Journal of Thermodynamics
Volume 2014 (2014), Article ID 496835, 15 pages
http://dx.doi.org/10.1155/2014/496835

Research Article

Towards the Equation of State for Neutral (C₂H₄), Polar (H₂O), and Ionic ([bmim][BF₄], [bmim][PF₆], [pmmim][Tf₂N]) Liquids

Vitaly B. Rogankov and Valeriy I. Levchenko

Department of Physics, Odessa State Academy of Refrigeration, Dvoryanskaya Street 1/3, Odessa 65082, Ukraine

Received 5 August 2014; Accepted 4 November 2014; Published 16 December 2014

Academic Editor: Pedro Jorge Martins Coelho

Copyright © 2014 Vitaly B. Rogankov and Valeriy I. Levchenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Despite considerable effort of experimentalists no reliable vapor-liquid coexistence at very small pressures and liquid-solid coexistence at high pressures have been until now observed in the working range of temperature / for ionic liquids. The measurements of high-pressure properties in low-temperature stable liquid are relatively scarce while the strong influence of their consistency on the phase equilibrium prediction is obvious. In this work we discuss the applicability of fluctuational-thermodynamic methodology and respective equation of state to correlate the properties of any (neutral, polar, ionic) liquids since our ultimate goal is the simple reference predictive model to describe vapor-liquid, liquid-liquid, and liquid-solid equilibria of mixtures containing above components. It is shown that the inconsistencies among existing volumetric measurements and the strong dependence of the mechanical and, especially, caloric derived properties on the shape of the functions chosen to fit the experimental data can be resolved in the framework of fluctuational-thermodynamic equation of state. To illustrate its results the comparison with the known experimental data for [bmim][BF₄] and [bmim][PF₆] as well as with the lattice-fluid equation of state and the methodology of thermodynamic integration is represented. It corroborates the thermodynamic consistency of predictions and excellent correlation of derived properties over the wide range of pressures /.

1. Introduction

Behavior of low-melting organic salts or ionic liquids (ILs) [1–6] in the region of phase transitions is qualitatively similar to that either for high-temperature nonorganic molten salts or long-hydrocarbon-chain organic solvents and, even, for polymer systems. Such characteristic features as negligible vapor pressure , undefined critical parameters , , for vapor-liquid ()-transition, split of liquid-solid (l,s)-boundary onto melting and freezing branches, existence of glassy states make the problem of metastability to be quite complex but vital for many potential uses of ILs. In particular, thermodynamic modeling and computer simulation of the phase behavior in mixtures formed by ILs with water and low-molecular organic solvents such as ethylene can be of great importance for the further tuning of their operational parameters. If one proceeds from a pure to a mixed fluid, it is especially advantageous to develop the same format of reference equation of state (EOS) and the common format of reference pair potential (RPP) for each component and mixture.

As a first step toward consistent modeling of the phase behavior of IL and its solution we demonstrate in this work how the fluctuational-thermodynamic (FT) EOS [7–12] and the relevant finite-range Lennard-Jones (LJ) RPP can be applied to model the underlying structure and properties of low-molecular (C₂H₄, H₂O) and imidazolium-based (1-butyl-3-methylimidazolium tetrafluoroborate ([bmim][BF₄]), 1-butyl-3-methylimidazolium hexafluorophosphate ([bmim][PF₆]), 2,3-dimethyl-1-propylimidazolium bis(trifluoromethylsulfonyl)imide([pmmim][Tf₂N])) solvents. For any pure component FT-model is based either on the measurable coexistence-curve input data , , (if they are achievable as for C₂H₄ and H₂O) or on the also measurable one-phase density of liquid at atmospheric pressure   ( MPa, T) for ILs. This methodology becomes purely predictive for density in any one-phase -regions including their metastable extensions. Only the measurable isobaric heat capacity data have to be added to the set of input data for prediction of other caloric properties (isochoric heat capacity , speed of sound , and Gruneisen parameter ) at higher pressures and lower or higher temperatures where is the hypothesized normal boiling temperature . Its existence itself is a debatable question because the thermal decomposition may be former .

Such approach was proposed recently [7, 8] to reconstruct the hypothetical -diagram of any ILs in its stable and metastable regions on the base of only standard reference data on density at [1–4] and one free parameter, an a priori unknown value of the excluded volume . To our knowledge this is first attempt to predict simultaneously the whole set of one-phase and two-phase properties for ILs without the fit at any other pressures including the negative ones. It was argued that the particular low-temperature variant of the most general FT-EOS [9–12] should be used to obtain the consistent prediction of volumetric properties and the standard response functions , , by the following equations:

where is the excluded molecular volume and is the -dependent effective cohesive energy. The derivative affects the thermal expansion and the thermal-pressure coefficient while the isothermal compressibility depends only on -value at the given pressure. The changeable sign of two thermal derivatives , offers a possibility to predict the properties of anomalous low-temperature substances (such as water, for example) too [7, 8].

Fortunately we have obtained now [13–19] a possibility to test our predictions not only by the direct experimental one-phase data [14, 16, 18, 19] on - and -surfaces. Another possibility is offered by comparison of the predictions obtained by FT-EOS for the critical parameters of ILs ([bmim][BF₄]: = 962,3 K, = 3503,9 kPa, = 438,565 kg·m⁻³ with those predicted here by the Sanchez-Lacombe EOS for lattice fluid (LF) [15]: = 885,01 K, = 2829 kPa, = 248,565 kg·m⁻³ as well as with those simulated by GEMC-methodology [6]: = 1252 K, = 390 kPa, = 181 kg·m⁻³. It seems that the relatively close location of -parameters predicted by both EOSs is some guarantee of their reliability while and from [6] are significantly overestimated and underestimated, respectively. Interestingly, the known descriptive factor of compressibility estimated by Guggenheim [20] in the vicinity of triple point for argon as is equal to close values for FT-EOS and for LF-EOS but only to very small value for result of GEMC-simulations if the common realistic estimate (see below) = 5,350646 mol·dm⁻³ at T = 290 K is used. Moreover, it will be shown that the characteristic dimensional parameters , , and another compressibility factor obtained by Machida et al. [14] by the fit to ()-experimental data for [bmim][BF₄] and [bmim][PF₆] provide the structural estimates of hard-core volume, number of lattice sites in a cluster, and energy of near-neighbor pair interactions which are surprisingly close to ones independently predicted by the FT-model of a continuum substance.

Taking into account the compatibility of above results it is important to consider the presumable similarity between the square-well fluid (which may be thought of as a continuum analogue of the lattice-gas (LG) or lattice-fluid (LF) systems) on the one hand and the LJ-fluid of finite-range interactions (RPP) on the other. This conceptual analogy has been pointed out long ago for the critical region by Widom [21] who suggested that it is the propagation of attractive correlations in the LG which determines the peculiarities of criticality. However, such unphysical LG-predictions at low temperatures of the ()-plane as the nonexistence of a ()-transition suggest that repulsive forces are not being treated properly by this RPP-model. In contrast with the discrete LG-model, it seems that both attractive and repulsive forces are being dealt with properly in the square-well continuum fluid because it exhibits both ()- and ()-transitions. The serious restriction of latter is however evident since any singularities of RPP imply an artificial jump of pair-distribution isotropic function at the point of cutoff radius for attractive interactions.

In this context only the shifted and smoothed at -point LJ-potential [5, 6] seems to be appropriate as RPP for a continuum system. Of course, the algebraic form of the respective reference EOS is essential too. In accordance with the statistical-mechanical arguments presented by Widom [21] there are the set of alternative forms including the original vdW-EOS and the LG-EOS in the well-known Bragg-Williams approximation which share the common restrictive feature. One may suppose that the probability of finding some prescribed value of the potential energy at an arbitrary point in the fluid is independent of at fixed  : . Another simplifying assumption is that such EOS supposes only two types of fluid structure, one of the excluded (or hard-core) volume where the singular hard-sphere branch of potential is infinite and one of free volume where the potential is uniform, weak, and unrestricted (an infinite-range rectilinear well). It should be directly proportional to density where is the total configurational energy and is the constant vdW-coefficient. These historical notes are important to explain how one can go beyond the above restriction of -independency by adoption of linear  -dependence for a generalized specific or molar energy (see also (8) below). Consider

Another aim of the developed FT-EOS follows from the possibility [7] to estimate the effective LJ-parameters without any fit. Indeed, their general T-dependent values,

are determined simply in the low-temperature range of all ILs where is constant in ((1)–(4)) while the compressibility factor of saturated liquid becomes negligible as well as the vapor pressure trends to zero. Taking into account this asymptotic behavior it is especially important to study the possible correlations of these parameters in the RPP-model of an effective LJ-potential for ILs as the functions of total molecular weight . This concept is unusual for the conventional consideration of a separate influence of the anion’s and cation’s components. It may provide, in principle, the useful insight the nature of -transition in ILs by effective capturing underlying pair interactions.

The distinction of both FT-EOS and LF-EOS [14] from the conventional hard-sphere reference EOS is crucial to provide the quantitative description of one-phase liquid. The formers include the quadratic in density contribution, which is dominating at high pressures along the isotherms. The latter considers this term as a small vdW-perturbation for the hard-sphere EOS. Such perturbation approach is not directly applicable to associating fluids such as water and alcohols for which presence of hydrogen bonding, anisotropic dipolar , or coulombic interactions in addition to isotropic dispersive attractions is inconsistent with the main assumption of the perturbation methodology that the structure of a liquid is dominated by repulsive forces [15].

The FT-model promotes the more flexible approach in which the above factors of attraction and clustering can be effectively accounted by the -dependence. It was firstly confirmed by Longuet-Higgins and Widom and, then, by many authors that a combination of Carnahan-Starling EOS, for example, with the vdW-perturbation is a reasonable approximation for the - and -phases but not the -phase. Guggenheim [20] has concluded its applicability only to a liquid when large clusters are more important than small clusters (i.e., at low temperatures ). In contrast with this observation, the general FT-EOS provides the adequate representation of entire subcritical range including the critical region and ()-phase transition [9–12]. It will be shown below by FT-model without undue complexity of calculations.

2. Universal FT-EOS for Any Low-Temperature Fluids

2.1. General Form of FT-EOS for Subcritical Temperatures

It is often claimed that the original van der Waals (vdW)-EOS with two constant coefficients , determined by the actual critical-point properties , is only an approximation at best and cannot provide more than qualitative agreement with experiment even for spherical molecules. However, it was proved recently [9–12] that the general FT-EOS with three -dependent coefficients,

is applicable to any types of fluids including ILs. The measurable volumetric data of coexistence curve (CXC) have been used for evaluation of -dependences without any fit. Consider

where the reduced slope of -function is defined by the thermodynamic Clapeyron’s equation:

This fundamental ratio of the -latent heat to the thermodynamic work of -expansion is the main parameter of FT-coefficients determined by ((8)-(9)). It should be calculated separately in each of high-temperature [9–12] v- and l-phases to obtain the reasonable quantitative prediction of one-phase thermophysical properties. The general FT-EOS is applicable to the entire subcritical range but it can be essentially simplified to the form of (1) if .

2.2. Particular Form of FT-EOS for Low Temperatures

An absence of input CXC-data , , for ILs is the serious reason to develop the alternative method for the evaluation of T-dependent FT-coefficients. The thermodynamically-consistent approach has been proposed in [7, 8] for the particular form of FT-EOS (1) applicable in the low-temperature range from the triple (or melting ) point up to the -point. Former one is usually known for ILs while the latter one is, as a rule, more than temperature of thermal decomposition  K. The methodology was tested on two low-molecular-weight substances (C₂H₄, H₂O) and two imidazolium-based ILs ([bmim][PF₆], [pmmim][Tf₂N]) with the promising accuracy of predictions even for the isothermal compressibility up to the pressure P = 200 MPa.

To illuminate the distinction between the particular (reference) and general form of FT-EOS let us discuss in brief the main steps of the proposed procedure. Its detailed analysis can be found elsewhere [7, 8]. The algorithm is as follows.

Step 1. At the chosen free parameter one determines the orthobaric molar densities ; to solve the transcendent equation:

for the reduced entropy (disorder) parameter and the respective molar heat of vaporization. Consider

Step 2. The universal CXC-function in (12) is determined by equalities

and it provides the possibility to estimate a preliminary value of ,

as well as to evaluate the orthobaric densities at any if  the function is known. Consider

Step 3 (-variant of -prediction [7, 8]). To calculate its values one must obtain two densities (at the assumption ) from equation

where , and the -function provides the preliminary estimate of for the low-temperature range (at the consistent assumption: ). Consider

Step 4. One substitutes from (18) in ((13a), (13b), (16a), (16b)) to calculate , , , respectively.

Step 5. A preliminary value of may be estimated then by the more restrictive assumption (used also in the famous Flory-Orwoll-Vrij EOS developed for heavy n-alkanes). Consider

Step 6 (-variant of -prediction). To control the consistency of methodology one may use instead of Step 3 (-variant) the same equation (17) with the approximate equality to solve (16b) at the a priori chosen -value for determination of alternative , and so forth, (Steps 4 and 5). Just this approach ( -variant) has been used below in the low-temperature range of [bmim][BF ₄].

Step 7. The self-consistent prediction of a hypothetical -diagram requires the equilibration of CXC-pressures by FT-EOS (1) with the necessary final change in -value from (19) to satisfy the equality

Only in the low-temperature range the distinction between the preliminary definition (18) and its final form (19) for -values is not essential at the prediction of vapor pressure .

3. Reference Equation of State, Effective Pair Potential, and Hypothetical Phase Diagram

To demonstrate universality of approach and for convenience of reader we have collected the coefficients of FT-EOS (1) for neutral (C₂H₄) and polar (H₂O) fluids [7, 8] in Table 1 and added in Table 2 to other ILs ([bmim][PF₆], [pmmim][Tf₂N] [7, 8]) the data for [bmim][BF₄] obtained in this work (Table 3). When temperature is low FT-model follows a two-parameter correlation of principle of corresponding states (PCS) on molecular level as well as a two-parameter correlation of PCS on macroscopic level.

Table 1: Coefficients of FT-EOS (1) for neutral (C₂H₄) and polar (H₂O) substances.

[dm³/mol], [J·dm³/mol²], [K]

C 2H 4 
H 2O 

= 0,04181
= 0,01658

103,99
1557,12
273,16
517,162

105
1459,76
278,15
527,171

110
1127,30
283,15
534,941

115
935,490
288,15
540,955

120
812,151
293,15
545,767

125
725,634
298,15
548,270

130
661,817
303,15
549,648

135
613,602
308,15
549,424

140
575,528
313,15
548,755

145
545,191
318,15
547,172

150
520,785
323,15
544,760

155
500,788
328,15
541,609

160
484,652
333,15
537,810

165
471,466
338,15
533,455

169,35
462,006
343,15
529,044

348,15
524,605

353,15
519,789

358,15
514,676

363,15
509,674

368,15
504,480

373,15
499,159

Table 2: Coefficients of FT-EOS (1) for ILs: [bmim][PF₆], [pmmim][Tf₂N], and [bmim][BF₄].

[dm³/mol], [J·dm³/mol²], [K]

[bmim][PF₆]
[pmmim][Tf₂N]
[bmim][BF₄]

= 0,1953
= 0,2711
= 0,178

285
8433,52
288,15
11724,8
290
9470,0

290
8214,24
295,15
12224,5
300
8900,9

295
8015,04
304,25
12279,0
310
8435,0

300
7833,44
313,65
11728,9
320
8047,5

305
7667,34
324,35
11308,4
330
7720,8

310
7514,97
335,15
10672,3
340
7442,2

315
7374,83
344,65
10051,1
350
7202,6

320
7245,64
350,15
9709,50

325
7126,31

330
7015,89

335
6913,55

340
6818,59

345
6730,37

350
6648,36

Table 3: Predicted hypothetical ()-transition in the low-temperature range for FT-model of [bmim][BF₄] based on the experimental data [1, 2] treated by FT-EOS (-variant of ()-prediction).

(K)
(mol/dm³)
(mol/dm³)
(kPa)
(J/mol)
(J·dm³/mol²)
(K)

290
5,350646
3,09E − 08
7,45E − 05
53082
9470,0
290

300
5,322170
2,1E − 07
5,24E − 04
49866
8900,9
300

310
5,293693
1,01E − 06
0,002
47230
8435,0
309,99

320
5,265215
3,74E − 06
0,009
45032
8047,5
319,99

330
5,236735
1,13E − 05
0,031
43175
7720,8
329,99

340
5,208254
2,92E − 05
0,082
41587
7442,2
339,99

350
5,179771
6,61E − 05
0,192
40216
7202,6
349,99

360
5,151287
0,000135
0,404
39022
6994,6
359,99

370
5,122802
0,000253
0,778
37974
6813,0
369,98

380
5,094315
0,000441
1,393
37048
6653,5
379,96

390
5,065828
0,000725
2,347
36225
6512,7
389,94

400
5,037338
0,001131
3,754
35488
6387,8
399,91

410
5,008848
0,001689
5,740
34826
6276,6
409,86

420
4,980355
0,002428
8,446
34228
6177,4
419,79

430
4,951862
0,003379
12,017
33684
6088,7
429,70

440
4,923367
0,004569
16,604
33188
6009,1
439,59

450
4,894871
0,006027
22,359
32733
5937,6
449,45

460
4,866373
0,007778
29,432
32314
5873,2
459,27

470
4,837873
0,009843
37,968
31926
5815,4
469,05

480
4,809372
0,012244
48,105
31566
5763,2
478,79

490
4,780869
0,014997
59,977
31230
5716,3
488,49

500
4,752365
0,018119
73,704
30914
5674,1
498,13

510
4,723859
0,021622
89,400
30618
5636,1
507,72

511
4,721008
0,021993
91,082
30589
5632,5
508,67

512
4,718157
0,022369
92,785
30561
5629,0
509,63

513
4,715307
0,022748
94,509
30532
5625,4
510,58

514
4,712456
0,023132
96,254
30504
5622,0
511,54

515
4,709605
0,023519
98,020
30476
5618,6
512,49

516
4,706754
0,023910
99,806
30448
5615,2
513,44

517
4,703903
0,024306
101,615
30420
5611,8
514,40

One the most impressed results of FT-methodology is shown in Figure 1 where the comparison between such different high- and low-molecular substances as ILs and C₂H₄, H₂O is represented. The results based on the coefficients of Tables 1 and 2 demonstrate that the proposed low-temperature model provides the symmetric two-value representation of vapor pressure similar to that observed for the ferromagnetic transition in weak external fields.

(a)

(b)

Figure 1: (a) Comparison of the predicted two-value vapor-pressures with the tabular ()-data for ethylene [23] and water [24]; (b) predicted two-value vapor-pressures for [bmim][PF₆] and [pmmim][Tf₂N].

To estimate the appropriate excluded molar volume (M = 225,82 g/mol) of FT-model we consider that it belongs to the range [,   cm³/mol]. The extrapolated to zero temperature T = 0 K “cold” volume = 162 cm³/mol follows from (27). The fixed value: = 178 cm³/mol () has been used in this work to demonstrate the main results of the proposed methodology. Such choice for [bmim][BF₄] on the ad hoc basis is in a good correspondence with the respective values: = 195,3 cm³/mol for [bmim][PF₆] and = 271,1 cm³/mol for [pmmim][Tf₂N] where the empirical relationship was also observed [7, 8]. Our estimates of the effective LJ-diameters by (6a) for ILs: σ([bmim][BF₄]) = 5,208 Å, σ([bmim][PF₆]) = 5,371 Å, and σ([pmmim][Tf₂N]) = 5,992 Å can be tested by comparison with the independently determined values [13] for anions ([BF₄]) = 4,51 Å; ([PF₆]) = 5,06 Å. We have verified Berthelot’s combining rule for spherical molecular ions (21a) and van der Waals’ combining rule for chain molecules (21b) usually considered by van der Waals’-type of EOS for mixtures [22]. Consider

The predicted by former rule of LJ-diameter for the same [bmim]-cation were close but still different, 5,906 Å and 5,682 Å. For the latter rule their values and distinction become even smaller, 5,757 Å and 5,651 Å. As a result, the chain rule (21b) seems preferable for ILs and its average value for [bmim] = 5,704 Å can be used to estimate the LJ-diameter of [Tf₂N]-anion: [Tf₂N] = 6,254 Å taking into account the equality: [bmim] = [pmmim] = 139 g/mol. The collected in Table 4 effective LJ-diameters are linear functions of in the set of ILs with different anions and cations if the molecular weight of latters is the same one.

Table 4: Effective LJ-diameters of FT-model for ILs determined by (6a), (6b), and (21b) on the base of estimates [7, 13] and the choice ₀ = 178 cm³/mol for [bmim][BF₄] in this work.

IL
(g/mol)
(Å)
/
/

bmimBF₄
225,82
5,208
139/86,82
5,757/4,51 [13]

bmimPF₆
284
5,371 [7]
139/145
5,651/5,06 [13]

pmmimTf₂N
419,1
5,992 [7]
139/280,1
5,704/6,254

Since the low-temperature compressibility factor is very small for all discussed liquids their dispersive energies (molecular attraction’s parameters) are comparable in accordance with (6b). However, the differences in cohesive energies (collective attraction's parameters) between the low-molecular substances (C₂H₄, H₂O) and ILs are striking as it follows from Tables 1 and 2. The physical nature of such distinction can be, at the first glance, attributed to omitted in the reference LJ-potential influence of intramolecular force-field parameters and anisotropic (dipole-dipole and coulombic) interactions. At the same time, one must account the collective macroscopic nature of -parameter. It corresponds to the scales which are compatible or larger than the thermodynamic correlation length . FT-model [9–12] provides an elegant and simple estimation of this effective parameter based on the concept of comparability between energetic and geometric characteristic of force field determined by the given RPP. Consider

Taking into account the above results and the coefficients from Tables 1–3 we have used (22) at T = 300 K () to compare the thermodynamic correlation length predicted for [bmim][BF₄] (a = 8900,9 J·dm³/mol²; = 178 cm³/mol;  ρ = 5,322294 mol/dm³) and at T = 298,15 K for water (a = 548,27 J·dm³/mol²; = 16,58 cm³/mol;  ρ = 55,444 mol/dm³) [24]. The dimensional and reduced values for former are, respectively,  ξ = 177,7 Å, = 34,12 while for latter ξ = 69,86 Å, = 29,45. No more need be said to confirm the universality of FT-model.

One may note that our estimates of correlation length are significantly larger than those usually adopted for the dimensional or reduced cutoff radius ( or ) of direct interactions at computer simulations. As a result, the standard assumption may become questionable in the comparatively small (mesoscopic) volumes of simulation . At this condition the simulated properties are mesoscopic although their lifetime may be essentially larger than its simulated counterpart. The key point here is the same as one near a critical point where the problem of consistency between the correlation length for statics and the correlation time for dynamics becomes crucial. In any case, the computer study of possible nongaussian nature of local fluctuations within the thermodynamic correlation volume may be quite useful. The relevant inhomogeneities in the steady spacial distributions of density and enthalpy can affect, first of all, the simulated values of volumetric and caloric derived quantities. Simultaneously, an account of internal degrees of freedom and anisotropy by the perturbed RPP may change the correlation length itself.

The above described by ((12)–(20)) FT-methodology has been used to reconstruct the hypothetical phase diagram (HPD) for [bmim][BF₄] shown in Figures 2, 3, and 4 and represented in Table 3. Both () (Figure 2) and () (Figure 4) projections contain also the branches of classical spinodal calculated by the LF (Sanchez-Lacombe)-EOS obtained in [14]. Its top is the location of a respective critical point. It seems that the relatively close -parameters predicted independently by FT-EOS and by LF-EOS (see Section 1) are reasonable.

Figure 2: Comparison of the GEMC-simulated (black triangles) ()-diagram [6] for [bmim][BF₄] with the HPD-predicted coexistence of orthobaric densities (lines with black squares); the characteristic (, ) points are emphasized as well as the distinction of respective rectilinear [6] and strongly-curved (HPD) diameters. The input low-temperature experimental -data [1, 2] are represented by white diamonds. Location of classical spinodal and its critical point (□) predicted by LF-EOS [14] is shown by dashed line.

Figure 3: Enthalpy of vaporization for [bmim][BF₄] calculated by different methodologies (GEMC-simulated [6] (▴, ▲); FT-EOS-predicted at (■▪▪); HPD-predicted (–■-■); tabular data for water [24] (–♦–♦)).

Figure 4: Comparison of the GEMC-simulated [6] pressures of condensation (▴, ▲) with the HPD-predicted pressures of boiling (–■-■) for [bmim][BF₄]. The characteristic () points are emphasized; tabular data for water [24] (–♦–). The location of spinodal predicted by LF-EOS [14] is shown by dashed line.

The FT-model provides a possibility to estimate, separately, the coordination numbers of LJ-particles in the orthobaric liquid - and vapor -phases. An ability to form the respective “friable” -clusters is defined by the ratio of effective cohesive and dispersive molar energies at any subcritical temperature. Consider

The term “friable” is used here to distinguish the clusters formed by the unbounded LJ-particles at the characteristic distance from the conventional “compact” ones with the bonding distance studied, in particular, by the GEMC-methodology [25] to model of molecular association. It is straightforwardly to obtain the low-temperature estimates based on the assumptions.

,

and to find the critical asymptotics based on the difference of classical and nonclassical () -dependent FT-EOS’ coefficients [9–12]. Consider

The crucial influence of excluded-volume in (24a) and its relative irrelevance in (24b) for -predictions are illustrated by Figure 5 where function is shown also for the entire l-branch based on the evaluated in the present work HPD. For comparison, the low-temperature ability to form the -clusters in liquid water [7, 24] is represented in Figure 5 too.

Figure 5: Comparison of two coordination numbers predicted by FT-EOS to characterize the clustering in an orthobaric liquid of [bmim][BF₄] at (■▪▪) and in the entire ()-range of HPD (–■-■); FT-EOS predictions for water are shown as –♦ at  K.

In according with ((25a), (25b)) the “friable” clusters can exist only as dimers in the classical critical liquid phase . It is not universal property in the meaning of scaling theory but it corresponds to the PCS-concept of similarity between two substances (H₂O and [C₄mim][BF₄], e.g.) if their -values are close. On the other side, the scaling hypothesis of universality is confirmed by the FT-EOS’ estimates in the nonclassical critical vapor phase. For the set of low-molecular-weight substances studied in [9] (Ar, C₂H₄, CO₂, H₂O); for example, one obtains by (25b) the common estimate which shows a significant associative near-mean-field behavior.

It is worthwhile to note here the correspondence of some FT-EOS’-estimates with the set of GEMC-simulated results. One may use the approximate estimate of critical slope [9] for [bmim][BF₄] based on the similarity of its -value with that for H₂O [24]. In such case, the respective critical excluded volume  cm³/mol becomes much more than vdW-value cm³/mol. Another observation seems also interesting. Authors [25] have calculated (see Figure 3 in [25]) for the “compact” clusters at = 0,7; 0,5; 0,45; the ()-diagram of simple fluids. One may note that only the value corresponds to the shape of strongly-curved diameter shown in Figure 2 for the HPD while the smaller values: ; 0,45 give the shape of HPD and the nearly rectilinear diameter strongly resembling those obtained by the GEMC-simulations [6] for the complex IL’s force-field. If this correspondence between the “friable” and “compact” clustering is not accidental one obtains the unique possibility to connect the measurable thermophysical properties with the both characteristics of molecular structure in the framework of FT-EOS.

4. Comparison with the Empirical Tait EOS and Semiempirical Sanchez-Lacombe EOS

The empirical Tait EOS is based on the observation that the reciprocal of isothermal compressibility for many liquids is nearly linear in pressure at very high pressures. Consider

where some authors [14, 19] omit the T-dependence in coefficient and ignore the value [14]. Such restrictions transform the Tait EOS into the empirical form of two-parameter () PCS because the sets of -values for different ILs become close one to another. For example, Machida et al. [14] have found the sets C = 0,09710 for [bmim][PF₆], C = 0,09358 for [bmim][BF₄], and C = 0,08961 for [bmim][OcSO₄] which is rather close to the set obtained by Matkowska and Hofman [19] C = 0,088136 for [bmim][BF₄] and C = 0,0841547 for [bmim][MeSO₄]. At the same time, Gu and Brennecke [3] have reported the much larger T-dependent values and for the same [bmim][PF₆].

Two other reasons of discrepancies in the Tait methodology is the different approximations chosen by authors for the reference input data and for the compound-dependent function . Some authors [4, 14, 18] prefer to fit the atmospheric isobars and with a second-order or even third-order polynomial equation while the others [1, 2, 16, 19] use a linear function for this aim.. As a result, the extrapolation ability to lower and higher temperatures of different approximations becomes restricted.

In this work we have used for [bmim][BF₄] the simplest linear approximation of both density and heat capacity,

taken from [1]. The extrapolated to zero of temperature value (0 K) = 1394,65 kg·m⁻³ [1] is in a good correspondence with that from [14] (0 K) = 1393,92 kg·m⁻³, in reasonable correspondence with that from [19] (0 K) = 1416,03 kg·m⁻³ and that from [16] (0 K) = 1429 kg·m⁻³ but its distinction from value (0 K) = 1476,277 kg·m⁻³ reported by authors [18] is rather large. The similar large discrepancy is observable between (0 K) = 273,65 J/mol·K from [1] and (0 K) = 464,466 J/mol·K from [18].

The different choices of an approximation function for (so authors [14] have used the exponential form while authors [19] have preferred the linear form) may distort the derivatives and calculated by the Tait EOS (26). The problem of their uncertainties becomes even more complex if one takes into account the often existence of systematic distinctions of as much as 0,5% between the densities measured by different investigators even for the simplest argon [23]. Machida et al. [14], for example, pointed out the systematic deviations measured densities from those reported by the de Azevedo et al. [18] and Fredlake et al. [1] for both [bmim][BF₄] and [bmim][PF₆]. Matkowska and Hofman [19] concluded that the discrepancies between the different sets of calculated - and -derivatives increase with increasing of and decreasing of due not only to experimental differences in density values but also result from the fitting equation used. The resultant situation is that the expansivity of ILs reported in literature was either nearly independent of T [18] or noticeably dependent of T [3, 19].

We can add to these observations that the linear in molar (or specific) volume Tait Eos (26) is inadequate in representing the curvature of the isotherm at low pressures. It fails completely in description of ()-transition where the more flexible function of volume is desirable. However, this has been clearly stated and explained by Streett for liquid argon [23] that the adjustable T-dependence of empirical EOS becomes the crucial factor in representing the expansivity and, especially, heat capacities , at high pressures even if the reliable input data of sound velocity were used.

From such a viewpoint, one may suppose that the linear in temperature LF-EOS proposed by Sanchez and Lacombe,

is restricted to achieve the above goal but can be used as any unified classical EOS common for both phases to predict the region of their coexistence. Such conjecture is confirmed by the comparison of FT-EOS with LF-EOS presented in Figures 2 and 4 and discussed below. The obvious advantage of former is the more flexible T-dependence expressed via the cohesive-energy coefficient . On the other hand, the LF-EOS is typical form of EOS (see Section 1) in which the constraint of T-independent potential energy is inherent [21].

One may consider it as the generalized variant of the well-known Bragg-Williams approximation for the ordinary LG presented here in the dimensional form

Such generalization provides the accurate map of phenomenological characteristic parameters , , which determine the constant effective number of lattice sites occupied by a complex molecule,

into the following set of molecular characteristic parameters for a simple molecule :

where is the volume of cell and is the coordination number of lattice in which the negative   is the energy of attraction for a near-neighbor pair of sites. In the polymer terminology from (31) is the segment interaction energy and is the segment volume which determines the characteristic hard core per mole (excluded volume in the vdW-terminology).

Another variant of described approach is the known perturbed hard-sphere-chain (PHSC) EOS proposed by Song et al. [15] for normal fluids and polymers

where is the pair radial distribution function of nonbonded hard spheres at contact and the term with reflects chain connectivity while the last term is the small perturbation contribution. Though the PHSC-EOS has the same constraint of the potential energy field authors [15] have introduced two universal adjustable - and -functions to improve the consistency with experiment. The vdW-type coefficients were rescaled as

where is the additional scaling function for . It provides the interconnection of molecular LJ-type parameters () with the phenomenological vdW-ones (). The resultant reduced form of PHSC is [15]

where the following characteristic and reduced variables are used:

It was compared with the simpler form of LF-EOS (29). Their predictions of the low-temperature density at saturation are comparable but, unfortunately, inaccurate (overestimated) even for neutral low-molecular liquids. The respective predictions of the vapor pressure are reasonable [15] excepting the region of critical point for both EOSs. Our estimates based on the LF-EOS [14] shown in Figures 2 and 4 are consistent with these conclusions.

The comparison of volumetric measurements and derived properties [14, 18] with the purely predictive (by the FT-EOS) and empirical (by the Tait EOS and LF-EOS) methodologies used for [bmim][BF₄] is shown in Figures 6–9. Evidently, that former methodology is quite promising. Machida et al. [14] have reported two correlations of the same -data measured for [bmim][BF₄] at temperatures from 313 to 473 and pressures up to 200 MPa. To examine the trends in properties of ILs with the common cation [bmim] the Tait empirical EOS was preliminarily fitted as the more appropriate model. The estimate of its extrapolation capatibilities for -surface in the working range () follows from the compatibility of experimental points (where those measured by de Azevedo et al. [18] in the range of temperature and pressure () were also included) with the thick curves in Figure 6. It is noticeable, for example, that the extrapolated Tait’s isotherm T = 290 K coincides practically with isotherm T = 298,34 K from [18] because the measured densities of latter source are systematically higher than those from [14]. Density data of Fredlake et al. [1] for [bmim][BF₄] (not shown in Figure 6) are also systematically shifted from measurements [14].

Figure 6: Comparison of experimental densities for [bmim][BF₄] (⚪-298,34 [18]; □-313,01 [18]; △-322,85 [18]; ♢-332,73 [6]; ■-313,1 [14]; ♦-332,6 [14]; ▲-352,6 [14]) with those calculated: (a) by the Tait EOS [14] (in a working range via the interval 10 K thick continuous curves); (b) by the Sanchez-Lacombe EOS [14] (thick dashed curves); (c) by the FT-EOS (thin continuous curves).

Figure 7: Comparison of isobaric expansivities for [bmim][BF₄] calculated: (a) by the Tait EOS [14]; (b) by the Sanchez-Lacombe EOS [14]; (c) by the FT-EOS.

Figure 8: Comparison of isothermal compressibilities for [bmim][BF₄] calculated: (a) by the Tait EOS [14]; (b) by the Sanchez-Lacombe EOS [14]; (c) by the FT-EOS.

Figure 9: Comparison of thermal pressure coefficient for [bmim][BF₄] calculated: (a) by the Tait EOS [14]; (c) by the FT-EOS.

The consequence of such discrepancies is also typical for any simple liquids (Ar, Kr, Xe) [23] at moderate and high pressures. It is impossible to reveal an actual T-dependence of volumetric (mechanical) derived functions , due to systematic deviations between the data of different investigators. In such situation an attempt “to take the bull by the horns” and to claim the preferable variant of EOS based exclusively on volumetric data may be erroneous. Indeed, since the Tait EOS is explicit in density while the LF-EOS—in temperature the direct calculation of , -derivatives for former and , -derivatives for latter are motivated. To illustrate the results of these alternative calculations we have used in Figures 6–9 the coefficients of LF-EOS reported by Machida et al. [14] for the restricted range of moderate pressures . The thick dashed curves represent the boundaries of working range where the extrapolation to T = 290 K is again assumed. One may notice the qualitative similarity of FT-EOS (the thin curves) and LF-EOS which can be hypothesized as an existence of certain model substance at the extrapolation to higher pressures . It demonstrates the smaller compressibility (Figure 8) and expansivity (Figure 7) than those predicted by the Tait EOS while the value of thermal-pressure coefficient for FT-EOS (Figure 9) becomes larger. It determines the distinctions in the calculated internal pressure. The choice of the FT-model’s substance as a reference system for the perturbation methodology provides the set of advantages in comparison with the LF-EOS.

It follows from Figure 6 that at moderate pressures the predictive FT-EOS is more accurate than the fitted semiempirical LF-EOS [14] although the discrepancies of both with the empirical Tait EOS [14] become significant at the lowest (extrapolated) temperature T = 290 K. The Tait’s liquid has no trend to ()-transition (as well as polymers) in opposite to the clear trends demonstrated by FT-EOS and LF-EOS. One may suppose [26] a competition between vaporization of IL (primarily driven by the isotropic dispersive attraction in RPP) and chain formation (driven mainly by the anisotropic dipolar interactions ) reflected by the Tait EOS fitted to the experimental data. Of course, such conjecture must be, at least, confirmed by the computer simulations and FT-model provides this possibility by the consistent estimate of RPP-parameters () at each temperature.

The differences of calculated expansivity in Figure 7 are especially interesting. FT-EOS predicts even less variation of it with temperature than that for the Tait EOS. This result and crossing of -isotherms are qualitatively similar to those obtained by de Azevedo et al. [18] although the pressure dependence of all mechanical and caloric derivatives (see Figures 10, 11, and 12) is always more significant for the FT-EOS predictions. It seems that the curvature of the -dependence following from the LF-EOS (29) is not sufficient to predict the behavior (Figure 7) correctly. The strong influence of the chosen input -dependence is obvious from Figures 7–9.

Figure 10: Comparison of predicted by FT-EOS isochoric heat capacity for [bmim][BF₄] (lines) with that evaluated by de Azevedo et al. [18] (points) on the base of speed velocity and density input data.

Figure 11: Comparison of predicted by FT-EOS isobaric heat capacity for [bmim][BF₄] (lines) with that evaluated by de Azevedo et al. [18] (points) on the base of speed velocity and density input data.

Figure 12: Comparison of predicted by FT-EOS ratio of heat capacities for [bmim][BF₄] (thin lines) with that evaluated by de Azevedo et al. [18] on the base of speed velocity and density input data (thick lines).

The prediction of caloric derivatives is the most stringent test for any thermal () EOS. It should be usually controlled [23] by the experimental ()-surface to use the thermodynamical identities,

in addition to the chosen input -dependence. de Azevedo et al. [18] applied this strategy to comprise the approximated by the Pade-technique measured speed of sound data for [bmim][PF₆] and [bmim][BF₄] (Figure 13) with the evaluated at high pressures heat capacities.

Figure 13: Comparison of predicted by FT-EOS speed of sound for [bmim][BF₄] (thin lines) with that (points) used by de Azevedo et al. [18] as the input data (see also Figure 5 for density used by de Azevedo et al. [18] as the input data at evaluations).

Our predictive strategy is based [17] on the differentiation of -dependence to evaluate directly the most subtle -surface in a low-temperature liquid

where the influence of the consistence for the chosen input - and -dependences (via (37) used for estimate of at the atmospheric pressure ) becomes crucial. The use of first derivative (even by its rough approximation in terms of finite differences: ) to calculate simultaneously by ((3), (4), (37), (40)) all volumetric and caloric derivatives is the important advantage over the standard integration of thermodynamic identities:

To illustrate such statement it is worthwhile to remind the situation described by the Streett [23] for liquid argon. Since isotherms of cross over for many simple liquids (Ar, Kr, Xe), this author concludes that the sign of changes also from positive to negative at the respective pressure. This conclusion is not valid because to change the sign of derivative it is enough to account for the exact equality

in which can be always positive. In this case one would expect the monotonous decrease of with increasing in accordance to ((41a), (41b)) while the presence of extremum (minimum or maximum of -dependence) seems to be artificial.

There is the variety of pressures reported by different investigators as a presumable cross-point for the same ILs. Machida et al. [14] have estimated it to be about 10 MPa on the base of Tait EOS for [bmim][PF₆] but have not found it (Figure 7) for [bmim][BF₄]. For latter our estimate by the FT-EOS is: P = 20,6 MPa. de Azevedo et al. [18] have reported the mild decrease of -dependence and the sharp decrease -dependence while a presumable cross-point is located between about 100 and 120 MPa for [bmim][BF₄]. Taking into account the above distinction in the evaluated -surface it is interesting to consider their consequences for caloric -- and -surfaces shown in Figures 10–12.

The remarkable qualitative and even quantitative (<8%) correspondence between the predicted by FT-EOS -values and those reported by de Azevedo et al. [18] follows from Figure 10. At the same time, although the discrepancies between -values [18] and those predicted by the FT-EOS are again within acceptable limits (<10%) the formers demonstrate the weak maximum and very small pressure dependence for [bmim][BF₄] (for [bmim][PF₆] this -dependence is monotonous as well as that predicted by the FT-EOS). It seems that the resultant ratio of heat capacity shown in Figure 12 which demonstrates the irregular crossing of isotherms [18] is questionable. It suggests that their pressure dependence either needs the more accurate approximation or reflects the realistic distinction of reference FT-EOS from the actual behavior of [bmim][BF₄].

The lock of noticeable variations in pressure is the common feature of integration methodology [18] based on the given ()- and ()-surfaces. The unavoidable accumulation of uncertainties at each stage of calculations in the set may cause the unplausible behavior of adiabatic exponent in liquid. The same is true for the set used in the FT-methodology. It is the most appropriate explanation of significant discrepancies for -dependence shown in Figure 13. Let us remind also that the precise mechanical measurements of speed velocity [18] in the very viscous IL cannot be attributed exactly to the condition of constant entropy.

Thus, strictly speaking, the measured ()-surface reflects the strong dispersive properties of media and must be less than its thermodynamic counterpart in the ideal (without a viscosity) liquid.

5. Conclusions

There are the structure-forming factors related to the above-discussed thermodynamic characteristic. Despite the certain discrepancies between the predicted and derived properties for FT-EOS and LF-EOS, both ones provide the close estimates of structure factors represented in Table 5.

Table 5: Comparison of excluded volumes ( [14] and ), characteristic interaction energy ( [14] and ), and effective number of bonded units ( [14, 15] and ) (see text).

Compound

(cm³/mol)

(cm³/mol)

(J/mol)

(J/mol)

bmimBF₄
175,3
178
5642
2661
17,6
17,0

bmimPF₆
196,2
195,3
5658
2661
18,8
18,1

Our aim here is to show that the thermodynamically-consistent predictions of thermodynamical properties by the FT-EOS yields also the molecular-based parameters which are, at least, realistic (see also Table 3). The estimate of average T-dependent well-depth by (6b) as well as estimate of average value by (24a) is related to the middle of temperature range: T = 320 K. The distinction of from the respective -parameters of LF-EOS [14] can be attributed to the difference between nonbonded interactions in the discrete (LF-EOS) and continuum (FT-EOS) models of fluid. Our estimate of cohesive-energy density by equality,

represented in Table 6 seems also physically plausible. Maginn et al. [5, 6] have determined it within the framework of GEMC-simulations by the knowledge of and the internal energy difference between an ideal-gas ion pair and the average internal energy of an ion pair in the liquid state.

Table 6: Comparison of internal pressure for [bmim][BF₄] based on the LF-EOS [14] and FT-EOS (this work) with the values estimated [16] by experimental data on speed of sound , density , and isobaric heat capacity .

[14]
(K)

(MPa)
[16]  
(K)

(MPa)
[FT]
(K)

(MPa)

(J/cm³)

313,1
482,78
283,15
459,91
290
743,76
555,15

332,6
471,50
288,15
459,31
310
573,16
486,41

352,6
460,35
293,15
459,20
330
463,91
437,85

372,7
450,11
298,15
457,79
350
388,61
401,57

392,8
440,00
303,15
455,76
370
333,84
373,35

412,9
429,23
308,15
453,73
390
292,27
350,67

432,6
419,35
313,15
451,20
410
259,66
331,94

452,3
408,84
318,15
448,06
430
233,40
316,13

472,2
399,19
323,15
446,64
450
211,77
302,52

328,15
444,61
470
193,64
290,60

333,15
443,09
490
178,19
280,01

338,15
441,68
510
166,76
270,46

343,15
439,75
517
162,41
267,31

Such definition indicates that cohesive energy densities of many ILs are on the order of 500–550 J/cm³ (see, for comparison, Table 6) and demonstrate a slight decrease as temperature increases.

Another relevant characteristic is the internal pressure determined by the derivative of molar (or specific) internal energy,

which is compared to ones calculated by different authors [14, 16] for [bmim][BF₄] in Table 6. As in the other cases, the FT-EOS predicts the much faster change of both cohesive energy density and internal pressure as temperature increases.

One should collect a large number of precise experimental measurements to reconstruct the thermodynamic surface of a substance. FT-methodology provides a possibility of preliminary reliable estimates of relevant macroscopic and molecular-based correlations. Its thermodynamic consistency provides the serious advantage in comparison with the purely empiric treatment of any volumetric measurements at the description of derived heat capacities.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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