Comparison of the scaling models for substance densities along saturation line

We discuss various thermodynamic equations used to represent the properties along the saturation line (fluid density, gas density, order parameter, mean diameter, etc) in a neighborhood of the critical temperature Tc. These properties are described scaling functions, depending on some parameters including the critical exponents α and β. Along with well-known models, we investigate a new model that represents the mean diameter as a sum of two scaling members in the critical region. The first term of this sum depends on the exponent, a and the second one depends on exponent, 2β. In the paper is given a methodological rationale for the new function representing the mean diameter. We have made numerical estimates for sulphur hexafluoride using the parameters involving with the above scaling equations.


Introduction
One of the most urgent problems is connected with a construction of thermodynamic functions (the liquid density (ρ l ), the gas density (ρ g ) of the order parameter (f s ), the mean diameter (f d ) etc) in a specified neighborhood of the critical temperature T c . The problem plays an important role in the scale theory of critical phenomena (MT). These thermodynamic functions depend on the critical exponents, α, β, as well as they must follow to a number of the MT conditions, including: (i) the indexes, α, β, are universal for all substances, (ii) α exponent is universal not only for the mentioned functions, but also for the saturation pressure (P ) and isochoric heat capacity (C v ).
Our analysis shows that the different values of the critical exponent α = (α 1 , α 2 , · · · ) used in the literature. For example, in common prior to 2000 scaling model of Wegner [1] uses the following series f s = (ρ l − ρ g ) (2ρ c ) −1 = B s0 τ β 1 + B s1 τ β 1 +∆ , where α = 0.109, β = 0.325-the exponents those follow from the MT and have an error estimated as ∼ 0.1% [2], ∆ = 0.5 is an amendment for the first non asymptotic member, (B si , B di )coefficients determined by a statistical processing of experimental data. It is important that equation (2) contains the singular member B d0 τ 1−α 1 that is the derivative df d /dT is singular (df d /dT ≈ B d0 τ 1−α 1 → −∞ when τ → 0, the parameter α 1 is small). Earlier, the first theoretical step in the construction of the functions (ρ g , ρ l , f s , f d · · · ) has been made by Landau [2,3]. He has used a special decomposition of the function g (the free energy density), which contains the arguments (the relative density (∆ρ = ρ/ρ c − 1), τ ). He has suggested the following expression of ∆ρ in the vicinity of the critical point ∆ρ g,l = ± (a/c ) τ β 2 + (b/(2c)) τ 1−α 2 where ∆ρ g and the sign "−" relate to gas branch, ∆ρ l and the sign "+" relate to the liquid branch, α 2 = 0 and β 2 = 0.5 are theoretical values of the critical exponents obtained by Landau [3]. Equation (3) contains other values of α 2 and β 2 in comparison to equations (1), (2). This equation reflects a Landau hypothesis, H L , in the following form: (i) equation (3) must contain six parameters (B d0 , B s0 , ρ c , T c , β 2 , α 2 ), while β 2 and α 2 are not dependent on substances, (ii) it includes a linear term (α 2 = 0) and does not contain a scaling member in f d with an index 1 > α > 0, (iii) it has positive coefficients B d0 > 0 and B s0 > 0.
It is shown in [4] that f d (3) mast include α 2 ≈ 0.1. The authors accepted that c = c 1 τ v and v ≈ 0.1 and got where (5) is accepted as the hypothesis, H ML , and meets the following conditions: (i) it contains only five parameters (B d0 , B s0 , ρ c , T c , β 2 ), (ii) it does not include a linear member, (iii) it has positive coefficients B d0 > 0, B s0 > 0.
These functions have been investigated in a great number of papers, where numerical data on parameters (B d0 , B s0 , ρ c , T c , β, α) are placed for a wide range of substances. So, in 1990 Anisimov [2] has developed a model that includes α 1 and β 1 , proposed by Wegner [1]. His model has a satisfactory accuracy in the interval τ low · · · τ high = 10 −4 · · · 10 −2 for H 2 O and is written in the form where 2 in a few works including [5][6][7]. In accordance with the H A , Anisimov [5] has got the numerical data of f d diameter written in the form Equation (8) satisfies the following conditions: (i) contains an additional singular term B d2 τ 2β 1 , (ii) includes a linear term, (iii) the exponents α 1 and β 1 meet the following condition: 1 > 1 − α 1 > 2β 1 (2β 1 = 0.65, 1 − α 1 = 0.89), (iv) the values of α 1 and β 1 chosen as theoretical ones.
These conditions lead to the conclusion that the second scaling member is dominant over others in some small region 0 < τ < τ A . Equations (6) and (8) take the following form in this range The derivative df d /dT is singular (df d /dT ≈ τ 2β 1 −1 for τ → 0) for equations (8) and (9). Equations (9) and (4) are similar in the shape: the exponent β 1 included in the diameter f d (equation (9) with factor of two. Numerical data on the parameters (B d0 , B d1 , B d2 ) included in equation (8) obtained in [5] for a number of substances, including SF 6 and N 2 in the range τ low · · · τ high = 10 −4 · · · 10 −2 . At the same time in their calculations, the authors of [5] attracted the experimental data on such properties along the saturation line as density ρ, specific heat C v and pressure P . M. Fisher introduced the term "complete scaling" within the H A hypothesis in the pioneering work [7]. Equation (8) has an aim to improve f d structure and to increase f d accuracy. It reflects current trends MT. Thus, the second scaling member first introduced in the model (8).
Note firstly, that the experimental critical exponents (α exp , β exp ) are presented in literature and obtained by statistical processing of the experimental data for a wide range of substances. These values significantly different from the theoretical values of α and β mentioned above. The values of α exp are given in references [2,4,8,9] and cover a whole range α exp = 0.10 · · · 0.14.
Fourthly, a wide range of values of α and β, presented in literature, talks about the important issue of "What are the values of the exponents and beta implemented in reality and how they should be used in the construction of scaling equations with respect to a new substance?".
Fifthly, we have a task to receive methodical substantiation of equation (8) and to make numerical estimates of the parameters (α, β, B d0 , B d1 , B d2 ), included in this equation, on the basis of experimental data on the density of a test substance.

2.
A correlation of the diameter, f d , and chemical potentials along the saturation line Let us consider the method of explaining the appearance of the additional singular term in equation (2), (3) and (7).
We are using differential equations of the thermodynamics to chemical potentials µ g , µ l on the saturation line in the form where v g , v l , s g , s l are the specific volumes and entropies along the saturation line. A summation of equations (10) and (11) yields We perform some transformation using equation (12) in order to obtain expressions: a) for the sum, ρ g + ρ l , and b) for the diameter, f d .
In the first step, we present the sum of the volumes as In the second step, we write the sum of the densities bringing equations (12) and (13) The third step will present the components of equation (14) those are the leading members of the scaling at small τ . For example, the terms, B s0 τ β , B d0 τ 1−α , are taken as an initial approximation of the terms those are included in the density. We write these components on the base of formulas presented in references [2,5,7,8,10] in forms The connection between the chemical potential and the specific heat has the form, and is taken into account in equation (19). The connection was studied in [5,7,10].
We assume that the densities have the form ρ l = ρ c 1 + B s0 τ β + B d0 τ 1−α and ρ g = ρ c 1 − B s0 τ β + B d0 τ 1−α and the complex, (ρ g ρ l ), can be written as In the fourth step, we write volumes v l and v g as v g = 1/ρ g = 1/(ρ c (1 + ∆ρ g )) = 1/ρ c 1 − ∆ρ g + ∆ρ 2 g + · · · ≈ (21) is the component obtained as a quadratic term of the Maclaurin series for the factor 1/(1 + ∆ρ l ); ∆ρ g = −B s0 τ β + B d0 τ 1−α , ∆ρ 2 g = B 2 s0 τ 2β + B d0 τ 2(1−α) + · · · . At the fifth step, we write the following complexes included in rquation (14) We write complexes s g (dP /dT ) −1 (ρ g ρ l ) and s l (dP /dT ) −1 (ρ g ρ l ), highlighting the constants and terms containing τ 2β function We derive an expression for the sum of (ρ g + ρ l ), using equations (22), (23) and separating first constant and terms containing τ 2β function as where the coefficients and parameters are related by the equation We write amount (ρ g + ρ l ), allocating first member comprising τ 2β function in the form Using equation (26), we present the expression for f d in the form This methodical approach shows that you can get a model of f d that is consistent with the models (4) and (9): on the following grounds: this model has a singular component comprising a function τ 2β .

Numerical data of the scaling equations for sulfur hexafluoride
It is of interest to make a comparative analysis of these equations using an example of the description of the properties (ρ l , ρ g , f s , f d ) for sulphur hexafluoride, which is selected as a test substance with very precise experimental (ρ g , ρ l , T )data [11] in the interval between τ low = 10 −4 to τ tr = 0.3. Along with this, it is necessary to solve the problem: to obtain experimental estimates for parameters of the scaling equations discussed.
At the first stage we consider the equations studied in [4,8] and consisted of scaling (F scale ) and regular (F reg ) parts with the numerical parameters in relation to SF 6 . These equations have a form where B si , B di , i = 0, 1, 2-coefficients related to F scale , B si , B di , i = 3, 4-coefficients related to F reg , α 4 , β 4 -indices defined on the basis of statistical treatment of experimental (ρ g , ρ l , T )-data. F scale structure meets MT. Equations (28) and (29) include the optimal values of the critical parameters (T c , ρ c , α 4 , β 4 , B s0 , B d0 ), which are calculated with empirical coefficients (B si , B di ) on the basis of experimental (ρ g , ρ l , T )-data and a nonlinear least square method (NLLSM) [4,8]. Note that in equation (29) is missing the linear term in accordance with the hypothesis H ML ; this hypothesis is used in several papers including [8,9]. The parameters of (28) and (29) (table 1) are obtained by us using experimental (ρ g , ρ l , T )-data [11] of SF 6 . Parameters (T c , ρ c , α 4 , β 4 , B s0 , B d0 ) is used as an initial approximation parameters of equation (27).
In accordance with NLLSM an initial approximation for the parameters (T c , ρ c , α 4 , β 4 , B s0 , B d0 ) was chosen in accordance with literature data including 1) α 4 = 1 − 2β 1 = 0.35 in accordance with the hypothesis H A , 2) β 4 = 0.325. The proposed equations reproduce the most  Figure 1. An additional function F and its members. 1-function F exp , built from experimental (ρ g , ρ l , T )-data [11], 3-term B s0 2 τ 2β 4 ρ c , 6-sum of terms B dexp τ 2β 4 ρ c + B d0 τ 1−α 4 ρ c ; 7-limit accurate (ρ g , ρ l , T )-data [11], covering the interval from τ low to τ tr with with a small root mean square deviation S = 0.34%, which is determined on the base of deviations of experimental data from corresponding values of the density obtained on the base of (28) and (29).
In the second stage of the analysis we have introduced an additional dimension function, F , in the form where F α and F β -terms of (30), for example, F α = B d0 τ 1−α 4 ρ c -a term corresponding to the first term of the equation (29). The analysis shows that the function F can be built with the help of equation (29); this variant is in satisfactory agreement with the function F exp (symbols 1, figure 1) calculated on the base of experimental (ρ g , ρ l , T )-data [12], and has an error δF exp ≈ 2δρ exp , here δρ exp -error of the data [12]. The deviation of the function F from F exp lies in the range ∆ρ = ±1.4 kg/m 3 at τ = τ low · · · 0.3. We have got F that coordinates satisfactory (within the estimate of δF exp ) to equation (29) and to the values of F exp . It allows us to conclude that f d (29) has a small error.
It possible to conclude from our comparison: (i) the term F α coinsides with the function F exp at τ = τ low · · · 0.01, (ii) the term, B 2 s0 τ 2β ρ c , is substantially higher than the function F exp at τ = τ low · · · 0.3, (iii) the term, B 2 s0 τ 2β , can not be included in the equation (29) along with the term B d0 τ 1−α 4 , since this amount is several times greater than f dexp .
The experimental values of F exp give some option for the empirical determination of the term F βexp , which can be complementary to the term F α and to obtain the value of the function F opt = F α + F βexp calculated and satisfactorily consistent with the values F exp . We have considered a term F βexp = B dexp τ 2β 4 ρ c (symbol 8), which meets the following conditions: (i) B dexp = 0.0518, (ii) sum B dexp τ 2β 4 ρ c + B d0 τ 1−α 4 ρ c represents the function F opt (symbol 6); it locates between the function F exp and the limit F high , (iii) the term F βexp (symbols 8) intersects the term F α (symbols 9) when τ = 3 × 10 −5 ; the term F βexp is a liding term compared to term F α at smaller τ ; it is possible to accept that there is a temperature limit Note, firstly, that the function F opt is higher than the term F α at τ = τ low · · · 0.01, but not beyond the margin ofthe error, ∆ρ = 1.4 kg/m 3 . Secondly, a quantitative characterization of the functions Thirdly, the comparison of the equations let us conclude: (a) the function F exp ((ρ g , ρ l , T )-points [12]) located systematically higher than the function F exp ((ρ g , ρ l , T )-points [11]); so, this deviation is 1.1 kg/m 3 at τ = 5 × 10 −4 ; (b) the value of F exp is F exp = 0.35 kg/m 3 for the data related to [11]; (c) the value of F exp is F exp = 1.45 kg/m 3 for the data related to [12]; (d) the value of F is F = 1.30 kg/m 3 for the data related to equation (8).
Fourthly, a good agreement between the function F opt and the function F exp allows us to assume that the diameter f dopt has a following form where B d0 and B dexp are the coefficients found on a basis of experimental data on the density at the saturation line. It is considered one more variant of F βexp in the form F βexp = −B dexp τ 2β 4 ρ c and the following conditions: where F opt is located between F exp and the boundary, F low = F exp − ∆ρ. It can be written F βexp = −F α and F opt = 0 at τ A = 3 × 10 −5 . There is some anomaly region, 0 < τ < τ A , where F opt < 0, compare relation (33) with condition (iv). Our evaluation has shown: (i) starting (ρ g , ρ l , T )-points [10], which are used in the work [5], have a considerably higher error (about an order of the magnitude) than the estimate of δρ exp related to (ρ g , ρ l , T )-data [12], (ii) the value of f dexp , wich are derived from (ρ g , ρ l , T )-points [12], deviate systematically from the values of f dexp related to (ρ g , ρ l , T )-data [10]; this cause leads toa low accuracy of f dopt related to model (8).

Conclusion
The proposed method gave the possibility to obtain the equation for the diameter, f dopt , which contains a term B dexp τ 2β 4 . For example, our test with SF 6 has given numerical estimates of the parameters α 4 and β 4 using statistical treaty of experimental (ρ g , ρ l , T )-data [12] in contrast to α and β those are related to equation (8) and chosen as theoretical values. Our analysis has shown that the equation f dopt (31) coincides with the experimental values of f dexp within the limits caused by the error of the reference data at temperatures τ = τ low · · · 0.01. A comparison of equations (31), (4) and (8) shows: (i) the coefficient B dopt coincides in the sign with the coefficient B d2 of equation (8), parameters (α 4 , β 4 ) coincide with the same parameters (α 1 , β 1 ) within (1 · · · 3)%; (ii) the coefficient B d0 of equation (31) is positive, and the coefficient B d0 of equation (8) is negative, which contradicts the hypothesis, H ML . Equation (31) provides an important condition, which is typical for equations (8) and (28), namely, the derivative df d /dτ is singular (df d /dτ ≈ B dexp τ 2β 4 −1 when τ → 0). The numerical data on the equation (31) explain the fact that the rate of 2β 1 ≈ 0.65 has not been calculated in any of the works in which the indexes of f d are defined on the basis of processing the experimental (ρ g , ρ l , T )-data. Our analysis shows that the term B dexp τ 2β 4 is small in comparison with the term B d0 τ 1−α 4 at temperatures τ low · · · 0.01. Therefore the diameter, f d , contains a term, B d0 τ 1−α 4 in the known equations; the degree, (1 − α 4 ) lies in the range of 0.85 · · · 0.90 and is found by processing the experimental (ρ g , ρ l , T )-data for a large number of substances. These values significantly exceed the degree included of 2β 1 = 0.65, which is the equation of diameter f d (9). Due to our oppenion, f d has the form f d ≈ B dexp τ 2β 4 > B d0 τ 1−α 4 at τ < τ A ; it will be possible to determine the term, B dexp τ 2β 4 , more accurately on the basis of experimental data to be obtained for the SF 6 at temperatures 0 < τ < 0.00003.