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... One of the more recent proposals for a Widom line is the R-Widom line, defined as the locus of maxima of the isotherms of the scalar curvature of TG [16,17]. There are several proposals for a geometrical description of thermodynamics based mainly on fluctuation theory [18][19][20][21][22][23][24][25]. ...

... Properties which are not so surprising since the curvature is proportional to a power of the correlation length near the critical point. Slightly above the critical temperature, the extreme values of R define a curve similar to those given by the maxima of the response functions in the ðP; TÞ plane [17], the R-Widom line. This curve also behaves linearly near the critical point and will be defined as the Widom line in this geometrical approach to supercritical thermodynamics. ...

... As was previously mentioned, two important characteristics related to the scalar curvature R are, the possibility to reproduce to a certain extent the coexistence curves and its sign [16,17,28,38], providing information about the kind of interaction. ...

The Thermodynamic Geometry (TG) of Mie fluids in the subcritical and supercritical region is studied using a third order thermodynamic perturbation theory equation of state (EOS). The R-crossing method of TG is applied to reproduce the coexistence curves related to Mie fluids and it is found that the validity of this methodology is range dependent. Besides, defining the R-Widom line, as the curve obtained from the extreme of the isotherms of the scalar curvature in the (P,T) plane, the behavior of this Widom line is analyzed varying the range and stiffness of Mie potentials and it is compared to the locus of the maxima of some response functions. A strong dependence of the R-Widom line is found with respect to stiffness and range potential for the Mie fluids. Besides, a kind of correspondence states principle it is found for the R-Widom line, and a Clausius-Clapeyron-type relation near the critical point in the supercritical region is fulfilled.

... Since the introduction of the Ruppeiner metric in 1979 7 , a substantial body of work [12][13][14][15] has been directed towards firming up its theoretical groundwork as well as developing its possible applications. The Ruppeiner metric has found many important applications in the field of thermal physics, such as in fluctuation theory 14 , finite-time thermodynamics [16][17][18] , phase transitions 15,[19][20][21][22][23][24][25][26] , and even black hole thermodynamics [25][26][27] , among others. In this work, we focus on phase transitions. ...

... Given a metric space, the immediate objects of interest are its geodesics and curvature. As a consequence, several studies [15][16][17][18][19][20][21][22][23][24][25][26][27] have centered on these two objects towards clarifying the physical significance of the abstract Ruppeiner metric space, and towards developing applications. ...

... Since correlation lengths are challenging to measure or compute, the Widom line is often obtained indirectly through the maxima of thermodynamic response functions instead, such as the isobaric heat capacity, isothermal compressibility, or thermal expansion coefficient. 20,30 The reason behind this is that all these response functions scale as powers of the correlation length near the critical point. So, the curves of these response function maxima should all asymptote to the Widom line at the critical point. ...

In the study of fluid phases, the Ruppeiner geometry provides novel ways for constructing the phase boundaries (known as the $R$-crossing method) and the Widom line, which is considered by many to be the continuation of the coexistence curve beyond the critical point. In this paper, we revisit these geometry-based constructions with the aim of understanding their limitations and generality. We introduce a new equation-of-state expansion for fluids near a critical point, assuming analyticity with respect to the number density, and use this to prove a number of key results, including the equivalence between the $R$-crossing method and the standard construction of phase boundaries near the critical point. The same conclusion is not seen to hold for the Widom line of fluids in general. However, for the ideal van der Waals fluid a slight tweak in the usual formulation of the Ruppeiner metric, which we call the Ruppeiner-$N$ metric, makes the Ruppeiner geometry prediction of the Widom line exact. This is in contrast to the results of May and Mausbach where the prediction is good only up to the slope of the Widom line at the critical point. We also apply the Ruppeiner-$N$ metric to improve the proposed classification scheme of Di\'osi et al. that partitions the van der Waals state space into its different phases using Ruppeiner geodesics. Whereas the original Di\'osi boundaries do not correspond to any established thermodynamic lines above (or even below) the critical point, our construction remarkably detects the Widom line. These results suggest that the Ruppeiner-$N$ metric may play a more important role in thermodynamic geometry than is presently appreciated.

... In addition, the Riemannian geometry has become increasingly important in the field of fluid thermodynamics. Evaluated fluid models include van der Waals and Lennard-Jones (LJ) [18,19,20,21]. In addition, studies were performed for a large number of real fluids [22,23,24]. ...

... The disappearance of the correlation hole in g RPA (r), characterized by an approach g RPA (r = 0) → 1 as T → ∞ and ρ → ∞, signals the occurrence of ideal gas-like behavior. At low temperatures, Eq. (19) can lead to an unphysical negative g RPA (r = 0). The corresponding threshold temperature T th (ρ) is frequently discussed as a limit that restricts the validity of the RPA [55]. ...

... Eq. (19) numerically, on setting g RPA (r = 0; ρ, T = T th ) = 0. The course of T th (ρ) is displayed in Figure 6 below. ...

The three-dimensional Gaussian core model (GCM) for soft-matter systems has repulsive interparticle interaction potential $\phi (r) = \varepsilon\, {\rm exp}\left[ -(r/\sigma)^{2} \right]$, with $r$ the distance between a pair of atoms, and the positive constants $\varepsilon$ and $\sigma$ setting the energy and length scales, respectively. $\phi (r)$ is mostly soft in character, without the typical hard core present in fluid models. We work out the thermodynamic Ricci curvature scalar $R$ for the GCM, with particular attention to the sign of $R$, which, based on previous results, is expected to be positive/negative for microscopic interactions repulsive/attractive. Over most of the thermodynamic phase space, $R$ is found to be positive, with values of the order of $\sigma^3$. However, for low densities and temperatures, the GCM potential takes on the character of a hard-sphere repulsive system, and $R$ is found to have an anomalous negative sign. Such a sign was also found earlier in inverse power law potentials in the hard-sphere limit, and seems to be a persistent feature of hard-sphere models.

... The LJ EOS of Quiñones-Cisneros et al. [65], Nicolas et al. [51], Cotterman et al. [35], Sun and Teja [60], Koutras et al. [66], and Stephan et al. [46] deviate by more than 20% from the exact values in the range T * < T * tr and 6 < T * . The LJ EOS of van Westen and Gross [56], Lafitte et al. [38], Thol et al. [41], Adachi et al. [64], May and Mausbach [62,63], Johnson et al. [28], Kolafa and Nezbeda [52], Boltachev and Baidakov [68], Ree [53], and Miyano [61] deviate by more than 20% from the exact values at T * < T * tr , too, but perform better at high temperatures. Excluding the vicinity of the Boyle temperature and extreme temperature conditions at T * < T * tr and 6 < T * , the LJ EOS of Mecke et al. [54,55], Johnson et al. [28], Boltachev and Baidakov [68], Adachi et al. [64], Miyano [61], Thol et al. [41], and Kolafa and Nezbeda [52] describe the exact second virial coefficient data within B * = ±2%-the LJ EOS of Refs. ...

... The agreement of the LJ EOS and exact values for the third virial coefficient C * is overall significantly less good than for the second virial coefficient. Only the LJ EOS of Johnson et al. [28], Kolafa and Nezbeda [52], Lafitte et al. [38], Mecke et al. [54,55], May and Mausbach [62,63], Thol et al. [41], and van Westen and Gross [56] qualitatively describe the trend of the third virial coefficient accurately. The LJ EOS of Kolafa and Nezbeda [52], Mecke et al. [54,55], and Thol et al. [41] describe the third virial coefficient qualitatively well up to T * = 100 . ...

... Several LJ EOS [41,52,56,61,67,69] produce reasonable Amagat curves over a wide temperature range, but yield distortions in the vicinity of the solid-fluid equilibrium. There are also some LJ EOS that produce Amagat curves exhibiting minor oscillations at high pressures [28,35,51,57,60,[62][63][64][65], i.e., a wrong curvature. ...

Equations of state based on intermolecular potentials are often developed about the Lennard-Jones (LJ) potential. Many of such EOS have been proposed in the past. In this work, 20 LJ EOS were examined regarding their performance on Brown’s characteristic curves and characteristic state points. Brown’s characteristic curves are directly related to the virial coefficients at specific state points, which can be computed exactly from the intermolecular potential. Therefore, also the second and third virial coefficient of the LJ fluid were investigated. This approach allows a comparison of available LJ EOS at extreme conditions. Physically based, empirical, and semi-theoretical LJ EOS were examined. Most investigated LJ EOS exhibit some unphysical artifacts.

... In two recent studies [6,7], the anomalous behavior of cold and supercooled liquid water was investigated by means of a relatively new approach, the thermodynamic metric geometry. This geometric concept has been systematically developed for atomic and molecular fluid systems using thermodynamic data obtained from experiments and computer simulations [8,9,10,11,12,13,14]. ...

... All the branches show power law behaviors, with MF critical exponent 2. Striking is the equality of the −R values in the high and low density phases, particularly for the ST2(I-MF) model. This equality is an example of the commensurate R theorem that was used to calculate the phase transition curves for the van der Waals and the Lennard-Jones models [8,9,10]. Fig. 3(c) looks very similar to Figure 1 for hydrogen in [8], and it is remarkable that we get such close equality for properties in fluids with such different densities. ...

... But this later method is usually problematic since the correlation length is not traditionally accessible in thermodynamics. A contribution of the geometry of thermodynamics is that it offers a thermodynamic link to the correlation length via the equality Eq. (20), and this is useful in calculating the Widom line [8,10,84,85,86]. ...

Liquid water has anomalous liquid properties, such as its density maximum at 4\degree C. An attempt at theoretical explanation proposes a liquid-liquid phase transition line in the supercooled liquid state, with coexisting low-density (LDL) and high-density (HDL) liquid states. This line terminates at a critical point. It is assumed that the LDL state possesses mesoscopic tetrahedral structures that give it solid-like properties, while the HDL is a regular random liquid. But the short-lived nature of these solid-like structures make them difficult to detect directly. We take a thermodynamic approach instead, and calculate the thermodynamic Ricci curvature scalar $R$ in the metastable liquid regime. It is believed that solid-like structures signal their presence thermodynamically by a positive sign for $R$, with a negative sign typically present in less organized fluid states. Using thermodynamic data from ST2 computer simulations fit to a mean field (MF) two state equation of state, we find significant regimes of positive $R$ in the LDL state, supporting the proposal of solid-like structures in liquid water. In addition, we review the theory, compute critical exponents, demonstrate the large reach of the MF critical regime, and calculate the Widom line using $R$.

... TG has been tested in many different systems: in phase coexistence for Helium, Hydrogen, Neon and Argon [8], for the Lennard-Jones fluids [9,10], for ferromagnetic systems and liquid-liquid phase transitions [11]; in the liquid-gas like first order phase transition in dyonic charged AdS black hole [12]; in the Hawking-Page transitions in Gauss-Bonnet-AdS black holes [13]. ...

... It is based on the continuity of the scalar curvature: knowing the thermodynamic quantities in the two phases, i.e. R, one can build up the transition curve by imposing the continuity of R. The RCM, coherent with Widom's microscopic description of the liquidgas coexistence region (i.e. with the idea that the correlation lengths of the two phases must be the same at the transition) has been tested in systems with different features: vapor-liquid coexistence line for the Lennard-Jones fluids [9,10], first and second order phase transitions of mean-field Curie-Weiss model (ferromagnetic systems), liquid-liquid phase transitions [11], phase transitions of cosmological interest as the liquid-gaslike first order phase transition in dyonic charged AdS black hole [12]. Another criterion, applied in the study of first order phase transitions in real fluids [18] and Lennard-Jones systems [10] is a first kind discontinuity in R. ...

... Therefore a change in sign of R is an indication of the balance between effective interactions, even when no transition occurs, and theoretical curves with R = 0 in pure fluids identify some anomalous behaviors observed in the experimental data of several substances (in particular, water) [18,29]. A transition from R > 0 to R < 0 has been also shown for the Lennard-Jones system [9,10] and Anyon gas [30,31]. For black holes [32], the the change in sign of the curvature occurs at the Hawking-Page transition temperature, therefore associated with the condition R = 0. ...

The formalism of Riemannian geometry is applied to study the phase transitions in Nambu -Jona Lasinio (NJL) model. Thermodynamic geometry reliably describes the phase diagram, both in the chiral limit and for finite quark masses. The comparison between the geometrical study of NJL model and of (2+1) Quantum Chromodynamics at high temperature and small baryon density shows a clear connection between chiral symmetry restoration/breaking and deconfinement/confinement regimes.

... Moreover the R-crossing method has been tested in systems with different features: in [16,21] it has been applied to construct the vaporliquid coexistence line for the Lennard-Jones fluids, finding striking agreement with other methods; in [22] the authors studied the geometry of the thermodynamics of first and second order phase transitions of mean-field Curie-Weiss model (ferromagnetic systems) and also of liquidliquid phase transitions. Another field of application of the R-crossing method is the study of phase transitions of cosmological interest: in [23] the authors studied the liquid-gas like fist order phase transition in dyonic charged AdS black hole and in [24] the Hawking-Page transitions in Gauss-Bonnet-AdS black holes. ...

... Moreover, the thermodynamic scalar curvature for the Lennard-Jones system exhibits a transition from R > 0 to R < 0 when the attraction in the intermolecular potential dominates [16,21]. A similar behavior has been found for quantum gases, but with a different meaning: R is positive for fermi statistical interactions and it is negative in the bosonic case [25]. ...

... Figure 2 shows the scalar curvature R evaluated by Eq. (14) and Eqs. (17)(18)(19)(20)(21)(22)(23). The black curves are based on lattice data with the condition n S = n Q = 0 (or equivalent for the isospin symmetric limit), whereas the red ones are for n S = 0 and n Q /n B = 0.4. ...

The thermodynamic geometry formalism is applied to strongly interacting matter to estimate the deconfinement temperature. The curved thermodynamic metric for Quantum Chromodynamics (QCD) is evaluated on the basis of lattice data, whereas the hadron resonance gas model is used for the hadronic sector. Since the deconfinement transition is a crossover, the geometric criterion used to define the \mbox{(pseudo-)critical} temperature, as a function of the baryonchemical potential $\mu_B$, is $R(T,\mu_B)=0$, where $R$ is the scalar curvature. The (pseudo-)critical temperature, $T_c$, resulting from QCD thermodynamic geometry is in good agreement with lattice and phenomenological freeze-out temperature estimates. The crossing temperature, $T_h$, evaluated by the hadron resonance gas, which suffers of some model dependence, is larger than $T_c$ (about $20\%$) signaling remnants of confinement above the transition.

... In particular, near a first order phase transition one expects that the values of scalar curvature evaluated in the two different phases coincide. This approach, called the R-Crossing Method (RCM), permits to determine the phase coexistence line for Helium, Hydrogen, Neon and Argon [5], for the Lennard-Jones fluids [6,7], for ferromagnetic systems and liquid-liquid phase transitions [8]. More recently, another field of application of the RCM is the study of the phase transitions of cosmological interest: the liquid-gas like fist order phase transition in dyonic charged AdS black hole [9] and in the Hawking-Page transitions in Gauss-Bonnet-AdS black holes [10]. ...

... Within the thermodynamic geometry approach, the physical meaning of the sign of R is still under debate. However, the picture is clear enough in the case of pure fluids [6,7,23] and of quantum gases, where R turns out to be positive for repulsive fermi interactions and negative in the bosonic case [11]. Particularly, in [13], the authors study a two dimensional ideal anyon gas of particles obeying fractional statistics finding that the sign of the scalar curvature of this system is a function of the parameter α that specify the particle content (α = 0 corresponds to bosons, α = 1 to fermions, and 0 < α < 1 to intermediate statistics). ...

The application of Riemannian geometry to the analysis of the equilibrium thermodynamics in Quantum Chromodynamics (QCD) at finite temperature and baryon density gives a new method to evaluate the transition temperature from the quark-gluon phase to the hadron phase. The results are in good agreement with the freeze-out temperature evaluated by the statistical hadronization model and with the estimates based on QCD lattice simulations.

... In order to further validate our results, we apply an equation of state to the real LJ systems we study. We have chosen the wellknown modified mBWR equation of state, which is the most widely used EoS for LJ fluids in the literature [57]. This equation was originally developed for pure LJ systems, but it can also be extended to LJ mixtures [56]. ...

... x is the reduced number density,T ¼ kBT εx is the reduced temperature, c 1 , c 2 , …, c 32 are the 32 linear parameters and g is the nonlinear parameter of the equation. All the values of the 33 parameters were taken from the work of May et al. [57]. In Fig. 9, the predictions of the mBWR equation of state for the density dependence to the mole fractions of the real systems are depicted with a red solid line. ...

The ability to extract thermodynamic properties of mixtures from molecular geometry and interactions is one of the major advantages of atomistic simulations, but, at the same time, can be a great challenge, especially for statistical properties such as the Gibbs energy of mixing (Δmix G). This challenge becomes even greater in the case of mixtures of complicated molecules or macromolecules. Kirkwood-Buff theory offers a promising avenue for estimating Δmix G from atomistic simulation of binary mixtures. In this work we perform molecular dynamics simulations of both ideal and real binary Lennard-Jones mixtures at various mole fractions. We estimate the Kirkwood-Buff integrals by two different methods and identify the most efficient one. Then we validate our methodology by comparing several thermodynamic properties of the ideal mixtures against the theoretical expressions of thermodynamics. Finally, we calculate the mixing thermodynamic properties for the real mixtures, namely the enthalpy, Gibbs energy, volume, and entropy of mixing, as well as their excess parts relative to an ideal solution. We compare our results against the predictions of the well-known modified Benedict-Webb-Rubin equation of state for Lennard-Jones systems and find good agreement.

... Recently, also experimental evidence has been found by Liu et al. [97]. Stanley [100]), the concept is interpreted differently by different researchers and furthermore applied to the gas-liquid critical point as well. Generally, all response functions 6 have been considered by Liu et al. [97] and Xu et al. [172]. ...

... The maximum density gradient (∂ρ/∂T) p was considered by Okamoto et al. [120]. Only Brazhkin et al. [26] found that Widom line and specific heat maxima should exactly coincide with the isochore for all fluids, a result different from the other authors, further discussed by May and Mausbach [100]. The latter also criticized that there exists no formal way of determining the Widom line as maximum of correlation length from response functions. ...

Although liquid propellant rocket engines are operational and have been studied for decades, cryogenic injection at supercritical pressures is still considered essentially not understood. This thesis intends to approach this problem in three steps: by developing a numerical model for real gas thermodynamics, by extending the present thermodynamic view of supercritical injection, and finally by applying these methods to the analysis of injection.
A new numerical real gas thermodynamics model is developed as an extension of the DLR TAU code. Its main differences to state-of-the-art methods are the use of a precomputed library for fluid properties and an innovative multi-fluid-mixing approach. This results in a number of advantages: There is effectively no runtime penalty of using a real gas model compared to perfect gas formulations, even for high fidelity equations of state (EOS) with associated high computational cost. A dedicated EOS may be used for each species. The model covers all fluid states of the real gas component, including liquid, gaseous, and supercritical states, as well as liquid-vapor mixtures. Numerical behavior is not affected by local fluid properties, such as diverging heat capacities at the critical point. The new method implicitly contains a vaporization and condensation model. In this thesis, oxygen is modeled using a modified Benedict-Webb-Rubin equation of state, all other involved species are treated as perfect gases.
A quantitative analysis of the supercritical pseudo-boiling phenomenon is given. The transition between supercritical liquid-like and gas-like states resembles subcritical vaporization and is thus called pseudo-boiling in the literature. In this work it is shown that pseudo-boiling differs from its subcritical counterpart in that heating occurs simultaneously to overcoming molecular attraction. In this process, the dividing line between liquid-like and gas-like, the so called Widom line, is crossed. This demarcation is characterized by the set of states with maximum specific heat capacity. An equation is introduced for this line which is more accurate than previous equations. By analyzing the Clausius-Clapeyron equation towards the critical limit, an expression is derived for its sole parameter. A new nondimensional parameter evaluates the ratio of overcoming molecular attraction to heating: It diverges towards the critical point but shows a significant pseudo-boiling effect for up to reduced pressures of 2.5 for various fluids.
It appears reasonable to interpret the Widom-line, which divides liquid-like from gas-like supercritical states, as a definition of the boundary of a dense supercritical fluid. This may be used to uniquely determine the radius of a droplet or the dense core length of a jet. Then, a quantitative thermodynamic analysis is possible. Furthermore, as the pseudo-boiling process may occur during moderate heat addition, this allows for a previously undescribed thermal jet disintegration mechanism which may take place within the injector.
This thermal jet break-up hypothesis is then applied to an analysis of Mayers and Branams nitrogen injection experiments. Instead of the constant density cores as predicted by theory, the majority of their cases show an immediate drop in density upon entering the chamber. Here, three different axial density modes are identified. The analysis showed that heat transfer did in fact take place in the injector. The two cases exhibiting a dense core are the cases which require the largest amount of power to reach the pseudo-boiling temperature. After this promising application of pseudo-boiling analysis, thermal break-up is tested numerically. By accounting for heat transfer inside the injector, a non dense-core injection can indeed be simulated for the first time with CFD.
Finally, the CFD model is applied to the A60 Mascotte test case, a reactive GH2/LOX single injector operating at supercritical pressure. The results are compared with experimental and other researchers numerical data. The flame shape lies well within the margins of other CFD results. Maximum OH* concentration is found in the shear layer close to the oxygen core and not in the shoulder, in agreement with experimental data. The axial temperature distribution is matched very well, particularly concerning position and value of the maximum temperature.

... Let me present results from fluid studies based on experimental fluid data [30,31], and on computer simulations in fluids and solids on particles interacting via an actual Lennard-Jones potential [32,33]. In each case R was determined by Eq. (6), differentiating f (T, ρ) obtained from fits to numerical experimental or computer data. ...

... The size of this cluster is given by the correlation length ξ, with |R| ∼ ξ 3 . Another critical point theme is shown in Fig. 4f, where we have equal values of R for the coexisting liquid and vapor phases, R l = R v , as the two phases have identical organized droplet sizes [30,31,32]. Fig. 4g shows a somewhat subtle vapor phase theme [30]. ...

I give a relatively broad survey of thermodynamic curvature $R$, one spanning
results in fluids and solids, spin systems, and black hole thermodynamics. $R$
results from the thermodynamic information metric giving thermodynamic
fluctuations. $R$ has a unique status in thermodynamics as being a geometric
invariant, the same for any given thermodynamic state. In fluid and solid
systems, the sign of $R$ indicates the character of microscopic interactions,
repulsive or attractive. $|R|$ gives the average size of organized mesoscopic
fluctuating structures. The broad generality of thermodynamic principles might
lead one to believe the same for black hole thermodynamics. This paper explores
this issue with a systematic tabulation of results in a number of cases.

... This is not unusual; for Lennard-Jones fluids these maxima also became negligible -depending on the actual quantity -above 1.5−2.5 T c and 2−10 p c . (Brazhkin et al, 2011, May andMausbach, 2012). For other quantities (like the speed of sound (c s )) a minimum is observed, which can be also easily tracked. ...

... It means that the effect is important only in the vicinity of the critical point. The other conclusion is that the anomalies exist not only in "fixed pressure -changing temperature" scenario, but also in "fixed temperature -changing pressure" one, although the location of the anomaly might differ (see for example May and Mausbach, 2012). This is important for studies dealing with fast pressure transients (like some fast LOCA). ...

Vapour pressure curves and stability lines can be extended beyond the critical points into the supercritical domain by so-called Widom lines, along which some thermodynamic property undergoes a rapid change
and liquid-like behaviour turns to vapour-like one. Knowledge about such lines is therefore important
for thermohydraulic calculations and design. There are several properties that can be chosen as defining property of a Widom line. In this short note we calculate and compare several kinds of Widom lines for water.

... This way, critical phenomena are related to distinctive signs of the scalar curvature, R TG , obtained from such metric: R TG = 0 means a system made of noninteracting components, while for R TG < 0 such components attract each other, and for R TG > 0 repel each other. Moreover, R TG diverges in a second order phase transition as the correlation volume, while it appears to have a local maximum at a crossover, as happens in quantum chromodynamics [23][24][25][26][27]. TG has been tested in many different systems: phase coexistence for helium, hydrogen, neon and argon [28], for the Lennard Jones fluids [29,30], for ferromagnetic systems and liquid liquid phase transitions [31]; in the liquid gas like first order phase transition in dyonic charged AdS BH [32]; in quantum chromodynamics (QCD) to describe crossover from Hadron gas and Quark Gluon Plasma [23][24][25][26][27]; in the Hawking Page transitions in Gauss Bonnet AdS [33], Reissner Nordstrom AdS and the Kerr AdS [34]. A list of results have been obtained by applying TG to BHs [35][36][37][38][39][40][41][42][43][44][45]. ...

We study the thermodynamics of spherically symmetric, neutral and non-rotating black holes in conformal (Weyl) gravity. To this end, we apply different methods: (i) the evaluation of the specific heat; (ii) the study of the entropy concavity; (iii) the geometrical approach to thermodynamics known as thermodynamic geometry ; (iv) the Poincaré method that relates equilibrium and out-of-equilibrium thermodynamics. We show that the thermodynamic geometry approach can be applied to conformal gravity too, because all the key thermodynamic variables are insensitive to Weyl scaling. The first two methods, (i) and (ii), indicate that the entropy of a de Sitter black hole is always in the interval $$2/3\le S\le 1$$ 2 / 3 ≤ S ≤ 1 , whereas thermodynamic geometry suggests that, at $$S=1$$ S = 1 , there is a second order phase transition to an Anti de Sitter black hole. On the other hand, we obtain from the Poincaré method (iv) that black holes whose entropy is $$S < 4/3$$ S < 4 / 3 are stable or in a saddle-point, whereas when $$S>4/3$$ S > 4 / 3 they are always unstable, hence there is no definite answer on whether such transition occurs. Since thermodynamics geometry takes the view that the entropy is an extensive quantity, while the Poincaré method does not require extensiveness, it is valuable to present here the analysis based on both approaches, and so we do.

... In addition, the Riemannian geometry has become increasingly important in the field of fluid thermodynamics. Evaluated fluid models include van der Waals and Lennard-Jones (LJ) [19][20][21][22] . In addition, studies were performed for a large number of real fluids [23][24][25] . ...

The three-dimensional Gaussian core model (GCM) for soft-matter systems has repulsive interparticle interaction potential ϕ(r)=εexp[−(r/σ)2], with r the distance between a pair of atoms, and the positive constants ε and σ setting the energy and length scales, respectively. ϕ(r) is mostly soft in character, without the typical hard core present in fluid models. We work out the thermodynamic Ricci curvature scalar R for the GCM, with particular attention to the sign of R, which, based on previous results, is expected to be positive/negative for microscopic interactions repulsive/attractive. Over most of the thermodynamic phase space, R is found to be positive, with values of the order of σ3. However, for low densities and temperatures, the GCM potential takes on the character of a hard-sphere repulsive system, and R is found to have an anomalous negative sign. Such a sign was also found earlier in inverse power law potentials in the hard-sphere limit, and seems to be a persistent feature of hard-sphere models.

... This way, critical phenomena are related to distinctive signs of the scalar curvature, R TG , obtained from such metric: R TG = 0 means a system made of noninteracting components, while for R TG < 0 such components attract each other, and for R TG > 0 repel each other. Moreover, R TG diverges in a second order phase transition as the correlation volume, while it appears to have a local maximum at a crossover, as happens in quantum chromodynamics [23][24][25][26][27]. TG has been tested in many different systems: phase coexistence for helium, hydrogen, neon and argon [28], for the Lennard Jones fluids [29,30], for ferromagnetic systems and liquid liquid phase transitions [31]; in the liquid gas like first order phase transition in dyonic charged AdS BH [32]; in quantum chromodynamics (QCD) to describe crossover from Hadron gas and Quark Gluon Plasma [23][24][25][26][27]; in the Hawking Page transitions in Gauss Bonnet AdS [33], Reissner Nordstrom AdS and the Kerr AdS [34]. A list of results have been obtained by applying TG to BHs [35][36][37][38][39][40][41][42][43][44][45]. ...

We study the thermodynamics of spherically symmetric, neutral and non-rotating black holes in conformal (Weyl) gravity. To this end, we apply different methods: (i) the evaluation of the specific heat; (ii) the study of the entropy concavity; (iii) the geometrical approach to thermodynamics known as \textit{thermodynamic geometry}; (iv) the Poincar\'{e} method that relates equilibrium and out-of-equilibrium thermodynamics. We show that the thermodynamic geometry approach can be applied to conformal gravity too, because all the key thermodynamic variables are insensitive to Weyl scaling. The first two methods, (i) and (ii), indicate that the entropy of a de Sitter black hole is always in the interval $2/3\leq S\leq 1$, whereas thermodynamic geometry suggests that, at $S=1$, there is a second order phase transition to an Anti de Sitter black hole. On the other hand, we obtain from the Poincar\'{e} method (iv) that black holes whose entropy is $S < 4/3$ are stable or in a saddle-point, whereas when $S>4/3$ they are always unstable, hence there is no definite answer on whether such transition occurs.

... Exploration of states is then usually done for ρ * and T * . A number of EOS are available for the LJ fluid, [11][12][13][14][15][16][17] which all have their merits. The reasons to formulate another, new EOS are as follows: ...

A large number (>30 000) of Monte Carlo simulations in range of 0.002–1.41 ρ* and T* ≤ 25 (* for reduced, dimensionless) was performed, producing a dense grid of state points for the internal energy U* and pressure p*. The dense grid in ρ* allows the direct integration to obtain the Helmholtz free energy F*. The results in U*, p*, and F* were used to fit an equations of state (EOS) for the Lennard-Jones fluid using the virial thermal coefficients B2–B6 taken from the literature and additional empirical coefficients (C7-C16), which correct the errors due to nonconverging behavior of virial thermal coefficients. Those additional coefficients have the same mathematical form as the virial thermal coefficients. The EOS allows an extrapolation to extreme conditions above T* > 100 and ρ* > 2.

... The hypothesis behind the application of the R-crossing method was first established in Ref. [7] and regarding fluid systems, it has been applied for instance for the Lennard-Jones fluid in Refs. [8,9] where it was tested the dependence of the curvature scalar R on the intermolecular strength by using the Weeks-Chandler-Anderson (WCA) ansatz. The method has also been applied for magnetic systems and an idealized liquid system in Ref. [10]. ...

The R-crossing method of thermodynamic geometry is applied to reproduce the coexistence curves of fluid systems described by hard-core Yukawa and hard-core Mye-Type interactions whose range can be var- ied. Connection between the range of the potential and the validity of the method is studied. Even when scaling relations suggest a dependence on the range, we found explicitly the quantitative dependence for two varying range potentials. Using the saturation pressures, it is possible to assign a percentage P(lambda) of measuring how far we can go below the critical point when reproducing the coexistence curve. It is found that there is a close relation between the range of the potential and P(lambda). Such relation assures that the larger the range of the potential, the deeper we can go below the critical point. Our results, together with the known low |R| limit, represent two independent criteria to restrict the applicability of the R-crossing method. Namely, at what extent we can accurately reproduce the coexistence curve when we know the range of the intermolecular potential of the thermodynamic system involved.

... There are a number of other empirical equations of state for the LJ fluid of varying quality and accuracy. [109][110][111][112][113] Thol et al. also developed an empirical multi-parameter equation of state for the truncated and shifted potential (LJTS, with r * cut = 2.5). 114 The LJT+LRC and LJTS potentials have qualitatively similar behavior, and their important temperatures and densities are summarized in Table 1. ...

Rosenfeld proposed two different scaling approaches to model the transport properties of fluids, separated by twenty-two years, one valid in the dilute gas, and another in the liquid phase. In this work, we demonstrate that these two limiting cases can be connected through the use of a novel approach to scaling transport properties and an empirical bridging function. This approach, which is empirical and not derived from theory, is used to generate reference correlations for the transport properties of the Lennard-Jones 12-6 potentials of viscosity, thermal conductivity, and self-diffusion. This approach, with a very simple functional form, allows for the reproduction of the most accurate simulation data to within nearly their statistical uncertainty. The correlations are used to confirm that for the Lennard-Jones fluid the appropriately scaled transport properties are nearly monovariate functions of the excess entropy from low-density gases into the supercooled phase and up to extreme temperatures. This study represents the most comprehensive meta-study of the transport properties of the Lennard-Jones fluid to date.

... The sign of R is positive/negative for microscopic interactions repulsive/attractive. The situation is particularly clear in the case of pure fluids [14,15,16]. ...

Black holes pose great difficulties for theory since gravity and quantum theory must be combined in some as yet unknown way. An additional difficulty is that detailed black hole observational data to guide theorists is lacking. In this paper, I sidestep the difficulties of combining gravity and quantum theory by employing black hole thermodynamics augmented by ideas from the information geometry of thermodynamics. I propose a purely thermodynamic agenda for choosing correct candidate black hole thermodynamic scaled equations of state, parameterized by two exponents. These two adjustable exponents may be set to accommodate additional black hole information, either from astrophysical observations or from some microscopic theory, such as string theory. My approach assumes implicitly that the as yet unknown microscopic black hole constituents have strong effective interactions between them, of a type found in critical phenomena. In this picture, the details of the microscopic interaction forces are not important, and the essential macroscopic picture emerges from general assumptions about the number of independent thermodynamic variables, types of critical points, boundary conditions, and analyticity. I use the simple Kerr and Reissner-Nordstrom black holes for guidance, and find candidate equations of state that embody a number of the features of these purely gravitational models. My approach may offer a productive new way to select black hole thermodynamic equations of state representing both gravitational and quantum properties.

... Again, with the van der Waals EoS the locus from Eq. (5) admits a closed-form representation; in other words, the relative maxima in the c p are located at the line P r = 4T r − 3, (and T r ≥ 1) (6) in the (P r , T r ) plane, or, equivalently, at v r = 1 (and T r ≥ 1) in the (T r , v r ) plane. Although this result has been known for a long time (see, e.g., Brazhkin To date, although the Widom line has been studied in simple model systems, such as, e.g., a van der Waals or a Lennard-Jones fluid [8,9] ...

Using cubic equations of state for a single-component fluid, we compute pseudocritical loci where the isobaric heat capacity is a relative maximum at constant pressure, or at constant temperature. These two loci, called the Widom line and the characteristic isobaric inflection curve (CIIC), are quite different from each other, as we show using the van der Waals equation, based on which the two loci admit a closed-form representation in the (P, T) plane. Similarly, the Redlich–Kwong (RK) equation leads to a closed-form representation for the CIIC in the (T,v) plane. With the Soave–Redlich–Kwong (SRK) and the Peng–Robinson (PR) equations we find almost coincident predictions for the above-mentioned pseudocritical loci; furthermore, comparing our results with a correlation obtained by regression of experimental data for CO2 and water shows that the increased complexity of the SRK and PR equations (as compared to RK) allows improved agreement with the experimental data.

... Several concepts are discussed in the literature concerning the demarcation between liquid-like and gas-like supercritical states 10,16,17 . Sciortino et al. 18 introduced the definition of a Widom line as the set of states with a maximum correlation length of the fluid 10,19 , but is often approximated as locus of maximum thermodynamic response functions, as they can be more readily evaluated from thermodynamic quantities. In pure fluids, the maxima of different response functions can be located far from each other 20,21 . ...

Recent experiments on pure fluids have identified distinct liquid-like and gas-like regimes even under supercritical conditions. The supercritical liquid-gas transition is marked by maxima in response functions that define a line emanating from the critical point, referred to as Widom line. However, the structure of analogous state transitions in mixtures of supercritical fluids has not been determined, and it is not clear whether a Widom line can be identified for binary mixtures. Here, we present first evidence for the existence of multiple Widom lines in binary mixtures from molecular dynamics simulations. By considering mixtures of noble gases, we show that, depending on the phase behavior, mixtures transition from a liquid-like to a gas-like regime via distinctly different pathways, leading to phase relationships of surprising complexity and variety. Specifically, we show that miscible binary mixtures have behavior analogous to a pure fluid and the supercritical state space is characterized by a single liquid-gas transition. In contrast, immiscible binary mixture undergo a phase separation in which the clusters transition separately at different temperatures, resulting in multiple distinct Widom lines. The presence of this unique transition behavior emphasizes the complexity of the supercritical state to be expected in high-order mixtures of practical relevance.

... For the geometrical framework described above this proposal translated to the crossing of the branches for the multiple valued thermodynamic scalar curvature as a function of its arguments. This was referred to as the R-Crossing Method and described the equality of the thermodynamic scalar curvature at a first order phase transition and the results for disparate fluid systems exhibited a remarkable correspondence with experimental data [38,[43][44][45][46]. This characterization has also been demonstrated for the first order phase transition in dyonic charged AdS black hole in a mixed canonical/grand canonical ensemble with a fixed magnetic charge and a varying electric charge of the black hole by two of the authors (PC and GS) in the collaboration [47]. ...

We investigate the thermodynamics and critical phenomena for four dimensional RN-AdS and Kerr-AdS black holes in the canonical ensemble both for the normal and the extended phase space employing the framework of thermodynamic geometry. The thermodynamic scalar curvatures for these black holes characterize the liquid-gas like first order phase transition analogous to the van der Waals fluids, through the R-Crossing Method. It is also shown that the thermodynamic scalar curvatures diverge as a function of the temperature at the critical point. *

... For the geometrical framework described above this proposal translated to the crossing of the branches for the multiple valued thermodynamic scalar curvature as a function of its arguments. This was referred to as the R-Crossing Method and described the equality of the thermodynamic scalar curvature at a first order phase transition and the results for disparate fluid systems exhibited a remarkable correspondence with experimental data [38,[43][44][45][46]. This characterization has also been demonstrated for the first order phase transition in dyonic charged AdS black hole in a mixed canonical/grand canonical ensemble with a fixed magnetic charge and a varying electric charge of the black hole by two of the authors (PC and GS) in the collaboration [47]. ...

We investigate the thermodynamics and critical phenomena for four dimensional RN-AdS and Kerr-AdS black holes in the canonical ensemble both for the normal and the extended phase space employing the framework of thermodynamic geometry. The thermodynamic scalar curvatures for these black holes characterizes the liquid-gas like first order phase transition analogous to the van der Waals fluids, through the $R$-Crossing Method. It is also shown that the thermodynamic scalar curvatures diverge as a function of the temperature at the critical point.

... The correlation procedure was improved with respect to the conventional method. Based on a purely simulated dataset, usually only pressure and energy values are correlated66 . For phosgene, also higher order Helmholtz energy derivatives according to Eq. (6) have been applied in the correlation procedure.Fig. ...

The Grüneisen parameter γ
G is widely used for studying thermal properties of solids at high pressure and also has received increasing interest in different applications of non-ideal fluid dynamics. Because there is a lack of systematic studies of the Grüneisen parameter in the entire fluid region, this study aims to fill this gap. Grüneisen parameter data from molecular modelling and simulation are reported for 28 pure fluids and are compared with results calculated from fundamental equations of state that are based on extensive experimental data sets. We show that the Grüneisen parameter follows a general density-temperature trend and characterize the fluid systems by specifying a span of minimum and maximum values of γ
G. Exceptions to this trend can be found for water.

... These curves are collectively called Widom lines, although this term was originally used for the correlation length anomaly only (Xu et al. 2005). Different properties have different Widom lines, so this group of curves forms a Widom region, as has been shown for various real and model liquids May and Mausbach 2012;Brazhkin et al. 2012Brazhkin et al. , 2014. In pressure-temperature space the Widom lines are located in a wedge-shaped region (called Widom region), pointing to the critical point, marking the region where physico-chemical quantities (e.g. ...

For supercritical fluids there is a wedge-shaped region called Widom region, where several physico-chemical quantities (e.g. compressibility, heat capacities, density, thermal expansivity, speed of sound) show anomalous behaviour. In this paper, several Widom lines of supercritical CO2 have been computed with the Wagner–Span reference equation of state. The locations of the Widom lines are compared with the P–T range of the Snøhvit, Sleipner, Nagaoka and Ketzin reservoirs, which are recently studied for their fitness for CO2 sequestration, and two natural CO2 storage analogues, Montmiral in France and Mihályi-Répcelak in Hungary. The potential consequences of leaking CO2 crossing any of the Widom lines are discussed.

... The original definition of the Widom line is the set of supercritical maximal thermodynamic correlation lengths [4]. However, this property is very hard to come by in a practical manner (May and Mausbach [21]). As a substitute, response functions -such as the heat capacity, the isothermal compressibility, and the thermal expansion -are used by several authors (e.g. ...

Recent publications in the open literature have shown that supercritical fluid states are not homogeneously distributed but, in fact, can be differentiated into two distinct regions with gas-like and liquid-like properties, respectively. These regions are divided by an extension of the coexistence line, commonly called Widom line. This paper shows that a supercritical analog to subcritical phase change, pseudoboiling, does exist when crossing this demarcation. The supercritical state transition does not occur in a phase equilibrium but takes place over a finite temperature interval. While subcritical vaporization requires energy to overcome intermolecular attraction, supercritical state transitions additionally require energy to increase the temperature. It could be shown that the attractive potential is the dominant energy sink up to a reduced pressure of 1.5 for argon, nitrogen, oxygen, and water. The effect reduces with growing pressure and becomes negligible for p/pcr > 3. Furthermore, a new equation for this Widom- or pseudoboiling line is given. It exhibits improved accuracy over previously published equations; performing a limit analysis of the Clapeyron equation allows to express its sole parameter purely in terms of thermodynamic variables. This parameter can then be evaluated from an equation of state or from fluid data - no nonphysical fitting is required.

... The Widom line serves as a line of dynamical crossover for fluid properties that seems to retain the memory of the distinct subcritical phases [33][34][35][36]. Thus our construction led to a complete unified geometrical framework for the characterization of subcritical, critical and supercritical phenomena based on the thermodynamic scalar curvature and has led to interesting further applications [37][38][39][40]. ...

We investigate phase transitions and critical phenomena of four dimensional
dyonic charged AdS black holes in the framework of thermodynamic geometry. In a
mixed canonical grand canonical ensemble with a fixed electric charge and
varying magnetic charge these black holes exhibit liquid gas like first order
phase transition culminating in a second order critical point similar to the
Van der Waals gas. We show that the thermodynamic scalar curvature R for these
black holes follow our proposed geometrical characterization of the R-crossing
Method for the first order liquid gas like phase transition and exhibits a
divergence at the second order critical point. The pattern of R crossing and
divergence exactly corresponds to those of a Van der Waals gas described by us
in an earlier work.

... 12 Originally, it has been defined as the locus of maximum correlation length. 33,50 Different researchers use different, more practical definitions; the ensemble of maximum specific isobaric heat capacities (Xu et al., 57 Santoro and Gorelli, 48 and Ruppeiner at al. 47 ) is a popular choice. Gorelli et. ...

In this paper, a new interpretation of cryogenic jet break-up in supercritical environments is introduced. It is firmly established that under these conditions a pure fluid will exhibit neither latent heat of vaporization nor surface tension. The jet undergoes a transition from a dense cryogenic fluid to an ideal gas as it mixes and blends with the surrounding warmer gas. Regarding the thermodynamic process, this transition is characterized by large changes in density and very small changes in temperature as energy is supplied. The state where density changes and the heat capacity are maximal is sometimes called 'pseudo-boiling' in the literature. However, no clear definition of this process is available, its very existence debated. In this paper, the first quantitative pseudo-boiling analysis is presented. It can be shown that pseudo-boiling exists along a line which effectively structures supercritical fluid states. An equation for this continuation of the coexistence line is given. Across this line, a continuous state transition can be identified. The temperature at pseudo-boiling replaces the critical temperature as relevant parameter at supercritical pressures. By introducing a suitable definition for a supercritical fluid boundary, super-critical jet break-up can be quantified thermodynamically. This suggests a novel, thermal, jet break-up mechanism. Experimental evidence from the literature is shown, further sup-ported by CFD simulations. The pseudo-boiling effect is found to play a role for injection conditions of reduced pressures smaller than 3, and reduced temperatures lower than 1.2.

... The relation between the scalar curvature and the correlation volume has been tested for a variety of classical models [3] and has also been applied to the theory of first order phase transitions in [5], and show excellent agreement with experimental results for liquid-gas systems. (For related recent results in liquidliquid phase transitions and for Lennard-Jones fluids, see [6], and [7]). It is by now established that for classical systems, R diverges on the spinodal curve, and its equality in co-existing phases indicates a first order phase transition. ...

We study information geometry of the Dicke model, in the thermodynamic limit.
The scalar curvature $R$ of the Riemannian metric tensor induced on the
parameter space of the model is calculated. We analyze this both with and
without the rotating wave approximation, and show that the parameter manifold
is smooth even at the phase transition, and that the scalar curvature is
continuous across the phase boundary.

Two boundary lines are frequently discussed in the literature, separating state regions dominated by repulsion or attraction. The Fisher-Widom line indicates where the longest-range decay of the total pair correlation function crosses from monotonic to exponentially damped oscillatory. In the context of thermodynamic metric geometry, such a transition exists where the Ricci curvature scalar vanishes, R=0. To establish a possible relation between these two lines, R is worked out for four simple fluids. The truncated and shifted Lennard-Jones, a colloid-like and the square-well potential are analyzed to investigate the influence of the repulsive nature on the location of the R=0 line. For the longer-ranged Lennard-Jones potential, the influence of the cutoff radius on the R=0 line is studied. The results are compared with literature data on the Fisher-Widom line. Since such data are rare for the longer-ranged Lennard-Jones potential, dedicated simulations are carried out to determine the number of zeros of the total correlation function, which may provide approximate information about the position of the Fisher-Widom line. An increase of the repulsive strength toward hard sphere interaction leads to the disappearance of the R=0 line in the fluid phase. A rising attraction range results in the nonexistence of the Fisher-Widom line, while it has little effect on the R=0 line as long as it is present in the fluid state.

The virial equation of state (VEOS) provides a rigorous bridge between molecular interactions and thermodynamic properties. The past decade has seen renewed interest in the VEOS, due to advances in theory, algorithms, computing power, and quality of molecular models. Now, with the emergence of increasingly accurate first-principles computational chemistry methods, and machine-learning techniques to generate potential-energy surfaces from them, VEOS is poised to play a larger role in modeling and computing properties. Its scope of application is limited to where the density series converges, but this still admits a useful range of conditions and applications, and there is potential to expand this range further. Recent applications have shown that for simple molecules, VEOS can provide first-principles thermodynamic property data that are competitive in quality with experiment. Moreover, VEOS provides a focused and actionable test of molecular models and first-principles calculations via comparison to experiment. This Perspective presents an overview of recent advances, and suggests areas of focus for further progress.

In this work, we utilize concepts from bifurcation theory to pinpoint hidden defects in accurate multi‐parameter simulation‐based equations of state for the Lennard‐Jones (LJ) fluid. The proposed bifurcation diagrams track the evolution of volume roots as temperatures vary at constant pressure. We critically evaluate four distinct types of LJ based equations of state: modified Benedict‐Webb‐Rubin equation (with three different parameter sets), Kolafa and Nezbeda, Mecke et al. and Thol et al. For each model, we mainly construct two bifurcation diagrams at subcritical and supercritical isobars. The unphysical behaviors associated with the studied equations involve spurious two‐phase separation regions, distorted volume‐temperature behavior, unphysical branches, unphysical turning points, and multiplicity in volume roots. Our proposed bifurcation diagram provides a reliable and simple technique to pinpoint hidden defects in equations of state based merely on temperature, volume and pressure without the need of their partial derivatives or thermodynamic potentials. This article is protected by copyright. All rights reserved.

Literature data on the thermophysical properties of the Lennard-Jones fluid, which were sampled with molecular dynamics and Monte Carlo simulations, were reviewed and assessed. The literature data were complemented by simulation data from the present work that were taken in regions in which previously only sparse data were available. Data on homogeneous state points (for given temperature T and density ρ: pressure p, thermal expansion coefficient α, isothermal compressibility β, thermal pressure coefficient γ, internal energy u, isochoric heat capacity c
v
, isobaric heat capacity c
p
, Grüneisen parameter Γ, Joule-Thomson coefficient μJT, speed of sound w, Helmholtz energy a, and chemical potential) were considered, as well as data on the vapor-liquid equilibrium (for given T: vapor pressure ps, saturated liquid and vapor densities ρ' and ρ″, respectively, enthalpy of vaporization Δhv, and as well as surface tension γ). The entire set of available data, which contains about 35 000 data points, was digitalized and included in a database, which is made available in the Supporting Information of this paper. Different consistency tests were applied to assess the accuracy and precision of the data. The data on homogeneous states were evaluated pointwise using data from their respective vicinity and equations of state. Approximately 10% of all homogeneous bulk data were discarded as outliers. The vapor-liquid equilibrium data were assessed by tests based on the compressibility factor, the Clausius-Clapeyron equation, and by an outlier test. Seven particularly reliable vapor-liquid equilibrium data sets were identified. The mutual agreement of these data sets is approximately ±1% for the vapor pressure, ±0.2% for the saturated liquid density, ±1% for the saturated vapor density, and ±0.75% for the enthalpy of vaporization-excluding the region close to the critical point.

Liquid water has anomalous liquid properties, such as its density maximum at 4 °C. An attempt at theoretical explanation proposes a liquid-liquid phase transition line in the supercooled liquid state, with coexisting low-density liquid (LDL) and high-density liquid (HDL) states. This line terminates at a critical point. It is assumed that the LDL state possesses mesoscopic tetrahedral structures that give it solidlike properties, while the HDL is a regular random liquid. But the short-lived nature of these solidlike structures makes them difficult to detect directly. We take a thermodynamic approach instead and calculate the thermodynamic Ricci curvature scalar R in the metastable liquid regime. It is believed that solidlike structures signal their presence thermodynamically by a positive sign for R, with a negative sign typically present in less organized fluid states. Using thermodynamic data from ST2 computer simulations fit to a mean field (MF) two state equation of state, we find significant regimes of positive R in the LDL state, supporting the proposal of solidlike structures in liquid water. In addition, we review the theory, compute critical exponents, demonstrate the large reach of the MF critical regime, and calculate the Widom line using R.

This paper focuses on predicting centrifugal compressor performance in the supercritical region of real gas. For this purpose, thermodynamic changes have been considered in the sub-regions of the supercritical space. It is known that some properties (e.g. compressibility or density) of supercritical fluids behave anomalously in a narrow temperature-pressure band, shaped by pseudocritical lines, which start at the critical point and extend to higher T and P values. To accurately predict the performance of supercritical carbon dioxide (sCO2) turbomachinery, the fluid behavior, in three regions (liquid-like, pseudocritical and vapour-like) created by pseudocritical lines, should be considered. For this purpose, computational fluid dynamics (CFD) is employed to calculate compressor performance in different regions of the supercritical space. The selected compressor geometry is the compressor impeller tested in the Sandia sCO2 compression loop facility. The results illustrate that operating points in the liquid-like region achieve the highest pressure rise. In addition, fluctuations in two fluid properties, density and speed of sound, have been observed wherever their pseudocritical lines have been crossed. However, the reason for these variations needs more investigation. The study considers the sudden changes occurring in the supercritical region and should lead to more accurate prediction of compressor performance,.

The representation of the Widom line as a line of maximums of the correlation length and a whole set of thermodynamic response functions above the critical point were introduced to describe anomalies observed in water above the hypothetical critical point of the liquid-liquid transition. The supercritical region for the gas-liquid transition was also described later in terms of the Widom line. It is natural to assume that an analogue of the Widom line also exists in the supercritical region for the first-order isostructural transition in crystals, which ends at a critical point. We use a simple semiphenomenological model, close in spirit the van der Waals theory, to study the properties of the new Widom line. We calculate the thermodynamic response functions above the critical point of the isostructural transition and find their maximums determining the Widom line position.

Sudden changes in the dynamical properties of a supercritical fluid model have been found as a function of pressure and temperature (T/T(c) = 2-5 and P/P(c) = 10-10(3)), striving with the notion of a single phase beyond the critical point established by thermodynamics. The sound propagation in the Terahertz frequency region reveals a sharp dynamic crossover between the gas like and the liquid like regimes along several isotherms, which involves, at sufficiently low densities, the interplay between purely acoustic waves and heat waves. Such a crossover allows one to determine a dynamic line in the phase diagram which exhibits a very tight correlation with a number of thermodynamic observables, showing that the supercritical state is remarkably more complex than thought so far.

According to textbook definitions, there exists no physical observable able to distinguish a liquid from a gas beyond the critical point, and hence only a single fluid phase is defined. There are, however, some thermophysical quantities, having maxima that define a line emanating from the critical point, named `the Widom line' in the case of the constant-pressure specific heat. We determined the velocity of nanometric acoustic waves in supercritical fluid argon at high pressures by inelastic X-ray scattering and molecular dynamics simulations. Our study reveals a sharp transition on crossing the Widom line demonstrating how the supercritical region is actually divided into two regions that, although not connected by a first-order singularity, can be identified by different dynamical regimes: gas-like and liquid-like, reminiscent of the subcritical domains. These findings will pave the way to a deeper understanding of hot dense fluids, which are of paramount importance in fundamental and applied sciences.

In the framework of the van-der-Waals model, analytical expressions for the
locus of extrema (ridges) for heat capacity, thermal expansion coefficient,
compressibility, density fluctuation, and sound velocity in the supercritical
region have been obtained. It was found that the ridges for different
thermodynamic values virtually merge into single Widom line only at $T<1.07
T_c, P<1.25P_c$ and become smeared at $T<2T_c, P<5P_c$, where $T_c$ and $P_c$
are the critical temperature and pressure. The behavior of the Batschinski
lines and the pseudo-Gruneisen parameter $\gamma$ of a van-der-Waals fluid were
analyzed. In the critical point, the van-der-Waals fluid has $\gamma=8/3$,
corresponding to a soft sphere particle system with exponent $n=14$.

Phase diagrams and critical constants for the long-range corrected, the truncated, and the truncated and shifted Lennard–Jones fluids with various values of the potential cutoff were computed from molecular simulations. Critical parameters were obtained from mixed-field finite-size scaling analysis. Multiple histogram reweighting was used to compute the phase envelop at temperatures well below criticality. For the long-range corrected fluid, the coexistence curve is systematically shifted to higher chemical potentials for a cutoff of 5.0σ compared with that for a cutoff of 2.5σ. The difference in the critical temperature for the truncated and truncated and shifted potentials decreases from 10 to 3.6% as the cutoff increases from 2.5σ to 3.5σ. The critical temperature for the long-range corrected fluid is about 1.4% larger than that for the truncated fluid with a cutoff of 5.0σ. The average absolute deviations of the coexistence densities between the truncated and long-range corrected fluid with rc=5.0σ are about 0.8 and 1% for the vapor and liquid branches, respectively. This indicates that the truncated Lennard–Jones fluid with a cutoff of 5.0σ is a reasonable quantitative approximation to the full Lennard–Jones fluid.

We review experimental results for the locus in the temperature–density plane of isothermal Cv (constant volume heat capacity) extrema of Ar, Kr and Xe, and published Percus–Yevick and simulation equations of state. It is likely that the locus of Cv maxima terminates at the critical point. We report new long (960 million Monte Carlo steps), 864 particle simulations of the heat capacity for the truncated Lennard-Jones potential (cutoff=2.5σ) near the liquid coexistence line, and establish directly that the locus of Cv minima intersects the coexistence line. On the basis of calculations and simulations for model systems, we induce that previously reported Cv extrema are caused by the interplay of three physical effects that we term, “aggregation,” “caging,” and “soft-core penetration.” We test our hypothesis by carrying out calculations for a one-dimensional, nearest-neighbor, infinite-chain, truncated interaction model with the following potentials: Lennard-Jones, Lennard-Jones with hard core, Lennard-Jones with hard core and no soft repulsion, square well, and the inverse twelfth. Using our physical understanding, we successfully explain the qualitative changes in the behavior of the Cv extrema as the interaction potential changes.

We have carried out a comprehensive study of a truncated Lennard‐Jones (TRLJ) system (potential cutoff =2.5σ) in the Percus–Yevick (PY) approximation using Baxter’s equations and algorithms with some degree of novelty. We have determined the liquid–vapor phase diagram from the ‘‘energy equation of state,’’ have determined the spinodal curve for the compressibility equation of state, and have made calculations of the energy, compressibility, and virial pressures. We have calculated heat capacities, examined their extrema for the energy and compressibility equations, and have contrasted them with previously published simulation data. Our ‘‘energy equation of state’’ is defined self‐consistently within the PY approximation and differs from the definition of other authors.

The thermodynamic excess properties for the Gaussian core model (GCM) fluid are calculated from an equation of state for the pressure and the internal energy. The equation of state is obtained from extensive Monte Carlo simulation data. Entropy–energy correlations as well as Rosenfeld's scaling laws for the temperature dependence of the excess entropy and internal energy are analysed. The predicted T−2/5 scaling of the excess entropy and T3/5 scaling of the internal energy at constant density is fairly well fulfilled for the GCM. It is shown that an excess entropy-based freezing criterion is approximately valid on the low-density side of the solid state region. Contrary to this, the freezing criterion is violated on the high-density (anomalous) side of the GCM. Finally, pressure–energy correlations are discussed by analysing the corresponding correlation and scaling coefficients. The results confirm the expectation that the GCM is not a strongly correlating liquid, and that therefore Rosenfeld's excess entropy scaling of transport coefficients fails for the GCM.

Precise descriptions of the thermodynamic properties of pure fluids require accurate vapor pressures and phase volumes as well as residual volumes, enthalpies and entropies. There is also the desirability of obtaining the density extrema in isothermal variations of the isochoric heat capacity, extrema of the isothermal compressibility and speed of sound, and densities where the reduced bulk modulus and isobaric expansivity are essentially independent of temperature (or have very weak maxima). While carefully fitted multiparameter equation of state models (EOS) show all of these qualities, cubic and other EOS based on 2- or 3-parameter corresponding states principles (CSP) usually do not. The common modifications to the attraction parameter and covolume dependence in generalized van der Waals models for improving vapor pressures and phase volumes do not improve descriptions of these extrema for the derivative properties. This paper describes characteristics of the derivative properties for methane from a highly accurate equation and compares them with results from several common EOS models. To obtain the extrema at all, the EOS covolume parameter must be at least temperature dependent, and most common models require density dependence. Accurate description is not possible with such models unless the covolume has a complex dependence on both temperature and density.

DOI:https://doi.org/10.1103/RevModPhys.68.313

We review the existing simulation data and equations of state for the Lennard-Jones (LJ) fluid, and present new simulation results for both the cut and shifted and the full LJ potential. New parameters for the modified Benedict-Webb-Rubin (MBWR) equation of state used by Nicolas, Gubbins, Streett and Tildesley are presented. In contrast to previous equations, the new equation is accurate for calculations of vapour-liquid equilibria. The equation also accurately correlates pressures and internal energies from the triple point to about 4·5 times the critical temperature over the entire fluid range. An equation of state for the cut and shifted LJ fluid is presented and compared with the simulation data of this work, and previously published Gibbs ensemble data. The MBWR equation of state can be extended to mixtures via the van der Waals one-fluid theory mixing rules. Calculations for binary fluid mixtures are found to be accurate when compared with Gibbs ensemble simulations.

Isochoric-, isobaric- and saturation-heat capacities are calculated from the modified Benedict-Webb-Rubin type equation of state (EOS) of Johnson et al. (1993) and from the van der Waals theory based EOS given by Mecke et al. (1995) for the Lennard-Jones fluid. The isochoric- and isobaric-heat capacities are also determined from NVT and NpT ensemble Monte Carlo simulations. A method is proposed for the determination of the saturation heat capacity using the appropriate fluctuation formulae in both ensembles. The EOS-based theoretical results show reasonable agreement with the simulation data. Comparing with real liquids the theoretical heat capacities based on EOSs, calculated along the vapour-liquid saturation curve, agree well with experimental data for argon and methane.

The statistical thermodynamics of a classical system composed of rigid molecules is considered in the molecular dynamics ensemble. Accepting Boltzmann’s S=kB ln W as the basic assumption of statistical mechanics, exact formalisms for two classical choices of W are derived. Since there are no restrictions on the order of thermodynamic derivatives, any measurable quantity is directly accessible in this ensemble. Explicit statistical analogs are given for the derivatives of the Helmholtz energy including an approximation for the chemical potential. Basic phase space functions are identified and their properties are explored. It is shown that the complete thermodynamics is governed by small perturbations of these functions from universal behavior.

Molecular dynamics calculations of the pressure and configurational energy of a Lennard-Jones fluid are reported for 108 state conditions in the density range 0·35 ρ* 1·20 and temperature range 0·5 T* 6 (where ρ* = ρσ3, T* = kT/ε). Particular attention is paid to the dense fluid region (ρ* 0·9), including state conditions in the subcooled liquid region. These new simulation results for P and U are combined with those of previous workers, together with low density values calculated from the virial series and values of the second virial coefficients themselves, to derive an equation of state for the Lennard-Jones fluid that is valid over a wide range of temperatures and densities. The equation of state used is a modified Benedict-Webb-Rubin equation having 33 constants. It fits the data well over the density range 0 ρ* 1·2 and for T* values ranging from 0·5 to 6·0 (the exact temperature range depending to some extent on the density considered). We also calculate for the same range of state conditions, certain two- and three-body integrals (J, K and L) that occur in perturbation theory for molecular fluids. An interpolation formula is presented for the estimation of these integrals.

Using standard Monte Carlo techniques for the canonical ensemble, we have simulated a model fluid of 864 particles interacting with a truncated Lennard-Jones pair potential over a wide range of thermodynamic conditions. In particular, we have calculated the excess heat capacity at constant volume. In agreement with previous PY calculations with the full Lennard-Jones pair potential and experimental results for inert gases, we have observed extrema with respect to density in the excess heat capacity. Our analysis of the radial distribution function suggests that these extrema are the result of competition between packing mechanisms and the increased pair distance correlation caused by the proximity of the critical point. We find that the locus of extrema of the isotherms for liquid densities terminates terminates below kT/Îµ = 0.92 and probably intersects the liquid-gas coexistence boundary near that temperature. 18 refs., 19 figs., 1 tab.

A new analytic equation of state for the Lennard-Jones fluid is proposed. The equation is based on a perturbed virial expansion with a theoretically defined temperature-dependent reference hard sphere term. The expansion is written for the Helmholtz free energy which guarantees the thermodynamic consistency of the pressure and internal energy. The equation covers much wider range of temperatures (up to seven times the critical temperature) than existing equations and is significantly more accurate and has less parameters than the best equation available to date, the modified Benedict-Webb-Rubin equation due to Johnson, Zollweg, and Gubbins (1993, Mol. Phys. 78: 591-618). As a side-product, highly accurate explicit analytic correlations of the hard sphere diameters, as given by both the hybrid Barker-Henderson and Weeks-Chandler-Andersen theories, have been obtained.Computer simulation data to be regressed by the equation have been compiled from several sources and critically assessed. It has been shown that many literature data for state points with a large compressibility are subject to large systematic finite-size errors. Additional simulations on a series of systems of different sizes have been therefore performed to facilitate the extrapolation to the thermodynamic limit in the region close to the critical point.

We investigate the phase behaviour of a single-component system in three dimensions (3D). The particles are interacting via a core-softened shoulder potential. Using standard N,V,T Monte Carlo and Gibbs-ensemble simulations, we obtain the complete gas–liquid phase behaviour, the coexistence line and the gas–liquid critical point in 3D for this potential for the first time. We develop an equation of state by means of a modified Benedict Webb Rubin (MBWR) equation. We then focus on thermodynamic anomalies of the liquid phase. We show for the range we have studied that anomalies like the maxima in the density and compressibility as a function of temperature known in the 1D and 2D case of that potential disappear in 3D, concluding that the occurrence of liquid-state anomalies in core-softened fluids strongly depends on the dimensionality of the system.

The locus of extrema (ridges) for heat capacity, thermal expansion coefficient, compressibility, and density fluctuations for model particle systems with Lennard-Jones (LJ) potential in the supercritical region have been obtained. It was found that the ridges for different thermodynamic values virtually merge into a single Widom line at T < 1.1T(c) and P < 1.5P(c) and become practically completely smeared at T < 2.5T(c) and P < 10P(c), where T(c) and P(c) are the critical temperature and pressure. The ridge for heat capacity approaches close to critical isochore, whereas the lines of extrema for other values correspond to density decrease. The lines corresponding to the supercritical maxima for argon and neon are in good agreement with the computer simulation data for LJ fluid. The behavior of the ridges for LJ fluid, in turn, is close to that for the supercritical van der Waals fluid, which is indicative of a fairly universal behavior of the Widom line for a liquid-gas transition.

The thermodynamic properties of pressure, energy, isothermal pressure coefficient, thermal expansion coefficient, isothermal and adiabatic compressibilities, isobaric and isochoric heat capacities, Joule-Thomson coefficient, and speed of sound are considered in a classical molecular dynamics ensemble. These properties were obtained using the treatment of Lustig [J. Chem. Phys. 100, 3048 (1994)] and Meier and Kabelac [J. Chem. Phys. 124, 064104 (2006)], whereby thermodynamic state variables are expressible in terms of phase-space functions determined directly from molecular dynamics simulations. The complete thermodynamic information about an equilibrium system can be obtained from this general formalism. We apply this method to the gaussian core model fluid because the complex phase behavior of this simple model provides a severe test for this treatment. Waterlike and other anomalies are observed for some of the thermodynamic properties of the gaussian core model fluid.

Thermodynamic fluctuation theory originated with Einstein who inverted the
relation $S=k_B\ln\Omega$ to express the number of states in terms of entropy:
$\Omega= \exp(S/k_B)$. The theory's Gaussian approximation is discussed in most
statistical mechanics texts. I review work showing how to go beyond the
Gaussian approximation by adding covariance, conservation, and consistency.
This generalization leads to a fundamentally new object: the thermodynamic
Riemannian curvature scalar $R$, a thermodynamic invariant. I argue that $|R|$
is related to the correlation length and suggest that the sign of $R$
corresponds to whether the interparticle interactions are effectively
attractive or repulsive.

A Riemannian geometric theory of thermodynamics based on the postulate that the curvature scalar R is proportional to the inverse free energy density is used to investigate three-dimensional fluid systems of identical classical point particles interacting with each other via a power-law potential energy gamma r(-alpha) . Such systems are useful in modeling melting transitions. The limit alpha-->infinity corresponds to the hard sphere gas. A thermodynamic limit exists only for short-range (alpha>3) and repulsive (gamma>0) interactions. The geometric theory solutions for given alpha>3 , gamma>0 , and any constant temperature T have the following properties: (1) the thermodynamics follows from a single function b (rho T(-3/alpha) ) , where rho is the density; (2) all solutions are equivalent up to a single scaling constant for rho T(-3/alpha) , related to gamma via the virial theorem; (3) at low density, solutions correspond to the ideal gas; (4) at high density there are solutions with pressure and energy depending on density as expected from solid state physics, though not with a Dulong-Petit heat capacity limit; (5) for 3<alpha<3.7913 , the solution goes from the low to the expected high density limit smoothly; (6) for alpha>3.7913 a phase transition is required to go between these regimes; (7) for any alpha>3 we may include a first-order phase transition, which is expected from computer simulations; and (8) if alpha-->infinity, the density approaches a finite value as the pressure increases to infinity, with the pressure diverging logarithmically in the density difference.

The calculation of thermodynamic state variables, particularly derivatives of the pressure with respect to density and temperature, in conventional molecular-dynamics simulations is considered in the frame of the comprehensive treatment of the molecular-dynamics ensemble by Lustig [J. Chem. Phys. 100, 3048 (1994)]. This paper improves the work of Lustig in two aspects. In the first place, a general expression for the basic phase-space functions in the molecular-dynamics ensemble is derived, which takes into account that a mechanical quantity G is, in addition to the number of particles, the volume, the energy, and the total momentum of the system, a constant of motion. G is related to the initial position of the center of mass of the system. Secondly, the correct general expression for volume derivatives of the potential energy is derived. This latter result solves a problem reported by Lustig [J. Chem. Phys. 109, 8816 (1998)] and Meier [Computer Simulation and Interpretation of the Transport Coefficients of the Lennard-Jones Model Fluid (Shaker, Aachen, 2002)] and enables the correct calculation of the isentropic and isothermal compressibilities, the speed of sound, and, in principle, all higher pressure derivatives. The derived equations are verified by calculations of several state variables and pressure derivatives up to second order by molecular-dynamics simulations with 256 particles at two state points of the Lennard-Jones fluid in the gas and liquid regions. It is also found that it is impossible for systems of this size to calculate third- and higher-order pressure derivatives due to the limited accuracy of the algorithm employed to integrate the equations of motion.

On the basis of NpT Monte Carlo simulations, a detailed analysis on the microscopic origins of some specific features of thermodynamic response functions of fluids is performed. Specifically, the residual isobaric heat capacity C(p) (res), the isobaric thermal expansivity alpha(p), and the isothermal compressibility kappa(T) for Lennard-Jones methane and optimized potential for liquid simulations (OPLS) methanol have been determined via standard techniques. For the former, data along the liquid, gas, and supercritical regions are presented, while a wide temperature range at a single supercritical pressure is covered for the latter. They have been obtained by computing the various pairwise fluctuations contributing to each property. Attention is mainly focused on isothermal and isobaric maxima found for both C(p) (res) and alpha(p), which have been rationalized at a molecular level using qualitative arguments. It is encountered that maxima emerge as a natural consequence of the destruction of fluid structure as temperature is increased or as pressure is decreased. The results for Lennard-Jones methane reveal the competition of energetic and volumetric effects, while those for OPLS methanol evidence that hydrogen-bonding is dominant as energetic effects are concerned. Further discussion on previous results and alternative approaches using equations of state as well as on closely related topics such as "maxima and critical phenomena" is included.

- G Ruppeiner
- A Sahay
- T Sarkar
- G Sengupta

G. Ruppeiner, A. Sahay, T. Sarkar, and G. Sengupta, e-print
arXiv:1106.2270.