Influence of the mass ratio on viscosity in Lennard–Jones mixtures: The one-fluid model revisited using nonequilibrium molecular dynamics

Fluid Phase Equilibria 234 (2005) 56–63

Influence of the mass ratio on viscosity in Lennard–Jones mixtures: The one-fluid model revisited using nonequilibrium molecular dynamics Guillaume Galli´ero a,∗ , Christian Boned a , Antoine Baylaucq a , Franc¸ois Montel b a

Laboratoire des Fluides Complexes (UMR-5150), Universit´e de Pau et des Pays de l’Adour, BP 1155, F-64013 Pau cedex, France b TOTAL, CSTJF, Avenue Larribau, F-64018 Pau, France Received 25 February 2005; received in revised form 24 May 2005; accepted 25 May 2005 Available online 5 July 2005

Abstract In the frame of the law of the corresponding states, a systematic study of the influence on viscosity of the mass ratio in mixtures has been performed. To achieve such a goal, the viscosity of Lennard–Jones binary, ternary and 10-components mixtures in various thermodynamic states, and for various molar fractions, has been evaluated thanks to nonequilibrium molecular dynamics simulations. To focus on the mass ratio effect alone, the components in the Lennard–Jones mixtures differ only in mass. It is shown that none of the tested one-fluid models for mass is able to provide an accurate estimation of the mixture viscosities. It has been found that the mass of the pseudo-compound equivalent to the mixture in the one-fluid approximation for viscosity is density dependent and weakly temperature dependent. A purely empirical density dependent model (adjusted on equimolar results) is proposed. This model provides results consistent with direct simulations on mixtures as well as with results for the zero-density systems, the deviations being in all cases smaller than 3%. © 2005 Elsevier B.V. All rights reserved. Keywords: Lennard–Jones; Mixture; Nonequilibrium molecular dynamics; One-fluid approximation; Viscosity

1. Introduction Recent advances in modelling pure compound viscosity have been realized [1–9]. This has been possible through combined theoretical and experimental efforts. Thus, it seems now possible to correctly estimate this transport property for a wide variety of pure compounds as well as on a large range of thermodynamic conditions. But when dealing with mixtures, the modelling of viscosity is, by far, a more complex problem which is still open despite the large number of approaches proposed in the literature [10]. Unfortunately, in academic and industrial fields where this property is needed (e.g. fluid mechanics, chemical engineering, petroleum industry, . . .), mixtures are involved most of the time. Classically, in order to estimate viscosity in mixtures, two possibilities are available: the first is based on the knowledge ∗

Corresponding author. Tel.: +33 5 59 40 76 46; fax: +33 5 59 40 76 95. E-mail address: [email protected] (G. Galli´ero).

0378-3812/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2005.05.016

of pure compounds viscosities for the same thermodynamic conditions which, combined with a mixing rule on these viscosities (e.g. Grunberg and Nissan [11], Wilke [12]), provide the viscosity of the mixture; the second relies on the so called “one-fluid model” and on the corresponding states law (the conformal solution theory [13]). In such models, it is assumed that it is possible to define a pseudo-compound, using mixing rules on the parameters of the model describing the molecules, which mimics the thermodynamic and/or transport properties of a given mixture [14]. As the main goal for a viscosity model is to be entirely predictive, it appears clearly that only the second scheme, which does not require pure compound viscosities, may reach this purpose. Nevertheless, in the one-fluid approach the formulation of the pseudo-compound parameters starting from those of pure components is not a trivial point. Once a molecular model is chosen, results given by a onefluid approximation are usually compared to experimental ones measured on real fluids. As a matter of fact, various

G. Galli´ero et al. / Fluid Phase Equilibria 234 (2005) 56–63

errors coming from the molecular model itself, the law of the corresponding states and the one-fluid approximation may compensate each other and therefore one-fluid model results may look acceptable despite some intrinsic errors. Hence, the evaluation of the effectiveness of the one-fluid model is in some sense difficult. It is nevertheless sometimes possible to achieve a direct test of the one-fluid model for some peculiar cases where the law of the corresponding states is fulfilled and either an analytical or an approached solution exists as in low density systems [15]. An alternative is to use molecular simulations applied on simple conformal spheres to provide new insights on the one-fluid model. Concerning the viscosity, molecular dynamics applied on a molecular model allows a direct comparison between the results on the mixtures and those on the pseudo-compound in the frame of the microscopic formulation of the corresponding states principle [16]. Hence, a direct test of the one-fluid approximation can be achieved. In two parameters intermolecular potentials (like the Lennard–Jones 12–6 one [17]), various one-fluid models exist [13,18]. Among them, the van der Waals one-fluid approximation applied on the two molecular parameters is the most widely used for thermodynamic properties with a reasonable success [19]. For transport properties, such as viscosity, there is a need of a one-fluid model for mass as well [13,18–20]. Theoretically based one-fluid models for mass have been proposed to reproduce the viscosity mass dependence in mixtures [10,13,18] and then compared with molecular dynamics simulations [13,18,19]. However, none of them were totally satisfactory. Furthermore, these relations where generally limited to a particular thermodynamic state (low density or liquid). In this work, molecular dynamics simulations have been performed to compute the viscosity of pure fluids and of mixtures where only the mass between the components differs, these fluids being modelled by 12–6 Lennard–Jones (LJ) spheres. By doing so, it is possible to test directly the efficiency of the one-fluid approximations on mass for various thermodynamic states, which has not been already achieved as far as we know. It is important to underline that, by choosing such mixtures, we put apart problems linked to the choice of a set of combining rules to define cross interaction parameters between unlike molecules, which complexify the problem [13,18,21], even if promising results using such approach have been reported on LJ mixtures recently [22]. It should be mentioned nevertheless, that, in order to provide an estimation of viscosity in real mixtures (where not only mass differs between molecules but also volume and energy), one-fluid rules on the volume and on the energy parameters have also to be introduced. Therefore, such study on the mass mixing rules is only a first step towards a comprehensive LJ fluid + one-fluid approximation scheme for application on real fluid mixtures. Different thermodynamic states and various mass ratios have been explored. To achieve such a goal in a reasonable time, a boundary driven nonequilibrium molecular dynam-

57

ics (NEMD) scheme has been used. Then, in mixtures, for each configuration the viscosity of the corresponding pseudocompound deduced from a one-fluid model has been evaluated and compared to the true value of the mixture. In a first part, we briefly describe the fluid model used and the molecular dynamics scheme chosen. Then, the microscopic formulation of the law of the corresponding states is described for pure fluids and for mixtures by introducing the one-fluid model. In this part, various one-fluid models for mass are derived from classical laws used in literature. In a second part, some molecular dynamics results on equimolar binary mixtures in various thermodynamic states are presented (the zero-density values are estimated by the Chapman–Enskog first order solution [13]). An empirical density dependent one-fluid model for mass is proposed as an alternative to the relative failure of the tested one-fluid models. Finally, results in nonequimolar binary mixtures as well as in ternary and 10-components mixtures are presented which emphasize the weakness of the tested one-fluid models.

2. Models 2.1. Interaction potential To model the fluid particle interactions, we have used the classical Lennard–Jones 12–6 potential:
   σ 12  σ  6 U = 4ε (1) − r r where ε is the potential strength, σ the atomic diameter and r is the intermolecular separation length. As in the simulated systems only the mass, m, differs, the σ and ε are the same for each molecule. The mass ratio between component i and the first component, the reference one, is noted αi throughout the text. 2.2. Law of the corresponding states The microscopic formulation of the law of the corresponding states postulates that, with an adequate rescaling, different fluids have superposing thermodynamic phase diagrams and have the same transport properties for a given set of thermodynamic conditions [23]. For the Lennard–Jones fluid this statement is valid if the appropriate reduced thermodynamic variables are used. In this work only mixtures where the mass between components differs will be treated, therefore the reduced LJ thermodynamic variables are simply: T∗ =

kB T , ε

ρ∗ =

Nσ 3 V

and

P∗ =

Pσ 3 ε

(2)

where T is the temperature, N the number of particles, V the volume of the simulation box and P is the pressure of the system. The reduced viscosity, which is a unique function

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G. Galli´ero et al. / Fluid Phase Equilibria 234 (2005) 56–63

of the reduced thermodynamic conditions for a LJ potential, is [23]: σ2 η∗ (T ∗ , ρ∗ ) = η √ mε

(3)

where m is the characteristic molecular weight of the studied fluid and η is the dynamic viscosity.

where i and j are the characteristic masses of these substances. It is then possible to deduce the one-fluid mass model for mass corresponding to a viscosity mixing rule. For the linear mixture viscosity model it is assumed that: ηmixing =

n 

xi η i

(8)

i=1

the corresponding one-fluid mass rule is, using Eq. (7): 2.3. One-fluid model for mass 1/2

In pure fluids m appearing in Eq. (3), is equal to the mass of the involved compound. But, in mixtures, this parameter must be defined thanks to a one-fluid model which lump the components into a single pseudo-compound “equivalent” to the mixture. This mass will be noted m1−fluid throughout the text. Usually, the pseudo-compound mass, in an n-components mixture, is defined as: m1−fluid =

n 

xi m i

(4)

i=1

where xi is the molar fraction of component i. This model will be referred in the text as to the linear one-fluid model. This relation is consistent in equilibrium systems for property such as molar volume [14], but is questionable when dealing with transport properties related to nonequilibrium states [13,19]. Despite this uncertainty, this relation is still used in various models relying on a one-fluid approximation to estimate viscosity in mixtures [3,10]. However, some alternatives to Eq. (4) exist. Using the mixing rules proposed in the low density state [10], as well as in liquid [19], for σ i = σ j and εi = εj , the pseudo-compound mass of the mixture studied in this work is defined as: 1/2 m1−fluid

=

n  n  i=1 j=1

1/2

xi xj mij

(5)

where mij =

2mi mj mi + m j

(6)

This one-fluid model will be referred as to the square root model. Other one-fluid models for mass may be deduced for these LJ mixtures from mixing rules on viscosity. The viscosity deduced from such mixing rules will be noted ηmixing . In Lennard–Jones fluids, the value of η* does not depend on the substance studied (pure fluid or mixture) but just on the reduced thermodynamic states, because the microscopic formulation of the corresponding states law is fulfilled in such systems. Therefore, following Eq. (3), as the components differ only in mass, the viscosity of a substance i is related to the one of a substance j (in the same reduced thermodynamic state): −1/2

ηj mj

−1/2

= ηi mi

(7)

m1−fluid =

n  i=1

1/2

xi m i

(9)

The logarithmic viscosity model, which is called the Grunberg and Nissan law [11]: ln ηmixing =

n 

xi ln ηi

(10)

i=1

can be rewritten in a one-fluid model for mass:  n   1/2 1/2 m1−fluid = exp xi ln (mi )

(11)

i=1

For the low density state, the Wilke model [12] (which is a simplification of the kinetic theory approach): ηmixing =

n  i=1

xi ηi n j=1 xi φij

(12)

where φij = (mj /mi )1/2 in the approximation of Herning and Zipperer [24] Eq. (12) can then be expressed in terms of a one-fluid model as: n xi mi 1/2 m1−fluid = ni=1 1/2 (13) i=1 xi mi Such relations, Eqs. (5), (9), (11) and (13), emphasize nevertheless a non trivial point underlying the one-fluid model: is the pseudo-compound mass dependent to the property studied? 2.4. Nonequilibrium molecular dynamics The code used is a homemade one in FORTRAN which uses the Verlet velocity algorithm to integrate the equation of motion [25]. To compute the viscosity a boundary driven NEMD scheme, which is simple to handle and provides reliable results [26], has been used. In this technique, the simulation box is divided in 32 slabs along the z direction. Then, the fluid is sheared using a net exchange of momentum along the direction x (perpendicular to z), which is performed between the central part of the simulation box, slabs 16 and 17, and the edge layers, slab 1 and 32, to conserve the periodic boundary conditions. To do so are stored the two particles in slab 1 and 32 with the largest negative x components of the momentum and the two particles in slabs 16 and 17 with the largest positive x components of the momentum. Then, x components of the velocity between the particles involved are

G. Galli´ero et al. / Fluid Phase Equilibria 234 (2005) 56–63

exchanged. The viscosity is simply deduced from the ratio between the momentum flux induced by the exchange and the shear rate (Newton’s law [26]). The exchange frequency needed in this algorithm has been taken equal to 300 to avoid any disturbances [27]. The usual periodic boundary conditions and minimum image convention have been applied [17]. Simulations have been performed in the NVT ensemble (i.e. at fixed T* and ρ* ). To keep the temperature stable during the simulation, we have used a Berendsen thermostat [28] with a time constant ·τ T = 1000δt* , where δt* is the time step in reduced units (δt* = δt(ε/(mσ 2 ))1/2 ). Simulations have been performed with a constant reduced time step (equal to 0.002). The use of a constant reduced time step allows a correct description of the dynamics of the system when changing the mass ratios between the species. In order to avoid finite size effects and to obtain a good accuracy on the value obtained, we have performed simulations on a 1500 particles system during 107 nonequilibrium time steps. A truncated potential with a 2.5σ cutoff radius has been used and long range correction has been introduced [17]. Using these numerical parameters the statistical errors produced on viscosity are around ±3%, except in dense phase where errors may reach 4%.

3. Results and discussion 3.1. Preliminary test First, a test of the ability of our own code to provide consistent results on pure fluid has been performed. Viscosity results have been compared with those coming from a reliable study using equilibrium molecular dynamics (EMD) on Lennard–Jones 12–6 particles [29]. Simulations have been performed at T* = 1 and ρ* = 0.7 and 0.9 and at T* = 2.5 for ρ* going from 0.3 to 0.9 with a step of 0.2. For these six points, we have obtained an average absolute deviation of 1.64%, a maximum deviation of 3.48% and a bias equal to −0.28%. For the same six points, we have compared our results on pressure with the one provided by the equation of state of Kolafa and Nezbeda [30]. We have obtained an average absolute deviation of 0.69%, a maximum deviation of 1.84% and a bias equal to 0.08%. The results are reasonable compared to the inherent statistical errors of the simulations. On mixtures, we have performed simulations on argon–krypton systems (xAr = 0.2, 0.4, 0.6 and 0.8) at T = 120 K with the parameters given in Fernandez et al. [22]. Compared to the results of Fernandez et al. [22] (which use EMD simulations), we have found an average absolute deviation of 4.27%, a maximum deviation of 9.6% and a Bias of −2.1%. Results are in fair agreement, except for xAr = 0.2 where the deviation reach −9.6%. Nevertheless, as seen in Fig. 3 in Fernandez et al. [22], it seems that this point, in their simulations, is lower than expected and therefore may explain the deviation noted compared to our result.

59

3.2. Binary equimolar mixtures In order to evaluate the efficiency of the one-fluid model on mass described previously, Eqs. (4), (5), (9) and (11), simulations have been performed on pure fluids as well as on various binary equimolar mixtures (x1 = x2 = 0.5) where only the mass between the compounds varies (i.e., σ 1 = σ 2 , ε1 = ε2 and α = m2 /m1 = 2, 5 and 10). This choice of mass ratios range is consistent with real fluid mixtures such as, for instance, a methane + n-decane one. As the dynamic of the system is closely linked to the thermodynamic conditions, these simulations have been carried out for a wide variety of states (subcritical and supercritical): T* has been taken equal to 1, 1.5, 2 and 2.5 and ρ* has been chosen equal to 0.1, 0.3, 0.5, 0.7 and 0.9. However, the three points corresponding to T* = 1 and ρ* = 0.1, 0.3 or 0.5 have been discarded, which means that 17 different state points have been studied. In fact, these three points correspond to a two-phase region as the critical point of the Lennard–Jones is located at Tc∗ ≈ 1.3 and ρc∗ ≈ 0.31. The results of the simulations for the 68 points (17 in pure fluids, η∗pure , and 51 in mixtures, η∗α = 2, 5, 10) are given in Appendix A. To extend the simulations results to the zero-density state for the same values of reduced temperatures, the Chapman–Enskog first order approximation of the kinetic theory has been used [15]. In addition, it has been verified that, for each configuration simulated, the reduced pressure is not dependent to the mass ratio value (i.e. within the statistical error of the one in pure fluid), which ensure a consistent numerical procedure. For each configuration, the viscosity given by a one-fluid model for mass, η1−fluid , deduced from Eq. (3), has been evaluated by: 
m1−fluid 1/2 ηpure (14) η1−fluid = mpure where m1−fluid is defined by a one-fluid model (Eqs. (4), (5), (9), (11) and (13)), ηpure is the viscosity of the reference pure fluid for the given set of T* and ρ* conditions (values are given in Appendix A) and mpure is its molecular weight. It is then possible, for each configuration and each onefluid model, to test the efficiency of the various one-fluid models by comparing η1−fluid to ηmix . It should be noted that a “perfect” one-fluid model for mass corresponds to η1−fluid = ηmix . Thus, for each reduced thermodynamic state and each mass ratio, the deviations induced by the one-fluid models have been calculated:
 η1−fluid Dev = 100 × 1 − (15) ηmix From these Q values of deviations (Q = 63, 51 by NEMD and 12 by Chapman–Enskog approach), the average absolute deviations (AAD) have also been estimated: Q

1  AAD = |Dev(i)| Q i=1

(16)

60

G. Galli´ero et al. / Fluid Phase Equilibria 234 (2005) 56–63 Table 1 Comparison between one-fluid models efficiency for mass when used for viscosity estimation in equimolar binary mixtures for α = 2, 5 and 10 and in various thermodynamic states (ρ* : 0–0.9 and T* : 1–2.5)

Fig. 1. Deviations on viscosity in equimolar mixtures, vs. reduced density, induced by the linear one-fluid model, Eq. (4), for different values of α (diamonds: 2, squares: 5 and circles: 10) and different reduced temperatures (white symbols: T* = 1, grey ones: T* = 1.5, dark grey ones T* = 2 and black ones T* = 2.5).

Along with the maximum deviations (Max): Max = max(|Dev(i)|)

(17)

and the bias: Q

1  Dev(i) Bias = Q

(18)

i=1

Fig. 1 clearly shows that the deviations on mixture viscosity induced by the linear one-fluid model for mass, Eq. (4), are not negligible, reaching −28.3% for α = 10 (which corresponds to a nearly 65% overestimation of the equivalent pseudo-compound mass). These deviations remain nevertheless limited in all cases when α = 2, being always lower than 4%. In addition, Fig. 1 shows that deviations are strongly density dependent and weakly temperature dependent. Furthermore, this figure shows that the trends in the deviations obtained from NEMD simulations are consistent with those computed at zero-density state, coming from the Chapman–Enskog expression.

One-fluid model

AAD

Max

Bias

Linear (Eq. (4)) Square root (Eq. (5)) Linear viscosity (Eq. (9)) Logarithmic (Eq. (11)) Low density (Eq. (13))

8.4 9.7 5.5 7.6 15.3

28.3 30.6 15.8 28.1 44.6

−7.5 9.7 −0.4 7.6 −15.3

Fig. 2 clearly shows that none of the one-fluid tested is able to provide consistent results on the whole range of reduced density for the worst case, i.e. α = 10. More generally, it is worth to emphasize that these results confirm that the deviations are strongly density dependent and weakly temperature dependent, whatever the one-fluid model. Such density dependence behaviour explains why none of the one-fluid model for mass described previously (which are independent of the density) is able to provide a good estimation of the mass of the equivalent pseudo-compound for all thermodynamic states tested as shown in Fig. 2 (for α = 10) and summarized in Table 1. A possible explanation of this trend may be that both contributions to the viscosity, the kinetic (predominant in low density systems) and the potential (predominant in dense phases), are related to two different one-fluid models as already stated by Mo and Gubbins [13]. Hence,

Fig. 3. Deviations on viscosity in equimolar mixtures, vs. reduced density, induced by the new one-fluid model, Eq. (19), for various configurations. Symbols are the same as in Fig. 1. Table 2 Comparison between one-fluid models efficiency for mass when used for viscosity estimation in nonequimolar supercritical binary mixtures (T* = 2 and ρ* = 0.5) for α = 5 and 10

Fig. 2. Deviations on viscosity in equimolar binary mixtures, vs. reduced density, induced by various one-fluid models for α = 10: () Eq. (4), ( ) Eq. (5), (
) Eq. (9), (♦) Eq. (11) and () Eq. (13).

One-fluid model

AAD

Max

Bias

Linear (Eq. (4)) Square root (Eq. (5)) Linear viscosity (Eq. (9)) Logarithmic (Eq. (11)) Low density (Eq. (13)) This work (Eq. (19))

13.3 6.8 4.5 4.4 23.1 0.9

25.9 12.5 8.9 8.4 46.9 2.4

−13.3 6.8 −4.5 4.4 −23.1 0.7

G. Galli´ero et al. / Fluid Phase Equilibria 234 (2005) 56–63

61

Table 3 Deviations between one-fluid models efficiency for mass when used for viscosity estimation in various multicomponent mixtures at T* = 2 and ρ* = 0.5 One-fluid model

αi = 1, 2, 5; xi = 1/3

αi = 1, 2, 10; xi = 1/3

αi = 1, 5, 10; xi = 1/3

αi = 2, 5, 10; xi = 1/3

αi = 1, 2, . . ., 9, 10; xi = 1/10

Linear (Eq. (4)) Square root (Eq. (5)) Linear viscosity (Eq. (9)) Exponential (Eq. (11)) Low density (Eq. (13)) This work (Eq. (19))

−9.4 3.8 −3.9 1.6 −14.7 −0.7

−20.2 8.7 −7.4 4.8 −34.7 −0.1

−15 6.8 −6.3 4.4 −24.6 −0.3

−7.9 4.3 −2.9 2.3 −13.1 0.1

−9.7 2.0 −5.1 0.5 −14.5 −2.0

the total viscosity (kinetic + potential) one-fluid model for mass cannot be expressed without a density dependence parameter. In order to get an efficient one-fluid model for mass when applied to the determination of mixture viscosity, an empirical alternative law is proposed in this work. To achieve such a goal, Eqs. (4) or (9) is modified to include a density dependent exponent on mass. Because this density dependence is obviously not linear, see Figs. 1 and 2, a quadratic dependence in the reduced density, ρ* , is introduced to represent the behaviour: a+bρ∗ +cρ∗2

m1−fluid

=

n  i=1

a+bρ∗ +cρ∗2

xi mi

(19)

where a, b and c are numerical parameters adjusted on molecular dynamics and Chapman–Enskog results on viscosity of mixtures. For our results on equimolar mixtures, the best fit gives, a = 1.22, b = −3.19 and c = 2.35. Using these parameters an AAD = 0.87%, a M × D = 2.24% and a Bias = −0.03% are obtained, which is far better than all the tested models. Results are shown in Fig. 3 (using the same scale as Fig. 1) and can be favourably compared with those displayed on Fig. 1 for the linear model. 3.3. Binary nonequimolar mixtures To further study the influence and the efficiency of the one-fluid models for mass on LJ viscosity, we have performed simulations on nonequimolar binary mixtures where only the mass differs between the two components. Simulations have been carried out for 9 molar fraction between 0.1 and 0.9 with a step of 0.1 at T* = 2 and ρ* = 0.5 and for ␣ = 5 and 10. For Eq. (19), the values for a, b and c, are those previously determined on equimolar mixtures. The results are given in Appendix A. Fig. 4 (α = 10) shows that the deviations induced by the one-fluid models are not negligible, except for the model proposed in this work, and that, usually, a maximum of absolute deviations is found for x1 ≈ 0.6–0.7. Table 2 summarizes the results obtained. The new model, which has been adjusted only on binary equimolar results, seems to adequately represent the molar fraction dependence of the equivalent pseudo-compound mass, as shown in Table 2.

Fig. 4. Deviations on viscosity in nonequimolar binary mixtures, vs. the molar fraction of the lightest component, induced by various one-fluid models for α = 10 at T* = 2 and ρ* = 0.5: () Eq. (4), ( ) Eq. (5), (
) Eq. (9), (♦) Eq. (11), () Eq. (13) and (+) Eq. (19).

3.4. Multicomponent mixtures It is interesting to analyse what occurs in n-component mixtures (n > 2), because in most of the real cases such multicomponent mixtures are involved. Therefore simulations have been performed in four equimolar (xi = xj ) ternary mixtures and in one equimolar 10-component mixture at T* = 2 and ρ* = 0.5. The definitions of the various mixtures are given in Table 3. The results provided in Table 3 indicate that the general trends found for binary mixtures are confirmed in the tested multicomponent mixtures. The model proposed in this work is able to provide very consistent results even in these complex cases. Moreover, it is interesting to note that even for a nearly “continuous” mixture, the 10-component one, deviations obtained using some models, such as the linear one which is widely used, are not negligible.

4. Conclusion This work provides new insights on the one-fluid model for equivalent mass applied on Lennard–Jones mixtures through molecular dynamics simulations on equimolar, nonequimolar, binary, ternary and multicomponent mixtures in various thermodynamic states. Lennard–Jones fluids studied here (only the mass differs between components)

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put apart problems linked to the choice of a set of combining rules to define cross interaction parameters between unlike molecules, which complexify the problem. That allows focusing on the role of the mixture equivalent mass in the one-fluid approximation. Our NEMD data show that, in the one-fluid approximation, the mass of the pseudo-compound equivalent to the mixture is density dependent and weakly temperature dependent. Hence, the usual one-fluid models for mass (Eqs. (4), (5), (9), (11) and (13)) applied to estimate viscosity of mixtures are shown to yield non-negligible deviations compared to the direct computation of the viscosity  of the mixtures. In particular, the linear law, m1−fluid = ni=1 xi mi , is shown to be inappropriate. Such results may partly explain why efficient viscosity models for pure compounds fail for mixtures where mass ratios between compounds are large, even if complex (and sometimes fitted) mass mixing rules have been proposed in the literature [7,10,31,32]. Finally, an empirical density dependent model adjusted on equimolar molecular dynamics results is developed in this work, Eq. (19). This empirical one-fluid model is shown to provide results that are consistent with molecular dynamics results in various mixtures in all cases (gas, liquid, supercritical) as well as with results for the zero-density systems, the maximum deviation being always lower than 3%. Nevertheless, a one-fluid model should include also rules on the volume and on the energetic parameters of the molecules when applied to treat real fluids. So, as we may suspect couplings between the various one-fluid rules (on mass, volume and energy) [13,19], the application to real fluids of the proposed empirical rule is certainly not straightforward and is only a first step towards a complete one-fluid scheme for LJ fluids.

ρ*

T*

η∗pure

η∗α=2

η∗α=5

η∗α=10

0.3 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.9 0.9 0.9 0.9

2.5 1.5 2 2.5 1 1.5 2 2.5 1 1.5 2 2.5

0.402 0.578 0.622 0.671 1.228 1.236 1.239 1.257 4.157 3.231 2.985 2.764

0.400 0.572 0.612 0.652 1.194 1.192 1.198 1.232 4.075 3.141 2.924 2.731

0.377 0.505 0.558 0.601 1.075 1.067 1.081 1.107 3.811 2.969 2.639 2.494

0.362 0.466 0.507 0.552 0.962 0.964 0.966 0.993 3.439 2.705 2.418 2.260

Viscosity in nonequimolar mixtures at T* = 2 and ρ* = 0.5 obtained by NEMD simulations (αi = mi /m1 , η* is defined thanks to Eqs. (3–4)) x1

η∗α=5

η∗α=10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.622 0.608 0.596 0.583 0.564 0.558 0.549 0.548 0.554 0.581 0.622

0.622 0.605 0.582 0.543 0.523 0.507 0.502 0.494 0.500 0.527 0.622

Viscosity in multicomponent mixtures at T* = 2 and obtained by NEMD simulations

ρ* = 0.5

η*

αi = 1, 2, 5; xi = 1/3 0.568

αi = 1, 2, 10; xi = 1/3 0.517

αi = 1, 5, 10; xi = 1/3 0.541

αi = 2, 5, 10; xi = 1/3 0.576

αi = 1, 2, . . ., 9, 10; xi = 1/10 0.567

Acknowledgement This work is a part of the ReGaSeq project managed by TOTAL and the Institut Franc¸ais du P´etrole. We thank B. Duguay and J.-P. Caltagirone for fruitful discussions and the Centre Informatique National de l’Enseignement Sup´erieur in Montpellier (France) which provided a large part of the molecular dynamic simulation computer time for this study.

Appendix A Viscosity in equimolar binary mixtures obtained by NEMD simulations (αi = mi /m1 , η* is defined thanks to Eqs. (3) and (4)) ρ*

T*

η∗pure

η∗α=2

η∗α=5

η∗α=10

0.1 0.1 0.1 0.3 0.3

1.5 2 2.5 1.5 2

0.178 0.223 0.263 0.314 0.362

0.177 0.225 0.263 0.307 0.354

0.176 0.221 0.260 0.289 0.338

0.177 0.222 0.262 0.275 0.323

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