The Maxwell crossover and the van der Waals equation of state

The well-known Maxwell construction[1] (the equal-area rule, EAR) was devised for vapor liquid equilibrium (VLE) calculation with the van der Waals (vdW) equation of state (EoS)[2]. The EAR generates an intermediate volume between the saturated liquid and vapor volumes. The trajectory of the intermediate volume over the coexistence region is defined here as the Maxwell crossover, denoted as the M-line, which is independent of EoS. For the vdW or any cubic[3] EoS, the intermediate volume corresponds to the unphysical root, while other two corresponding to the saturated volumes of vapor and liquid phases, respectively. Due to its unphysical nature, the intermediate volume has always been discarded. Here we show that the M-line, which turns out to be strictly related to the diameter[4] of the coexistence curve, holds the key to solving several major issues. Traditionally the coexistence curve with two branches is considered as the extension of the Widom line[5,6-9]. This assertion causes an inconsistency in three planes of temperature, pressure and volume. It is found that the M-line is the natural extension of the Widom line into the vapor-liquid coexistence region. As a result, the united single line coherently divides the entire phase space, including the coexistence and supercritical fluid regions, into gas-like and liquid-like regimes in all the planes. Moreover, along the M-line the vdW EoS finds a new perspective to access the second-order transition in a way better aligning with observations and modern theory[10]. Lastly, by using the feature of the M-line, we are able to derive a highly accurate and analytical proximate solution to the VLE problem with the vdW EoS.

In the field of supercritical fluid study, one of the most important achievements is the finding of the dynamically heterogeneous structures, namely liquidlike and gas-like regimes, demarcated by the Widom line [5][6][7][8][9] . The definition of the Widom line is the locus of the maximum of isobaric heat capacity, ( ⁄ ) = 0, or equivalently, the locus of maximum correlation length 8 in the supercritical fluid region. Up to now, the Widom line has been considered as the extension of the coexistence curve. However, there exists a deep-level inconsistency here. In the pressure-temperature (~) plane the Widom line is a C 1 continuation (equalities of the zeroth and first derivatives) of the equilibrium pressure at the critical point, hence divides the entire phase space into two regions. In contrast, in the pressure-volume (~) and temperature-volume (~) planes, the coexistence curve splits into the liquid and vapor branches and the continuity of density (volume) is at the C 0 level. The inconsistency in three planes raises a question on the assertion that the Widom line is the extension of the coexistence curve (or other way around).
On the other hand, for over one and half centuries, thermodynamics theory and experiments have recognized the existences of superheated liquid and supercooled vapor phases 1,2,11,12 . Supercooled vapor and superheated liquid have been predicted by an equation of state (EoS), such as the van der Waals EoS as subjected to the Maxwell construction. Experimental measurements confirmed the existences of the supercooled vapor and superheated liquid states for various substances 12 . Therefore, both theory and experimental measurements suggest that the vaporliquid coexistence phase can also be divided into liquidlike (rich) and vapor-like (rich) regions. The problem is how to determine the demarcation.
One of the most important applications of EoS is the vapor-liquid equilibrium (VLE, the first-order transition) calculation. As the vdW EoS or any cubic EoS is employed, the EAR or the Gibbs free energy will generate a transcendental equation, which makes the VLE calculation inevitably iterative. The numerical solutions cause great amount of repetitive human efforts and machine time. In addition, in some cases, derivative properties, such as ⁄ along the coexistence curve, are required. Such tasks can become difficult or tedious by numerical solutions. Apparently, analytical solutions would be very useful for both theoretical and practical applications.
Another major application of the van der Waals EoS is for accessing the second-order phase transition. By using the critical constants (pressure and temperature) a reduced form of the EoS can be obtained and this sets up the foundation for the corresponding principle theory (see Rowlison's review in Ref 1). But there is a drawback. According to the classic vdW EoS' theory (the mean field theory 10,13 ), the second-order transition is continuous at the molecular level since liquid and vapor phases (molecules) become indistinguishable at the critical point. However, modern fluctuation theory asserts that the critical behavior is governed by fluctuations of extensive properties 10 . It is the fluctuations, not details of molecular interactions, that determines the critical behavior. The density fluctuation can be so high that becomes equivalent to dynamic nanoclusters of the sizes comparable to wavelength of light, which causes the critical opalescence 13 . While the modern theory can successfully explain the phenomenon, the classic van der Waals theory fails to do so. Can the vdW EoS do better in this regard?
The three seemly unrelated issues mentioned above can be brought together with the Maxwell construction 1 , the EAR, which was devised for the VLE calculation with the vdW EoS to ensure that the equilibrium properties (volumes and pressure) are obtained. At modern time, the equilibrium conditions for pressure and chemical potential (the Gibbs free energy) are mostly used for the same purpose. As shown below, these two methods are equivalent except that the EAR generates an intermediate volume. The trajectory of the intermediate volume is named here as the Maxwell crossover or M-line, which holds the key to addressing the three major issues.

The Maxwell construction and the van der Waals EoS
The Maxwell construction is depicted in Figure 1 and can be written analytically as: where the reduced volume is defined as = ⁄ , temperature, = ⁄ , pressure, = ⁄ , and the subscript "c" refers to the critical point. The intermediate volume, (Figure 1), is so determined such that area FEDF = area DCBD and we have the following "extended" pressure equilibrium condition: The trajectory of in the two phase coexistence region is the Maxwell crossover. It should be emphasized that the Maxwell construction is applicable to any EoS, not necessarily a cubic one. The modern time VLE conditions 3 are composed of Eq.(2) (without the ( ) term) and ( ) = ( ) where is the Gibbs free energy or the chemical potential. Since ( ) ≠ ( ) ( ) (see Figure 2b) while the Mline satisfies Eq. (2), the state at is not at equilibrium and thermodynamic properties expressed in terms of are not equilibrium properties.
In fact, as shown by Figure 1, the entire section EC represents the unstable region. The integration in Eq.(1) also cross the unstable region. For this reason, the Maxwell construction received some controversial critics. The criticizing can be found in some classic text books of thermodynamics 11,14 and generations of researcher have been influenced by them until recently [15][16][17] . The classic proof 18 of the rule also involves cyclic arguments across the unstable region and hence it is inconvincible. Because the Maxwell construction is the foundation of this entire work, here we provide a different proof. As criticizing the rule, Tisza suggested 11 that the integration could be carried out around the critical point and always staying in the absolute stable domain. As a matter of fact, in molecular thermodynamics a state function is usually calculated with the ideal gas as a reference. For example, the Gibbs free energy can be expressed as 19 : Where is the gas constant, are the internal energy and entropy of the reference state ( → ∞, → ∞ ), respectively, which are temperature-dependent only. The integration in Eq.(3) is carried out from current state to the ideal gas state via a reversible (equilibrium) path. Applying Eq.(3) to the equilibrium liquid and vapor states, respectively, then using the pressure equilibrium condition, Eq.(2) and the Gibbs free energy condition ( ) = ( ) , we immediately obtain Eq.(1). The reversible cycle, ( , , ) → ( ) → ( , , ) , takes advantages of a state function and completely avoids the unstable region∎.
The above arguments are only phenomenological, but physically sound. The proof also shows that the Maxwell construction is equivalent to the combination of the pressure and chemical potential equilibrium conditions. After all, the integration across the unstable region does lead to the correct result since we are dealing with the state functions. Now we are ready to move forward with the M-line. As shown in Figure 1, when the vapor phase (A) is compressed/cooled to point B and as the experimental conditions are carefully controlled 11,12 the system can overpasses B until C while keeping in vapor state. This metastable region (from B to C) is known as the supercooled vapor. The liquid system starts from point G, upon decompressing/heating, overpasses point F until E (under control) and the region from F to E is known as the superheated liquid. The system on the left side of point D is richer in liquid and the right side richer in vapor. Therefore, the Maxwell crossover divides the coexistence phase into two regions, namely liquid-like (rich) and vapor-like (rich) regions. Consequently, the M line is physically the natural continuation of the Widom line into the coexistence phase, or the other way around. Now we use the vdW EoS to materialize the Mline and explore it's relation with the Widom line.
The details on the vdW EoS and various relations are provided in the Supplementary Information (SI). From the pressure equilibrium condition, we have a quadratic relation: where the notation in the subscript " | " refers to and , respectively, corresponding to "±" on the right hand side of the equation, and For imposing the chemical potential equilibrium condition, we use an equation derived from the pressure and chemical potential equilibrium conditions that only involves the volumes 15 : By replacing from Eq.(5) in Eq. (7), we see that the exact VLE calculation with the vdW EoS is reduced to solving one-unknown ( ) transcendental equation, which turns out to be stable and can be easily solved with the Excel Solver. At the same time, we have the solution for from Eq.(5) at the given temperature. By solving the transcendental equation along the entire coexistence curve we have the M-line.
On the other hand, if is given, and can be calculated from it. By rewriting Eq.(4) with being replaced by , we obtain the following solutions: where is defined the same way as Eq.(6) by replacing with . Now we can easily derive some very useful results. With some simple algebra, from Eq.(5) and (8) we have where the reduced density, = 1 ⁄ . Meanwhile the saturated volumes have a simple relation with the equilibrium pressure: These remarkable results shows that is directly related to the equilibrium properties. For instance, the diameter of the coexistence curve is defined as 4 Using the diameter as a tool to study the coexistence curve (hence VLE) has been a long time effort 4,20 . We will see that using is a better choice than using the diameter. In general, the diameter is related to the critical exponents 20 , = 1 + | | + | | + + ⋯, where = ( − ) ⁄ , and being the critical exponents. The implication is that also holds the information for the second-order phase transition. By the way, we can obtain a mean-field order parameter 13 : With the saturated volumes, and , we can calculate all other thermodynamic properties, such as heat capacity, isothermal compressibility etc. The strict solutions from Eq.(5), (7) and (8) for , and are only numerical values. Now we propose a procedure to derive a highly accurate and analytical approximate solution to the VLE problem with the vdW EoS.

Analytically approximate solution to the VLE problem with the vdW EoS
In aligning with the early work of Gibbs 21 , Lekner 15 proposed a parametric solution to the VLE problem with the vdW EoS. The solution uses an entropy function (along the coexistence curve) as the parameter and all other thermodynamic properties are expressed in terms of this parameter. This type of solution has limited values in practical applications since the entropy function still needs to be solved. However, it does provide a tool for some theoretical analysis. For example, based on this parametric solution 15 , Johnston 22 was able to derive series expansions for various properties starting from the critical point. Here we define an entropic quantity in terms of : This quantity turns out to be a smooth function of temperature ( Figure 2a). For comparison, the same plots for liquid and vapor phases are also presented. The M-phase refers to the metastable phase represented by the M-line. The advantage of using is obvious.   . Figure 2a illustrates the smooth feature of the function, . The parameters are so determined such that the equilibrium condition " = " holds globally ( Figure 2b).
For the vdW EoS our goal is to obtain a highly-accurate analytical solution to replace the exact solution for any purposes. To this end, we divide the entire temperature range into two regions: 0 < ≤ , and < ≤ 1. The analytical function to be determined for will only apply to the "high" temperature range, < ≤ 1, to maximize the accuracy. The value of is selected together with other coefficients to best meet the equilibrium condition " = " in the entire temperature range, and the result is = 0.35. Figure  2b shows the agreements.
For the low temperature range, as shown in the SI, we can derive the following highly accurate analytical solutions: Eq. (12) and (13) allow us to calculate the volumes (hence pressure etc.) at any low temperature, < . Now for the temperature range, < ≤ 1 , we propose the following function: The reasons for using a function with 7 coefficients are: (1) we need a highly accurate equation for our purpose; (2) we have all the required data since there totally 7 equations available at the critical point and at , respectively. The functional form, Eq. (14), is a matter of choice, and one can select other type of function for the same purpose. The coefficient obtained are listed in Table 1. The details of the calculations are provided in Methods and the SI. are calculated with Eq. (8). For consistency and high accuracy, the equilibrium pressure is calculated by the following equation for the entire temperature range: which can be obtained from the reduced van der Waals EoS and the Maxwell construction, Eq.(1). It is straightforward to prove that the same result can be obtained from the pressure and chemical potential equilibrium conditions. With analytical solutions we can easily perform some calculations that may be difficult with the numerical solutions. For example, application of the Clapeyron equation 18 region, the vdW EoS has only one root. Therefore at each point when temperature and pressure are given the volume can be easily found. For further exploring the physical features of the M-line in comparison with equilibrium properties, we adopt a metric for quantitatively measuring the strength and type of interactions in a thermodynamic system, namely, the Riemann scalar curvature. An in-depth review of the applications of this quantity can be found in Ref 24 . In Ref 17 , an analytical function of the curvature for the van der Waals EoS has been derived and provided in the SI for convenient access. Figure 3 presents the phase diagrams in all three planes, which leads to our primary conclusion: the Maxwell crossover is the logical extension of the Widom line into the coexistence region. Figure 3a illustrates the (~) plane where the Widom line is defined. Figure S1 19) 23 ; (2) calculate the volume at the pressure and temperature using the vdW EoS (one root). The zone between vapor and the vapor spinodal is vapor-rich region and the zone between liquid and the liquid spinodal is liquid-rich region.  Figure  3b while the plot is against the density corresponding to each pressure at the same temperature. The zone between the liquid spinodal (the trajectory of point C in Figure 1) and liquid curves is the metastable liquid-rich region and the zone between the vapor spinodal (the trajectory of point E in Figure  1) and vapor curves is the metastable vapor (gas)-rich region.  Figure  S1 (see SI) presents the details on the extensions of the Widom line into the subcritical region. In both planes, we immediately see that the coexistence curve, composed of liquid and vapor branches, is not a smooth (C 1 ) continuation of the Widom line since ⁄ | ≠ ⁄ | and ⁄ | ≠ ⁄ | . In the slope calculations, the diameter required is from ref 22 . The superheated liquid zone and supercooled vapor zone are generated from the trajectories of point E and point C in Figure 1, respectively, by using the condition: ⁄ = 0. As mentioned, these two zones are not only found by theoretical predictions from the EoS, but also found in experiments for many substances 12 . Therefore, the division of liquid-like and vapor-like regions in the coexistence phase is physically sound. Figure 3c and Figure S1 show that in the vicinity of the critical point, the M line asymptotically approaches to the Widom line extension. This is consistent with the observations from Figure 2b and Figure S1. Figure 3c suggests that the rectilinear law of the diameter 4,20 may be related to this pseudo-stable phase near the critical point. This law breaks as the diameter (or M-line) goes deeply into the coexistence region where heterogeneous structures dominate the coexistence phase.

Results and conclusions
In summary, from Figure 3, Figure 2b and Figure S2 (SI), we see that for pressure, volume and all zeroth-order properties, such as chemical potential, enthalpy, entropy, the continuity between the M-line and the Widom line is smooth (at level). The second full derivatives, such as ⁄ and the first partial derivatives, such as the specific heat, = ( ⁄ ) , isothermal compressibility, = − ( ⁄ ) , diverge as illustrated in Figure 4.       26 that "negative heat capacity in nanoclusters is an artifact of applying equilibrium thermodynamic formalism on a small system trapped in a metastable state differing from true thermodynamic equilibrium." Defined in the heterogeneous region, the M line, not surprisingly, exhibits features of a heterogeneous nanocluster system and is physically coherent with the Widom line. A basic difference is that in the subcritical region, the system is heterogeneous and the nanocluster structure is "static" whereas at the critical point and in the supercritical region the phase is thermodynamically homogeneous and the nanocluster structure is dynamic. In the later case, thermodynamics produces positive heat capacity. The same applies to the isothermal compressibility shown in Figure S3. For the M-line, the isothermal compressibility exhibits negative values.

Discussions
The Maxwell construction (the EAR), hence the M-line, was proposed almost one and half centuries ago 1 and the Widom line was defined in 1972 5 . These two lines are now finally united. The feature of the heterogeneous nanocluster structure embedded in the coexistence phase and in the supercritical fluid is the physical background for one single line to divide the entire phase space into liquid-like and vapor/gas-like regions. There is one thing worth mentioning here. In deeply supercritical fluid region, the Widom line may cease to exist and the Frenkel takes the place 9 . But this subject is beyond the scope of current work.
By virtue of the special feature of the M-line, we are able to propose a procedure for developing analytically approximate solutions to the VLE calculations with the vdW EoS. This procedure can be adopted for other cubic EoS as well, such as the Soave-Redlish-Kwong EoS 28 , which will be reported in the succeeding paper 29 . The vdW or SRK EoS discussed here is for demonstration purpose. Other non-cubic EoS can also be adopted for the same goal. For instance, the well-known Carnahan-Starling EoS 30 for the hard sphere fluid can be adopted as the repulsive term and the final EoS will have three roots as well.
As mentioned, the classic van der Waals theory fails to explain the density-fluctuation phenomenon 10,13 with the molecular-level continuity. Now with the M-line we have a different perspective to address the secondorder transition within the framework of a cubic EoS.
Since the heterogeneous nanocluster feature is embedded in the M line the transition at the critical point can be seen as "continuous" at level of nanocluster structures. There is a basic difference between the EoS view and the modern fluctuation theory. The negative heat capacity shows that the phase is heterogeneous and the nanocluster is "static" [25][26][27] . Therefore, along the M line the second-order transition can be considered as the change of nanocluster structure from "static" to dynamic at the critical point. Consequently, the response functions, such as , change signs from "-" to "+". The simple vdW EoS subjected to the Maxwell construction seems even more powerful than we already knew. While the VLE problem (first-order phase transition) can be solved with the two physical roots, the information of the second-order transition is embedded in the third root. As presented, there may exist a narrow pseudoequilibrium subcritical region and for future work, It would be instructive to study the region by using computer simulations or experiments.

Evaluations of the coefficients of Eq.(14)
At low temperature end, , Eq. (12) and (13) can provide accurate saturated volumes and from Eq.(8) we have at . For improving the accuracy, instead of directly using the volume, we impose the following conditions from the chemical potential condition at : where the subscript "0" refers to the temperature at . Eq. (16) This will provide the ~ curve. For volume calculations, the vdW EoS is written as: Eq. (19) and (22) will provide the Widom line in the ( ~ ) and ( ~ ) planes. The derivatives of saturated volumes and pressure are provided in the SI.

Thermodynamic properties
All thermodynamic properties along the coexistence curve can be derived analytically from Eq. (14). For example, enthalpy in reduced form reads 22 : The isobaric heat capacity are given by 22 : From Eq.(23) we can calculation the latent heat: ∆ = − . Finally, the well-known Clapeyron equation reads 18,22 : An fundamental thermodynamic consistency testing is to prove that Eq. (25) and (23)

Relations related to the vdW EoS
The vdW EoS can be written in a reduced form 22 : In the reduced form, the two constants, and , appeared in the attractive and repulsive terms, respectively, are related to the critical constants, and . By applying the pressure equilibrium condition to the vdW EoS we have: From which one can obtain: By re-arranging Eq.(S3), we have Eq.(4) (in the article).
In terms of the density Eq.(S3) can be written as With Eq.(9) we have The Helmholtz free energy is given by 22 : where is a constant and is simply taken as 1 without losing generality. Iso-thermal compressibility, is given by the following 22 : Finally, the full derivative of the equilibrium pressure, Eq. (15): Where and are provided by Eq.(S39) and (S40).
For the entire temperature range, the agreement between the analytical solutions and exact solutions is excellent, as depicted partly in Figure 5. Taking the saturated pressure as an example, the worst region is around ~0.46 , and the analytical solution gives (0.46) = 0.0154511 while exact solution is (0.46) = 0.0154512. Overall, the accuracies of the analytical solutions are above 5-digits for the equilibrium pressure, which can serve any purpose. Table S1 lists calculation results for liquid and vapor volumes at few temperature points. It can be seen that except in the neighbourhood of = 0.46, the accuracy of the analytical solutions, Eq. (14) and (8), is very high. and finally: At the critical point: By the way, at → ∞, the system becomes an ideal gas: The Riemann scalar curvature for the vdW EoS The Riemann scalar curvature ( ℛ ) 17 starts with the Helmholtz free energy density function and is defined as: ( , ) = ( , )⁄ ( 16) where is the molar volume, and ( , ) is given by Eq. (S4). The value of ℛ is then calculated by: where: And finally for the vdW EoS 17 : where is the constant in the original vdw EoS and as mentioned in the main text, = 3/2.

Determination of the coefficients of Eq.(14)
Based on the parameter solution of the VLE problem with the vdW EoS 15 , Johnston 22 has derived the following serial expansion at the critical point: The derivatives of the function: At the critical point: Finally, we have the first-order derivatives:  Eq. (23). The Widom line is calculated with the van der Waals EoS where the relation between and is given by Eq. (19) (ref. 23). The full derivatives of the two lines (w.r.t. temperature and hence volume) is equal, while partial derivatives, such as heat capacity, diverge at the critical point. Figure S3. Plots of iso-thermal compressibility coefficients for difference phases, Eq.(S5). Homogeneity is reflected by the signs of , similar to the heat capacity case. The supercritical fluid behaves like a liquid due to high density.

The SRK EoS and the M-line
The Soave-Redlich-Kwong (SRK) EoS is given by 28 : Fro the Maxwell construction and Eq.(S46): On the other hand, the pressure equilibrium condition reads: and the chemical potential equilibrium condition becomes: where the compressibility coefficient is defined as = ⁄ . From solving Eq.(S48) and (S49), we get the saturated volumes for liquid and vapor phases and from Eq.9S47) we have the equilibrium pressure. It is easy to confirm that Eq.(S47) can also be obtained from Eq.(S48) and (S46). The temperature function ( ) = ( ) is defined as 28 : Due the addition of the acentric factor, , the application of the SRK EoS should be on individual substance base. All solutions are exact and obtained from Eq.(S48) and (S49). Here we use ethane as the example. We can see that from SRK EoS is dependent on temperature, which is a major difference from the vdW EoS. The heat capacity of the ideal gas at the same temperature ( ) and the critical constants for ethane are from ref (S1). The experimental data for ethane is from ref (S2). The experimental data for isobaric heat capacity is from ref S3 .   (Table S1). The experimental data are from ref s2 . The spinodal curves (dotted lines) are illustrative only.  Figure S5. Experimental data are from ref (S3). The general behavior at the critical point is the same as that observed for the van der Waals EoS, Figure 4b. Positive heat capacity shows a thermodynamicstable homogeneous system while negative heat capacity implies a heterogeneous system, or a "static" nanocluster system.