# Scaling Model of Low-Temperature Transport Properties for Molecular and Ionic Liquids

Authors: Vitaly B.  Rogankov
Journal: Journal of Thermodynamics
 (2015)

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## Abstract

The universal scaling concept is applied to the low-temperature range ofany liquid states and substances located between the melting (Tm) and normal boiling (Tb) points far away from the critical region. The physical reason to develop such approach is the revealed collapse of all low-temperature isotherms onto thesingle universal one argued by the model of fluctuational thermodynamics (FT) proposed recently by author. The pressure reduced by the molecular parameters of the effectiveshort-range Lennard-Jones (LJ)potential depends here only on the reduced density. To demonstrate the extraordinary predictive abilities of the developed low-temperature scaling model it has been applied to the prediction of equilibrium and transport (kinetic and dynamic viscosity, self-diffusion, and thermal conductivity) properties not only formolecular liquids but also for molten organic salts termedionic liquids (ILs). The best argument in favor of the proposed methodology is the appropriate consistency with the scarce experiments prediction of transport coefficients for ILs on the base of universal scaling function constructed for thesimplest LJ-like liquid argon. The only input data of any substance for prediction are the linear approximations of T-dependent density and isobaric heat capacity taken from the standard measurements at atmospheric pressure.


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## Body

## 1. Introduction

The concept of scaling description at low temperatures based on the chosen effective form of a pair interparticle potentialϕ(r) is not completely novel. In particular, Hoover et al. [1] used the oversimplified (even in this specific region of a liquid state) so-calledsoft-sphere (ss) purely repulsive function(1)ϕr≈εσrnwith constant ss-parameters ε,σ and exponent n ranging from about six n≳6 for metals to about twelve for rare gases n≲12. Its limitative value n→∞ for a primitive hard-sphere (hs) model leads thescaling variablex introduced by the above authors ρ=N/V;D=3,(2)x=ρσ32εkBTD/n,to be only ρ∗=ρσ3-dependent. Another notable feature here is the implied PCS (principle of corresponding states) nature of such ss-scaling. Indeed one has to fix the exponent n in (1) and (2) to select the group of similar substances.Nevertheless the useful consequence of scaling transformation has been noticed by Hoover et al. [1] because any equilibrium or transport reduced properties (e.g., dynamic viscosity ηρ-2/3m0kBT-1/2 or self-diffusion Dρ1/3m0/kBT1/2, m0 mass of particle) of different ss-systems with fixed exponent n should coincide at the different ρ and T being expressed in terms of x-variable from (2). Of course, the additional serious restriction of ss-model (common with hs-model) is an absence of gas- (vapor-) liquid (g,l) transition at any value of repulsive exponent.It was shown recently [2, 3] in the framework of FT-model [4–6] that the aforementioned oversimplifications of ss- and/or hs-model are not necessary if one considers the more realistic LJ-type potential ϕl,n/m with the effective T-dependent parameters σ(T),ε(T):(3)ϕl,n/mr=lεTσTrn-σTrm.The advantage of such approach in comparison with the conventional estimates of their constant LJ-counterparts σ, ε (mainly, from the experimental second-virial-coefficient’s data) follows from the exact T-dependence derived by FT-model for any subcritical temperature:(4)εT=kBT1-ZlT=kBT-PσTρlT,σ3T=3AσT-24πρdTAσT-1=3bT2π,where Zl=Pσ(T)/ρl(T)kBT1/3, AσT=T/PσdPσ/dT, ρd=ρg+ρl/2, and b(T) is the T-dependent parameter of excluded volume.The determination of both is not trivial task for ILs [7] in which the vapor-pressure curve Pv(T) is inaccessible by the standard lab. vacuum (~10−3, Pa) experiment. However, just this restriction leads to the significant simplification of (4) at low temperatures T≤Tb as is shown in Figure 1 for the well-studied simple and complex substances.Figure 1
T-dependent well-depth of Lennard-Jones-type potential predicted by FT-model [3–6] on the base of experimental CXC-data for simple (Ar), normal (CO2), and polar (H2O) fluids; the low-temperature range [Tm, Tb] is specified by condition T∗≈1.One may note that in parallel with the well-known condition for compressibility factor measured along theg-branch of CXC: Zg=Pσ(T)/ρg(T)kBT=1 its value Zl along the l-branch becomes here negligible or close to zero Zl≈0. It leads to the interconsistent low-temperature values of LJ-parameters:(5a)T∗=kBTε≈1,(5b)ρ∗=ρσ03=32πρb0.This is main assumption used in the present work to formulate the phenomenology of scaling transport coefficients and the respective equilibrium properties following from the generalized reduced form of LJ-EOS discussed below. It is interesting to compare the particular value of ss-scaling variable from (2) at hs-limit n→∞: xhs=ρσ3/2 with that from LJ-model in (5b). For the former T value is irrelevant while for the latter it determines by (5a) the well-depth energy ε.


## 2. Theoretical Basis of Global Asymmetry between the Nonordered (Gas) and Condensed (Liquid) States

FT-model is described in detail elsewhere [2, 6]. For the convenience of readers its main concepts and results are discussed, in brief, below. In the field of equilibrium properties the essential feature is the well-founded rejection from the vdW-concept of aunified (i.e., common for both coexisting g- and l-phases)EOS. This crucial step leads to the different alternative forms oflocal EOS for the above phases. They are based, however, on the same mathematical form accepted for an effective pair potential. The choice here of LJ-form requires, for information, at least, the superficial comparison of FT-model’s results with the enormous number of simulated LJ-data described [8] just by the unified, purely empirical (so-called, fundamental) EOS for free energy. Despite the widespread belief of its universal applicability the arguments in favor ofglobal indubitable asymmetry betweeng- andl-states [2] are represented below in terms of respective localthermal EOSs. Another theoretical consequence of the discussed duality is the necessity to modify significantly the description of transport coefficients (see Section 3) without the permanent appeals to the well-established PCS formalism of a diluted nonordered g-phase and its properties [9, 10].Striking asymmetry between the exact thermal EOSs of LJ-fluid may be easily revealed by the FT-model’s approach. Thermal form of EOSP(ρ,T) has the certain restrictions in the representation of caloric properties and, in particular, it needs the knowledge of so-called ideal gas heat capacities if the concept of a unified EOS is used. However, opposite to the conventional methodology of a fundamental (caloric) EOS, its exact form follows immediately for the given PCS-type of LJ-fluid (l,n/m) (with the unspecified here integers) from the application of virial theorem [20]. One should note for that the evident formal property of underlying LJ-potential from (3). It is a sum of repulsive and attractive contributions with the different degrees of mathematical homogeneity (n and m). This observation leads straightforwardly to the equivalent forms of thermal EOS for acondensed LJ-fluid (l,n/m) (P∗=Pσ3/ε,Z=P∗/ρ∗T∗):(6)PLJ∗=ρ∗T∗+lnDρ∗n/D+1-mDρ∗m/D+1,(7)ZLJ-1=lT∗nDρ∗n/D-mDρ∗m/D.The important consequence of the latter is the irrelevant role of reduced T∗ along Boyle’s line Z=1 (so-called Zeno-line) where the reduced density ρ∗ becomes constant in the whole range of l-states:(8)ρLJ∗=mnZ=1D/n-m.This result can be consistent with the known and empirically confirmed linearity of Zeno-line expressed, for example, in terms of dimensional variables for the vdW-EOS:(9)ρBvdW=1b1-kBbTaonly if the reduced LJ-density ρLJ∗=ρσ3 implies the inverse ρ-dependence of cubic LJ-diameter σ3(T) while the vdW-coefficient a is linear in T.At low temperatures where the specific conditions are acceptedTLJ∗≈1 by (5a) and σLJ=σ0 by (5b)all LJ-isochores collapse to Boyle’s line in accordance with (8). The scaling PLJ∗ρ∗-form of reduced pressure follows immediately from (7):(10)PLJ∗TLJ∗≈1=ρ∗+lnDρ∗n/D+1-mDρ∗m/D+1,which collapses here at Z=1 (PLJ∗=ρLJ∗) to Boyle’s line too. Such coincidence of isobars with isochores is a sign of singular incompressible behavior of liquid in the low-temperature region. It is noteworthy that Maslov [21, 22] has revealed recently the similar phenomenon in the low-temperature, so-calledsoft LJ-fluid’s states located below Zeno-line in the (T,ρ)-plane. He termed it “the jamming effect.”The described specific features of FT-model’s LJ-liquid in the low-temperature region legalizes the simplest linear choice of input density data (Section3) measured for any substances at atmospheric pressure P0 and known even for ILs:(11)ρ0T,P0=ρ001-TT0,where “cold” density ρ00 and “hot” temperature T0 are the substance-dependent parameters (see, for comparison, vdW-equality (9)). However, the two other input T-dependencies chosen below at fixed P0 are less motivated and linear ones only for simplicity:(12)CP0T,P0=CP01+TTP,cs0T,P0=Cs01-TTs,where CP is the isobaric heat capacity of liquid (not that for ideal gas as it is in an unified EOS) and cs is the adiabatic speed of sound in a liquid too.To be convinced of the global(g,l)-asymmetry [2, 3] existing for LJ-fluid one should consider the known virial expansion [23] for nonordered gas truncated after the second virial contribution B(T)ρ:(13)ZLJg-1≈2πρ∗3lT∗D/nγ1-Dn-lT∗D/mγ1-Dm.Its comparison to (7) leads to the physically well-founded conclusion about the dominance of ρ∗ in l-phase but the dominance of inverse reduced temperature l/T∗ in g-phase. Moreover, at Boyle’s condition BTB=0ZLJg=1 the reduced density ρ∗ is irrelevant while the reduced temperature T∗ becomes constant:(14)TLJ∗=lγn,Dγm,Dmn/Dn-m.Such exact estimate can be quite useful at the development of non-mf (mean-field) LJ-theory [2] but in the present work has only passing interest.


## 3. Continual Phenomenological Model of Transport Coefficients in the Low-Temperature Liquid

The use of such standard notions [23], well-defined in the diluted g-phase as the statistical mean free-path of hs-particle between two collisions l and its mean transport velocity u, becomes less informative due to the global (g,l)-asymmetry described in Section 2. It may be rejected in favor of the respective continual scales l- and u- for l-phase determined accurately by its thermodynamic equilibrium parameters ρ, T (D- is thetransport scale (ability) of continuum):(15)D-=l-u-=ρ-1/3kBTm01/2.Hoover et al. [1] have restricted the proposed scaling ss-description of transport coefficients only by the mechanical transport of mass and momentum by self-diffusion Dsd and kinematic velocity ν=η/ρm0 represented in terms of x-variable from (2):(16a)DsdD-=Dsd∗x,(16b)νD-≡ηρm0D-=ν∗x.It is natural to add here thetemperature-conductivityχ=λ/ρm0CP expressed in terms of thermal conductivity:(17)χD-≡λρm0CPD-=χ∗x.The certain similarity of scaling functions in (16b) and (17) with the known [9, 10] (inverse here) definitions of the Reynolds’ (Re¯) and Pecle’s (Pe¯) numbers for aconvective flow can be noticed taking into account the choice of characteristic thermodynamic scales by (15).There are two interconnectedkinetic andfield scales of velocityu- and cs which both are necessary to reduce the momentum flux of pressure:(18a)Z=Pρm0u-2,(18b)Eu¯=Pρm0cs2=Pβs.Former Z is namely the compressibility factor discussed in Section 2 while the latter Eu¯ is the generalized Euler’s number here (βs is the adiabatic compressibility). One has to take into account that the spontaneously arisen in homogenous (ρ,T)-continuum local Gaussian (small and independent) fluctuations are diffusible with the speed of sound cs(ρ,T). Thus the transport process of any type depends simultaneously on two (molecular and wave) physical mechanisms [24].To the best of my knowledge, the contribution of wave transport by the adiabatic fluctuations of pressure in the transport properties of liquids was ignored until now [21, 22]. Its physical nature is different from the molecular-based mechanism and based on the alternative concept of continuum. Hence, two independent scales offast andslow transport should be taken into account for any fluctuational-relaxation processes. Of course, it is hardly solvable problem to separate them and to determine accurately their relative contributions in the transport coefficients. In principle, they may be either comparable or essentially asymmetric (Section 5) for different liquids.To make a start of consideration it was assumed below that the equal fractions of both contributions are appropriate for the thermal conductivityλ and the dynamic viscosity η:(19a)λ=λP+λs2,(19b)η=ηP+ηs2,where subscripts of isobar P and of isoentrop s specify, respectively, the above types of transport. Then such concept of superposition was realized in the explicit phenomenological forms by the acceptance of two adjustable (but, presumably, universal for all substances) constant coefficients A and B:(20)λCP=x2/3m0εT1/22σ02χ∗x+AkBcsTm0CPTu-T,(21)η=x2/3m0εT1/22σ02ν∗x+exp⁡Bcs2Tu-2T,(22)Pr¯=ηCPλ=ν∗x+exp⁡Bcs2/u-2χ∗x+AkBcs/m0CPu-,where the generalized Prandtl’s number Pr¯ here is essential to estimate the transport ability of a liquid, in total.It is possibly the most remarkable consequence of the proposed correlations here that theuniversal coefficientsA=2 and B=0.0192for a variety of quite different,simple, and complex molecular and/or ionic compounds have been reasonably estimated, exclusively, from the tabular data [11] forthe simplest LJ-like argon. Besides, here the chosenon the ad hoc basis correlations are based in (20)–(22) on the semiempirical PCS-functions recommended for a liquid state [9, 10]. The explicit purely empirical form of two scaling functions from (16a), (16b), and (17),(23a)χ∗x=53x-1,(23b)ν∗=1.11-2π/3x,is based exclusively on the argon’s data [11] too. In other words, the main aim was the verification of the low-temperature scaling concept but not the best fit of experimental data (see below) on the transport coefficients. More accurately, the above-supposed correlations (20)–(22) take a cue from their successful use in prediction of λ(T) and η(T), separately, within a group of thermodynamically similar substances. To go beyond the PCS-framework [9, 10] of molecular similarity, the more restrictive concept of scaling (which is applicable, supposedly, to the different PCS-groups) has been used. Hence, the reasonable agreement of the purely predictable coefficients below (Sections 4 and 5) (without any PCS-reference to values λ0 and η0 at fixed T0) with the experimental data λ(T), η(T)should bear testimony to general applicability of scaling methodology far away from a critical point. Thus, one may, of course, improve the chosen correlations (20)–(22) here, to fit better the universal scaling parameters A, B within the possibly modified equations (23a) and (23b). However, the dual molecular-wave structure of all aforementioned universal functions should be the same. The generalized Mach number (Ma¯) following from the ratio of Z/Eu¯ in (18a) and (18b), (24)1Ma¯=csu-,is independent on the equilibrium parameters P, ρ and has the different degrees (linear and quadratic) in the proposed equation (20) for transport of heat and (21) for transport of momentum, respectively.


## 4. Problem of Input Low-Temperature Data

To avoid any subjectivity of PCS-estimates one has to use the common approximation scheme at the representation of the selected input (i.e., reference) data on each compound. Two individual coefficients of linear equations (11)-(12) determined for each respective property ρ0T,P0, CP0T,P0, cs0T,P0 from the experimental data [11–19] are reported in Table 1 for Ar, N2, C2H4, and C2H6 and the set of seven ILs. For the latter the working temperature range [290, 370 K] of a stable liquid without its upper limit Tb (it is unknown, as a rule) was considered. The possible slight curvature of measured data ρ0T,P0 often leads to the speculations about the sign of thermal expansivity αPT,P0 [25]. For (11) it is fixed and positive, at least, at atmospheric pressure:(25)αP0=-1ρ∂ρ∂TP0=1T0-T.This simple hyperbola gives, however, the good correspondence with the above experimental data. On the other hand, a variety of ρ0T,P0-data for ILs has been measured in the different ranges: ΔT=Th-Tl (h—higher, l—lower), with the different levels of accuracy, and some authors [13] have used the quadratic polynomials(26)ρ0T,P0=ρ00+ρ0′T+ρ0′′T2to obtain the best fit. The comparison of such approximation with the different variants of a linear one in (11) reported in Table 2 demonstrates the relatively steady behavior of the “cold” density ρ00 for both popular ILs: [C4mim][BF4] and [C4mim][PF6].Table 1
Linear fit of experimental input data [11–19] by (11)-(12).
Substance
Reference
ρ
0
0, kg/m3
T
0, K
C
p
0, J/kg⋅K
T
P, K
c
s
0, m/s
T
s, K
Argon (CXC)
[11]
1919,23
320,27
783,57
231,83
1397,00
214,92
Nitrogen
[11]
1150,07
258,92
1914,17
1190,19
1783,33
147,59
Ethane
[11]
754,25
663,95
2079,02
1102,60
2628,97
372,36
Ethylene
[11]
790,58
600,97
1928,57
540,00
2635,43
336,95
[bmim][BF4]
[12]
1394,63
2170,63
1211,79
1048,41
2223,66
997,61
[bmim][PF6]
[14]
1601,25
1976,85
1082,85
1018,31
2064,86
957,79
[pmmim][Tf2N]
[12]
1702,36
2045,31
639,07
278,70
2458,12
889,41
[EtMePyr][EtSO4]
[19]
1379,84
2235,86
2494,16
1033,08
[BuEtPyr][EtSO4]
[19]
1307,73
2270,37
2413,62
975,86
[BuMePyr][MeSO4]
[19]
1347,04
2226,92
2519,83
973,19
[E3MNPyr][MeSO4]
[19]
1352,03
2244,84
2663,49
1013,29Table 2
Coefficients of first- or second-order polynomial fits for atmospheric isobarP0  in different ΔT-intervals.
Compound
Δ
T [exp.], K
ρ
0
0, g/cm3
-
ρ
0
′
⋅
10
+
4, g/cm3 K
ρ
0
′
′
⋅
10
+
8, g/cm3 K2
Source
[C4mim][BF4]
295.5–343.9
1.39464
6.42562
—
[12]
[C4mim][BF4]
283.2–343.2
1.429
7.0
—
[15]
[C4mim][BF4]
283–353
1.41603
7.19854
—
[17]
[C4mim][BF4]
314–344
1.40686
6.81333
—
[18]
[C4mim][BF4]
298–333
1.47628
11.0351
65.02034
[13]
[C4mim][PF6]
298.2–343.2
1.60151
8.08889
—
[14]
[C4mim][PF6]
283.2–343.2
1.619
8.0
—
[15]
[C4mim][PF6]
313–343
1.60987
8.37
—
[18]
[C4mim][PF6]
298–333
1.62070
8.79556
8.481401
[13]The influence of discussed approximations on the “hot” temperatureT0 from (11) is more pronounced due to the rather large differences in the negative coefficient ρ0′ of (26). The coefficients ρ00, T0 reported in Table 3 and calculated without any fit from ρ0Tl- and ρ0Th-points provide the maximum deviations in the range of ±0.9%.Table 3
Recommended values of excluded molar volumeb0 determined by the extrapolated linear correlations for low-temperature densities of 39 liquids.
Number
Substance
M, g/mol
v
0, cm3/mol
ρ
0
0, kg/m3
T
0, K
b
0, cm3/mol
1st group (light and heavyn-alkanes)
1
C2H4
28
39,78
703,96
600,97
43,75
2
C2H6
30
35,42
847,05
663,95
38,96
3
C6H14
86
92,41
930,67
1005,47
101,65
4
C7H16
100
105,22
950,38
1044,95
115,74
5
C8H18
114
120,10
949,21
1127,99
132,11
6
C9H20
128
134,42
952,26
1189,77
147,86
7
C10H22
142
148,37
957,06
1235,91
163,21
8
C11H24
156
162,39
960,66
1277,25
178,63
9
C12H26
170
176,39
963,75
1313,68
194,03
10
C13H28
184
189,97
968,57
1338,26
208,97
11
C14H30
198
203,85
971,30
1365,63
224,24
12
C15H32
212
217,56
974,44
1387,10
239,32
13
C16H34
226
231,42
976,59
1409,23
254,56
14
C17H36
240
245,01
979,54
1424,79
269,51
15
C18H38
254
258,95
980,88
1445,12
284,85
16
C19H40
268
272,58
983,20
1458,76
299,84
17
C20H42
282
286,39
984,66
1474,05
315,03
2nd group (friable ILs and light substances)
1
N2
28,00
24,35
1149,90
258,92
26,79
2
Ar
39,94
20,81
1919,27
320,27
22,89
3
1-Methilimidazole
82,12
63,82
1286,74
1503,96
70,20
4
Toluene
92,14
78,88
1168,10
1140,22
86,77
5
[bmim][dca]
205,00
167,90
1220,96
2225.86
184,69
6
[NBuPy][BF4]
223,02
158,71
1405,20
2195,70
174,58
7
[C8mim][BF4]
282,13
219,38
1286,03
1968,41
241,32
8
[C8mim][PF6]
340,28
236,55
1438,51
2004,15
260,21
9
[E3MN][MeSO4]
343,26
253,89
1352,00
2244,84
279,28
10
[EtMePyr][EtSO4]
353,25
256,01
1379,83
2235,86
281,61
11
[BuMePyr][MeSO4]
395,29
293,45
1347,04
2226,92
322,80
12
[BuEtPyr][EtSO4]
437,34
334,43
1307,72
2270,37
367,87
3rd group (compact ILs)
1
[bmim][BF4]
225,82
161,80
1395,67
2170,63
177,98
2
[bmmim][BF4]
239,82
162,38
1476,91
1135,72
178,62
3
[bmim][PF6]
284,00
177,36
1601,26
1976,85
195,10
4
[bmim][triflate]
288,10
192,40
1497,40
2258,09
211,64
5
[bmmim][PF6]
298,00
181,24
1644,23
1241,47
199,36
6
[emim][Tf2N]
391,00
216,92
1802,51
1903,16
238,61
7
[emmim][Tf2N]
405,00
230,37
1758,04
1936,14
253,41
8
[pmmim][Tf2N]
419,00
246,13
1702,35
2045,31
270,74
9
[bmim][Tf2N]
419,10
246,13
1702,76
1912,71
270,74
10
[bmim][methide]
550,00
297,58
1848,24
1924,63
327,34Since the ultimate goal of developed here methodology is the scaling prediction of transport coefficients, the search for the best fit of input data is not the task of current work.Most likely such fit cannot affect essentially the predicted values of self-diffusivity, thermal conductivity, and viscosity of liquids if the chosen scaling form of their description is inherently consistent. Nevertheless, to determine the dependences of input data on the scaling variablex=ρ∗ from (5b) one needs the reliable estimate of excluded low-temperature volume b0 (see Table 3).Its value for any ILs can be reasonably estimated [6, 7] just from the long-range extrapolated “cold” volume v0=1/ρ00T→0,P0. The choice of this work emphasizes that the “cold” volume for the variety of molecular and ionic liquids [7] should bealways slightly less than the respective mf-estimate of bc0 following from the unified vdW-EOS (the mf-estimate is impossible for ILs):(27)bc0=13ρc>b0≈1.1v0,where coefficient 1.1 has [6, 7] certain physical meaning.Indeed, the important, nearly linear correlations ofv0(M) for the wide groups of compounds have been revealed as is shown in Figure 2 and represented in Table 3 for 39 compounds.Figure 2
Linear correlations of cold molar volumev0(M) represented for (1)–(3) groups of thermodynamically similar substances (Table 3) (special value for reference argon is shown by ⟐).There are three (apparent to the naked eye) groups of thermodynamically similar, high-molecular-mass substances: (1) heavy normaln-alkanes CnH2(n+1) with n≥6; (2) “friable” ILs (mainly, pyrrolidinium- and ammonium-based ones which can be extrapolated to the low-molecular-mass values for Ar, N2, Li, and H2O as well as above n-alkanes to the low-molecular-mass C2H6 and C2H4); (3) “compact” ILs (only imidazolium-based ones). The terms offriable andcompact ILs used here are related exclusively to the interrelation between the “cold” volumes at the equal large molecular masses M≥200g/mol. The average percent deviations of recommended v0 values (Table 3) from their linear correlations shown in Figure 2 are as follows: for 1st group: 0.22%, for 2nd group: 3.96%, and for 3rd group: 2.36%.All input linear functions of (11)-(12) expressed in terms of x,(28)Cp=Cp01+T0TP1-x1.1,cs=cs01-T0Ts1-x1.1,u-=kBT0m01-x1.1,can be now directly used for predictions by (20)–(22). In spite of the dramatic differences between v0 values from Table 3 the proposed scaling methodology has to be applicable to all different PCS-types of compounds (Section 5).


## 5. Comparison of Predicted Transport Coefficients with Experimental Ones

### 5.1. Thermal Conductivity

The spectacular confirmation ofuniversality, which demonstrates the proposed scaling approach, is represented forthermal conductivity in Figure 3. In contrast with the standard PCS approach [9, 10] any input values of this quantity itself λ0T0 taken at the certain fixed temperature T0 have not been used. Besides, only low-temperature data for argon [11] measured within the quite narrow range of temperature [84, 88 K] are necessary to predict, for example, thermal conductivity of ILs and other compounds in the much wider ranges of temperature shown in Figure 3.Figure 3
Comparison of heat conductivities predicted (curves) by universal (argon-based) scaling function (20) with experimental data (points) at P0≈0,1 MPa.Unfortunately, the experimental data on thermal conductivity (and viscosity) are often scarce and scattered (white symbols) even for the simple molecular liquids. All measured values for ILs (black symbols) have been taken from the work of Nieto de Castro et al. [25]. I have not found in literature the necessary input data on speed of sound cs(T) for [pmmim][Tf2N] and used those (Table 2) for the structurally close [bmim][Tf2N] with the same molecular mass and common anion. The predicted data in Figure 3 are reasonably compatible with the available literature data on λT,P0 and the maximum deviations are in the range from −6 to 9%.It is also important and interesting to calculate the relative percentage contribution of molecular-based mechanism in the overall transport of heat and momentum by equations following from (20) and (21):(29)kλ%=CPχ∗100CPχ∗+2kB/m0cs/u-,(30)lη%=ν∗100ν∗+exp0.0192cs/u-2,where coefficients A=2 and B=0.0192 are adjusted to the argon data. Such calculations are represented in Table 4.Table 4
Percentage contribution of molecular-based mechanism into the transport of heat (kλ%, (29)) and momentum (lη%, (30)).
(a)
Argon
Nitrogen
Ethylene
T, K
k, %
l, %
T, K
k, %
l, %
T, K
k, %
l, %
83,804
26,00
73,45
63,16
33,91
72,32
104
39,43
70,28
84
25,96
73,46
64
33,77
72,52
110
39,80
71,02
85
25,73
73,50
66
33,38
72,94
120
40,25
72,30
86
25,48
73,54
68
32,91
73,31
130
40,46
73,40
87
25,22
73,57
70
32,37
73,63
140
40,43
74,29
88
24,95
73,60
72
31,73
73,90
150
40,11
74,97
74
30,99
74,13
160
39,45
75,45
76
30,13
74,31
169,13
38,47
75,74
77,35
29,48
74,42
(b)
[bmim][BF4]
[bmim][PF6]
[pmmim][Tf2N]
T, K
k, %
l, %
T, K
k, %
l, %
T, K
k, %
l, %
290
71,52
20,87
290
72,49
14,37
290
77,97
9,95
300
71,90
23,80
300
72,81
17,04
300
78,50
12,57
310
72,25
26,82
310
73,12
19,89
310
79,00
15,52
320
72,59
29,86
320
73,40
22,85
320
79,48
18,74
330
72,91
32,89
330
73,67
25,88
330
79,93
22,18
340
73,21
35,86
340
73,91
28,93
340
80,36
25,77
350
73,50
38,75
350
74,14
31,96
350
80,78
29,43
360
73,77
41,53
360
74,35
34,94
360
81,17
33,09
370
74,03
44,19
370
74,54
37,83
370
81,54
36,70One may note that the contribution of molecular mechanism gives roughly one-third in totalλ(T) for simple molecular substances (Ar, N2, C2H4) but becomes more than two-thirds for ILs ([bmim][BF4], [bmim][PF6], [pmmim][Tf2N]). The reverse interrelation of contributions is observable for the transport of momentum (see below). It is more than two-thirds for simple molecular liquids but becomes relatively small for ILs especially near their melting temperatures. Moreover, the contribution of molecular mechanism in η(T) for all pyridinium-based (friable) ILs (not represented in Table 4) is less than 3% (i.e., negligible) even for the highest temperature of the measured range. Hence, the natural conclusion is that the transport of momentum in such ILs has the dominantly wave mechanism.


### 5.2. Viscosity

Taking into account the results of Table4 it is especially interesting to discuss the scaling prediction of η(T), namely, for the pyridinium-based ILs (Figures 5 and 6) and to compare it with the results for simple molecular liquids (Figure 4).Figure 4
Comparison of viscosities for molecular liquids predicted (curves) by universal (argon-based) scaling function (21) with experimental data (points) at P0≈0,1 MPa.Figure 5
Experimental [19] (points) and predicted (curves) by scaling (argon-based) universal equation (21) viscosities for pyridinium-based ILs with the common anion [EtSO4]. The extrapolated η(T) values in the metastable range between melting Tm and freezing Tf temperatures are shown by dashed curves.
(a)
(b)Figure 6
Experimental [19] (points) and predicted (curves) by scaling (argon-based) universal equation (21) viscosities for pyridinium-based ILs with the common anion [MeSO4]. The extrapolated η(T) values in the metastable range between melting Tm and freezing Tf temperatures are shown by dashed curves (both of the above temperatures are unobservable experimentally [19] for [BuMePyr][MeSO4]).
(a)
(b)Again the spectacular predictive abilities of the scaling approach as well as its universality are evident. All measuredη(T)-data represented by black symbols in Figures 5 and 6 for the pairs of ILs with the common anions ([EtSO4] and [MeSO4], resp.) have been taken from the work of Gonzále et al. [19]. This work contains also the necessary input data on density ρ0T,P0 and speed of sound csT,P0. Again the reasonable (<30%) overall level of accuracy was demonstrated, though for some ILs ([EtMePyr][MeSO4]) and ([EtMePyr][EtSO4]) the deviations from experiment [19] were large and systematic while for the others ([BuEtPyr][EtSO4] and [BuMePyr][MeSO4]) were close to the experimental uncertainties.At the same time, our attempt to predict the viscosity for twocompact imidazolium-based ILs ([bmim][BF4] and [bmim][PF6]) with the common cation was not so successful. The use of input data on density ρT,P0 and speed of sound csT,P0 reported in the work [13] predicts by (21) the significantly underestimated quantities of ηT,P0 (located between the limits 0,006 and 0,001 Pa·s). Such predicted values are larger than those for molecular liquids (located (see Figure 4) for C2H4 and C2H6 between the limits of 0,0015 and 0,0002 Pa·s and for Ar and N2 between the limits of 0,0003 and 0,00015 Pa·s) butmuch lower than ones reported by different authors for [bmim][PF6] (e.g., 0,173; 0,2179; 0,272; 0,450 Pa·s [26–28]). Of course, the accuracy of the above scattered experimental data must be improved because here the presumable sharp jump between the average viscosities of friable and compact ILs may be the illuminating factor at the choice of IL for practical usage. High sensitivity of results to the input csT value has been confirmed, namely, for the viscosity (21), while the thermal conductivity (20) remains relatively insensitive to its value.


### 5.3. Prandtl’s Number

The remarkable predictive abilities of the scaling approach are again confirmed by Figure7(a) for molecular liquids and, partially, for the imidazolium-based ILs by Figure 7(b). Taking into account the relative failure of proposed methodology at the prediction of viscosity for [bmim][BF4] and [bmim][PF6], it is interesting to test its ratio Pr¯ at the respective, reasonably predicted λ(T) value for them as is shown in Figure 3. The result of such test is rather unwonted. The scaling approach predicts theabsolutely realisticPr¯ values for both [bmim][BF4] and [bmim][PF6] (compatible with that (experimental) for [pmmim][Tf2N]) shown in Figure 7(b). The only reason of such observation follows from the definition of Pr¯-number by (22). To compensate the presumably underestimated dynamic viscosity η one needs the significantly overestimated CP values for the prediction of realistic Pr¯. This trend is confirmed, partially, in the measurements of isobaric heat capacity of ILs by different authors [29]. The comparison of Pr¯ for molecular liquids (Figure 7(a)) with that for ILs leads to the simple practical conclusion. The latter is preferable, in total, as the thermal storage media while the imidazolium-based ILs can be, additionally, useful as the potential heat-transfer liquids too.Figure 7
Experimental (points) and predicted (curves) by the universal argon-based scaling function (22) Prandtl’s numbers for the wide range of temperatures in the molecular (a) and ionic (b) liquids at P=0,1 MPa.
(a)
(b)


## 5.1. Thermal Conductivity

The spectacular confirmation ofuniversality, which demonstrates the proposed scaling approach, is represented forthermal conductivity in Figure 3. In contrast with the standard PCS approach [9, 10] any input values of this quantity itself λ0T0 taken at the certain fixed temperature T0 have not been used. Besides, only low-temperature data for argon [11] measured within the quite narrow range of temperature [84, 88 K] are necessary to predict, for example, thermal conductivity of ILs and other compounds in the much wider ranges of temperature shown in Figure 3.Figure 3
Comparison of heat conductivities predicted (curves) by universal (argon-based) scaling function (20) with experimental data (points) at P0≈0,1 MPa.Unfortunately, the experimental data on thermal conductivity (and viscosity) are often scarce and scattered (white symbols) even for the simple molecular liquids. All measured values for ILs (black symbols) have been taken from the work of Nieto de Castro et al. [25]. I have not found in literature the necessary input data on speed of sound cs(T) for [pmmim][Tf2N] and used those (Table 2) for the structurally close [bmim][Tf2N] with the same molecular mass and common anion. The predicted data in Figure 3 are reasonably compatible with the available literature data on λT,P0 and the maximum deviations are in the range from −6 to 9%.It is also important and interesting to calculate the relative percentage contribution of molecular-based mechanism in the overall transport of heat and momentum by equations following from (20) and (21):(29)kλ%=CPχ∗100CPχ∗+2kB/m0cs/u-,(30)lη%=ν∗100ν∗+exp0.0192cs/u-2,where coefficients A=2 and B=0.0192 are adjusted to the argon data. Such calculations are represented in Table 4.Table 4
Percentage contribution of molecular-based mechanism into the transport of heat (kλ%, (29)) and momentum (lη%, (30)).
(a)
Argon
Nitrogen
Ethylene
T, K
k, %
l, %
T, K
k, %
l, %
T, K
k, %
l, %
83,804
26,00
73,45
63,16
33,91
72,32
104
39,43
70,28
84
25,96
73,46
64
33,77
72,52
110
39,80
71,02
85
25,73
73,50
66
33,38
72,94
120
40,25
72,30
86
25,48
73,54
68
32,91
73,31
130
40,46
73,40
87
25,22
73,57
70
32,37
73,63
140
40,43
74,29
88
24,95
73,60
72
31,73
73,90
150
40,11
74,97
74
30,99
74,13
160
39,45
75,45
76
30,13
74,31
169,13
38,47
75,74
77,35
29,48
74,42
(b)
[bmim][BF4]
[bmim][PF6]
[pmmim][Tf2N]
T, K
k, %
l, %
T, K
k, %
l, %
T, K
k, %
l, %
290
71,52
20,87
290
72,49
14,37
290
77,97
9,95
300
71,90
23,80
300
72,81
17,04
300
78,50
12,57
310
72,25
26,82
310
73,12
19,89
310
79,00
15,52
320
72,59
29,86
320
73,40
22,85
320
79,48
18,74
330
72,91
32,89
330
73,67
25,88
330
79,93
22,18
340
73,21
35,86
340
73,91
28,93
340
80,36
25,77
350
73,50
38,75
350
74,14
31,96
350
80,78
29,43
360
73,77
41,53
360
74,35
34,94
360
81,17
33,09
370
74,03
44,19
370
74,54
37,83
370
81,54
36,70One may note that the contribution of molecular mechanism gives roughly one-third in totalλ(T) for simple molecular substances (Ar, N2, C2H4) but becomes more than two-thirds for ILs ([bmim][BF4], [bmim][PF6], [pmmim][Tf2N]). The reverse interrelation of contributions is observable for the transport of momentum (see below). It is more than two-thirds for simple molecular liquids but becomes relatively small for ILs especially near their melting temperatures. Moreover, the contribution of molecular mechanism in η(T) for all pyridinium-based (friable) ILs (not represented in Table 4) is less than 3% (i.e., negligible) even for the highest temperature of the measured range. Hence, the natural conclusion is that the transport of momentum in such ILs has the dominantly wave mechanism.


## 5.2. Viscosity

Taking into account the results of Table4 it is especially interesting to discuss the scaling prediction of η(T), namely, for the pyridinium-based ILs (Figures 5 and 6) and to compare it with the results for simple molecular liquids (Figure 4).Figure 4
Comparison of viscosities for molecular liquids predicted (curves) by universal (argon-based) scaling function (21) with experimental data (points) at P0≈0,1 MPa.Figure 5
Experimental [19] (points) and predicted (curves) by scaling (argon-based) universal equation (21) viscosities for pyridinium-based ILs with the common anion [EtSO4]. The extrapolated η(T) values in the metastable range between melting Tm and freezing Tf temperatures are shown by dashed curves.
(a)
(b)Figure 6
Experimental [19] (points) and predicted (curves) by scaling (argon-based) universal equation (21) viscosities for pyridinium-based ILs with the common anion [MeSO4]. The extrapolated η(T) values in the metastable range between melting Tm and freezing Tf temperatures are shown by dashed curves (both of the above temperatures are unobservable experimentally [19] for [BuMePyr][MeSO4]).
(a)
(b)Again the spectacular predictive abilities of the scaling approach as well as its universality are evident. All measuredη(T)-data represented by black symbols in Figures 5 and 6 for the pairs of ILs with the common anions ([EtSO4] and [MeSO4], resp.) have been taken from the work of Gonzále et al. [19]. This work contains also the necessary input data on density ρ0T,P0 and speed of sound csT,P0. Again the reasonable (<30%) overall level of accuracy was demonstrated, though for some ILs ([EtMePyr][MeSO4]) and ([EtMePyr][EtSO4]) the deviations from experiment [19] were large and systematic while for the others ([BuEtPyr][EtSO4] and [BuMePyr][MeSO4]) were close to the experimental uncertainties.At the same time, our attempt to predict the viscosity for twocompact imidazolium-based ILs ([bmim][BF4] and [bmim][PF6]) with the common cation was not so successful. The use of input data on density ρT,P0 and speed of sound csT,P0 reported in the work [13] predicts by (21) the significantly underestimated quantities of ηT,P0 (located between the limits 0,006 and 0,001 Pa·s). Such predicted values are larger than those for molecular liquids (located (see Figure 4) for C2H4 and C2H6 between the limits of 0,0015 and 0,0002 Pa·s and for Ar and N2 between the limits of 0,0003 and 0,00015 Pa·s) butmuch lower than ones reported by different authors for [bmim][PF6] (e.g., 0,173; 0,2179; 0,272; 0,450 Pa·s [26–28]). Of course, the accuracy of the above scattered experimental data must be improved because here the presumable sharp jump between the average viscosities of friable and compact ILs may be the illuminating factor at the choice of IL for practical usage. High sensitivity of results to the input csT value has been confirmed, namely, for the viscosity (21), while the thermal conductivity (20) remains relatively insensitive to its value.


## 5.3. Prandtl’s Number

The remarkable predictive abilities of the scaling approach are again confirmed by Figure7(a) for molecular liquids and, partially, for the imidazolium-based ILs by Figure 7(b). Taking into account the relative failure of proposed methodology at the prediction of viscosity for [bmim][BF4] and [bmim][PF6], it is interesting to test its ratio Pr¯ at the respective, reasonably predicted λ(T) value for them as is shown in Figure 3. The result of such test is rather unwonted. The scaling approach predicts theabsolutely realisticPr¯ values for both [bmim][BF4] and [bmim][PF6] (compatible with that (experimental) for [pmmim][Tf2N]) shown in Figure 7(b). The only reason of such observation follows from the definition of Pr¯-number by (22). To compensate the presumably underestimated dynamic viscosity η one needs the significantly overestimated CP values for the prediction of realistic Pr¯. This trend is confirmed, partially, in the measurements of isobaric heat capacity of ILs by different authors [29]. The comparison of Pr¯ for molecular liquids (Figure 7(a)) with that for ILs leads to the simple practical conclusion. The latter is preferable, in total, as the thermal storage media while the imidazolium-based ILs can be, additionally, useful as the potential heat-transfer liquids too.Figure 7
Experimental (points) and predicted (curves) by the universal argon-based scaling function (22) Prandtl’s numbers for the wide range of temperatures in the molecular (a) and ionic (b) liquids at P=0,1 MPa.
(a)
(b)


## 6. Conclusions

The predictive abilities and universality of the proposed scaling model of low-temperature transport coefficients are obvious. This model is testable by the MD-simulations based on the thermodynamically predicted LJ-parametersσ0 and ε(T) here. The derived exact form of EOS for LJ-liquid leads to the potential possibility to construct thecrossover formalism connecting critical (Tc)and normal boiling (Tb)regions by the FT-model’s concepts and results.



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*Source: 208486-2015-12-06.xml*