Relationship between Viscosity Coefficients and Volumetric Properties

J. Phys. Chem. B 2008, 112, 5563-5574

5563

Relationship between Viscosity Coefficients and Volumetric Properties Using a Scaling Concept for Molecular and Ionic Liquids Alfonso S. Pensado,†,‡ Agı´lio A. H. Pa´ dua,‡ Marıá J. P. Comun˜ as,† and Josefa Ferna´ ndez*,† Laboratorio de Propiedades Termofı´sicas, Departamento de Fı´sica Aplicada, Facultade de Fı´sica, UniVersidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain, and Laboratoire de Thermodynamique des Solutions et des Polyme` res, CNRS UMR 6003, UniVersite´ Blaise Pascal Clermont-Ferrand, 63177 Aubie` re, France ReceiVed: December 14, 2007; In Final Form: February 14, 2008

In this work, a scaling concept based on relaxation theories of the liquid state was combined with a relation previously proposed by the authors to provide a general framework describing the dependency of viscosity on pressure and temperature. Namely, the viscosity-pressure coefficient (∂η/∂p)T was expressed in terms of a state-independent scaling exponent, γ. This scaling factor was determined empirically from viscosity versus TV γ curves. New equations for the pressure- and temperature-viscosity coefficients were derived, which are of considerable technological interest when searching for appropriate lubricants for elastohydrodynamic lubrication. These relations can be applied over a broad range of thermodynamic conditions. The fluids considered in the present study are linear alkanes, pentaerythritol ester lubricants, polar liquids, associated fluids, and several ionic liquids, compounds selected to represent molecules of different sizes and with diverse intermolecular interactions. The values of the γ exponent determined for the fluids analyzed in this work range from 1.45 for ethanol to 13 for n-hexane. In general, the pressure-viscosity derivative is well-reproduced with the values obtained for the scaling coefficient. Furthermore, the effects of volume and temperature on viscosity can be quantified from the ratio of the isochoric activation energy to the isobaric activation energy, EV /Ep. The values of γ and of the ratio EV/Ep allow a classification of the compounds according to the effects of density and temperature on the behavior of the viscosity.

Introduction The dependences of the transport coefficients of liquids on density and temperature are of considerable interest and importance because they provide information on not only the structure of liquids but also on the dynamic processes occurring in the condensed phase. This is of fundamental importance in assessing theoretical models and their foundation. In recent years, essential studies have been performed on the dynamics of glass-forming materials as well as of simple liquids.1-6 Nevertheless, the physics underlying the glass transition is still poorly understood. Models generally interpret the transition in terms of either a jamming of molecular motions due to a reduction in the volume available for the molecules to rearrange or a slowing down of the molecular motions due to a decrease in the energy available to each molecule. The same interpretations are found underlying models describing dynamic properties in broad ranges of temperature and pressure. Neither of these descriptions is completely satisfactory. Two essential properties in the field of the dynamics of glass-forming materials are the structural relaxation time and the viscosity. Thus, at constant pressure, there is a remarkable increase in shear viscosity as the temperature decreases. At the same time, some of the motions inside the fluid slow down in the same proportion. This phenomenon is linked to structural relaxation. When the pressure is increased at a constant temperature, analogous increases in * To whom correspondence should be addressed. Tel.: 34981563100 ext. 14046; fax: 34981520676; e-mail: [email protected]. † Universidade de Santiago de Compostela. ‡ Universite ´ Blaise Pascal Clermont-Ferrand.

viscosity and structural relaxation time occur. From measurements of the dielectric relaxation time, τ, Roland et al.,7-10 among others,11-13 have found the existence of a scaling relation

τ(T,V) ) u (TVγ)

(1)

where V(T,p) is the specific volume at temperature T and pressure p, and γ is an empirically determined constant, characteristic of a given material. The power-law form TVγ enables a precise superposition of the relaxation times of a compound over broad temperature and pressure ranges. Casalini, Roland and co-workers6-10,14 have determined values of the exponent γ, finding values between 0.16 and 8.5 for more 50 compounds, mainly glass-forming liquids and polymers. These authors remarked that the scaling relation only fails for some H-bonded compounds, such as water and oligomeric polypropylene glycol.15 The constant γ reflects the intensity of the intermolecular and intramolecular interactions. The repulsive part of the intermolecular potential dominates the local liquid structure,16,17 so that for local properties, such as the structural relaxation in liquids or the local segmental relaxation in polymers, the interaction potential can be approximated by a spherically symmetric two-body interaction17-19

U(r) )

(σr )

m

-

a rn

(2)

where and σ are the characteristic energy and length scales of the system, r is the intermolecular distance, and m and n are

10.1021/jp711752b CCC: $40.75 © 2008 American Chemical Society Published on Web 05/01/2008

5564 J. Phys. Chem. B, Vol. 112, No. 18, 2008

Pensado et al.

the exponents of the repulsive and attractive terms, respectively. For liquids composed of symmetric molecules, Roland et al.15 identified m ) 3γ, whereas for liquids of nonsymmetric molecules, the parameter γ reflects not only the magnitude of the intermolecular forces but also the contribution from internal molecular modes, such as vibrations and torsions.15 Besides, γ is also affected by strong attractive interactions. Coslovich and Roland20 recently compared empirically determined values of γ with the actual steepness of the intermolecular repulsive potential for m - 6 Lennard-Jones liquids. Very recently, Casalini et al.21 derived this scaling by a revision of the entropy model of the glass transition dynamics due to Avramov.3 The modified Avramov equation proposed by these authors accounts well for the variation of relaxation times with T and V for a variety of organic liquids and polymers. On the other hand, several authors11,15,22,23 extended this thermodynamic scaling of the relaxation times, relating it to the viscosity, η. That is, for a given compound, η is a unique function of TVγ

η(T,V) ) η(T,p) ) f(TVγ) ) f(TF-γ)

(3)

where F is the density. The first applications of this scaling principle11,22,23 were performed for o-terphenyl OTP (γ ) 4), glycerol (γ ) 1.8), and salol (γ ) 5.2). Subsequently, Roland et al.15 verified this equation for several molecular and ionic liquids. These authors found decreasing γ values in going from van der Waals fluids (γ ) 8 for n-octane) to ionic liquids (γ ) 2.25 for 1-methyl-3-octylimidazolium tetrafluroborate), although for some strongly H-bonded materials, such as water, they observed that the superposition fails. In a previous work,24 we determined the exponent γ for several pure and mixed pentaerythritol esters using experimental density and viscosity data from Fandinõ et al.,25-28 Pensado et al.,29,30 and Lugo et al.31 The anomalous viscosity-pressure dependence of water, ZnCl2, and some glass-forming silicate liquids is another interesting behavior that has been analyzed in the course of many experimental and theoretical investigations.32-40 Schmelzer et al.37 proposed a simple equation that relates the pressure and temperature derivatives of density and viscosity, assuming that viscosity is only dependent on density. Subsequently, Pensado et al.29 deduced a more general relation for Newtonian fluids, considering an additional term that accounts for the effect in the viscosity of structural adjustments (changes of conformations, coordination number, structural order in the liquids, etc.) due to pressure changes. To select lubricants with appropriate temperature and pressure dependences of viscosity, Spikes derived a similar equation that involves the internal pressure.41 In the present work, we combined the thermodynamic scaling of viscosity with the relation of Pensado et al.29 to derive an expression for the viscosity-pressure derivative (∂η/∂p)T that includes the scaling exponent γ. This new equation can be applied over the same broad range of thermodynamic conditions as eqs 1 and 3. Accordingly, this new relation can be used to estimate the viscosity of a polymer melt when injected into a mold at high pressure and high temperature, in elastohydrodynamic lubrication contacts (for which pressure can go up to 1-4 GPa), or the viscosity of magma at the extreme conditions found below the Earth’s surface. Furthermore, this equation corrects an equation that Schmelzer et al.37 had derived to explain viscosity anomalies for Newtonian fluids at thermodynamic equilibrium. The new equation was applied to linear alkanes (from n-hexane to n-octadecane), branched alkanes (9-octylpentadecane and squalane), pentaerythritol tetraalkanoates (pentanoate, heptanoate, nonanoate, and 2-ethylhexanoate), polar

liquids (diethylene glycol dimethyl ether (DEGDME), triethylene glycol dimethyl ether (TriEGDME), tetraethylene glycol dimethyl ether (TEGDME), dimethyl carbonate, and diethyl carbonate), associated fluids (methanol, ethanol, 2-methoxyethanol, 2-ethoxyethanol, and 2-isopropoxyethanol), and seven ionic liquids: 1-butyl-3-methylimidazolium hexafluorophosphate ([bmim]PF6), 1-butyl-3-methylimidazolium bis(trifluorosulfonyl)imide ([bmim]Tf2N), 1-butyl-3-methylimidazolium tetrafluoroborate ([bmim]BF4), 1-methyl-3-octylimidazolium hexafluorophosphate ([omim]PF6), 1-methyl-3-octylimidazolium tetrafluoroborate ([omim]BF4), 1-methyl-3-hexylimidazolium hexafluorophosphate ([hmim]PF6), and 1-methyl-3-hexylimidazolium bis(trifluorosulfonyl)imide ([hmim]Tf2N). In addition, for these compounds, the relation between the value of the scaling constant γ and the ratio EV/Epsthe ratio of the isochoric activation energy, EV ) (R(∂ ln η)/∂T-1)V) to the isobaric activation energy, Ep ) (R(∂ ln η)/∂T-1)p)swas analyzed. Pressure-Viscosity Coefficients and Derived Volumetric Properties The petrochemical industry requires reliable viscosity data at high pressures for hydrocarbon mixtures, so as to validate their process designs.42 Viscosity measurements of electrolyte solutions under high pressure have been important in the design of power generating plants and other facilities making use of geothermal brine or seawater and also in the study of dynamic properties of ions or solvent molecules at high pressures.43 The high pressure viscosity also may be needed to estimate the diffusion rate of molecules in a fluid.44 Knowledge of the viscosity of polymer solutions is extremely important for many processes involving polymer formation and solution processing.45 The variation of the viscosity with pressure is especially important for lubricants because, in most mechanical applications, films of fluid are compressed between sliding or rolling surfaces under very high loads. In the elastohydrodynamic regime of lubrication (EHL), the shear viscosity of the lubricant and the pressure-viscosity coefficient, R ) (1/η)(∂η/∂p)T are the main parameters that characterize the thickness of the fluid film that protects the mechanical device from high friction and premature wear. Fluids of higher R values produce thicker lubrication films, so that rolling-element bearings, gears, and rotors are better protected at high pressures. On the other hand, the EHL film thickness is affected by the operating temperature of the components. Environmental factors can influence the actual operating temperatures. In addition, if the equipment will need to make a cold start, it is also important for the viscosity at the starting temperature to be low enough for the machine to start up easily.46 A temperature increase of a few degrees can induce an important viscosity decrease, significantly changing the film thickness. It follows that it is important to know how much the viscosity changes with temperature. Thus, for the selection of a lubricant, the operating temperatures should be taken into account because the ability of a lubricant to form an adequate separating layer between the rubbing surfaces over a wide range of temperature and load conditions is highly beneficial. Hence, these requirements imply a lubricant with a low temperature-viscosity coefficient, β ) -(1/η)(∂η/∂T)p. Hence, the pressure and temperature viscosity coefficients are of considerable technological importance. Pressure-viscosity coefficients, R, are also important from a fundamental point of view. The question as to how the shear viscosity of fluids evolves with pressure is not still clearly solved. Most pure compounds and mixtures show a typical liquid behavior, with the viscosity increasing with applied pressure.

Scaling Concept for Molecular and Ionic Liquids

J. Phys. Chem. B, Vol. 112, No. 18, 2008 5565

For other fluids, such as water, ZnCl2, and certain glass-forming silicate liquids, the viscosity presents an anomalous behavior since it decreases with increasing pressure under certain conditions. To explain this anomalous behavior, Schmelzer et al.37 assumed that the free volume only determines the value of the viscosity and that the free volume is uniquely linked with the total volume of the system. They obtained the following relation for simple Newtonian liquids:

∂V ∂T ∂T ∂p T = )∂p η ∂p V ∂V ∂T p

( ) ( )

( ) ( )

(4)

After some algebra for simple Newtonian liquids for which η(F) was assumed, i.e. (∂η/∂T)V ) 0 and (∂η/∂p)V ) 0 led Schmelzer et al.37 to derive the following equation:

κT(T,p)

(∂η∂p) ) - R (T,p)(∂T∂η) T

(5)

p

p

where κT and Rp are the isothermal compressibility and the isobaric thermal expansion coefficient, respectively. Hence, this equation permits us to determine the dependence of viscosity on pressure, provided that the temperature dependence of viscosity, the isothermal compressibility, and the isobaric thermal expansion coefficient of the liquid are known. Recently, Pensado et al.,29 assuming that the viscosity is an explicit function of the temperature and of the volume η(V,T), derived the following equation:

κT(T,p)

1 (∂η∂p) ) - R (T,p)(∂T∂η) - R (T,p)V (∂V∂p) (∂T∂η) T

p

p

p

η

p

η

(7)

Taking into account the following expression:

∂η ∂η ∂T p )∂p T ∂p ∂T η

( )

( ) ( )

together with the next exact relation for Newtonian fluids29

(∂T∂p) ) (∂T∂p) + (∂F∂p) (∂T∂F)

(6)

This equation separates the pressure-viscosity derivative in two contributions: one due to the change of density as a result of pressure variation and another due to structural changes (changes of conformations, coordination number, structural order of the liquids, etc.) induced by the pressure increase. The second contribution arises from the explicit temperature dependence of the viscosity. The first contribution is positive except in the few cases in which Rp is negative, whereas the second term is usually negative because (∂V/∂p)η is negative. Pensado et al.29 showed that for 2,3-dimethylpentane, which could be considered a van der Waals liquid with a small specific volume, the second contribution is almost negligible, that is, eq 5 is closely obeyed. On the contrary, for water, squalane, and pentaerythritol esters, eq 5 is not verified, the second term in the right-hand side becoming very important, especially in the case of water for which the first contribution is negligible. Unfortunately, eq 6 is not predictive since to determine (∂V/ ∂p)η, it is necessary to know the viscosity as a function of pressure and density. Nonetheless, for liquids that verify eq 3, the viscosity is constant when TF-γ is constant, hence

(∂T∂F) ) γTF

Figure 1. Dynamic viscosity94 of water against density95 from 0.1 to 900 MPa at several temperatures: (0) 278.15 K, (2) 279.15 K, (() 280.15 K, (b) 281.15 K, (9) 282.15 K, and (×) 283.15 K. The horizontal line is indicated to allow the reader to evaluate the signs of (∂T/∂F)η.

(8)

η

F

T

(9)

η

and eq 7, it is easy to obtain the following relation:

κT(T,p)γT

(∂η∂p) ) - (1 + R (T,p)γT)(∂T∂η) ) T

p

p

-

κT(T,p)

∂η (10) 1 ∂T p + Rp(T,p) γT

(

( )

)

Usually, γ is positive, and therefore, viscosity-pressure derivatives obtained with eq 10 are lower that those obtained with eq 5. That is, an explicit temperature dependence of the viscosity leads to a weaker dependence of viscosity with pressure. Eq 10 can be reduced to eq 5 when Rp(T,p)γT . 1. The higher γ is, according to eq 3, the more important the explicit density dependence of the viscosity in comparison with its explicit temperature dependence is (i.e., the relations η(F), (∂η/∂T)V ) 0, (∂η/∂p)V ) 0, and (∂V/∂p)η ) 0, as well as eqs 4 and 5, are more accurate). On the contrary, for lower γ values, eqs 6 or 10 should be used instead of eq 5. In the previous study by Roland et al.15 on the thermodynamic scaling of the viscosity, the superposition principle failed for water. This can be explained through eq 7. The variation of density with temperature at constant viscosity should be either positive (γ > 0) or negative (γ < 0) in the entire temperature and viscosity range. This is not verified for water. As can be seen in Figure 1, the viscosity of water presents a minimum as a function of the density, so the derivative (∂F/∂η)T may be positive or negative depending on the conditions. Accordingly, at densities lower than that of the viscosity minimum, (∂F/∂T)η < 0, whereas at densities higher than that of the minimum, (∂F/


Pensado et al.

∂T)η > 0. Thus, according to eq 7, it can be concluded that for fluids that, like water, present an extreme in the curve η(F), it is not possible to use a unique γ value to represent the scaling of viscosity. As γ is not uniquely defined for water, eq 10 cannot be used to explain the anomaly in the viscosity of water. The isothermal compressibility and isobaric thermal expansion coefficient are related with the internal pressure, π(T,p), which describes the sensitivity of the internal energy U to a change in volume through the relations

π(T,p) )

() ( ) ∂U ∂p

)T

T

κT(T,p)

Rp(T,p)

-p

(11)

Combining eqs 10 and 11 it easy to obtain

κT(T,p)γT

(∂η∂p) ) - (1 + κ (T,p)γ(π(T,p) + p))(∂T∂η) T

T

p

(12)

Using the definitions of the pressure-viscosity coefficient, R ) (1/η)(∂η/∂p)T, and the temperature-viscosity coefficient, β ) -(1/η)(∂η/∂T)p, eq 12 can be written in the form

κT(T,p)γT R ) β (1 + κT(T,p)γ(π(T,p) + p))

(13)

This equation is of considerable technological interest in the search for appropriate lubricants for elastohydrodynamic lubrication. As we have already commented, higher R values produce thicker lubricant films, and low β values help the lubricant to form an adequate separating layer between the rubbing surfaces over a wide range of temperatures. Hence, it can be considered that lubricants with a high R/β ratio are more appropriate. Eq 13 could be considered a useful tool to find suitable lubricants. Thus, a high R/β ratio could be obtained with lubricants of low internal pressure and high scaling coefficient and isothermal compressibility. A common measure of the effects of volume and temperature on viscosity41 is the ratio of the isochoric activation energy, EV ) (R(∂ ln η)/∂T-1)V) to the isobaric activation energy, Ep ) (R(∂ ln η)/∂T-1)p). Equivalent definitions in terms of the relaxation time also are used. The following relation can be derived for the ratio EV /Ep:

EV ∂p ∂T )1Ep ∂T V ∂p η

( )( )

(14)

Williams47 obtained an equivalent equation for relaxation times. If volume dominates the viscosity behavior, then from eq 4, one obtains (∂T/∂p)V ) (∂T/∂p)η, and from eq 14, one can deduce EV/Ep ) 0. On the contrary, if viscosity is only temperaturedependent, then (∂T/∂p)η ) 0, and EV/Ep ) 1. Hence, the ratio EV/Ep ranges from 0 to 1. From eq 14, the following relation can be obtained:

EV 1 ) Ep 1 - (Rp(T,p)/Rη(T,p))

(15)

where the isoviscous thermal expansion coefficient is given by Rη(T,p) ) -(1/F)(∂F/∂T)η ) (1/V)(∂V/∂T)η. Casalini and Roland48 derived the equivalent relation for relaxation times. Using the scaling exponent γ, it can be deduced from eq 7 that

Rη(T,p) ) -

1 γT

(16)

From eqs 15 and 16, it can be obtained that

EV 1 ) Ep 1 + γTRp(T,p)

(17)

As expected, this equation is analogous to that previously obtained9 for the ratio of the isochoric activation energy, EV ) (R(∂ ln τ)/∂T-1)V) to the isobaric activation energy, Ep ) (R(∂ ln τ)/∂T-1)p) of the relaxation times. Hence, if the scaling exponents for viscosities and relaxation times are the same (eqs 1 and 3), then the corresponding EV/Ep ratio also should be the same. There are only a few studies comparing γ values obtained from relaxation times with those obtained from viscosities. For o-terphenyl,22 glycerol,11 salol,23 and di-n-butylphthalate,15 γ values obtained from both properties are similar. To derive eq 17 for relaxation times, an exponential function for the relaxation times τ in terms of the product TVγ was assumed by Casalini and Roland.9 In our derivation of eq 17, there was no assumption apart from the scaling superposition of viscosities. Hence, the equivalent relation of eq 17 for relaxation times also could be derived in a similar way. Furthermore, Roland and Casalini7 derived eq 16 for the thermal expansion coefficient at a constant relaxation time, Rτ, at the glass transition temperature γRτ(Tg). These authors concluded that the product TgRτ(Tg) is independent of pressure, this constancy previously being known empirically as the Boyer-Spencer rule.49 From eq 16, and its equivalent for relaxation times Rτ(T,p) ) -(1/γT), we can extend the results of Roland and Casalini7 by concluding that for all the materials for which superposition of viscosities or relaxation times is verified, the two products TRη(T,p) and TRτ(T,p) are pressureand temperature-independent. Besides, from eq 7 and from Rτ(T,p) ) -(1/γT), it can be concluded that both thermal pressure coefficients at constant viscosity or relaxation time should be pressure-independent. Results and Discussion In Table 1, we present, for the fluids studied, the pressure and temperature intervals used to determine the values of γ and the average values of the ratio (∂η/∂p)Teq5/(∂η/∂p)Texp, where (∂η/ ∂p)Teq5 represents the predictions using eq 5, and (∂η/∂p)Texp represents the values determined from experimental viscosities. The compounds were chosen to represent molecules of different sizes and with diverse intermolecular interactions, taking into account the availability of density and viscosity data in broad pressure and temperature ranges. For several of these compounds, we determined the scaling factor γ. To obtain γ for each fluid, η was plotted against 1000FγΤ-1. The following equation, derived by Casalini and Roland50 from entropy considerations, was used to fit the scaled viscosity plots:

η(T,V) ) η0 exp

[( ) ] A TVγ

φ

(18)

Different γ values, stepped 0.05, were used to obtain the best regression statistics. The effects of volume and temperature on the viscosity were analyzed for several compounds using the values of γ, (∂η/ ∂p)Teq5/(∂η/∂p)Texp, and EV/Ep. When the volume was dominant in the behavior of viscosity, γ ) ∞, (∂η/∂p)Teq5/(∂η/∂p)Texp) 1, and EV/Ep ) 0. On the contrary, if the thermal energy governs the viscosity behavior, γ ) 0, (∂η/∂p)Teq5/(∂η/∂p)Texp) ∞, and EV /Ep ) 1. Simple Fluids and Alkanes. For simple van der Waals liquids with a relatively high specific volume, the experimental



TABLE 1: Database Used for Studied Fluids, Scaling Exponents (γ), Values of Average Ratio (Dη/Dp)Teq5/(Dη/Dp)Texptl, and EW/Ep compound n-hexane n-heptane n-octane n-decane n-dodecane n-octadecane 9-octilpentadecane squalane PEC5 PEC7 PEC9 PEB8 [bmim]BF4 [bmim]PF6 [bmim]Tf2N [hmim]PF6 [hmim]Tf2N [omim]BF4 [omim]PF6 methanol ethanol 2-methoxyethanol 2-ethoxyethanol 2-isopropoxyethanol DEGDME TriEGDME TEGDME dimethyl carbonate diethyl carbonate a

p (MPa)

T (K)

ref

γ

(∂η/∂p)Teq5/ (∂η/∂p)Texptl

0.1-250 0.1-249 0.1-373 0.1-254 0.1-192 0.1-92 0.1-533 0.1-60 0.1-60 0.1-60 0.1-60 0.1-60 0.1-300 0.1-249 0.1-249 0.1-299 0.1-240 0.1-40 0.1-224 0.1-176 0.1-60 0.1-100 0.1-100 0.1-100 0.1-100 0.1-60 0.1-60 0.1-60 0.1-60 0.1-60

303.15-348.15 303.15-348.15 283.15-353.15 303.15-348.15 298.15-473.15 323.15-473.15 273.15-372.04 273-373 273-373 273-373 283-373 273-373 273.15-353.15 273.15-353.15 273.15-353.15 273.15-353.15 273.15-353.15 288.15-353.15 273.15-353.15 273.15-353.15 273.15-333.15 273.15-353.15 293.15-353.15 293.15-353.15 293.15-353.15 293.15-353.15 283.15-353.15 283.15-353.15 293.15-353.15 283.15-353.15

84 85 86 84 87 87 88 25, 26, 30 28, 29 27, 29 27, 30 25, 30 66, 67 63 63, 67 65, 83 65, 68 69 64 64 89 89 90, 91 90, 91 90, 91 90, 92 93 93 93 93

13 11 8b 7 6.5 6.3 4.2 4.2b 5.3c 5.4c 4.5c 3.6c 2.25 2.9b 3.4 2.85 2.75 2.45 2.25b 2.4b 1.85 1.45 3.1 3.2 3.4 6.1 5.3 4.4 10.1 10.3

1.3 1.3 1.7 1.6 1.6 1.6 2.2 1.9 1.8 1.8 1.9 2.2 3.1 2.9 2.9 3.0 2.8 2.9 4.0 2.7 2.7 3.4 2.1 2.9 1.9 1.5 1.7 1.8 1.3 1.3

At 313.15 K and 0.1 MPa. b Reported by Roland et al.15

c

Ev/Epa 0.15 0.18 0.25 0.30 0.33 0.48 0.49 0.43 0.43 0.48 0.54 0.66 0.64 0.61 0.62 0.66 0.66 0.69 0.69 0.58 0.66 0.51 0.49 0.47 0.33 0.34 0.44 0.20 0.21

Reported by Fandinõ et al.24

Figure 2. Dynamic viscosity vs density at different temperatures: (0) 2,3-dimethylpentane96 and (O) carbon dioxide.97

viscosities at different pressures and temperatures fall in the same curve η(F), as can be seen in Figure 2 for 2,3dimethylpentane and carbon dioxide. Hence, explicit dependence on temperature is negligible. For this type of fluid, eq 5 was verified with a high accuracy. Thus, the predicted pressureviscosity derivative using this equation for 2,3-dimethylpentane ranged from 0.9 to 1.2 times that of the experimental data.29 When volume exclusively controls the dynamics, eq 5 is verified, and so the ratio among the predicted and experimental viscosity derivatives is 1. For liquids that verify approximately η(F), the scaling factor γ is large and not well-defined. Thus, γ values ranging from

20 to 29 in the case of 2,3-dimethylpentane give similar regression statistics. For n-hexane and n-heptane, we obtained the best results with γ values of 13 and 11, respectively. According to Roland et al.15 (see eq 2), this corresponds to repulsive exponents of 39 and 33 in the intermolecular potential. But, even with lower scaling factors, superpositions of similar quality still were obtained. Thus, for simple fluids, temperature does not directly affect the viscosity, but only indirectly through the density. When the number of covalent bonds increases, this situation changes due to contributions from the internal degrees of freedom of the molecules. For longer alkanes, eq 5 predicts pressure-viscosity derivatives larger than the experimental ones and the scaling factor decreases, as can be seen in Figures 3 and 4. In this case, 3γ reflects the steepness of the overall effective potential, which is composed of the intermolecular repulsive potential combined with stretching and bending intramolecular components. Roland et al.15 have shown from molecular simulations of 1,4polybutadiene that the steepness of the repulsive portion of the effective Lennard-Jones potential decreases from 12 to 8.5 when a backbone stretching potential is added. Hence, when the number of covalent bonds increases, the scaling factor decreases in agreement with our results for alkanes (Figure 3a). Figure 3b shows the quality of the superposition curves for n-hexane, n-heptane, n-octane, n-decane, n-dodecane, and n-octadecane. The slope of the master curve is lower for the longer alkanes (n-dodecane and n-octadecane) than for the shorter ones (n-hexane and n-heptane). We have not found γ values for these compounds in the literature except for n-octane, for which a value of 8 was determined by Roland et al.15 In Figure 3a, it can be seen that this value confirms the expected trend of γ with the size of the alkane. Figure 4a shows the pressure-viscosity derivatives obtained using eq 10 for n-hexane, n-heptane, n-octane, and n-decane at


Pensado et al.

Figure 3. (a) Scaling factor γ vs the number of carbon atoms of the lineal alkanes CnH2n+2 and (b) superpositioning of the viscosity values: (0) n-hexane, (2) n-heptane, (() n-octane, (b) n-decane, (9) n-dodecane, and (×) n-octadecane.

Figure 4. (a) (∂η/∂p)T derivative obtained directly from experimental data (symbols) and from eq 10 (line) vs pressure and (b) EV/Ep vs temperature: (0) n-hexane, (2) n-heptane, (() n-octane, (b) n-decane, (9) n-dodecane, and (×) n-octadecane.

313.15 K together with the values obtained from the correlations of the experimental viscosity values. With the γ values calculated, eq 10 reproduces well the experimental curves. Equation 5 reproduces reasonably well the experimental viscosity-pressure derivative for n-hexane, whereas for n-octadecane, the predicted values are an average of 1.6 times the experimental values. From these results, we can conclude that for shorter alkanes (2,3-dimethylpentane and n-hexane) and for carbon dioxide, volume is the dominant variable controlling the viscosity-pressure dependence but that temperature plays a more important role when the size of the alkanes increases. Consequently, we found using eq 17 that the EV/Ep values increase with the number of carbons in the alkanes as can be seen in Figure 4b. This is in agreement with the recent results obtained by Roland et al.51 concerning several samples of polystyrene. For this polymer, these authors found a systematic increase in the EV/Ep ratio with increasing molecular weight. As the number of carbon atoms in the alkanes increases, the presence of molecular flexibility softens the potential (i.e., the scaling factor decreases). Thus, we found for the branched alkanes 9-octylpentadecane (C23H48) a γ value of 4.2. This value

coincides with that of squalane (C30H62) determined by Roland et al. In Figure 5a, new viscosity values, reported in the present work, at 0.1 MPa from 273 to 373 K, together our previous viscosity values29 from 303.15 to 353.15 K up to 60 MPa, are plotted versus 1000Fγ/T. The new viscosity data can be found in the Supporting Information. Both sets of viscosity values superpose with γ ) 4.2. The predicted viscosity-pressure derivatives for 9-octylpentadecane (C23H48) and squalane (C30H62) using eq 5 are, on average, 2.1 and 1.9 times the experimental values (i.e., both temperature and volume affect dynamics in the liquid phase with comparable contributions). Polar Liquids. In a previous article,24 we determined γ values for four pure pentaerythritol esters (PE): pentaerythritol tetrapentanoate (PEC5, C25H44O8), pentaerythritol tetraheptanoate (PEC7, C33H60O8), pentaerythritol tetranonanoate (PEC9, C41H76O8), and pentaerythritol tetra-2-ethylhexanoate (PEB8, C37H68O8), and for three mixtures of PEs. The pentaerythritol ester molecules have four identical acid chains. The highest value of γ (γ ) 5.4) was obtained for PEC7, and the lowest for a PEB8 + PEC5 mixture (γ ) 3.0). These results indicated that both density and temperature exert a significant influence



Figure 5. (a) Superpositioning of the viscosity values and (b) (∂η/∂p)T derivative obtained directly from experimental data (symbols) and from eq 10 (line) vs pressure: (0) squalane, (2) PEC5, (() PEC7, (b) PEC9, and (9) PEB8.

on viscosity. Roland et al.15 found for viscosity and relaxation times a γ value of 3.2 for di-n-butylphthalate (DBP; C16H22O4, C6H4-(COOC4H9)2). According to these authors, DBP exhibits the lowest γ value found to date for any molecular glass-former lacking hydrogen bonds. As DBP is quite polar (µ ) 2.4 D), Roland et al. concluded that the polarity in DBP could soften the effective repulsive interactions in a similar manner as the effect of intramolecular terms. The number of bonds in the PEs studied here was larger than in di-n-butylphthalate, but, unfortunately, we do not know the dipole moments of the PEs for comparison. Fandinõ et al.24 also found that for the three pure PEs with linear carboxylic acid chains, the sequence of γ is inverse to that of the glass transition temperatures, Tg, at 0.1 MPa, that were measured by Shobha and Kishore.52 The dependence of Tg on molecular weight for several complex esters was explained by these authors using the concept of segmental motion. As the molecular weight increases beyond 600 g mol-1, the flexibility of the chain will not be favored in branched compounds due to steric effects restricting the segmental motion, and hence, Tg increases gradually with the molecular weight.52 In Figure 5a, we included new experimental viscosity values for the four pure PEs at 0.1 MPa, which had not been used in the study by Fandinõ et al.24 to determine γ. The new viscosity values for the four pure PEs are gathered in Table S1 of the Supporting Information. The samples of PEC5, PEC7, and PEC9 used in the present study were not from the same batch as those used previously by Pensado et al.29,30 to determine the viscosities at high pressures. The new experimental values for the four PEs were used to estimate the glass transition temperature, Tg, at 0.1 MPa. Traditionally, the glass transition temperature has been kinetically defined as the temperature at which the viscosity reaches a value of 1015 mPa s, so it does not coincide exactly with the temperature at which momentum transport vanishes. The values obtained for Tg can differ from those determined by calorimetry, due to the different time scales involved53,54 but also due to the suitability of the model. We first performed these calculations with the Andrade equation (i.e., considering Arrhenius behavior). The values obtained from the Arrhenius plot at atmospheric pressure for PEC5, PEC7, and PEC9 were around 120 K lower than the calorimetric Tg data.52 For squalane, the value obtained at 0.1 MPa from the Arrhenius plot was around 90 K lower

than that obtained by calorimetry.35,55 Nevertheless, using the VFT correlations,56-58 the estimated glass transition temperatures at atmospheric pressure for the pentaerythritol esters were quite close to the experimental ones: for PEC5, PEC7, and PEC9, the estimated Tg values at 0.1 MPa were 186, 189, and 189 K, and the experimental calorimetric values52 were 190, 188, and 202 K, respectively. For PEB8, the estimated Tg value at 0.1 MPa was 200 K, but no calorimetric Tg value was found. For squalane, the estimated Tg value at atmospheric pressure was 192 K, which is a little higher than the value of 167.4 K obtained by Richert et al.35 from dielectric relaxation measurements and is also higher than the value of 171.3 K determined from calorimetric measurements.35,55 Hence, in all these cases, the VFT equation considerably improves the estimation of Tg. This is because these compounds present an intermediate behavior in which two control parameters, density and temperature, have comparable effects,11 so they do not follow Arrhenius behavior. As indicated in a previous work,29 in the case of PEs, the pressure-viscosity derivative predicted assuming that viscosity is only a function of density (eq 5) is almost double the experimental results. That is, the variation of temperature necessary to keep the viscosity constant upon a pressure change is half that required to keep the density constant (see eq 4). In Figure 5b, the pressure-viscosity derivative for the pentaerythritol esters at 323.15 K is plotted against pressure, and it can be seen that using the scaling factor (eq 10), the experimental values of (∂η/∂p)T are well-reproduced. For these compounds, the relation EV/Ep ranges from 0.40 to 0.62, in agreement with the assumption of comparable effects of temperature and volume on viscosity. We also estimated the scaling factor for three polyalkylene glycol dimethyl ethers CH3O-(CH2CH2O)n-CH3: diethylene glycol dimethyl ether (DEGDME, n ) 2), triethylene glycol dimethyl ether (TriEGDME, n ) 3), and tetraethylene glycol dimethyl ether (TEGDME, n ) 4) and two organic carbonates CH3(CH2)mO-CO-O(CH2)mCH3: dimethyl (DMC, m ) 0) and diethyl (DEC, m ) 1) carbonate. The superposition curves are shown in Figure 6a. For polyalkylene glycol dimethyl ethers, the values of γ decrease when the size of the molecules increases; thus, a value of γ ) 6.1 is found for DEGDME versus 4.4 for TEGDME. This is due to an increase both of the dipole moment (µ ) 1.97 D for DEGDME,59 µ ) 2.16 D for


Pensado et al.

Figure 6. (a) Superpositioning of the viscosity values and (b) EV/Ep vs temperature for (0) TEGDME, (2) TriEGDME, (() DEGDME, (b) DEC, and (9) DMC.

Figure 7. (a) Superpositioning of the viscosity values and (b) (∂η/∂p)T derivative obtained directly from experimental data (symbols) and from eq 10 (line) vs pressure for (0) methanol, (2) ethanol, (() 2-methoxyethanol, (b) 2-ethoxyethanol, and (9) 2-isopropoxyethanol.

TriEGDME,60 and µ ) 2.46 for TEGDME61) and of the number of bonds. The predicted (∂η/∂p)T values using eq 5 are on average 1.5, 1.7, and 1.8 times the experimental values for DEGDME, TriEGDME, and TEGDME, respectively. For these three fluids, the relation between EV/Ep goes from 0.33 (for DEGDME) to 0.44 (for TEGDME), increasing with the dipole moment as can be seen in Figure 6b. In the case of dimethyl and diethyl carbonates, the values of γ are high and very close, around γ ) 10 (Figure 6a). Furthermore, the predicted (∂η/∂p)T values are around 1.3 times the experimental values for both carbonates. The EV/Ep ratio is around 0.2 (Figure 6b). As can be concluded from the values of γ and EV/Ep and from the predicted (∂η/∂p)T data indicated in Table 1, the behavior of these carbonates is very similar to that of n-heptane. The number of bonds in these carbonate molecules is lower than that in n-heptane, but this is likely to be compensated by their dipole moments (µ ) 0.86 D for DMC62 and 0.88 for DEC62). Eq 10 with the obtained γ value reproduces the experimental values of (∂η/∂p)T within an average

absolute deviation (AAD) of 4% for carbonates and of 8% for polyethers, respectively. Associated Liquids (H-Bonded Materials). The scaling factor was determined for two alkanols (methanol and ethanol) and three alkoxyethanols (2-methoxyethanol, 2-ethoxyethanol, and 2-isopropoxyethanol). The superposition curves are shown in Figure 7a. The values of γ are 1.85 for methanol, 1.45 for ethanol, and around 3 for the analyzed alkoxyethanols (3.1, 3.2, and 3.4 for MEGME, MEGEE, and iso-MEGPE, respectively). The γ values for ethanol and methanol are similar to those found previously by Roland et al.15 for glycerol (γ ) 1.6) from relaxation times. The scaling parameters of these two alkanols are the lowest among those determined in the present work (Table 1). Casalini and Roland found for sorbitol that γ ) 0.13 from relaxation times. These low values are due to hydrogen-bonded liquids being strongly associated, and thus, the volume exerts a weak effect on their dynamics. Sorbitol has the lowest γ value due to its six hydroxyl groups. According to Roland et al.,15 for some associated fluids and polymers, the influence of



Figure 8. (a) Superpositioning of the viscosity values and (b) (∂η/∂p)T derivative obtained directly from experimental data (symbols) and from eq 10 (line) vs pressure for (0) [bmim]PF6, (2) [hmim]PF6, (() [omim]PF6, (b) [bmim]BF4, (9) [omim]BF4, (×) [bmim]Tf2N, and (+) [hmim]Tf2N.

volume increases with the molecular weight due to a decrease in H-bonding (γ increases). This is the same trend that we found for alkoxyethanols. Nevertheless, methanol and ethanol do not follow this sequence probably due to the influence of internal degrees of freedom. The predicted (∂η/∂p)T values using eq 5 range from 1.9 to 3.4 times the experimental values for methanol, ethanol, MEGDME, MEGEE, and iso-MEGPE, respectively. In Figure 7b, the pressure viscosity derivative at 323.15 K is plotted against pressure, and it can be seen that using the scaling factor (eq 10), the (∂η/∂p)T experimental values are well-reproduced. The EV/Ep ratio is 0.58 for methanol, 0.66 for ethanol, and around 0.5 for the three alkoxyethanols (Table 1). These values are in agreement with the trends found for γ. Room-Temperature Ionic Liquids. Organic salts that have low melting points (usually defined as below 100 °C) have come to be known collectively as room-temperature ionic liquids or simply ionic liquids. Since they are salts, they tend to have extremely low vapor pressures, and it is this feature that has garnered them much recent attention as potential replacements for volatile organic solvents in a wide variety of chemical reactions, separations, and manufacturing processes. For example, ionic liquids have been considered as solvents for reactions, as absorption media for gas separations, as the separating agent in extractive distillation, as lubricants, as heat transfer fluids, for processing biomass, and as the working fluid in a variety of electrochemical applications (batteries, solar cells, etc.). Understanding basic thermophysical properties of ionic liquids is vital for design and evaluation. For instance, melting points, glass transition temperatures, and thermal decomposition temperatures are needed to set the feasible temperature operating range for a particular fluid. Density (as a function of temperature) is needed for equipment sizing. Viscosity is one of the criteria to select solvents for extraction or to choose lubricants. Nevertheless, there are few experimental viscosity data at high pressures of ionic liquids. Harris et al.63-66 measured viscosities for 1-butyl-3-methylimidazolium hexafluorophosphate ([bmim]PF6), 1-methyl-3-octylimidazolium hexafluorophosphate ([omim]PF6), 1-methyl-3-octylimidazolium tetrafluoroborate ([omim]BF4),1-methyl-3-hexylimidazoliumhexafluorophosphate([hmim]PF6), 1-butyl-3-methylimidazolium bis(trifluorosulfonyl)imide, ([bmim]Tf2N), and 1-butyl-3-methylimidazolium tetrafluoroborate [bmim]-

BF4 between 0 and 80 °C and up to maximum pressures around 250 MPa. In addition, Tomida et al.67,68 measured the density and viscosity of [bmim]BF4, [bmim]PF6, [hmim]PF6, and [omim]PF6 from 293.15 to 353.15 K and up to 20 MPa. Kandil et al.69 recently measured the density and viscosity of 1-methyl3-hexylimidazolium bis(trifluorosulfonyl)imide, [hmim]Tf2N, from 288 to 433 K and up to 50 MPa. Roland et al.15 determined the scaling exponents for [bmim]PF6 (γ ) 2.9), [omim]BF4 (γ ) 2.25), and [omim]PF6 (γ ) 2.4) using the densities and viscosities reported by Harris et al.,63,64 who had measured the viscosities and taken density values extrapolated from the experimental data of Gu and Brennecke70 between 298.2 and 343.2 K and up to 200 MPa. For [bmim]PF6, we found a better superpositioning with γ ) 3.4 instead of 2.9 using only the data of Harris et al. or together with the Tomida et al. values. For [omim]PF6 and [omim]BF4, we found with the data of Harris et al. the same γ values as Roland et al. had found. The results of Tomida et al. for [omim]PF6 fall in the corresponding superposition curve (as can be seen in Figure 8a). The scaling coefficients (γ)also were determined for the other four ionic fluids, [bmim]BF4, [hmim]PF6, [bmim]Tf2N, and [hmim]Tf2N, using the database presented in Table 1, yielding values of 2.25, 2.75, 2.85, and 2.45, respectively. The superposition curves are plotted in Figure 8a. The low γ values found for all these ionic liquids are due to the strong coulomb forces present. For each cation, the sequences found for the scaling parameter with the anion were γ [bmim]PF6 > γ [bmim]Tf2N > γ [bmim]BF4; γ [hmim]PF6 > γ [hmim]Tf2N; and γ [omim]PF6 > γ [omim]BF4. These trends are in agreement with the previous γ values found by Roland et al. with the octylimidazolium salts [omim]BF4 and [omim]PF6. This is logical since, as these authors and Tsunekawa et al.71 discussed, the tetrafluoroborate anion is more compact than the hexafluorophosphate anion; hence, the electrostatic interactions are stronger in salts containing BF4-. For the electrical conductivity, σ, opposite trends with the anion were found.72-75 The influence of the anion sizes in the electrical conductivity of the ionic liquids was quite complex, as Vila et al.73 remarked. These authors compared the σ values for [bmim]PF6 and [bmim]BF4, finding that the ionic liquids with the larger anion PF6 present a lower conductivity, in spite of the fact that both anions PF6 and BF4 are very similar

5572 J. Phys. Chem. B, Vol. 112, No. 18, 2008 in structure. Vila et al. concluded that the anion size has two opposite effects on the electrical conductivity: the decrease of the surface electrical charge density and the effect of size for dynamical movement. The molar volumes of [bmim]BF4, [bmim]PF6, and [bmim]Tf2N at 25 °C and 0.1 MPa were approximately76 189, 208, and 292 cm3 mol-1, respectively. Hence, a consideration of anion size effects also leads us to expect a more marked difference on the values of γ and σ for ionic liquids with the anion Tf2N-. In addition, a possible important factor is that the Tf2N- anion is highly flexible and adopts a number of conformations,77,78 as opposed to PF6- and BF4- that are conformationally rigid. This flexibility results in a destructuring effect of the ionic liquid, contributing to lowering very significantly the viscosity of ionic liquids when compared with the other two smaller anions. Changing the length of the alkyl side chains in the cations has nontrivial effects on the transport properties.79-81 Besides the change in the conformational flexibility of the cations, longer alkyl side chains induce a nanometer-scale segregation between regions of high charge density (the imidazolium headgroups and the anions) and regions of low charge density (the nonpolar alkyl side chains). This kind of self-organization is more pronounced for longer side chains, and it also corresponds to a larger importance of van der Waals interactions (that increase as the side chain increases) when compared to the Coulomb terms (that tend to remain almost constant). These structural effects have consequences on the dynamics of the liquids.82 It would seem reasonable to expect that the values of γ decrease with increasing side chain since we are progressively moving toward a more complex liquid, showing persistent microscopic structures and slower dynamics. This trend is in agreement with the fact that electrical conductivities for Tf2N-, PF6-, and BF4imidazolium salts decrease with increasing side chain, as several authors have72,73,75 remarked. Widegren et al.,72 who studied the electrical conductivity of Tf2N- imidazolium salts, stated that the electrical conductivity is related to the self-difussion of the species, so the largest cation has the lowest conductivity. Umecky et al.75 observed that the apparent cationic transference number, Dcation/(Dcation + Danion) where Dcation and Danion are the self-diffusion coefficients of the cation and anion, decreases with increasing alkyl chain length, mainly due to the fact that the friction for the translational motions can work more effectively for the bulky spherical PF6- anion than for the plane imidazolium cations. Similar results also were found by Vila et al.73 The results of the literature and those of this work seem to indicate relationships between the electrical conductivity values and the scaling parameter for ionic liquids, although the effect of the cation and the anion on both properties can be quite different. Nevertheless, for a better understanding of the influence of the molecular structure of the ionic liquids in the dynamic properties, more experimental information is needed. The previous arguments for molecular liquids concerning the effect of intramolecular modes do not seem to transpose in a straightforward manner to ionic liquids. The predicted pressure-viscosity derivative using eq 5 ranges from 2.9 to 4 times that of the experimental values (Table 1). In Figure 8b, it can be seen as to how the γ values determined by Roland et al. allow a reproduction of the experimental (∂η/ ∂p)T values using eq 10. In Figure 9, it can be seen that at 0.1 MPa, the EV/Ep ratio slightly decreases with increasing temperature for all the ionic liquids analyzed. At 100 MPa, this ratio increases with temperature for [bmim]Tf2N. This is due to the isotherms of the isobaric thermal expansion coefficient, Rp,

Pensado et al.

Figure 9. EV/Ep vs temperature at 313.15 K for several ionic fluids: (0) [bmim]PF6, (2) [hmim]PF6, (() [omim]PF6, (b) [bmim]BF4, (9) [omim]BF4, (×) [bmim]Tf2N, and (+) [hmim]Tf2N.

Figure 10. EV/Ep vs γ at 0.1 MPa and 313.15 K for ([0]) PE lubricants, (2) polar polyethers, (() carbonates, (b) n-alkanes, (9) ionic liquids, (×) alkanols, and (O) associated glycols.

presenting a crossing point around 25 MPa. The Rp values were determined through a fit of a Tait equation to the experimental density data reported by Gomez de Azevedo83 and by obtaining the corresponding analytical derivatives. This procedure was used in previous studies.24-29 At pressures higher than 25 MPa, Rp decreases when the temperature rises, and at lower pressures, Rp increases when the temperature increases. In accord with eq 17, the curve of the EV/Ep ratio versus temperature can change its slope. Conclusion In Figure 10, the ratio EV/Ep was plotted against the scaling parameter, γ, for all the fluids analyzed in the present work. In Figure 10, the good correlation between both factors can be seen clearly. The fluids with lower γ values reach the highest

Scaling Concept for Molecular and Ionic Liquids EV/Ep ratio. This is the case of ionic and associated fluids, mainly methanol and ethanol. On the contrary, short alkanes and carbonates have the highest γ values and the lowest EV/Ep ratio. Hence, the values of γ and of the ratio EV/Ep allow a classification of the compounds according to the effects of density and temperature on viscosity. The worse predictions of (∂η/∂p)T with eq 5 were obtained for the fluids with lower γ values, attaining 4 times the experimental values (Table 1). For the alkanes studied, there was a strong dependence of the scaling factor on the number of covalent bonds. The superposition curves exhibit, for all the fluids except squalane, a slope with 1000FγΤ-1 greater than that of the exponential function. Until now, this behavior had only been reported for n-octane and toluene by Roland et al.15 The other compounds studied showed the usual behavior found in the literature for relaxation times and viscosities, that is, an increasing or constant slope of the superposition curve. Acknowledgment. This work was supported by the Spanish Science and Technology Ministry (CTQ2005-09176-C02-01/ PPQ) and Xunta de Galicia (PGIDIT05TAM20601PR and PGIDIT06PXIC20640PN). The participation of A.S.P. was made possible by a grant from DXID, Xunta de Galicia. Supporting Information Available: New viscosity values at atmospheric pressure for the four pure PEs (Table S1). This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Angell, C. A.; Ngai, K. L.; McKenna, G. B.; McMillan, P. F.; Martin, S. W. J. Appl. Phys. 2000, 88, 3113. (2) Angell, C. A. Science (Washington, DC, U.S.) 1995, 167, 1924. (3) Avramov, I. J. Non-Cryst. Solids 2000, 262, 258. (4) Avramov, I.; Vassilev, T.; Penkov, I. J. Non-Cryst. Solids 2005, 351, 472. (5) Sastry, S. Nature (London, U.K.) 2001, 409, 164. (6) Roland, C. M.; Hensel-Bielowka, S.; Paluch, M.; Casalini, R. Rep. Prog. Phys. 2005, 68, 1405. (7) Roland, C. M.; Casalini, R. J. Non-Cryst. Solids 2005, 351, 2581. (8) Roland, C. M.; Paluch, M.; Casalini, R. J. Polym. Sci., Polym. Phys. 2004, 42, 4313. (9) Casalini, R.; Roland, C. M. Colloid Polym. Sci. 2004, 283, 107. (10) Casalini, R.; Roland, C. M. Phys. ReV. E: Stat., Nonlinear, Soft Matter Phys. 2004, 69, 62501/1. (11) Dreyfus, C.; Le-Grand, A.; Gapinski, J.; Steffen, W.; Patkowski, A. Eur. J. Phys. 2004, 42, 309. (12) Tarjus, G.; Kivelson, D.; Mossa, S.; Alba-Simionesco, C. J. Chem. Phys. 2004, 120, 6135. (13) To¨lle, A. Rep. Prog. Phys. 2001, 64, 1473. (14) Casalini, R.; Capaccioli, S.; Roland, C. M. J. Phys. Chem. B 2006, 110, 11491. (15) Roland, C. M.; Bair, S.; Casalini, R. J. Chem. Phys. 2006, 125, 124508/1. (16) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 4969. (17) Shell, M. S.; Debenedetti, P. G.; Nave, E. L.; Sciortino, F. J. Chem. Phys. 2003, 118, 8547. (18) March, N. H.; Tosi, M. P. Introduction to Liquid State Physics; Scientific World: Singapore, 2002. (19) Hoover, W. G.; Ross, M. Contemp. Phys. 1971, 12, 339. (20) Coslovich, D.; Roland, C. M. J. Phys. Chem. B 2008, 112, 1329. (21) Casalini, R.; Mohanty U.; Roland, C. M. J. Chem. Phys. 2006, 125, 14505/1. (22) Alba-Simionesco, C.; Cailliaux, A.; Alegria, A.; Tarjus, G. Europhys. Lett. 2004, 68, 58. (23) Casalini, R.; Roland, C. M. Phys. ReV. B: Condens. Matter Mater. Phys. 2005, 71, 14210. (24) Fandinõ, O.; Comunãs, M. J. P.; Lugo, L.; Lo´pez, E. R.; Fernańdez, J. J. Chem. Eng. Data 2007, 52, 1429. (25) Fandinõ, O.; Pensado, A. S.; Lugo, L.; Comunãs, M. J. P.; Fernańdez, J. J. Chem. Eng. Data 2005, 50, 939. (26) Fandinõ, O.; Pensado, A. S.; Lugo, L.; Comunãs, M. J. P.; Fernańdez, J. J. Chem. Eng. Data 2006, 51, 2274.

J. Phys. Chem. B, Vol. 112, No. 18, 2008 5573 (27) Fandinõ, O.; Pensado, A. S.; Lugo, L.; Lo´pez, E. R.; Fernańdez, J. Green Chem. 2005, 7, 775. (28) Fandinõ, O.; Garcıá, J.; Comunãs, M. J. P.; Lo´pez, E. R.; Fernańdez, J. Ind. Eng. Chem. Res. 2006, 45, 1172. (29) Pensado, A. S.; Comunãs, M. J. P.; Fernańdez, J. Ind. Eng. Chem. Res. 2006, 45, 9171. (30) Pensado, A. S.; Comunãs, M. J. P.; Lugo, L.; Fernańdez, J. Ind. Eng. Chem. Res. 2006, 45, 2394. (31) Lugo, L.; Canet, X.; Comunãs, M. J. P.; Pensado, A. S.; Fernańdez, J. Ind. Eng. Chem. Res. 2007, 46, 1826. (32) Kushiro, I. J. Geophys. Res. 1976, 81, 6347. (33) Scarfe, C. M.; Mysen, B. O.; Virgo, D. Magmat. Processes: Physicochem. Princ. 1987, 59. (34) Bottinga, Y.; Richet, P. Geochim. Cosmochim. Acta 1995, 59, 2725. (35) Richert, R.; Duvvuri, K.; Duong, L.-T. J. Chem. Phys. 2003, 118, 1828. (36) Mori, S.; Ohtani, E.; Suzuki, A. Earth Planet. Sci. Lett. 2000, 175, 87. (37) Schmelzer, J. W. P.; Zanotto, E. D.; Fokin, V. M. J. Chem. Phys. 2005, 122, 74511/1. (38) Gupta, P. K. J. Am. Ceram. Soc. 1987, 70, 486. (39) Kushiro, I.; Yoder, H. S. J.; Mysen, B. O. J. Geophys. Res. 1971, 81, 6351. (40) Cleave, B.; Koronaios, P. J. Chem. Soc., Faraday Trans. 1997, 93, 1601. (41) Spikes, H. A. Tribol. Trans. 1990, 33, 140. (42) Iglesias-Silva, G. A.; Estrada-Baltazar, A.; Hall, K. R.; Barrufet, M. A. J. Chem. Eng. Data 1999, 44, 1304. (43) Sawamura, S.; Takeuchi, N.; Kitamura, K.; Taniguchi, Y. ReV. Sci. Instrum. 1990, 61, 871. (44) King, H. E.; Herbolzheimer, E.; Cook, R. L. J. Appl. Phys. 1992, 71, 2071. (45) Kiran, E. Polymer formation, modifications and processing in or with supercritical fluids. Supercritical Fluids: Fundamentals and Applications; Kiran, E., Sengers, J. H. M. L., Eds.; Kluwer Academic Publishers: Netherlands, 1994; pp 541-588. (46) Lansdown, A. R. Lubrication and Lubricant Selection; Professional Engineering Publishing: London, 2004. (47) Williams, G. Trans. Faraday Soc. 1965, 61, 1564. (48) Casalini, R.; Roland, C. M. J. Chem. Phys. 2003, 119, 4052. (49) Boyer, R. F.; Spencer, R. S. J. Appl. Phys. 1944, 15, 398. (50) Casalini, R.; Roland, C. M. J. Non-Cryst. Solids 2007, 353, 3936. (51) Roland, C. M.; McGrath, K. J.; Casalini, R. J. Non-Cryst. Solids 2006, 352, 4910-4914. (52) Shobha, H. K.; Kishore, K. J. Chem. Eng. Data 1992, 37, 371. (53) Angell, C. A.; Smith, D. L. J. Phys. Chem. 1982, 86, 3845. (54) Grocholski, B.; Jeanloz, R. J. Chem. Phys. 2005, 123, 204503. (55) Wang, L. M.; Velikov, V.; Angell, C. A. J. Chem. Phys. 2002, 117, 10184. (56) Vogel, H. Z. Phys. 1921, 22, 645. (57) Tamman, G.; Hesse, W. Z. Anorg. Allg. Chem. 1926, 156, 245. (58) Fulcher, G. S. J. Am. Ceram. Soc. 1925, 8, 339. (59) Zhao, H. Process for the Preparation of Triazine Carbamates. U.S. Patent 6204381, 2001. (60) Abboud, J. L. M.; Notario, R. Pure Appl. Chem. 1999, 71, 645. (61) Hanke, E.; vonRoden, K.; Kaatzea, U. J. Chem. Phys. 2006, 125, 084507. (62) MacClellan, A. L. Tables of Experimental Dipole Moments; Rahara Enterprises: San Francisco, CA, 1989. (63) Harris, K. R.; Woolf, L. A.; Kanakubo, M. J. Chem. Eng. Data 2005, 50, 1777. (64) Harris, K. R.; Kanakubo, M.; Woolf, L. A. J. Chem. Eng. Data 2006, 51, 1161. (65) Harris, K. R.; Kanakubo, M.; Woolf, L. A. J. Chem. Eng. Data 2007, 52, 1080. (66) Harris, K. R.; Kanakubo, M.; Woolf, L. A. J. Chem. Eng. Data 2007, 52, 2425. (67) Tomida, D.; Kumagai, A.; Qiao, K.; Yokoyama, C. Int J. Thermophys. 2006, 27, 39. (68) Tomida, D.; Kumagai, A.; Kenmochi, A.; Qiao, K.; Yokoyama, C. J. Chem. Eng. Data 2007, 52, 577. (69) Kandil, M. E.; Marsh, K. N.; Goodwin, A. R. H. J. Chem. Eng. Data 2007, 52, 2382. (70) Gu, Z.; Brennecke, J. F. J. Chem. Eng. Data 2002, 47, 339. (71) Tsunekawa, H.; Narumi, A.; Mitsuru, H.; Akio, F.; Yokoyama, H. J. Phys. Chem. B 2003, 107, 10962. (72) Widegren, A.; Widegren, J. A.; Saurer, E. M.; Marsh, K. N.; Magee, J. W. J. Chem. Thermodyn. 2005, 37, 569. (73) Vila, J.; Varela, L. M.; Cabeza, O. Electrochim. Acta 2007, 52, 7413. (74) Kanakubo, M.; Harris, K. R.; Tsuchihashi, N.; Ibuki, K.; Ueno, M. Fluid Phase Equilib. 2007, 261, 414.

5574 J. Phys. Chem. B, Vol. 112, No. 18, 2008 (75) Umecky, T.; Kanakubo, M.; Ikushima, Y. J. Mol. Liquids 2005, 119, 77. (76) Hu, Y. F.; Xu, C. M. Chem. ReV. 2008, in press. (77) Canongia-Lopes, J. N. A.; Pa´dua, A. A. H. J. Phys. Chem. B 2004, 108, 16893. (78) Deetlefs, M.; Hardacre, C.; Nieuwenhuyzen, M.; Pa´dua, A. A. H.; Sheppard, O.; Soper, A. K. J. Phys. Chem. B 2006, 110, 12055. (79) Canongia-Lopes, J. N. A.; Pa´dua, A. A. H. J. Phys. Chem. B 2006, 110, 3330. (80) Tokuda, H.; Hayamizu, K.; Ishii, K.; Susan, M. A. B. H.; Watanabe, M. J. Phys. Chem. B 2005, 109, 6103. (81) Triolo, A.; Russina, O.; Bleif, H. J.; DiCola, E. J. Phys. Chem. B 2007, 111, 4641. (82) Hu, Z.; Margulis, C. J. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 831. (83) Gomez-de-Azevedo, R.; Esperanç a, R.; Szyslowski, J. M. S. S.; Visak, Z. P.; Pires, P. F.; Guedes, H. J. R.; Rebelo, L. P. N. J. Chem. Thermodyn. 2005, 37, 888. (84) Oliveira, C. M. B. P.; Wakeham, W. A. Int. J. Thermophys. 1992, 13, 773. (85) Assael, M. J.; Papadaki, M. Int. J. Thermophys. 1991, 12, 801. (86) Harris, K. R.; Malhotra, R.; Woolf, L. A. J. Chem. Eng. Data 1997, 42, 1254.

Pensado et al. (87) Caudwell, D. R.; Trusler, J. P. M.; Vesovic, V.; Wakeham, W. A. Int. J. Thermophys. 2004, 25, 1339. (88) ASME. Viscosity and Density of oVer 40 Lubricating Fluids of Known Composition at Pressures to 150 000 psi and Temperatures to 425 °F; American Society of Mechanical Engineers: New York, 1953. (89) Lemmon, E. W.; Huber, M. L.; McLinden, M. O. REFPROP, Reference Fluid Thermodynamic and Transport Properties; National Institute of Standards and Technology: Gaithersburg, MD, 2002. (90) Comunãs, M. J. P.; Reghem, P.; Baylaucq, A.; Boned, C.; Fernańdez, J. J. Chem. Eng. Data 2004, 49, 1344. (91) Reghem, P.; Baylaucq, A.; Comunãs, M. J. P.; Fernańdez, J.; Boned, C. Fluid Phase Equilib. 2005, 236, 229. (92) Comunãs, M. J. P.; Baylaucq, A.; Plantier, F.; Boned, C.; Fernańdez, J. Fluid Phase Equilib. 2004, 222-223, 331. (93) Comunãs, M. J. P.; Baylaucq, A.; Boned, C.; Fernańdez, J. Int. J. Thermophys. 2001, 22, 749. (94) Kestin, J.; Sengers, J. V.; Kamgar-Parsi, B.; Sengers, J. M. H. L. J. Phys. Chem. Ref. Data 1984, 13, 175. (95) Wagner, W.; Pruss, A. J. Phys. Chem. Ref. Data 2002, 31, 387. (96) Pensado, A. S.; Comunãs, M. J. P.; Lugo, L.; Fernańdez, J. J. Chem. Eng. Data 2005, 50, 849. (97) Pensado, A. S.; Pa´dua, A. A. H.; Comunãs, M. J. P.; Fernańdez, J. J. Supercrit. Fluids 2008, 44, 172-185.

Relationship between Viscosity Coefficients and Volumetric Properties

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