Long-range correlations in the statistical theory of fluid criticality

Fluid Phase Equilibria 506 (2020) 112417

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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d

Long-range correlations in the statistical theory of fluid criticality V.N. Bondarev Department of Theoretical Physics, Odessa I.I. Mechnikov National University, 2 Dvoryanskaya St., 65026, Odessa, Ukraine

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 June 2019 Received in revised form 11 November 2019 Accepted 14 November 2019 Available online 18 November 2019

Using the approach formulated in the previous papers of the author, a consistent procedure is developed for calculating non-classical asymptotic power terms in the total correlation function hc ðrÞ of a “critical” fluid. Analyzing the Ornstein-Zernike equation with taking account of the contribution e h4c ðrÞ in the direct correlation function cc ðrÞ allows us to find, for the first time, the values of transcendental exponents n0 ¼ 1:73494::: and n00 ¼ 2:26989:::which determine the asymptotic terms next to the leading one in hc ðrÞ. It g (it was is shown that already the simplest approximation based on only two asymptotic terms, r 6=5 n0 , leads to the functions h ðrÞ and c ðrÞ, which agree (at least, qualitatively) with the found earlier) and rg c c corresponding correlation functions of the Lennard-Jones fluid (argon) in the near-critical state. The obtained results open a way for consistent theoretical interpretation of the experimental data on the critical characteristics of real substances. Both the theoretical arguments and analysis of published data on the experimentally measured critical exponents of real fluids lead to the conclusion that the known assumption of the similarity of the critical characteristics of the Ising model and the fluid in the vicinity of critical point (the “universality hypothesis”) should be questioned. © 2019 Elsevier B.V. All rights reserved.

Keywords: Fluid criticality Theory Correlation functions Non-classical critical exponents Universality hypothesis

1. Introduction The total pair correlation function (PCF) hðrÞ and the direct PCF cðrÞ play fundamental role in the determination of the thermodynamic characteristics of statistical media [1e3]. Therefore, the search for approaches that allow one to calculate the transformations of PCFs in wide ranges of temperature and density, including the regions of phase transitions, is an actual problem. The effects in the vicinity of the gas-liquid critical point where the singular character of PCFs becomes apparent extremely brightly are of special interest [1e3]. However, the conventional approaches (such as the hyper-netted-chain and the Percus-Yevick approximations [2,3]) based on the integral equations of the statistical theory of liquids lead, in fact, to pure classical (by van der Waals [4]) values of the critical exponents or to values noticeably differing from those observed experimentally (see review [5]). Numerous attempts to obtain non-classical critical exponents from the equations of modern theory of fluids were unsuccessful for a long time. Comparative analysis of the experimentally observed critical singularities of the thermodynamic functions of different media led in due time to the formulation of the so-called “universality hypothesis” (about it see, e.g. Ref. [6]). According to

E-mail addresses: [email protected], [email protected]. https://doi.org/10.1016/j.fluid.2019.112417 0378-3812/© 2019 Elsevier B.V. All rights reserved.

this hypothesis, any fluid in the near-critical state belongs to the same universality class O(1) as the three-dimensional (3D) Ising magnet. Hence, it followed that to find non-classical critical exponents for fluid it is enough to study the critical characteristics of the 3D Ising (lattice gas) model by its numerical solution [5,7]. On the other hand, the renormalization group and ε-expansion methods [5,7,8] up to the recent past were the only analytical ones for finding the non-classical critical exponents which are compatible with the experimental data for fluids (see, e.g. Refs. [6,9]). This, however, does not remove the question of whether it is possible to build a non-classical theory of the critical point directly from the fundamental equations of the theory of fluids. In paper [10] (and further in Ref. [11]), based on the formally closed equations for hðrÞ and cðrÞ, the author has proposed an approach allowed to give the independent analytical calculation of the nonclassical critical exponents of one-component non-Coulomb fluid. The exponents [10] have the form of rational fractions satisfying all known scaling relations, including the hyper-scaling relation. The values of the exponents [10,11] (except the exponent h, see below) are in rather good agreement with the corresponding values found by the numerical solution of the 3D Ising model [5e7]. Besides, the approach [10] was applied in Ref. [11] to the calculation of the next to the leading terms in the expressions for the correlation lengths (the so-called “correction-to-scaling” terms, following the terminology of [12]). Good quantitative agreement of the theory [10,11]

2

V.N. Bondarev / Fluid Phase Equilibria 506 (2020) 112417

and experiment opened a way to accurate processing of the experimental data on the critical properties of real substances (see Refs. [13e15]). It is significant that the theory [10,11] allows further development and deepening in a direction which, in fact, has not been investigated until now. The point is that in addition to the mentioned “correction-to-scaling” terms, there is another source of the correction terms in the singular parts of the thermodynamic functions of the “critical” fluid. Really, let us represent, as usually (see, e.g. Ref. [4]), the total PCF near the critical point as follows:

hðrÞ ¼ hc ðrÞer=x :

(1)

Here, x is the mentioned correlation length which tends to infinity when approaching the critical point, while the asymptotic (r/∞) behavior of the 3D “critical” total PCF hc ðrÞ is determined by expression1

g hÞ ; hc ðrÞr ð1þ

(2)

where, according to Ref. [10], h ¼ 1/5.2 It is seen from (1) and (2) that next to the leading critical singularities of the thermodynamic quantities of the fluid will be determined not only by the “correction-to-scaling” terms in x but also by next to the leading powertype asymptotic terms in hc ðrÞ. It is important that the theory [10,11] really allows us to find such terms in hc ðrÞ and to calculate corresponding critical exponents, that could hardly be done by the renormalization group and ε-expansion methods [5,7,8]. Thus, the purpose of the present paper is to go beyond the limits of the approximation in which the “critical” PCF hc ðrÞ of real fluid is described by the only exponent h. In Section 2 (and also in Appendices A and B), being based on the fundamental equations for PCFs, we develop an approach which allows us to obtain, for the first time, the explicit form of the power-type asymptotic terms, which follow the leading one in hc ðrÞ. Using these results, as well as exact conditions on the “critical” PCFs, in Subsection 2.1 we find self-consistent expressions for these PCFs and demonstrate the possibility of applying them to interpreting (at least, qualitatively) experimental PCFs for real fluid (argon) near the critical point. Further (Section 3), we argue that the approach developed in this paper (and in [10,11,13e15]) allows us to describe experimental data on the critical features of the thermodynamic functions of real fluids more accurately than the 3D Ising model. This means that the mentioned “universality hypothesis”, which relates the “critical” fluid and the lattice gas (the Ising model) to one and the same universality class, hardly takes place. At last, in Section 4, we formulate the conclusions from the obtained theoretical results and note that their comprehensive experimental verification should be of fundamental interest.

2. Basic equations and asymptotic expansions of PCFs The theory [10] is based on two formally exact equations of the statistical theory of liquids closed at the level of hðrÞ and cðrÞ PCFs. One of these equations obtained for the first time in Refs. [16,17] has the form:

1 Here and below, considering the situation at the critical point itself, we mark the corresponding values by the subscript “c”. 2 Recall that the corresponding classical value h ¼ 0 contradicts the hyper-scaling relation in 3D space [4]. This means that just the “non-classical” results obtained in the present paper using the approach [10,11] (see also [13e15]) must be put into the basis of a consistent theory of liquid-gas criticality that allows experimental verification.

ln½hðrÞ þ 1 ¼ hðrÞ  cðrÞ þ BðrÞ 

WðrÞ ; kB T

(3)

where W(r) is the energy of pair interaction between atoms separated by a distance r; T is the temperature, and kB is the Boltzmann constant. The second equation connecting hðrÞ and cðrÞ is the wellknown OrnsteineZernike equation [1e3]:

ð hðrÞ  cðrÞ ¼ r dr0 cðr 0 Þhðjr0  rjÞ;

(4)

where r is the atom number density. All information concerning many-particle correlations is contained in the so-called bridge function BðrÞ represented by a complicated diagrammatic series in the form of integrals from the products of the total PCFs (see Refs. [16,17]). Equations (3) and (4) determine the equilibrium statistical characteristics of a one-component fluid. Having in mind nonCoulomb media, as W(r) one assumes the Lennard-Jones potential

hs12 s6 i ; WðrÞ ¼ 4ε  r r

(5)

where ε and s are characteristic constants (for argon usually take ε/kB z 120 K and s z 3.40 Å [18]). A key point of the theory [10] is the representation of the asymptotic behavior of B(r) in the form of a local expansion:

BðrÞ ¼ a2 ðT; rÞh2 ðrÞ þ a3 ðT; rÞh3 ðrÞ þ a4 ðT; rÞh4 ðrÞ þ :::

(6)

with coefficients aj ðT; rÞ (j ¼ 2, 3, 4, …) which depend on the thermodynamic state of the medium. In particular cases, the representation (6) leads to the well-known approximations, e.g., the hyper-netted-chain approximation [2,3] at aj ðT; rÞ≡0 or the meanspherical approximation [5,19] at aj ðT; rÞ≡ð1Þjþ1 =j. Refusal of such harsh conditions on the coefficients in (6) leads to the theory [10,11] in which a non-classical asymptotic behavior of the “critical” PCFs follows from the fundamental equations (3) and (4). In this case, the two critical values, Tc and rc, must satisfy the two conditions [10]:

1 1 a2 ðTc ; rc Þ ¼  ; a3 ðTc ; rc Þ ¼ ; 2 3

(7)

e2 ðrÞ and h e3 ðrÞ in the which lead to the vanishing of the terms h c c asymptotic limit of equation (3). Then the first non-vanishing asymptotic term on hc ðrÞ in equation (3) is ða4c þ1 =4Þh4c ðrÞ, where we have designated a4c ≡a4 ðTc ; rc Þ. As the result, one can arrive at the above-mentioned value h ¼ 1/5 [10] for real 3D fluid. Note, that when finding the critical exponents, potential W(r) of the form (5) should not at all be taken into account [10]. Meanwhile, the critical values Tc and rc will be determined just by the characteristic constants of this potential.3 Thus, the asymptotic form of equation (3) at the critical conditions is as follows:

    1 4 1 5 WðrÞ hc ðrÞ þ a5c  h ðrÞ þ :::  cc ðrÞ ¼ a4c þ ; 4 5 c kB Tc

(8)

3 In paper [15] the author has demonstrated how these critical values can be restored in the framework of the non-classical theory of criticality when using the known numerical calculations of the virial coefficients for the Lennard-Jones fluid; see also [20].

V.N. Bondarev / Fluid Phase Equilibria 506 (2020) 112417

where we have written the first non-vanishing terms of the expansion in powers of hc ðrÞ. We shall see (Appendices A and B) e4 ðrÞ that for our purposes it is sufficient to consider only the term h c in (8). n in h ðrÞ with n≡1 þ h ¼ 6/5 [10] deThe asymptotic term rg c termines the leading divergent contribution into the isothermal compressibility of the “critical” fluid, kT ¼ r1 ðvr=vpÞT , where p is the pressure. From (A2) and (A3) (see Appendix A) it follows that n0 under the the terms in hc ðrÞ with the asymptotic behavior rg condition 6=5 < n0 < 3 will also lead to the divergent contributions into kT . So, we represent the “critical” total PCF at r/ ∞ in the form: 0

A∞ A hc ðrÞ ¼ 6=5 þ n∞0 r r

(9) 0

with some constants A∞ and A∞ ; as for the exponent n0 , we shall consider it as the smallest permissible value from the interval 6= 5 < n0 < 3. Substituting (9) into (8) we find the asymptotic form of the “critical” direct PCF up to the main term containing the exponent n0 :

cc ðrÞ ¼



a4c þ

1 4



0

!

A4∞ A3∞ A∞ þ 4 18=5þn : 0 r 24=5 r

(10)

In this expression, in contrast to the similar expression (20) from Ref. [11], we take into account also an additional asymptotic term g 0 decaying faster than the main term r 24=5 g . It is important r 18=5n g 0 in (10) de0 to note that if n <12/5, the asymptotic term r 18=5n creases more slowly than the last two terms in the expression (8). Therefore, below we will find the exponent n0 in the range 6= 5 < n0 < 12=5. Somewhat bulky calculations in Appendix A lead to equality:

n0 ¼ 1:73494::: :

(11)

Thus, in the framework of the theory [10,11] we succeeded in finding the numerical value of the exponent n0 , determining the asymptotic behavior of the next to the leading term in the singular part of the “critical” PCF (9). The found exponent (11) is transcendental, unlike the leading exponent n≡1 þ h ¼ 6=5 [10]. Since n0 < 12= 5 g 0 in (10) decreases slower than the last the asymptotic term r 18=5n two terms in (8) and therefore is really the most important after the g .4 leading term r 24=5 The terms written in (9) do not exhaust the singular ones in hc ðrÞ that lead to the divergent contributions into the isothermal compressibility of the “critical” fluid. Notable interest is the calculation of one more power term decreasing “faster” than those written in (9). Adding a term with exponent n00 > n0 into the “critical” total PCF we can present its asymptotic form as follows: 0

hc ðrÞ ¼

0

0 A∞ A∞ A∞ þ 0 þ n0 0 ; r/∞; n r r 6=5 r

(12)

0

0 where A∞ is a new constant. The calculation of the exponent n00 given in Appendix B leads to the value:

n00 ¼ 2n0 

6 ¼ 2:26989::: 5

The value of n00 is still less than 12/5, so the corresponding asymptotic contribution into cc ðrÞ will decrease “slower” than the last two terms in (8). Note, that the exponent n00 as well as n0 is transcendental. One can make sure that the terms in (12) with the found exponents n ¼ 6=5 [10], n0 (see (11)), and n00 (see (13)) exhaust all the power terms, the origin of which is due to the term e h4c ðrÞ in (8).

2.1. The self-consistent PCFs of the Lennard-Jones fluid at criticality: an example Now we apply the obtained results to construct approximate PCFs of the Lennard-Jones fluid directly at the critical point. To do this, we use the asymptotic expression (6) for the “critical” bridge function, as well as the critical conditions (7), and approximately represent Bc ðrÞ over the entire interval 0 < r < ∞ in the form:

1 1 Bc ðrÞ ¼  h2c ðrÞ þ h3c ðrÞ þ a4c h4c ðrÞ: 2 3

(14)

We shall also use the so-called cavity PCF yðrÞ, coupled with hðrÞ by the relation (see, e.g. Ref. [21]):

1 þ hðrÞ ¼ ½1 þ f ðrÞ yðrÞ:

(15)

f ðrÞ≡eWðrÞ=kB T

Here  1 is the well-known Mayer function (see, e.g. Ref. [3]), and also yðr /∞Þ/1. In contrast to the Mayer function defined by the interatomic potential (5), the function vc ðrÞ≡yc ðrÞ  1, like hc ðrÞ, is long-range at the critical point, i.e. g when r / ∞. Therefore, in order to find an approximate vc ðrÞr 6=5 form of the function hc ðrÞ that has a correct behavior at r / 0 and r / ∞, it is enough to specify the correct asymptotic behavior of the function vc ðrÞ. Having in mind expression (9) and taking into account the value (11) for the next to the leading exponent of the asymptotic decay of hc ðrÞ, we represent the simplest approximate expression for the function vc ðrÞ at 0 < r < ∞ as follows:

vc ðrÞ ¼ 

A∞ r 2 þ r02

0

3=5 þ 

A∞ r 2 þ r02

n0 =2 :

(16)

The presence of constant r 20 in expression (16), firstly, eliminates a non-physical divergence of function vc ðrÞ at the origin, and secondly, ensures the appearance of non-singular (integrable at large r) terms that correct the short-range behavior of this function. 0 The constants A∞ > 0, A∞ , and r 20 entering in (16) will be determined from the integral conditions to which the correlation functions must satisfy. Two of these conditions were formulated in Ref. [11], starting from the requirement that the first three density derivatives of the fluid pressure at the critical point vanish (and also the finiteness of the fourth derivative required in the theory [10,11,15]). In the above notation, these two conditions have the form: ∞ ð

dr r 4

dWðrÞ ½1 þ fc ðrÞvc ðrÞ ¼ 0; dr

(17)

dr r 5

dWðrÞ ½1 þ fc ðrÞ vc ðrÞ ¼ 0: dr

(18)

0

(13)

together with definite relation (B9) allowing us to express the 0 0 0 constant A∞ > 0 through the constants A∞ > 0 and A∞ . e5 ðrÞ. in equation (8) at h ¼ 1/5 decreases in It is curious to note, that the term .h c the asymptotic region like r6 , i.e. as the Lennard-Jones potential (5). 4

3

∞ ð 0

As the third condition, it is natural to choose the requirement of the divergence of the isothermal compressibility (A3) at the critical point of the fluid. This requirement, taking into account (A1), reads:

4

V.N. Bondarev / Fluid Phase Equilibria 506 (2020) 112417

4prc

∞ ð

dr r 2 cc ðrÞ ¼ 1:

(19)

0

With the help of (3), (6), and (7), one can represent the approximate PCF cc ðrÞ entering the critical condition (19) in the form:

1 1 cc ðrÞ ¼ ln½1 þ vc ðrÞ  þ hc ðrÞ  h2c ðrÞ þ h3c ðrÞ þ a4c h4c ðrÞ: 2 3 (20) In this case, the definition (15), as well as the relation (A17) should be taken into account. In what follows we shall use the dimensionless quantities: r* ¼ rs3 , T * ¼ kB T=ε, A*∞ ≡ A∞ = s6=5 > 0, 0*

0

0

A ∞ ≡A∞ =sn and the values of the critical parameters r*c ¼ 0.3449, T *c ¼ 1.3303 found in Ref. [15]. Substituting the values of integrals (A8) into (A16), we obtain for r*c ¼ 0.3449 the following relation

between a4c and the required constant A*∞ > 0:

a4c ¼

  0:5364 5 1  : 4 A*∞

(21)

It is clear from the structure of expression (20) that the desired function (16) must be such that the inequality vc ðrÞ > e 1 (following from the general inequality vðrÞ > e 1, see (3)) would be satisfied in the whole domain 0 < r < ∞. Below we will use the dimensionless variable b r ≡r=s and the dimensionless constant b r 0 ≡ r0 = s. Then equations (17) and (18) can be written as a system of linear homogeneous equations with respect to the unknown quantities A*∞ > 0* 0 and A ∞ : 0*

A*∞ g1 ðb r 0 Þ þ A ∞ g1 ðb r 0 Þ ¼ 0;

(22)

    0* 0 A*∞ g2 b r 0 þ A ∞ g2 b r 0 ¼ 0:

(23)

0

The coefficients in these equations depend on b r 0 and have the form:

  ∞   ð rÞ 1 2 1 þ fc ðb r 0 ¼ db g1 b r 3 1 6  2 ; 2 3=5 b b b r r r þb r0 0

  ∞   ð 0 1 2 1 þ fc ðb rÞ g1 b r 0 ¼ db r 3 1 6  ; 2 2 n0 =2 b b b r r r þb r0

(24)

0

  ∞   ð 1 2 1 þ fc ðb rÞ g2 b r 0 ¼ db r 2 1 6  ; 2 2 3=5 b b r r b r r þb 0

Fig. 1. The total PCF hc ðbr Þ calculated by our theory at criticality (r*c ¼ 0.3449, T *c ¼ 1.33; solid red line) in comparison with the total PCF hðbr Þ extracted from the experimental data [22] for argon at near-critical conditions (dashed lines, between which the measurement error interval is enclosed; r* z 0.3476, T * ¼ 1.35).

0

  ∞   ð 0 1 2 1 þ fc ðb rÞ g2 b r 0 ¼ db r 2 1 6  2 : 2 n0 =2 b b b r r r0 r þb

(25)

0

Equating the determinant of system (22) and (23) to zero, we obtain the condition for the nontrivial solvability of this system:

        0 0 r 0 g2 b r 0  g2 b r 0 g1 b r 0 ¼ 0: g1 b

take into account (21) and use the requirement (19), then as a result we obtain an equation for determining the constant A*∞ > 0, after 0*

which we also find the constant A ∞ . Within this scheme, using a simple numerical procedure, the following values of the dimensionless constants were found: b r 0 ¼ 1.5247, A*∞ ¼ 0.6627, 0*

A ∞ ¼ 1.1732. In addition, based on representations (14) and (20) 5

with the found value a4c ¼ ð0:5364=A*∞ Þ  1=4 ¼ 0.09746, one can construct the desired approximate expression for the “critical” bridge function Bc ðrÞ. The obtained results are enough to calculate the “critical” PCFs hc ðrÞ and cc ðrÞ in the framework of the developed approximation. Unfortunately, there is no data in the literature concerning the form of PCFs for fluids at criticality, which is associated with large (and, in fact, insurmountable) difficulties in carrying out real experiments or computer simulations directly at the critical point. So, most informative seems the comparison of the obtained theoretical results with the experimental data for real fluids in the near-critical region. Apparently, the old papers [22,23] dedicated to the study of x-ray scattering on argon are most suitable for this (see also [24]). The PCFs of argon hðrÞ [22] and cðrÞ [23] were recovered by the experimental data processing both on isochore at mass density 0.536 g/cm3 (close to the critical value, 0.531 g/cm3 [25]), and on isotherm at temperature T z 148 K (close to the critical value, 150.65 K [25]).5 In Fig. 1, the solid (red) line is built by the results of our calculation of hc ðb r Þ in the approximation (14)e(16). By the dashed lines (between which the measurement error interval is enclosed) we reproduce the values of hðb r Þ obtained experimentally [22] for argon at mass density 0.536 g/cm3 and temperature T z 153 K close to the corresponding critical values. In this case, the height and position of the main maximum of the calculated function hc ðb r Þ are in satisfactory agreement with those found in the experiments [22]. On the

(26)

This condition is the equation for finding b r 0 . Knowing b r 0 and 0*

using equation (22), one can obtain a relation A ∞ ¼  A*∞ g1 ðb r 0 Þ= 0 r 0 Þ between the remaining two quantities. If we substitute this g1 ðb

relation and the found value of b r 0 into expressions (16) and (20),

5 Using the experimental [25] values of the critical parameters for argon and accepting the values of the dimensionless critical parameters r*c ¼ 0.3449, T *c ¼ 1.3303 found in Ref. [15], one can restore the effective characteristic constants of the Lennard-Jones potential: ε/kB z 113.3 K and s z 3.51 Å (cf. the corresponding values shown after Eq. (5)). In this connection, it is useful to mention the discussion [23] concerning the state-dependent “constants” of the potential functions of real substance (argon).

V.N. Bondarev / Fluid Phase Equilibria 506 (2020) 112417

Fig. 2. The direct PCF cc ðbr Þ calculated by our theory at criticality (r*c ¼ 0.3449, T *c ¼ 1.33; solid red line) in comparison with the direct PCF cðbr Þ extracted from the experimental data [23] for argon at near-critical conditions (dashed lines, between which the measurement error interval is enclosed; r* z 0.3476, T * ¼ 1.35).

other hand, the short-range “fine structure” of hc ðb r Þ is lost in this crude approximation. In addition, an analogous “fine structure” of hðb r Þ was established in the molecular-dynamics simulation of PCFs of the Lennard-Jones fluid in a near-critical state [26]. Consequently, one can also expect a similar behavior of the “critical” PCF hc ðb r Þ in the short-range area. Thus, the further task of the developed theory should be the construction of a more perfect approximation for vc ðrÞ, which would make it possible to reproduce the non-monotonous behavior of hc ðb r Þ at b r > 1.5. Note, however, that the “critical” PCF hc ðb r Þ, decreasing at r/∞ as r 6=5 [10], will be long-range one (corresponding to the divergent correlation length), whereas the asymptotic behavior of any “near-critical” PCF hðb r Þ will inevitably carry information on the interatomic potential W(r), which is assumed to be short-range. Fig. 2, in the same notations and at the same values of the mass density and temperature, demonstrates the results of our calculation of cc ðb r Þ also using the approximation (14)e(16) (solid red line) in comparison with the experimental (with taking account of measurement error) dependence of cðb r Þ for argon [23]. It can be seen that the general course of the theoretical dependence cc ðb r Þ is in satisfactory agreement with the experimental data given [23]. Thus, already within the framework of the simplest approximation, our theory makes it possible to grasp some characteristic features (including the locations and heights of the main maxima) of the “critical” PCFs of the Lennard-Jones fluid. Herewith, we used only the theoretically calculated values of the two critical exponents (the leading [10] and the next to the leading (11)), describing the asymptotic decay of hc ðb r Þ, and the exact integral conditions for PCFs. The improvement of approximation (14)e(16) using asymptotic form (12) (with one more exponent (13)) should lead to closer agreement of the PCFs hc ðb r Þ and cc ðb r Þ with the corresponding experimental ones from Refs. [22,23] in the short-range area (the author is going to return to this question later). As for the “critical” pressure pc, then in the adopted rough approximation it turns out to be negative: p*c ≡pc s3 =εz 0.059 (whereas the calculation [15] using the virial expansion gives positive value: p*c z 0.136). The origin of the formally negative p*c sign becomes clear given the following circumstance. The fact is that the oversimplified approximation based on (16) leads to an overestimated contribution of the long-range correlations in p*c , since at large enough distances the interatomic interaction has the

5

character of attraction (cf. [15], where the question of the negative sign of the singular contribution to the “critical” pressure is discussed). It is important, that the results obtained in this paper allow us to refine the theoretical description of the “critical” PCFs by using a term with one more critical exponent n00 (see (13)). It can be expected that due to the appearance of an additional term in the expression for vc ðrÞ it will be possible to reproduce in more detail the experimentally observed features of the structure of PCFs at criticality. The same applies to the “critical” pressure, the calculation of which in a more perfect approximation can lead to a value consistent with that found, for example, in [15]. It is useful to mention the recent work [27], in which the results of numerical simulation of the functions 1 þ hðb r Þ, yðb r Þ, and Bðb r Þ for the Lennard-Jones fluid have been demonstrated at temperatures close to T *c but at densities r* ¼ 0.177 (much lower than r*c ) and r* ¼ 0.462 (much higher than r*c ). Therefore, one can only perform a qualitative comparison of our results with the data of [27]. For example, the heights of the main peak of the function hðb r Þ at both indicated values of the density [27] were close to each other and found to be in numerical agreement with the height of the corresponding peak for our PCF hc ðb r Þ represented in Fig. 1. However, the structure of hðb r Þ at b r > 1.5, according to Ref. [27], is more pronounced than that of our hc ðb r Þ. As for the “critical” bridge function Bc ðrÞ, the calculations by Eq. (14) show that for all r it remains negative and is in qualitative agreement with the bridge functions from Ref. [27]. Note, that although the “critical” bridge function (14) is a long-range one, its asymptotic decay follows the law  r 12=5, i.e. turns out to be much “faster” than the decay of hc ðrÞ. 3. Discussion of the results Here, it is appropriate to touch upon the question of how adequately it is possible to describe the critical characteristics of real fluids using existing theoretical approaches. To date, extensive material has been accumulated in the literature on the critical properties of pure liquids and liquid mixtures. Beginning in the 70s of the 20th century, to interpret the experimentally observed critical features of the thermodynamic quantities of real fluids, the results of the numerical analysis of 3D lattice gas model (isomorphic to the Ising magnet) became to be actively used. However, the “universality hypothesis”, according to which the “critical” fluid can be described by the 3D Ising model, remained unproved. Below, based on known experimental data, we will argue that real fluids in the vicinity of the liquid-gas critical point should be placed into a special universality class that does not coincide with the universality class of the 3D Ising model. At the same time, as an alternative to the lattice gas model, we will use the results of our approach [10,11], based on the fundamental equations of the theory of liquids. Although the critical exponents [10] turned out to be rather close to the corresponding exponents of the 3D Ising model there are numerical differences between them (especially for the exponent h, see above).6 This result can be considered as an additive argument in favor of the conclusion that real “critical” fluid and the 3D Ising magnet can hardly belong to one and the same universality class.7

6 The differences between the two-dimensional (2D) critical exponents calculated using theory [10], on the one hand, and the corresponding Onsager's exponents for the 2D Ising model [4], on the other hand, are more striking than for the 3D case. A detailed discussion of this question we leave for future. 7 In this regard, it is worth mentioning the paper [28] containing the next statement: “However, nothing is proven about liquid-gas criticality from the universality conjecture”.

6

V.N. Bondarev / Fluid Phase Equilibria 506 (2020) 112417

For today, one can point out quite a few experimental evidences that the critical exponents of real fluids in the asymptotic region (immediately adjacent to the critical point) are not in precise accordance with those found by the numerical solution of the 3D Ising model. Furthermore, attempts to fit the experimental data using formulas which contain, along with the leading Ising term, a few more Wegner [12] “correction-to-scaling” terms often lead to less accurate description than approximations by formulas containing sometimes the only one term of non-Ising type. In addition, formulas with non-Ising exponents are capable to give the quantitative description of experiments in a much wider area of thermodynamic variables than analogous formulas with the 3D Ising exponents. Let us give some of the most reliable evidences in favor of the non-Ising interpretation of the experiments on the critical properties of real fluids.8 a) Exponent a characterizes the temperature singularity of the heat capacity both along the critical isochore and along the liquid-gas coexistence curve: CVejtja , t≡ðT =Tc  1Þ. The 3D Ising model calculations give a ¼ 0.109 [6]; the theory [10] leads to the value a ¼ 1/8. In paper [29], an attempt of numerical processing of the most accurate (in a microgravity environment during the German Spacelab Mission D-2) CV measurements in the near-critical region of SF6 had been made.9 The asymptotic region was estimated in Ref. [29] as jtj < 1:6  104 , whereas the whole temperature interval of measurements was 3  106 < jtj < 102 . Later, in Ref. [30], the experimental data [29] were reanalyzed and described with high accuracy up to jtjy102 using the simple power law (see also [6]). The conclusion of [30] reads: “These heat capacity measurements were found to be inconsistent with renormalization group predictions”. Furthermore, according to Ref. [6], “all data” of [29] “were actually within the asymptotic region”. Serious arguments were presented in Ref. [31] in favor of the existence of the so-called “Yang-Yang anomaly” (the divergence of the second temperature derivatives of both the pressure and the chemical potential in the expression for CV [32]) in the vicinity of 3He critical point. Then, according to Ref. [31], “… a revision of the conventional scaling theory for fluids based on the analogy between fluids and latticegas model will be required”.10 It is also useful to mention the article [35], in which the measured value a ¼ 0.124 ± 0.006 was given for the heat capacity exponent of the liquid-liquid mixture nitrobenzene-dodecane near its critical consolute point. In addition, in work [36] a path is outlined, allowing to establish a trend to the divergence of the isochoric heat capacity of a fluid near the liquid-gas critical point based on the results of molecular dynamics simulation, including also the effects of three-body interaction. b) Exponent b at t < 0 characterizes the critical singularity of the e    liquid-gas coexistence curve: rl;g  rc tjb , where rl and rg are the fluid densities at the liquid and gas branches, respectively. The calculations by the 3D Ising model [6] give b z 0.325; according to the theory [10] b ¼ 3/8.

8 An extensive collection of measured critical exponents for pure liquids and mixtures is given in books [9,25]. 9 Unfortunately, the outlined in [6] program on microgravity study of other thermodynamic characteristics, such as the coexistence curve, the isothermal compressibility, etc., apparently, still has not been implemented to “critical” fluids. 10 Moreover, the “Yang-Yang anomaly” was detected repeatedly in other systems (see, for example, papers [33,34]).

The experimental coexistence curves of Ne, HD, N2, and CH4 were analyzed [37] in the interval 1:3  104 < jtj < 2  102 using the single-term fitting with b ¼ 0.355. As the result, the conclusion [37] reads: “We draw attention to the disadvantages of using the extended scaling equations with the Ising exponent b for describing coexistence curves of simple one-component fluids”. For a number of “critical” fluids the exponent b is at the level of b ¼ 0.34e0.36 [9,25,38e45] reaching for Heptane even the value b ¼ 0.385 ± 0.016 [46] far exceeding the Ising value. One should also mention paper [47] devoted to molecular dynamics simulation of the liquid-gas criticality. It is essential, that without the use of the so-called “finite size scaling” approach, the non-Ising exponent b ¼ 0.35e0.36 for a fluid with the Lennard-Jones interaction was obtained in Ref. [47]. In our papers [13,14], in addition, it has been demonstrated the possibility of accurate processing of the experimental data for a number of substances just with the exponent b ¼ 3/8 found in Ref. [10]. c) Exponent n determines the divergence of the fluid correlation length on the critical isochore: xejtjn . The value n ¼ 5/8 in the theory [10] is very close to the corresponding value n z 0.623e0.63 for the 3D Ising model [5]. In paper [10] the author has shown that the light-scattering experiments near the critical point of 3He can be well described quantitatively by n ¼ 5/8, at least when jtj < 3  103 . The last value agrees on the order of magnitude with the above restriction on the asymptotic region near the critical point of real fluids. d) Regarding the critical exponents g and d which determine, respectively, the divergence of the isothermal compressibility kTejtjg and the form of the critical isotherm jp  pc jejr  rc jd in the pressure e density plane one can find significant discrepancies in the literature. Often, the experimental data processing (see, for example, [40,43,48]) led to a conclusion that these exponents differ from the corresponding values g z 1.24 and d z 4.8 for the 3D Ising model [6], being closer to the values g ¼ 9/8 and d ¼ 4 found in the theory [10]. Papers [49e53] reported on the measurements of non-Ising critical exponents for noble fluids: g z 1.10e1.18, d z 3.8e4.21, as well as b ¼ 0.354e0.361. It is interesting, that in the longstanding paper [54] dedicated to the study of critical properties of xenon, carbon dioxide, and hydrogen, it was found that “the critical isotherm is of the fourth degree”, more precisely, d z 4.1 ± 0.2. e) Exponent h determines, according to (2), the asymptotic (at large distances r) behavior of the total PCF of the 3D “critical” fluid. Remind that, according to the theory [10], exponent h ¼ 1/ 5 is noticeably greater than the value h z 0.035 [5] found in the framework of the 3D Ising model. The published experimental estimates for h are in a fairly wide range e from h z 0.05 to h ¼ 0.15 [40,55e59], h ¼ 0.2 ± 0.1 (see the discussion of this question in [60]) and even higher [61]. Also it is useful to cite the conclusion of [56]: “Our experiment clearly demonstrates that the ordinary lattice-gas model cannot be used to calculate the critical exponent h for a fluid”. Thus, the value h ¼ 1/5 derived in Ref. [10] falls within the boundaries of the above experimental data. So, the analysis of the experimental data obtained by long-term studies of critical properties of real fluids, gives good reason to doubt the “universality hypothesis”, which attributes the fluid near the liquid-gas critical point to the universality class O(1) of the 3D Ising model. In fact, this means that “critical” fluids should be placed in a

V.N. Bondarev / Fluid Phase Equilibria 506 (2020) 112417

separate universality class, the singular properties of which were investigated in papers [10,11,13e15] (as well as in the present paper) using an approach based on fundamental equations of the modern theory of liquids.

7

than using the 3D Ising model exponents, should be considered as an additive evidence against the “universality hypothesis” in the context of real fluids criticality. Therefore, the implementation of new precision experiments in the close vicinity of the liquid-gas critical point of real substances is of fundamental interest.

4. Summary of results and final remarks Author contribution statement Now, let us summarize the results obtained in the present paper. A regular procedure is developed which allows one calculating the next to the leading non-classical asymptotic terms in the “critical” PCFs hc ðrÞ and cc ðrÞ of real fluid. Based on the approach formulated in the author's papers [10,11], it is shown that not only the leading asymptotic term in hc ðrÞ but also the following two terms giving the divergent contributions to the isothermal compressibility of fluid e4 ðrÞ in the asymptotic expansion are determined just by the term h c of cc ðrÞ. Studying the analytical properties of the Ornstein-Zernike equation we, for the first time, succeeded in finding the transcendental values of the next to the leading critical exponents n0 ¼ 1:73494::: and n00 ¼ 2:26989::: for hc ðrÞ. n0 As a useful application of the obtained results, the new term rg g 6=5 (in addition to the term r [10]) is taken into account to construct a simple approximate form of hc ðrÞ, and with it, cc ðrÞ. It is significant that for finding the characteristic parameters defining the form of hc ðrÞ and cc ðrÞ, only strict integral conditions on the “critical” PCFs are used. It is shown that, already in the accepted simple approximation, the theory is capable to reproduce satisfactorily some characteristic features of PCFs (in particular, the positions and heights of the main maxima) of argon in the nearcritical region. It is important that the theory allows for further improvement of the approximate description of the “critical” PCFs, n00 into h ðrÞ. in particular, by including the new term rg c Moreover, the results of the present work, which relate directly to the liquidegas critical point, should be taken into account when calculating the thermodynamic characteristics on the whole phase diagram of the fluid. Indeed, such calculations involve solving equations (3) and (4) for PCFs with a particular closure procedure, and then using these solutions to obtain the fluid equation of state, i.e. finding pressure as a function of r and T. In this case, a necessary condition for constructing a suitable expression for hðrÞ should be the requirement that at the critical point, itself, the PCF hc ðrÞ has a correct (non-classical) asymptotic decay, as established above. This requirement substantially narrows the class of fundamentally admissible constructions for hðrÞ, although the very finding of the explicit form of suitable PCFs (and therefore the equation of state and the coexistence curve of the fluid) can be not an easy task. Thus, the use of the results obtained in the present paper opens a way for the consistent theoretical interpretation of the experimentally observed peculiarities of the thermodynamic quantities of real fluids in the vicinity of the liquid-gas critical point. The literature data analysis performed in the present paper leads to the conclusion that the experimentally measured values of the critical exponents for real fluids, usually, do not coincide with the corresponding values calculated in the framework of the 3D Ising model. Taking into account the arguments given, it is necessary once again to consider the question about the similarity of the critical characteristics of the Ising magnet and of fluid near the liquid-gas critical point (the “universality hypothesis”). Note that a definite answer to this question could be given by the “microgravity” measurements of the critical exponents b, g, d for fluids. Although up to the present time such measurements have been carried out only for the specific heat of SF6 (exponent a), their results [28,29] clearly show the existence of the extended temperature asymptotic region in which “correction-to-scaling” terms are insignificant. The fact that using the critical exponents found in the papers [10,11] one achieves (see Refs. [13,14]) better agreement with the experiment

All contribution into the content of the paper “Long-range correlations in the statistical theory of fluid criticality” is my property as the sole author of the article. Declaration of competing interest The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The author wishes to thank Dr. S.I. Kosenko and Dr. V.V. Zavalniuk for help in the numerical calculations. This work was supported by the Ministry of Education and Science of Ukraine. Appendix A. The calculation of critical exponent n0 We need the Fourier representation of the OrnsteineZernike equation (4) (see, e.g. Ref. [3])

~ hðkÞ ¼

~cðkÞ ; 1  r~cðkÞ

(A1)

where the Fourier-component of the total PCF of the 3D fluid is

~ hðkÞ ¼

ð

dr hðrÞeikr ¼

4p k

∞ ð

dr r sinðkrÞhðrÞ

(A2)

0

and we can write similar expression for ~cðkÞ. The equation for the isothermal compressibility has the following form (see, e.g. Ref. [3]):

kT ¼



1 1 ~ þ hð0Þ : kB T r

(A3)

Equation (A1) at the critical point, itself, can be rewritten as follows:

~ c ðkÞ ¼ ~cc ð0Þ~cc ðkÞ ðkÞ: h ~cc ð0Þ  ~cc

(A4)

~ c ðkÞ in the limit k/0, we represent To analyze the behavior of h hc ðrÞ in the form: 0

hc ðrÞ ¼

Ac ðrÞ Ac ðrÞ þ n0 ; r r 6=5

(A5) 0

where we have introduced the functions Ac ðrÞ and Ac ðrÞ which formally provide the proper behavior of hc ðrÞ at any r; at that, ac0 0 cording to (9), Ac ðr /∞Þ/A∞ , Ac ðr /∞Þ/A∞ . Now, let us substitute (A5) into (A2) and pass to new variable of integration z ¼ kr:

~ c ðkÞ ¼ 4p h k

∞ ð 0

dr r sinðkrÞhc ðrÞ

8

V.N. Bondarev / Fluid Phase Equilibria 506 (2020) 112417

¼ 4p

∞ ð



0 0 Ac ðz=kÞ A ðz=kÞ : dz z sin z k9=5 6=5 þ kn 3 c n0

z

z

0

(A6)

0

~ c ðkÞ ¼ 4pA∞ I k9=5 þ 4pA0 In0 kn0 3 h 6=5 ∞

(A7)

which contains the positive integrals with the Euler's gammafunction [62]: ∞ ð

dz

sin z

0

z

s1

ps ; 1 < s < 3: ¼ Gð2  sÞsin 2

dz s

At k/0, the value of the integral in (A6) will be determined by 0 the behavior of the functions Ac ðz =kÞ and Ac ðz =kÞ at ðz =kÞ/ ∞, i.e. 0 by their asymptotic values A∞ and A∞ . Thus, the main contributions ~ c ðkÞ at k/0 in the interval 6=5 < n0 < 12=5 will be given by the to h expression:

Is ≡

∞ ð

(A8)

z

1 cc ðrÞ ¼ a4c þ 4

(A9)

# " 4 0 Aс ðrÞ A3c ðrÞAc ðrÞ þ 4 18=5þn0 r 24=5 r

(A10)

with the asymptotic behavior (10) and then let us study the difference



1 ~cc ð0Þ  ~cc ðkÞ ¼ 4p a4c þ 4 " 

0

A4c ðrÞ A3c ðrÞAc ðrÞ þ 4 0 r 24=5 r 18=5þn

∞ ð

#

z

¼

Is ; 1 < s < 3: sðs  1Þ

(A13)

1 ¼ r2c ½~cc ð0Þ  ~cc ðkÞ; ~ c ðkÞ h

(A14)

where we have omitted the terms inessential for further analysis. Substituting expressions (A7) and (A12) into (A14), we obtain with necessary accuracy the next equation at k/0: 0

1 A In0 0 k9=5  ∞ kn þ3=5 4pA∞ I6=5 A∞ I6=5

!

# 0  "  25A4∞ I14=5 9=5 4A3∞ A∞ I8=5þn0 1 n0 þ3=5 k þ 0 k ¼ 4p a4c þ r2c ; 4 126 ðn þ 8=5Þðn0 þ 3=5Þ (A15)

Equating the coefficients at k9=5 in this equation, we find, first of all, the expression for the constant A∞ :

"

A∞

63 ¼ 200p2 r2c I6=5 I14=5 ða4c þ 1=4Þ

#1=5 :

(A16)

From this expression, taking account of (A9) the important condition on the admissible values of a4c follows:

1 a4c >  : 4

(A17)

0 þ3=5 Furthermore, equating the terms kng in equation (A15) and taking into account conditions (A9) and (A17), we find that the only 0 permissible value of A∞ is

0 6 7 A∞ ¼ 0; < n0 < : 5 5

sinðkrÞ dr r 1  kr 2

0

(A11)

at k/0. Here, the next cases arise: 6=5 < n0 < 7=5 and 7= 5 < n0 < 12=5. In the first of them, 6=5 < n0 < 7=5, the coefficients at two main terms in ~cc ð0Þ  ~cc ðkÞ when k/0, will be determined just by the 0 asymptotic values A∞ and A∞ . Then, in analogy with the derivation of (A7), we obtain from (A11):

"  1 25A4∞ I14=5 9=5 ~cc ð0Þ  ~cc ðkÞ ¼ 4p a4c þ k 4 126

# 0 4A3∞ A∞ I8=5þn0 n0 þ3=5 k þ 0 : ðn þ 8=5Þðn0 þ 3=5Þ

 sin z

6 7 < n0 < : 5 5

Really, the isothermal compressibility kT (see (A3)) must be positive for any real equilibrium system [4]. On the other hand, the singular part of kT directly at the critical point can be considered as ~ c ðk /0Þ=Tc where the leading singularity will be deterthe limit h mined by the first term in the right-hand side of expression (A7). As the result, condition (A9) follows. Now, let us consider the function



1

Then, let us rewrite equation (A4) as follows:

The analysis of expression (A7) allows us to claim that

A∞ > 0::



Thus, the terms with 6=5 < n0 < 7=5 obviously cannot be present in the asymptotic expansion of the “critical” total PCF hc ðrÞ. So, let us consider ~cc ð0Þ  ~cc ðkÞ at 7=5 < n0 < 12=5. Now, for k/0 g 0 in the integral from the term with the asymptotic behavior r 18=5n (A11) will already converge at the upper limit. Therefore, at small k, this integral will begin with an analytical term f k2 . In this situation, a 2 coefficient at k will be determined by the behavior of the integrand in the whole region of variation of r, but not only at large distances r. However it is important that already the next term in ~cc ð0Þ  ~cc ðkÞ is non-analytic. To determine this term let us turn to formula (A11) at 7=5 < n0 < 12=5 and consider separately the expression

Q ðk; n0 Þ ≡ qðn0 Þk2 þ (A12)

Note, that when finding (A12) we have reduced the integrals to the form (A8) with the help of the repeated integration by parts:

(A18)

∞ ð

" dr r 2 1 

sinðkrÞ k2 r 2  kr 3!

#

0

A3c ðrÞAc ðrÞ 0 r 18=5þn

0

(A19) in which we have specially isolated the analytical term f k2 with the coefficient

V.N. Bondarev / Fluid Phase Equilibria 506 (2020) 112417

1 qðn Þ ¼ 3! 0

∞ ð

fulfillment of the equality

dr

0 A3 ðrÞAc ðrÞ r 4 c 18=5þn 0 : r

(A20)

0

The remaining integral in (A19) at 7=5 < n0 < 12=5 will lead again 0 þ3=5 to non-analytic term kng in the limit k/0. A coefficient at 0 kn þ3=5 will again be determined by the asymptotic values A∞ and 0 A∞ (cf. (A12)), and thus at small k we obtain: 0

Q ðk; n0 Þ ¼ qðn0 Þk2 þ A3∞ A∞ k3=5þn

0

∞ ð 0

 2 sin z z :  1  0 6 z z8=5þn dz

(A21) Taking the integral in (A21) by parts several times, we can reduce it to the form (A8) for the interval 7=5 < n0 < 12=5: ∞ ð



dz 0

0

z8=5þn

1

sin z

z



z2 6



"  4 1 25A∞ I14=5 9=5 ~cc ð0Þ  ~cc ðkÞ ¼ 4p a4c þ k þ 4qðn0 Þk2 4 126 # 0 4A3∞ A∞ In0 2=5 n0 þ3=5 k  0 : ðn þ 8=5Þðn0 þ 3=5Þðn0  2=5Þðn0  7=5Þ

(A23)

Now, substituting expressions (A7) and (A23) into equation (A14) and retaining the main terms at k/0 we come to the equation (cf. (A15)):



0

k

¼ 4p a4c þ



A In0 0  ∞ kn þ3=5 A∞ I6=5

!

"

4 1 2 25A∞ I14=5 9=5 k rc þ 4qðn0 Þk2 4 126

# 0 4A3∞ A∞ In0 2=5 12 n0 þ3=5 7 k  0 ; < n0 < : 5 5 ðn þ 8=5Þðn0 þ 3=5Þðn0  2=5Þðn0  7=5Þ (A24) From here we again obtain expression (A16). Besides, from (A24) the integral condition follows:

7 12 : qðn0 Þ≡0; < n0 < 5 5

Appendix B. The calculation of critical exponent n00 00

To find the exponent n00 and the form of the constant A∞ in the asymptotic expression for the “critical” total PCF (12) let us 00 continue expansion (A5), introducing a new function Ac ðrÞ, in order to guarantee the correct behavior of hc ðrÞ at all r: 0

(A22)

(A25)

This condition could play a useful role in the modeling of the 0 functions Ac ðrÞ and Ac ðrÞ at all r (however, now this is not our task). 0 þ3=5 At last, equating the coefficients at the terms kng in (A24) and using (A16), one can see that now, in contrast to (A18), non-zero 0 values for the constant A∞ are, in principle, admissible. The 0 necessary condition for the existence of the values A∞ s 0 is the

(A26)

As not complicated numerical calculation shows, the transcendental equation (A26) in the pointed out interval of n0 really has unique solution which is given by Eq. (11).

hc ðrÞ ¼

Pay attention to the negativity of this integral. As the result, taking n0 from the interval 7=5 < n0 < 12=5 and keeping the main terms in the limit k/0 we obtain the following expression:

9=5

    In0 2=5 25I14=5 8 3 2 n0 þ n0  n0 n0 þ ¼ 5 5 5 In0 504I6=5  7 7 12 ; < n0 < :  5 5 5

¼

In0 2=5 :  0 ðn þ 8=5Þðn0 þ 3=5Þðn0  2=5Þðn0  7=5Þ

1 4pA∞ I6=5

9

0

Ac ðrÞ Ac ðrÞ Ac0 ðrÞ þ n0 þ n0 0 ; r r r 6=5 00

(B1)

00

where Ac ðr /∞Þ/A∞ . Being based on (B1) and acting as when ~c ðkÞ deriving expression (A7), we find the main contributions into h at k/0:

~ c ðkÞ ¼ 4pA∞ I k9=5 þ 4pA0 In0 kn0 3 þ 4pA00 In00 kn00 3 ; h 6=5 ∞ ∞

(B2)

where the last term will be also divergent if n00 < 3. We shall make sure that the desired value of n00 will be less than 12/5, so it will e4 ðrÞ in the again be enough to restrict ourselves by the term h c “critical” direct PCF (8), based on approximation (B1). Assuming to ~c ðkÞ1 in (A14), we have from (B2) the next equation to use ½h within the main contributions at k/0:

2 0 00 0 00 1 4k9=5  A∞ In kn0 þ3=5  A∞ In kn00 þ3=5 ¼ ~ 4 p A I A I A I ∞ ∞ ∞ 6=5 6=5 6=5 hc ðkÞ 3 !2 0 0 A∞ In0 þ k2n 3=5 5: A∞ I6=5 1

(B3)

Further, to within the main terms consistent with (B1) and the asymptotic dependence (12), one can represent the “critical” direct PCF (8) in the form:

( 4 0

00 1 Ac ðrÞ A3c ðrÞ Ac ðrÞ Ac ðrÞ cc ðrÞ ¼ a4c þ þ 4 þ 0 00 4 rn r 24=5 r 18=5 r n )

0 A2c ðrÞ Ac ðrÞ 2 : þ 6 12=5 0 rn r 

(B4)

Note, that the last term in the curly brackets of (B4) must be conserved because it has the same asymptotic behavior as the term g 00 (see below). Now, the singular contributions into r 18=5n ~cc ð0Þ  ~cc ðkÞ in the limit k/0 will be determined not only by the 0 00 constants A∞ and A∞ but by the constant A∞ also. Using the explicit form (B4) and acting as when obtaining (A23), we find to within the main terms at k/0:

10

V.N. Bondarev / Fluid Phase Equilibria 506 (2020) 112417

"  4 1 25A∞ I14=5 9=5 ~cc ð0Þ  ~cc ðkÞ ¼ 4p a4c þ k þ 4qðn0 ; n00 Þk2 4 126 0

0

00

00

4A3∞ A∞ In0 2=5 kn þ3=5 4A3∞ A∞ In00 2=5 kn þ3=5   0 ðn þ 8=5Þðn0 þ 3=5Þðn0  2=5Þðn0  7=5Þ ðn00 þ 8=5Þðn00 þ 3=5Þðn00  2=5Þðn00  7=5Þ #  0 2 0 6 A∞ A∞ I2n0 6=5 k2n 3=5 7 12 ; < n0 < n00 < :  5 ð2n0 þ 2=5Þð2n0  3=5Þð2n0  8=5Þð2n0  13=5Þ 5

Here 4qðn0 ; n00 Þk2 (сf. (A23)) is an analytical contribution due to terms with the exponents n0 and n00 in (B4) (for our purposes, we will not need the explicit form of this contribution). Substituting expressions (B3) and (B5) into (A14), we come to the equation:

2 0 00 0 00 1 4k9=5  A∞ In kn0 þ3=5  A∞ In kn00 þ3=5 þ 4pA∞ I6=5 A∞ I6=5 A∞ I6=5

0

A∞ In0 A∞ I6=5

!2

terms of the corresponding powers of k in both sides of equation (B6). Then, taking into account (A16) and omitting intermediate calculations, we obtain the expression:

3 k2n 3=5 5 0

 "  4 1 2 25A∞ I14=5 9=5 ¼ 4p a4c þ k rc þ 4qðn0 ; n00 Þk2 4 126 0

0

(B5)

(B6) 00

00

4A3∞ A∞ In0 2=5 kn þ3=5 4A3∞ A∞ In00 2=5 kn þ3=5   0 ðn þ 8=5Þðn0 þ 3=5Þðn0  2=5Þðn0  7=5Þ ðn00 þ 8=5Þðn00 þ 3=5Þðn00  2=5Þðn00  7=5Þ #  0 2 0 3 A∞ A∞ I2n0 6=5 k2n 3=5 7 12  ; < n0 < n00 < : 5 8ðn0 þ 1=5Þðn0  3=10Þðn0  4=5Þðn0  13=10Þ 5

Now, one should equate the coefficients at the equal powers of k in both sides of equation (B6). In this case, as well as earlier, equalities (A16) and (A26) will follow, but as an integral condition we obtain, instead of (A25):

7 12 qðn0 ; n00 Þ≡0; < n0 < n00 < : 5 5

(B7)

However the situation with the rest of the terms in equation (B6) turns out to be rather curious. When trying to equate the terms 00 þ3=5 in both sides of (B6) and assuming the existence of nonkng 00 trivial solution for the constant A∞ , we immediately come to the 00 equation for n , exactly coinciding in form with equation (A26) for n0 . This means that, taking into account the last term in (B1), nothing new is obtained, that is, this term should be eliminated, 00 0 3=5 setting A∞ ¼ 0. Further, if we shall try to equate the terms k2ng in (B6), then due to their certainly different signs, we should come again to the trivial solution (A18), but now for the interval 7= 5 < n0 < 12=5, what is absurd in view of condition (11) for the existence of 0 00 A∞ s0. In reality, a non-trivial solution for A∞ can be found noncontradictorily if we require the equality of the exponents

n00 þ

3 3 ¼ 2n0  ; 5 5

(B8)

which is equivalent to equation (13) of the main text. Under the condition (B8), we must equate the resulting coefficients at the

 0 2 C1 A∞ A∞ ¼ ; C2 A∞ 00

(B9)

where we have designated:

C1 ≡ þ

In0

!2

I6=5 189I2n0 8=5 ; 100ðn0 þ 1=5Þðn0  3=10Þðn0  4=5Þðn0  13=10ÞI14=5 (B10)

C2 ≡

I2n0 6=5 I6=5



63I2n0 8=5 : 50ðn0 þ 1=5Þðn0  3=10Þðn0  4=5Þðn0  13=10ÞI14=5 (B11)

Using the explicit values of n0 from (11) and of the integrals (A8) we find C1 ¼ 1:99953::: and C2 ¼ 1:42476:::, so that the ratio figuring in (B9) is C1 =C2 ¼ 1:40341:::. Turn our attention to the fact 00 00 that A∞  0 (сf. (A9)); moreover, from (B9) it follows that A∞ can 0 become zero only simultaneously with A∞ . Note, at last, that the 0 values of constants A∞ and A∞ can be set independently, whereas 00 A∞ will be expressed through them by (B9). Thanks to this “genetic”

V.N. Bondarev / Fluid Phase Equilibria 506 (2020) 112417

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