical isochore. Furthermore, response-function ratios provide a test for thermodynamic consistency at the critical point and manifest nonuniversal features sharply distinguishing liquid-liquid from standard gas-liquid criticality. The whole pattern of behavior revealed by simulations is qualitatively the same as the one of a three-state Ising model of water exhibiting a low-temperature liquid-liquid critical point. Exact solutions for the two-state components of such a three-state model are also provided.

Water's two-critical-point scenario in the Ising paradigm.

We present a spin-1, three-state Ising model for the unusual thermodynamics of fluid water. Thus, besides vacant cells, we consider singly occupied cells with two accessible volumes in such a way that the local structures of low density, energy, and entropy associated with water's low-temperature "icelike" order are characterized. The model has two order parameters that drive two phase transitions akin to the standard gas-liquid transition and water's hypothesized liquid-liquid transition. Its mean-field equation of state enables a satisfactory description of results from experiments and simulations for the ST2 and TIP4P/2005 force fields, from the phase diagram, the density maximum, or the deeply "stretched" states to the behavior of thermodynamic response functions at low temperatures at which water exists as a supercooled liquid. It is concluded that the model may be regarded as a most basic prototype of the so-called "two-critical-point scenario."

The isobaric heat capacity of liquid water at low temperatures and high pressures.

Isobaric heat capacity for water shows a rather strong anomalous behavior, especially at low temperature. However, almost all experimental studies supporting this statement have been carried out at low pressure; very few experimental data were reported above 100 MPa. In order to explore the behavior of this magnitude for water up to 500 MPa, a new heat flux calorimeter was developed. With the aim of testing the experimental methodology and comparing with water results, isobaric heat capacity was also measured for methanol and hexane. Good agreement with indirect heat capacity estimations from the literature was obtained for the three liquids. Experimental results show large anomalies in water heat capacity. This is especially true as regards its temperature dependence, qualitatively different from that observed for other liquids. Heat capacity versus temperature curves show minima for most studied isobars, whose location decreases with the pressure up to around 100 MPa but increases at higher pressures.

Heat capacity singularity of binary liquid mixtures at the liquid-liquid critical point.

The critical anomaly of the isobaric molar heat capacity for the liquid-liquid phase transition in binary nonionic mixtures is explained through a theory based on the general assumption that their partition function can be exactly mapped into that of the Ising three-dimensional model. Under this approximation, it is found that the heat capacity singularity is directly linked to molar excess enthalpy. In order to check this prediction and complete the available data for such systems, isobaric molar heat capacity and molar excess enthalpy near the liquid-liquid critical point were experimentally determined for a large set of binary liquid mixtures. Agreement between theory and experimental results-both from literature and from present work-is good for most cases. This fact opens a way for explaining and predicting the heat capacity divergence at the liquid-liquid critical point through basically the same microscopic arguments as for molar excess enthalpy, widely used in the frame of solution thermodynamics.

Thermal properties of ionic systems near the liquid-liquid critical point.

Isobaric heat capacity per unit volume, C(p), and excess molar enthalpy, h(E), were determined in the vicinity of the critical point for a set of binary systems formed by an ionic liquid and a molecular solvent. Moreover, and, since critical composition had to be accurately determined, liquid-liquid equilibrium curves were also obtained using a calorimetric method. The systems were selected with a view on representing, near room temperature, examples from clearly solvophobic to clearly coulombic behavior, which traditionally was related with the electric permittivity of the solvent. The chosen molecular compounds are: ethanol, 1-butanol, 1-hexanol, 1,3-dichloropropane, and diethylcarbonate, whereas ionic liquids are formed by imidazolium-based cations and tetrafluoroborate or bis-(trifluromethylsulfonyl)amide anions. The results reveal that solvophobic critical behavior-systems with molecular solvents of high dielectric permittivity-is very similar to that found for molecular binary systems. However, coulombic systems-those with low permittivity molecular solvents-show strong deviations from the results usually found for these magnitudes near the liquid-liquid phase transition. They present an extremely small critical anomaly in C(p)-several orders of magnitude lower than those typically obtained for binary mixtures-and extremely low h(E)-for one system even negative, fact not observed, up to date, for any liquid-liquid transition in the nearness of an upper critical solution temperature.

The critical behavior of the dielectric constant in the polar + polar binary liquid mixture nitromethane + 3-pentanol: an unusual sign of its critical amplitude in the one-phase region.

Dielectric constant measurements have been carried out in the one- and two-phase regions near the critical point of the polar + polar binary liquid mixture nitromethane + 3-pentanol. In the two-phase region, evidence for the |t|(2ß) singularity in the coexistence-curve diameter has been detected, thus confirming the novel predictions of complete scaling theory for liquid-liquid criticality. In the one-phase region, an "unusual" negative sign for the amplitude of the |t|(1-α) singularity has been encountered for the first time in an upper critical solution temperature type of binary liquid mixture at atmospheric pressure. Mass density measurements have also been carried out to provide additional information related to such experimental finding, which entails an increase of the critical temperature T(c) under an electric field.

On the isobaric thermal expansivity of liquids.

The temperature and pressure dependence of isobaric thermal expansivity, α(p), in liquids is discussed in this paper. Reported literature data allow general trends in this property that are consistent with experimental evidence to be established. Thus, a negative pressure dependence is to be expected except around the critical point. On the other hand, α(p) exhibits broad regions of negative and positive temperature dependence in the (T, p) plane depending on the nature of the particular liquid. These trends are rationalized here in terms of various molecular-based equations of state. The analysis of the Lennard-Jones, hard sphere square well and restricted primitive model equations allows understanding the differences in the α(p) behavior between liquids of diverse chemical nature (polar, nonpolar, and ionic): broader regions of negative temperature and positive pressure dependencies are obtained for liquids characterized by larger ranges of the interparticle potential. Also, using the statistical associating fluid theory (SAFT) allowed the behavior of more complex systems (basically, those potentially involving chain and association effects) to be described. The effect of chain length is rather simple: increasing it is apparently equivalent to raise the interaction range. By contrast, association presents a quite complex effect on α(p), which comes from a balance between the dispersive and associative parts of the interaction potential. Thus, if SAFT parameters are adjusted to obtain low association ability, α(p) is affected by each mechanism at clearly separate regions, one at low temperature, due to association, and the other to dispersive forces, which has its origin in fluctuations related with vapor-liquid transition.

Heat capacity anomalies along the critical isotherm in fluid-fluid phase transitions.

The behavior of the isochoric heat capacity of pure fluids and the isobaric heat capacity at constant composition of binary mixtures along isothermal paths of approach to liquid-gas and liquid-liquid critical points is studied. From the complete scaling formulation of fluid-fluid criticality, explicit expressions for the critical amplitudes of the leading /Y-Y(c)/(-alpha/beta) (where Y can be the density or the mole fraction) contributions are found to reveal previously discovered features of the scaling function, whereas the nature of the most important asymmetry-related terms is characterized. Data for pure toluene and for the binary mixture nitromethane-isobutanol are described within experimental uncertainty using the /Y-Y(c)/(-alpha/beta) singularity plus a linear term. Extensive data for mixtures allow proper visualization of the topological features of the heat capacity-density-temperature surface in the critical region.

Thermodynamic consistency near the liquid-liquid critical point.

The thermodynamic consistency of the isobaric heat capacity per unit volume at constant composition C(p,x) and the density rho near the liquid-liquid critical point is studied in detail. To this end, C(p,x)(T), rho(T), and the slope of the critical line (dT/dp)(c) for five binary mixtures composed by 1-nitropropane and an alkane were analyzed. Both C(p,x)(T) and rho(T) data were measured along various quasicritical isopleths with a view to evaluate the effect of the uncertainty in the critical composition value on the corresponding critical amplitudes. By adopting the traditionally employed strategies for data treatment, consistency within 0.01 K MPa(-1) (or 8%) is attained, thereby largely improving the majority of previous results. From temperature range shrinking fits and fits in which higher-order terms in the theoretical expressions for C(p,x)(T) and rho(T) are included, we conclude that discrepancies come mainly from inherent difficulties in determining the critical anomaly of rho accurately: specifically, to get full consistency, higher-order terms in rho(T) are needed; however, the various contributions at play cannot be separated unambiguously. As a consequence, the use of C(p,x)(T) and (dT/dp)(c) for predicting the behavior of rho(T) at near criticality appears to be the best choice at the actual experimental resolution levels. Furthermore, the reasonably good thermodynamic consistency being encountered confirms that previous arguments appealing to the inadequacy of the theoretical expression relating C(p,x) and rho for describing data in the experimentally accessible region must be fairly rejected.

Heat capacity of associated systems. Experimental data and application of a two-state model to pure liquids and mixtures.

The predictions from a recently reported (J. Chem. Phys. 2004, 120, 6648) two-state association model (TSAM) have been tested against experimental data. The temperature, T, and pressure, p, dependence of the isobaric heat capacity, C(p), for three pure alcohols and the temperature dependence at atmospheric pressure of the excess heat capacity, C(p)(E), for four alcohol + ester mixtures have been measured. The branched alcohols were 3-pentanol, 3-methyl-3-pentanol, and 3-ethyl-3-pentanol, and the mixtures were 1-butanol and 3-methyl-3-pentanol mixed with propyl acetate and with butyl formate. These data, together with literature data for alcohol + n-alkane and alcohol + toluene mixtures, have been analyzed using the TSAM. The model, originally formulated for the C(p) of pure liquids, has been extended here to account for the C(p)(E) of mixtures. To evaluate its performance, quantum mechanical ab initio calculations for the H-bond energy, which is one of the model parameters, were performed. The effect of pressure on C(p) for pure liquids was elucidated, and the variety of C(p)(E)(T) behaviors was rationalized. Furthermore, from the C(p) data at various pressures, the behavior of the volume temperature derivative, (deltaV/deltaT)(p), was inferred, with the existence of a (deltaV/deltaT)(p) versus T maximum for pure associated liquids such as the branched alcohols being predicted. It is concluded that the TSAM captures the essential elements determining the behavior of the heat capacity for pure liquids and mixtures, providing insight into the macroscopic manifestation of the association phenomena occurring at the molecular level.

Griffiths-Wheeler geometrical picture of critical phenomena: experimental testing for liquid-liquid critical points.

An experimental approach to the verification of specific relations between thermodynamic properties as predicted from the Griffiths-Wheeler theory of critical phenomena in multicomponent systems is developed for the particular case of ordinary liquid-liquid critical points of binary mixtures. Densities rho(T) , isobaric heat capacities per unit volume C(p)(T) , and previously reported values of the slope of the critical line (dT/dp)c for five critical mixtures are used to check the thermodynamic consistency of C(p) and rho near the critical point. An appropriate treatment of rho (T) data is found to provide the key solution to this issue. In addition, various alternative treatments for C(p)(T) data provide values for both the critical exponent alpha and the ratio between the critical amplitudes of the heat capacity A+/A- that are in agreement with their widely accepted counterparts, whereas two-scale-factor universality is successfully verified in one of the systems studied.