0372-400Х (Edition in Ukrainian)
2071-0186 (Edition in English)
2071-0194 (in electronic form)
Abbreviated key title: Ukr. J. Phys.
Current reviews Ukr. J. Phys. Reviews 2015, Vol. 10, N 1, p.33-97
Institute for Condensed Matter Physics, Nat. Acad. of Sci. of Ukraine
(1, Svientsitskoho Str., Lviv 79011, Ukraine; e-mail: firstname.lastname@example.org)
Phase Transitions in the Vicinity of the Vapor–Liquid Critical Point
Original Author's Text: Ukrainian
Abstract: A method of integration of the grand partition function for simple systems of interacting particles is elaborated, by considering the theory of vapor-liquid critical point as an example.
An interparticle interaction potential consists of two terms: i) a short-range potential, which is modeled by the interaction in a system of hard spheres and characterizes the impenetrability of particles and ii) a potential of attraction. The attractive potential can be described by various functions such as the van der Waals attraction , the Morse exponents or novel specific approximations for particular physical systems. The necessary conditions, which the taken attractive potential must satisfy, are the existence of the Fourier transform and . Based on the given potential, we developed a method of evaluation of the grand partition function (GPF) in the context of the vapor-liquid critical point.
Since the hard-sphere potential and the van der Waals attraction potential have parameters, which depend oppositely on the density, the calculation of the GPF is carried out within an extended phase space. This phase space consists of the subspace of individual particle coordinates for the description of the short-range repulsion and of the subspace of collective variables for the description of the long-range attraction. The latter variables are related to the Fourier-transform of the attractive potential and are functions of the wave vector .
The overcompleteness of the phase space is removed by introducing the products of corresponding generalized functions. The obtained expression is identical to the initial form of the GPF.
As a result of integration in the phase space of particle coordinates, the GPF is reduced to a functional integral over the set . The integrand has the form of the exponential function. The exponent consists of the sum of the terms quadratic in for the van der Waals attraction and of the infinite series (of the entropy type) of products k1 ...k with closed sums over . The coefficients at the products are the semiinvariants of the reference system M2, M3, ..., M. The problem becomes incredibly complicate. But the Nature gives us a chance.
The essence of the method. We have used a certain reversibility between the spaces of the Cartesian coordinates r and the wave vectors k. Since the interaction between the hard spheres occurs at very small distances of the order of the diameter of a hard sphere, the Fourier transform of these interactions shows their properties at large and is insensitive at small . In contrast, for the description of the van der Waals attraction, the large distances and, therefore, small , are essential.
Therefore, in the functional of the GPF, the integrals over with small values of are of significance. In these regions of , the semiinvariants M2, M3, M4, ... behave monotonously and are close to their values at = 0. This issue was noticed in Ref. 39. This fact was the basis of the mathematical theory for the description of vapor-liquid systems in a vicinity of the critical point.
The first zero of the Fourier transform of the attractive potential, which occurs at = , corresponds to the middle of the shell-like curves for all semiinvariants M2, ..., M; these curves start from their values at = 0. For the semiinvariant M2(0), they are equal to the mean value of quadratic fluctuations of the number of particles, for M3(0) – of the mean value of cubic fluctuations of the number of particles, and so on. As a result, all semiinvariants are constant values, which are proportional to the average number of particles .
The Fourier transform of the van der Waals attraction potential is negative for all ≤ . The main effects related to the van der Waals interaction are just focused in the region 0 ≤ ≤ .
The calculation of the functional of the GPF is restricted to the integrals over k in the region ≤ . An estimate of the integrals for > gives correction, whose value is less than one percent.
It has been proved that the basic measure density, with respect to which all moments converge, is the fourth-order measure density. After the elimination of the cubic term, the resulting Hamiltonian is similar to the Hamiltonian of the Ising model in an external field. The field is characterized by the generalized chemical potential . Furthermore, we integrate only the fourth-order measure density. The presence of the factor , the certain number of particles, in front of the Hamiltonian gives evidence for the trajectories, along which the measure density or the probability has maximal values. Therefore, we consider the Euler system of equations for each variable k including the variable 0. Analysis of the Euler equations shows that the maximum of the integrand in the GPF functional is achieved by the solutions k = 0 for all = 0 and 0 = max0 for = 0. This important fact says that the main problem of statistical thermodynamics, i.e., the calculation of the GPF, splits into two macroscopic problems: – integration over all variables k with = 0, – one-fold integration with respect to 0.
In the first problem, the calculation does not contain the chemical potential. The chemical potential or the generalized chemical potential , where = and denotes the dimensionless density, is present in the second problem.
The first problem is a typical Ising problem. Here, the excellent results have been achieved after calculations of the fourth-order and sixth-order densities of measures. Starting from the first principles, all thermodynamic functions (i.e., the free energy, entropy, all second derivatives, critical exponents, critical amplitudes, critical temperature, and critical density) have been calculated. The obtained results agree with the data obtained by specific methods for certain groups of thermodynamic quantities.
The second problem is the integration with respect to the macroscopical variable 0, which describes, at the temperatures below the critical one, , a phase transition of the first order. Here, we have used a new approach for this calculation. Since in front of the Hamiltonian in the integrand, we have the number of particle , that is a large number, the integration is carried out by the steepest descent method. As a result, we immediately obtain an expression for the generalized chemical potential as a function of the most probable trajectories for the isotherms of the system. The solution of two equations with respect to the parameter for an enveloping gives an information about the probability of the system to be either in the vapor or liquid state. In the thermodynamic limit, the curves for the probabilities approach the Heaviside function and acquire the values 1 for the vapor phase and 0 for the liquid phase (if the argument is in the region /2 < ≤), or the values 0 for the vapor phase and 1 for the liquid phase (if the argument is in the region 0 ≤ /2), or 1/2 for the both phases (if = /2). Just the point = = 1/2 gives, on the isotherm ), the value of a jump of the density and its position.
It has been found that there are two trajectories for isotherms (, ), at which the integrand has maximum. One corresponds to the vapor state and another one corresponds to the liquid state. Moreover, the main maximum of the vapor phase is in correspondence to the relative maximum of the liquid phase, and, vice versa, the main maximum of the liquid phase is in correspondence to the relative maximum of the vapor phase.
The isotherm plot has a horizontal part at the transition from the vapor phase to the liquid state and vice versa. In the thermodynamic limit, the main maxima on the curves of the vapor and liquid states coincide with the corresponding parts of van der Waals’ curves.
The length of the horizontal part of the isotherm coincides with the distance between the end of the main maximum of the vapor isotherm and the beginning of the main maximum of the liquid isotherm, that is, with the length of the horizontal part on van der Waals’ isotherm. In the thermodynamic limit, the horizontal part of the isotherm consists of three parts: two external ones and one internal one. The coordinates of the ends of these parts are situated symmetrically with respect to straight diameter. For the processes, which occur on the line of the phase transition (, ) = 0, the external parts correspond to the appearance of a nucleus of the liquid phase under stretching the vapor or a bubble of the vapor phase under a tension of the liquid (nucleation processes). The internal part under 96 stretching of the vapor gives a jump of the density from the vapor (metastableї) phase to the liquid phase (droplet); both states (vapor and liquid) have the probability 1/2. Vice versa, under a tension of the liquid or, after all, under the removal of the residual bubble as the final act of the phase transition from vapor to liquid.
Let us underline that the transition-jump occurs between the states, which have the identical probabilities of their existence. The transition cannot occur between the states which correspond to the limiting points along the horizontal part of the van der Waals isotherm, because if the probability of existence is 1 for one point, then it is 0 for another one, and vice versa.
The states at the beginning and the end of the jump of the density under compressing the gas, which characterizes the internal part of the isotherm, correspond to the densities of a completely supersaturated vapor and a completely superdiluted liquid. This is confirmed in a series of novel experiments.
We have constructed binodals and spinodals. The bimodal-1 is asymmetric with respect to the straight diameter,and the distance to the liquid branch is always longer than the distance to the vapor branch with only exception when temperatures are very close to the critical temperature. The binodal-2 characterizes the beginning of the nucleation process and coincides with the van der Waals binodal. The spinodal-1 restricts the points, where the density jump from vapor to liquid takes place. All curves are flattened as approaching = 0, since the change of the height, while approaching = 0, varies as 5/2, whereas the change of the width varies as /2.
Thus, we have described the main features of the first-order phase transition at < based on the analysis of the onefold integral with respect to 0 in the expression for the GPF. The contribution of all other integrals was assumed to be fixed and equal to the values of these integrals at = .
Some not principal problems are out of the scope of this study; their solutions must dress the backbone description of the vapor-liquid critical point suggested in this paper.
Key words: critical point, phase transition of the first order, grand partition function, fourth-order measure density, collective variables, equation of state, binodals and spinodals.