Probing the link between residual entropy and viscosity of molecular fluids and model potentials

Significance

We confirm, based on a large database of experimental measurements, Rosenfeld’s hypothesis from 1977 that the viscosity (a transport property) and the residual entropy (a thermodynamic property) are intimately connected in dense fluid phases. This study also provides a means to estimate viscosity with knowledge of only thermodynamic property information or to characterize a fluid’s full liquid viscosity surface based upon a very small number of high-accuracy experimental measurements.

Abstract

This work investigates the link between residual entropy and viscosity based on wide-ranging, highly accurate experimental and simulation data. This link was originally postulated by Rosenfeld in 1977 [Rosenfeld Y (1977) Phys Rev A 15:2545–2549], and it is shown that this scaling results in an approximately monovariate relationship between residual entropy and reduced viscosity for a wide range of molecular fluids [argon, methane, ${\mathrm{C}\mathrm{O}}_2$, ${\mathrm{S}\mathrm{F}}_6$, refrigerant R-134a (1,1,1,2-tetrafluoroethane), refrigerant R-125 (pentafluoroethane), methanol, and water] and a range of model potentials (hard sphere, inverse power, Lennard-Jones, and Weeks–Chandler–Andersen). While the proposed “universal” correlation of Rosenfeld is shown to be far from universal, when used with the appropriate density scaling for molecular fluids, the viscosity of nonassociating molecular fluids can be mapped onto the model potentials. This mapping results in a length scale that is proportional to the cube root of experimentally measurable liquid volume values.

In 1977 Rosenfeld (1) postulated a quasi-universal relationship between reduced transport properties and the reduced residual entropy. This analysis was based on the analysis of simulation data for hard spheres, the one-component plasma, and the Lennard-Jones 12-6 model potential in the liquid phase only. This scaling, here referred to as the Rosenfeld scaling, was of the form

$$\frac{\eta }{{\eta }^{\mathrm{R}}}=f\left(-\frac{s^{\mathrm{r}}}{R}\right),$$ [1]

where the reducing viscosity ${\eta }^{\mathrm{R}}$, in the same units as $\eta$, is given by

$${\eta }^{\mathrm{R}}={\rho }_{\mathrm{N}}^{2/3}\sqrt{m{k}_{\mathrm{B}}T},$$ [2]

which is obtained by scaling the viscosity in units of Pa$\cdot$s [with dimensions of mass/(length $\times$ time)] by the appropriate dimensional scaling parameters [for Newtonian dynamics, mass is $m$, time is ${\rho }_{\mathrm{N}}^{-1/3}\sqrt{m/\left(k_{\mathrm{B}}T\right)}$, and length is ${\rho }_{\mathrm{N}}^{-1/3}$] (2, 3). The parameter ${\rho }_{\mathrm{N}}$ is the number density, not to be confused with the molar density $\rho$; $m$ is the mass of one particle or molecule in kilograms; $k_{\mathrm{B}}$ is the Boltzmann constant in J$\cdot$mol⁻¹; $T$ is the temperature in kelvins; and $-s^{\mathrm{r}}/R$ is the reduced residual entropy. For more on the selected unit system and nomenclature, see SI Appendix, section 1.

Twenty-two years later, in 1999, Rosenfeld (4) proposed the “universal” correlation for viscosity given by

$$\frac{\eta }{{\eta }^{\mathrm{R}}}=0.2\exp \left(-0.8\frac{s^{\mathrm{r}}}{R}\right).$$ [3]

Empirical equations of a similar form have been obtained for a growing body of fluids and intermolecular potentials in dense phases (5–9).

Over the last few years, a theoretical basis for the scaling effects that Rosenfeld saw four decades ago has been developed with isomorph theory (2, 3, 10–14). This theory stipulates that the viscosity scaled in the manner of Eq. 1 should be invariant along lines of constant residual entropy if there is a high degree of correlation between fluctuations in the virial of the system and fluctuations in its intermolecular potential energy. A fluid that follows this behavior, even in some of its phase space, is referred to as a Roskilde-simple (R-simple) fluid (3). No molecular fluids are truly perfectly correlating in the R-simple sense, and furthermore, this R-simple scaling may apply only in part of the liquid domain, but this is a powerful theoretical tool to understand the dynamic behavior of molecular fluids. The recent review of Dyre (14) summarizes the state of the art in residual entropy scaling of transport properties.

Density scaling and residual entropy scaling are directly connected by isomorph theory (15–20). The reduced dynamic properties of fluids that can be modeled with inverse-power pair potentials scale with ${\rho }^{n/3}/T$, where $n$ is the exponent of the inverse-power pair potential (15, 19, 20); for the inverse-power pair potential (Model Potentials, Inverse-Power Pair Potential section), there is a one-to-one relationship between ${\rho }^{n/3}/T$ and the residual entropy (SI Appendix, Fig. S8).

Rosenfeld scaling can also be applied to dynamic properties like diffusivity; there are now a number of studies focused on Rosenfeld scaling of diffusivity from molecular simulation (for instance, refs. 21–26) due to the ease with which self-diffusion can be extracted from the results of molecular simulations. Studies considering the entropy scaling of experimental diffusivity measurements are growing in number as well (25, 27–29). As is highlighted by refs. 25 and 27, and also seen in this work in the case of viscosity, one of the limitations of the Rosenfeld scaling applied to self-diffusion is that unique curves are in general obtained for each species studied. The residual entropy corresponding-states approach proposed in this work should also apply to self-diffusion, allowing for harmonization of the self-diffusion studies that have been carried out thus far.

The Rosenfeld scaling of viscosity has been comparatively less studied. Abramson (30–35) was one of the first to consider the Rosenfeld scaling of his experimental viscosity data at very high pressures. Since then, modified Rosenfeld scaling of viscosity (reducing by the dilute-gas viscosity rather than Eq. 2) has also been successfully investigated (36–42).

This work investigates the hypothesis that the Rosenfeld-scaled viscosity should in general be invariant along lines of constant residual entropy, as is proposed by isomorph theory. A comprehensive study of this hypothesis based upon viscosities obtained from experimental measurements of molecular fluids and molecular simulation of model potentials is carried out here. The nearly monovariate relationship for nonassociating fluids between reduced viscosity and residual entropy in the liquid phase, where simple fluids are approximately R-simple, is shown. Furthermore, this monovariate scaling is shown to apply surprisingly well to hydrogen-bonding fluids approaching the melting line. Network forming (hydrogen bonding) tends to destroy the R-simple character of the fluid and is expected to result in a nonmonovariate scaling between reduced viscosity and residual entropy.

The model potentials show the same monovariate dependency of reduced viscosity on the residual entropy as the molecular fluids and deviate from this behavior in the same ways. The scaling of the molecular fluids and the model potentials collapse by a residual entropy corresponding-states approach. In this case, the residual entropy (the measure of structure of the fluid phase) is the parameter that must be corresponding for dynamic states to be equivalent.

Thermodynamic and transport properties have traditionally been considered independently. This work shows that residual entropy is the scaling parameter that connects the thermodynamic and transport properties of dense fluids.

While empirical viscosity models are usually complicated functions of temperature and density (43, 44), much simpler functional forms can be developed in terms of one variable, the residual entropy. This scheme offers the practical promise of a different approach for correlating the viscosity of fluids. Only a single variable (the residual entropy) is involved, and thus far fewer experimental data points would be required compared with a function depending on temperature and density. We have begun to apply residual entropy scaling with promising results.

Molecular Fluids

The term “molecular fluid” is used in this work to differentiate from model intermolecular potentials; model potentials are useful theoretical models but are not experimentally accessible in a laboratory. The study of molecular fluids in this section is indebted to the work of the experimental transport property community; without their tireless work, this study would not have been possible.

Fluid Selection.

Molecular fluids for this study were selected according to the availability of (i) a significant body of high-quality experimental viscosity data covering most of the liquid, gas, and supercritical states and (ii) a well-constructed equation of state for the thermodynamic properties that yields high-fidelity predictions of the residual entropy over the entire fluid range.

Unfortunately, there are not many fluids (perhaps 30) that meet these requirements. The selected molecular fluids represent the following classes: (i) a monatomic gas (argon), (ii) nonpolar molecules (methane, carbon dioxide, and sulfur hexafluoride), (iii) halogenated refrigerants [1,1,1,2-tetrafluoroethane (R-134a) and pentafluoroethane (R-125)] with electrostatic interactions due to polarity, and (iv) strongly associating fluids (methanol and water).

Table 1 lists the equations of state that were used in this work. All of the equations of state are multiparameter reference equations. The NIST REFPROP (National Institute of Standards and Technology Reference Fluid Properties) thermophysical property library (45) was used to carry out all of the calculations. The equations of state in REFPROP have been critically assessed and deemed to be the most reliable for the given fluid and all have been published in the literature.

Table 1.

Equations of state used in this work

Common name	EOS refs.	$T_{max}$, K	$p_{max}$, MPa
Argon	(46)	2,000	1,000
Methane	(47)	625	1,000
${\mathrm{S}\mathrm{F}}_6$	(48)	625	150
${\mathrm{C}\mathrm{O}}_2$	(49)†	2,000	800
R-134a	(50)	455	70
R-125	(51)	500	60
Methanol	(52)	620	800
Water	(53)	2,000	1,000

^†The equation of state of Giordano et al. (54) is used above 800 MPa instead of that of Span and Wagner (49).

If temperature and pressure are known for the experimental state point, the density is iteratively obtained from the equation of state (EOS). With the exception of carbon dioxide, for which the EOS of ref. 54 (see SI Appendix, section 2.B for the use of this EOS) was used above the maximum pressure of the EOS of ref. 49, measurements at pressures above the stated maximum pressure of the EOS were excluded to avoid errors associated with extrapolation.

In SI Appendix, Fig. S4 shows the coverage of the experimental viscosity data available for the studied fluids and the limits of the EOSs for these fluids. SI Appendix, Fig. S4 demonstrates that there is significant disparity in data coverage, even among the best-studied fluids.

Evaluation of $s^{\mathrm{r}}$.

The state-of-the-art EOSs for molecular fluids are Helmholtz-energy explicit with temperature and density as independent variables. In these formulations, the molar Helmholtz energy $a$ is expressed as a sum of the ideal-gas $a^0=RT{\alpha }^0$ and residual $a^{\mathrm{r}}=RT{\alpha }^{\mathrm{r}}$ contributions, given as

$$\alpha \left(\tau ,\delta \right)=\frac{a}{RT}={\alpha }^0\left(\tau ,\delta \right)+{\alpha }^{\mathrm{r}}\left(\tau ,\delta \right),$$ [4]

where the independent variables are the reciprocal reduced temperature $\tau =T_{\mathrm{c}}/T$ and the reduced density $\delta =\rho /{\rho }_{\mathrm{c}}$, and $T_{\mathrm{c}}$ and ${\rho }_{\mathrm{c}}$ are the critical temperature and molar density, respectively.

Expressed in terms of derivatives of $\alpha$, the molar entropy $s=-{\left(\partial a/\partial T\right)}_{\rho }$ is given by

$$\frac{s}{R}=\tau \left[{\left(\frac{\partial {\alpha }^0}{\partial \tau }\right)}_{\delta }+{\left(\frac{\partial {\alpha }^{\mathrm{r}}}{\partial \tau }\right)}_{\delta }\right]-{\alpha }^0-{\alpha }^{\mathrm{r}},$$ [5]

and the residual entropy $s^{\mathrm{r}}$ is the part of Eq. 5 that is based only on ${\alpha }^{\mathrm{r}}$ and its derivatives, resulting in

$$\frac{s^{\mathrm{r}}}{R}=\tau {\left(\frac{\partial {\alpha }^{\mathrm{r}}}{\partial \tau }\right)}_{\delta }-{\alpha }^{\mathrm{r}}.$$ [6]

For additional details of the use of multiparameter EOSs, the reader is directed to the literature (55–57).

The residual entropy should not be confused with the term “excess” entropy (27, 58), which refers to differences of mixture thermodynamics from ideal-solution behavior.

The residual entropy is defined as the part of the entropy that arises from the interactions among particles or molecules. This contribution is negative due to repulsive and attractive interactions that increase the structure beyond that of the noninteracting ideal gas (19). To illustrate this property, Fig. 1 shows contours of the reduced residual entropy for ordinary water, where $-s^{\mathrm{r}}/R$ is evaluated from the EOS of Wagner and Pruss (53). In the zero-density limit, $-s^{\mathrm{r}}/R$ is zero (no increase in structure caused by molecular interactions), and as the density increases, so does $-s^{\mathrm{r}}/R$. The maximum value for $-s^{\mathrm{r}}/R$ is found along the melting line at the maximum pressure of the EOS; this can be intuitively understood as the state within the fluid domain where the fluid is most structured.

Fig. 1.

Contours of the residual entropy $s^{\mathrm{r}}/R$ for water from the EOS of Wagner and Pruss (53). The dashed curve is the line of maximum pressure of the EOS, the solid red curve is the melting curve, and the solid black curve is the vapor–liquid coexistence curve (the binodal).

Data Analysis.

Experimental viscosity data were curated for a selection of fluids that experience more complex interactions than the simple model fluids investigated by Rosenfeld (1). For each experimental data point, the molar density was determined, taken either directly from the measurement or from an iterative thermodynamic calculation of the EOS given $T$ and $p$. The residual entropy was then evaluated at the specified molar density and temperature as described in Evaluation of s^r.

Fig. 2 shows the experimental viscosity data for the eight molecular fluids under study, with the Rosenfeld universal relationship overlaid for each fluid. The viscosity is reduced in the same manner as proposed by Rosenfeld (1). The data for each of the fluids in these scaled coordinates have a characteristic, and roughly similar, shape. The data for other molecular fluids (investigated but not discussed in this paper) also have the same shape.

Fig. 2.

Overview of relationship between reduced viscosity and residual entropy for the molecular fluids from a total of 12,987 experimental data points. The dashed line represents the universal scaling law of Rosenfeld (4). The data are vertically stacked by multiplying by increasing powers of 10.

For the fluids that are Lennard-Jones–like (e.g., argon or methane), the universal correlation of Rosenfeld captures the correct qualitative relationship between the viscosity and the residual entropy at liquid-like conditions ($-s^{\mathrm{r}}/R\gtrsim 1$) at moderate densities. As the intermolecular interactions qualitatively increase in intensity (i.e., for the associating fluids), the Rosenfeld universal relationship does not agree with the experimental data either qualitatively or quantitatively in the liquid-like phase.

Costigliola et al. (59) and others (19, 60, 61) suggest that water (and other associating fluids) should not have a monovariate viscosity scaling in terms of residual entropy in the liquid phase due to the presence of hydrogen-bonding networks. Fig. 2 shows that water does in fact demonstrate an approximate collapse of the reduced viscosity surface with monovariate dependency on the reduced residual entropy, with the exception of states approaching the melting line where the analysis of Ruppeiner (62) and Ruppeiner et al. (63) (Liquids section) suggests a means of identifying the presence of hydrogen-bonding networks from a high-accuracy EOS.

Liquids.

For liquid-like states ($-s^{\mathrm{r}}/R\gtrsim 1$), the experimental data for each nonassociating molecular fluid (aside from some scatter in the experimental measurements) collapse onto master curves—a monovariate functional dependence. The curvature in semilog coordinates differs, depending on the intermolecular interactions. In the case of argon, methane, ${\mathrm{S}\mathrm{F}}_6$, ${\mathrm{C}\mathrm{O}}_2$, and the refrigerants R-134a and R-125, the liquid-like scaling is roughly linear in semilog coordinates. The experimental data for these fluids (particularly for ${\mathrm{S}\mathrm{F}}_6$ and ${\mathrm{C}\mathrm{O}}_2$ and less so for R-134a and R-125) extend to the melting line (SI Appendix, Fig. S4).

For the associating fluids methanol and water, a more complicated functional dependence is seen, particularly at large values of $-s^{\mathrm{r}}/R$. While the reduced viscosity data are still a nearly monovariate function of the residual entropy, the curvature of the data increases at higher values of $-s^{\mathrm{r}}/R$. The pronounced increase in curvature can be ascribed to the presence of transient structures in the fluid caused by hydrogen-bonding networks in the bulk liquid phase. This pronounced curvature in the Rosenfeld-scaled viscosity is consistent with the behavior identified by other authors for diffusivity (25) and viscosity (34) of water. The thermodynamic states where these networks are present can be identified by states with positive Riemannian curvature (62–65).

To assess the monovariability of the relationship between the reduced viscosity and the residual entropy, polynomial correlations for each fluid at states from $-s^{\mathrm{r}}/R=0.5$ up to the melting curve of the fluid were developed in this work. The correlations are of the form

$$\ln \left[\frac{\eta }{{\rho }_{\mathrm{N}}^{2/3}\sqrt{m{k}_{\mathrm{B}}T}}\cdot {\left(-\frac{s^{\mathrm{r}}}{R}\right)}^{2/3}\right]=\sum _i{c}_i{\left(-\frac{s^{\mathrm{r}}}{R}\right)}^i.$$ [7]

Multiplication of the reduced viscosity by ${\left(-s^{\mathrm{r}}/R\right)}^{2/3}$ was used to remove the divergence in the dilute-gas limit, consistent with the theory of Rosenfeld (4, 14) that the reduced viscosity should be proportional to ${\left(-s^{\mathrm{r}}/R\right)}^{-2/3}$ in dilute gases. The coefficients of the polynomial fits are in SI Appendix, Table S9. Fig. 3 shows the deviations between the experimental data points and calculations from the fits from Eq. 7. Even though the absolute deviations are as much as 35% for the hydrogen-bonding fluids in the compressed liquid due to the breakdown in monovariability caused by hydrogen bonding, the average absolute deviation (AAD) for each fluid is less than 4.3% for $-s^{\mathrm{r}}/R>0.5$. This demonstrates that the relationship between reduced viscosity and residual entropy is indeed approximately monovariate (except for the associating fluids).

Fig. 3.

Deviations from monovariability between data and fits from Eq. 7. The deviation term is given by $\mathrm{\Delta }\eta =100\times \left({\eta }_{\mathrm{fi}\mathrm{t}}/{\eta }_{\exp }-1\right)$ and dashed lines represent $\pm 10$%. The correlation is fitted for $-s^{\mathrm{r}}/R>0.5$; values below the range of the fit are impacted by the extrapolation behavior of the fit and nonmonovariate scaling.

The practical implication of this is that if an EOS of sufficient accuracy is available for a fluid, a correlation of viscosity in terms of residual entropy can describe the viscosity surface.

These liquid-like data do not directly refute the analysis of Rosenfeld, who merely proposed a monovariate relationship between reduced viscosity and residual entropy. The results in Figs. 2 and 3 indicate that this monovariate relationship is present, even if the relationship might be different for each fluid. As shown in Residual Entropy Corresponding States, the mapping onto the results of model intermolecular potentials can reduce the data so that nonassociating fluids can all be collapsed onto a single curve with one adjustable parameter.

Gas.

In the gaseous domain where $-s^{\mathrm{r}}/R\lesssim 1$, there is a pronounced deviation from monovariate scaling, as is visible in Fig. 2 and more readily seen in the detailed view of this region in SI Appendix, Fig. S1, which demonstrates two deficiencies in the scaling proposed by Rosenfeld: (i) The scaling diverges at zero density (where $-s^{\mathrm{r}}/R=0$) and (ii) the gaseous region does not reduce to a monovariate dependence of reduced viscosity with $-s^{\mathrm{r}}/R$.

Other authors (36–42, 66) have proposed alternative residual-entropy–based schemes that are more successful at scaling the viscosity in the dilute gas limit, but they introduce significant deviations from monovariate scaling in the compressed liquid phase for small nonassociating molecules and in the gaseous phase for associating molecules. Examples of these difficulties for argon and water are shown in SI Appendix, Fig. S5. For that reason, alternative entropy scalings are not discussed in this work.

Isomorph theory describes why the reduced viscosity should not be a monovariate function of the residual entropy in the gas phase. In this region, the motion of the molecules is predominantly ballistic, aside from the infrequent interactions between molecules via collision. Therefore, the fluid should not be R-simple, isomorph scaling should be invalid, and reduced viscosity should not be a monovariate function of residual entropy.

Model Potentials

Model intermolecular potentials, and the simulation results that are obtained from these potentials, have much to teach us about transport properties of molecular fluids. After some general information, this section covers four model potentials.

The viscosity of single-site models in molecular simulations is in general given in the form ${\eta }^{*}=\eta {\sigma }^2/\sqrt{m\epsilon }$ (67), in terms of the reduced temperature $T^{*}=T/\left(\epsilon /k_{\mathrm{B}}\right)$ and reduced density ${\rho }^{*}={\rho }_{\mathrm{N}}{\sigma }^3$, where $\epsilon$ is an energy scale, $\sigma$ is a length scale, and ${\rho }_{\mathrm{N}}$ is the number density in m⁻³; for more information on working in molecular simulation units, see ref. 68.

Hard Sphere.

The hard-sphere model potential is a particularly simple one; rigid spherical particles have ballistic trajectories until they collide with another particle. The reduced viscosity of the hard-sphere potential, as well as its associated residual entropy, can be obtained as a function of the packing fraction $\zeta =\pi {\rho }^{*}/6$. The parameter $\zeta$ is not a function of temperature, but only a function of density. SI Appendix, Fig. S7 is a graphical representation of the scaling for the hard sphere, and the curve of the reduced viscosity vs. the residual entropy is also shown in Fig. 4. The shape of the viscosity vs. the $-s^{\mathrm{r}}/R$ curve in scaled coordinates bears significant but imperfect resemblance to that of the argon data in Fig. 2. The hard-sphere model has been used to develop a theoretical understanding of transport properties (5, 69–71). The other potentials studied here provide higher-fidelity representations of intermolecular interactions and are more suitable reference models for the corresponding-states approach described below.

Fig. 4.

Overlaid data for each of the model potentials studied in this work [blue $△$, Lennard-Jones 12-6 potential (67, 85–90); black $⊲$, IPP potential with $n=12$ (74); red $○$, repulsive WCA potential; yellow $△$, Lennard-Jones data from Ashurst and Hoover (84) (zero shear-rate extrapolation) considered by Rosenfeld (1); orange dashed curve, hard sphere (Enskog theory plus correction of ref. 91); black dashed line, correlation from Rosenfeld (4)]. A larger version of this figure is shown in SI Appendix, Fig. S14.

Inverse-Power Pair Potential.

Real molecules are not rigid; they are more like a rubber ball than a billiard ball. As a result, it is more reasonable to treat molecules as soft spheres than as hard spheres. The inverse-power pair (IPP) potential is a repulsive potential commonly used to model fluids with soft repulsive interactions given by $U=\epsilon {\left(\sigma /r\right)}^n$, where $r$ is the distance in meters and $n$ is an integer power. The density and temperature are not independent for the IPP potential (11, 72, 73); they are linked via the scaling variable $\gamma ={\rho }_{\mathrm{N}}{\sigma }^3{\left(T^{*}\right)}^{-3/n}$.

The ratio of viscosity $\eta$ to ${\rho }_{\mathrm{N}}^{2/3}\sqrt{m{k}_{\mathrm{B}}T}$ is then

$$\frac{\eta }{{\rho }_{\mathrm{N}}^{2/3}\sqrt{m{k}_{\mathrm{B}}T}}=\frac{{\eta }^{*}/{\left(T^{*}\right)}^{n\prime }}{{\gamma }^{2/3}},$$ [8]

in which $n\prime =\left(2/n\right)+\left(1/2\right)$. The simulation for the IPP potential is carried out at specified pairs of ${\rho }^{*}={\rho }_{\mathrm{N}}{\sigma }^3$ and $T^{*}=T{k}_{\mathrm{B}}/\epsilon$, for which the simulation data are expressed in terms of ${\eta }^{*}/{\left(T^{*}\right)}^{n\prime }$ as a function of the scaling variable $\gamma$ (ref. 74 and SI Appendix, section 3.C.2). For the $n=12$ IPP potential, the residual entropy is obtained from integration of the convergent virial expansion given by ref. 72. For other values of $n$, the asymptotically convergent approximation of ref. 73 is used (see SI Appendix, section 3.C.3 for further description of this method).

Lennard-Jones.

Real fluids interact by both attraction and repulsion (as well as long-range electrostatic interactions); the potential should capture this. The canonical example of a fluid with both attraction and repulsion is the Lennard-Jones 12-6 potential; it is given by

$$U=4\epsilon \left[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right].$$ [9]

A number of researchers have carried out molecular simulation on the Lennard-Jones 12-6 potential and evaluated viscosities through application of the Green–Kubo formalism (75). The coverage of the simulation results for the Lennard-Jones fluid is shown in SI Appendix, Fig. S10. The most accurate EOS for the Lennard-Jones 12-6 potential is the one recently developed by Thol et al. (76), which is valid up to $T^{*}=9$ and $p^{*}=65$, where $p^{*}=p{\sigma }^3/\epsilon$. Due to the availability of the multiparameter EOS for the Lennard-Jones 12-6 potential (76), the same methodology for the Lennard-Jones 12-6 potential is applied as with the molecular fluids in Inverse-Power Pair Potential—for a given set of $T^{*}$, ${\rho }^{*}$, and ${\eta }^{*}$ from one simulation, the residual entropy is evaluated from the EOS as described in Eq. 6.

Weeks, Chandler, and Andersen.

Weeks, Chandler, and Andersen (WCA) (77) proposed a means of deconstructing potentials into reference and attractive contributions. The reference part of the WCA deconstruction of the Lennard-Jones 12-6 potential results in a fully repulsive potential that has dynamic behavior similar to that of the Lennard-Jones 12-6 potential with shorter-ranged interactions. This reference potential is obtained by truncating the Lennard-Jones 12-6 potential at the location of its minimum value at $r=2^{1/6}\sigma$ and shifting the curve upward by $\epsilon$, or

$$U=\left\{\begin{array}{cc}\hfill 4\epsilon \left[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right]+\epsilon ,\hspace{1em}\hfill & \hfill r\le 2^{1/6}\sigma \hfill \\ \hfill 0,\hspace{1em}\hfill & \hfill r>2^{1/6}\sigma .\hfill \end{array}\right.$$ [10]

We refer to the reference part of the WCA deconstruction of the Lennard-Jones 12-6 potential as the “repulsive WCA potential” for concision.

The repulsive WCA potential retains the same intermolecular force as the Lennard-Jones 12-6 potential (the force between particles is the negative of the derivative of the potential with respect to position) within $r\le 2^{1/6}\sigma$.

The transport properties of the repulsive WCA potential are similar to those of the Lennard-Jones 12-6 potential, while having thermodynamic properties that are more straightforward to evaluate because the EOS reduces to a quasi-monovariate function of the effective packing fraction without a liquid phase or a critical point (78).

Analogously to the soft-sphere potential, an effective packing fraction (with implicit temperature dependence) is defined by Heyes and Okumura (79) by ${\zeta }_{\mathrm{e}}=\pi {\rho }^{*}{\left({\sigma }_{\mathrm{e}}/\sigma \right)}^3/6$, with the effective particle diameter given by ${\sigma }_{\mathrm{e}}/\sigma ={\left[2/\left(1+\sqrt{T^{*}}\right)\right]}^{1/6}$. Alternative effective particle diameter models are described in the literature (80–83). The residual entropy of the repulsive WCA potential is obtained by integration of the empirical compressibility factor model proposed by Heyes and Okumura (79) (SI Appendix, section 3.D). There is currently a scarcity of high-accuracy tabulated viscosity data for the repulsive WCA potential; however, sufficient data exist to develop the empirical correlation given in SI Appendix. Of particular interest are the new simulation results from Krekelberg provided in SI Appendix, section 3.D.1 with permission.

Overview.

Fig. 4 presents the simulation results for all of the model potentials included in our study. These data comprise the corpus of data for the Lennard-Jones 12-6 potential, simulation results for the IPP potential with $n=12$, results for the repulsive WCA potential, and the curve for the hard-sphere potential.

The universal scaling of Rosenfeld does not reproduce all of the Lennard-Jones simulation data in the liquid phase. In the work of Rosenfeld (4), he described good agreement with simulation results for the universal correlation. In reality, the correlation was compared with a single dataset comprising four data points at zero shear rate from Ashurst and Hoover (ref. 84, table VI); the present data coverage of results on the Lennard-Jones fluid is far more comprehensive. Rosenfeld’s curve of universal scaling might not have been quite right, but with the appropriate caveats, most repulsive-dominated potentials are remarkably consistent in the Rosenfeld scaling framework.

Residual Entropy Corresponding States

Figs. 2 and 4 demonstrate a remarkable similarity for the nonassociating fluids. The primary difference between fluids and potentials is the scaling of the residual entropy. Therefore, a means of connecting the molecular fluids and the model potentials is needed. This link is formed through the use of residual entropy corresponding states.

It is possible to map from experimental units into simulation units of $T^{*}$, ${\rho }^{*}$, etc., by adjusting the parameters $\epsilon /k_{\mathrm{B}}$ and $\sigma$. Carrying out the appropriate cancellation results in

$$\frac{\eta }{{\rho }_{\mathrm{N}}^{2/3}\sqrt{m{k}_{\mathrm{B}}T}}=\frac{{\eta }^{*}}{{\left({\rho }^{*}\right)}^{2/3}\sqrt{T^{*}}}$$ [11]

and, therefore, scaling properties from number density to ${\rho }^{*}$, from temperature to $T^{*}$, and from viscosity to ${\eta }^{*}$ will not change the Rosenfeld-reduced viscosity. On the other hand, modifying $\epsilon /k_{\mathrm{B}}$ and $\sigma$ adjusts the residual entropy.

At this point, it is necessary to determine (i) the most appropriate reference potential and (ii) a set of values for $\sigma$ for the mapping from a molecular fluid to a reference potential.

While the Lennard-Jones potential is appealing as a model potential, its use as the reference system for molecular fluids is problematic because (i) the Lennard-Jones 12-6 potential behaves like a molecular fluid and has a liquid phase (see the relevant phase diagrams in SI Appendix, Figs. S12 and S13) and also no convenient scaling variable such as the $\gamma$ of the IPP potential; (ii) the EOS for the Lennard-Jones 12-6 potential has areas in the unstable region between the spinodals where nonphysical residual entropies are obtained (SI Appendix, Fig. S11); and (iii) no highly accurate viscosity correlation for the Lennard-Jones 12-6 potential exists, although several empirical viscosity correlations of poorer accuracy are available in the literature (92, 93).

For these reasons, scaling onto the repulsive WCA potential was chosen; the repulsive WCA potential has a compressibility factor that is a monovariate function of the thermodynamic scaling parameter ${\zeta }_{\mathrm{e}}$, and the simulation results for the viscosity of the repulsive WCA potential lie within the range of results from the Lennard-Jones simulations (Fig. 4). Mapping the properties onto the $n=12$ IPP potential was slightly less successful, as described in SI Appendix, Fig. S19. The mapping onto the hard-sphere potential was also carried out with the same methodology. The hard-sphere mapping was not successful, as shown in SI Appendix, Fig. S22.

The value of $\epsilon /k_{\mathrm{B}}$ was set equal to the critical temperature of the molecular fluid divided by 1.32 [$T_{\mathrm{c}}^{*}=1.32$ for the Lennard-Jones EOS (76); the repulsive WCA potential is fully repulsive and therefore does not have a critical point] and $\sigma$ was left as an adjustable parameter. In this way, corresponding states between the Lennard-Jones analog (the repulsive WCA potential) and the molecular fluid are enforced. Values of $\epsilon /k_{\mathrm{B}}=T_{\mathrm{c}}$ and $\epsilon /k_{\mathrm{B}}=T_{\mathrm{c}}/0.7$ were also considered, as described in SI Appendix; there is a very weak dependence of $\sigma$ on $\epsilon /k_{\mathrm{B}}$.

Each fluid was mapped onto the residual entropy of the reference potential. To do this, a 1D optimization of $\sigma$ was carried out to minimize the difference between the Rosenfeld scalings at liquid-like states. The approach is as follows: (i) For a given molecular fluid experimental data point for which $-s^{\mathrm{r}}/R>1$, calculate the reduced quantity $\eta /\left[{\rho }_{\mathrm{N}}^{2/3}\sqrt{m{k}_{\mathrm{B}}T}\right]$. (ii) At the same value of reduced viscosity for the repulsive WCA correlation, calculate the corresponding value of ${\zeta }_{\mathrm{e}}$ for the repulsive WCA potential; the correlation is monotonic. (iii) From ${\zeta }_{\mathrm{e}}$, calculate ${\rho }^{*}$ for the given ${\sigma }_{\mathrm{e}}/\sigma$ from ${\rho }^{*}\hspace{0.17em}=\hspace{0.17em}6{\zeta }_{\mathrm{e}}{\left({\sigma }_{\mathrm{e}}/\sigma \right)}^3/\pi$ and then obtain $\sigma ={\left({\rho }^{*}/{\rho }_{\mathrm{N}}\right)}^{1/3}$. The median value of $\sigma$ among all of the experimental data points for which a value of $\sigma$ is successfully obtained is retained; the median $\sigma$ was used to avoid the influence of outliers. It may not be possible to obtain the value for $\sigma$ if $\eta /\left[{\rho }_{\mathrm{N}}^{2/3}\sqrt{m{k}_{\mathrm{B}}T}\right]$ is below the minimum value of $\eta /\left[{\rho }_{\mathrm{N}}^{2/3}\sqrt{m{k}_{\mathrm{B}}T}\right]\approx 0.57$ that can be achieved for the repulsive WCA potential. Once the value of $\sigma$ has been determined for a molecular fluid, this value can then be used to scale all of the experimental data into the “simulation” units of $T^{*}$, ${\rho }^{*}$, and ${\eta }^{*}$.

This approach was carried out for the eight molecular fluids discussed in this study, and the obtained values of $\sigma$ are given in Table 2. Fig. 5 shows the scaled experimental data for the nonassociating fluids; the results for associating fluids are shown in SI Appendix, Fig. S24. In the case of the nonassociating fluids, the qualitative agreement is surprisingly good; with the appropriate scaling, all experimental data can be closely mapped onto a master curve given by the repulsive WCA correlation. The repulsive WCA model potential does not perfectly match the Rosenfeld-scaled experimental data mapped onto the repulsive WCA potential, and it is evident that although the majority of the data in the liquid phase can be predicted within 20% (the dashed lines), the curvature of the mapped experimental data does not perfectly match the curvature of the correlation.

Table 2.

Optimized values of $\sigma$ for the eight molecular fluids included in this study

Fluid	$v_{\mathrm{N},\hspace{0.17em}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}}^{1/3}$	$v_{\mathrm{N},\hspace{0.17em}\mathrm{0.8}{\mathrm{T}}_{\mathrm{c}}}^{1/3}$	$v_{\mathrm{N},\hspace{0.17em}\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{1/3}$	${\sigma }_{\mathrm{I}\mathrm{P}\mathrm{P}}$	${\sigma }_{\mathrm{W}\mathrm{C}\mathrm{A}}$
Water	3.104	3.334	4.529	3.084	2.973
Argon	3.615	3.855	4.985	3.676	3.476
${\mathrm{C}\mathrm{O}}_2$	3.966	4.081	5.387	3.943	3.724
Methane	3.901	4.226	5.471	3.957	3.733
Methanol	3.892	4.311	5.739	4.178	3.950
R-134a	4.745	5.204	6.917	5.095	4.830
${\mathrm{S}\mathrm{F}}_6$	5.094	5.250	6.888	5.244	4.995
R-125	4.909	5.314	7.030	5.265	4.992

Units of all variables are 10⁻¹⁰ m (Ǻ).

Fig. 5.

Scaled experimental data mapped onto the repulsive WCA potential for the nonassociating fluids argon, methane, ${\mathrm{C}\mathrm{O}}_2$, ${\mathrm{S}\mathrm{F}}_6$, R-134a, and R-125. The residual entropies are evaluated for the repulsive WCA potential. The solid line is the correlation for the repulsive WCA potential, and the dashed lines show $\pm 20\%$.

Fig. 6 shows the deviations between the experimental viscosities and the viscosities calculated by the fitted values of $\sigma$ for each of the nonassociating fluids described in Fig. 5. Within the recommended range of validity ($0.5\lesssim -s^{\mathrm{r}}/R\lesssim 3.5$) of the repulsive WCA potential, the deviations are in general less than 10% within the bulk of the range, except at larger values of $-s^{\mathrm{r}}/R$, where the curvature of the repulsive WCA potential results begins to move the correlation away from the experimental data (Fig. 5). Within the center of the region of validity, the absolute deviations are in general less than 5%.

Fig. 6.

Deviations between Rosenfeld-scaled experimental data mapped onto the repulsive WCA potential and experimental data for the nonassociating fluids argon, methane, ${\mathrm{C}\mathrm{O}}_2$, ${\mathrm{S}\mathrm{F}}_6$, R-134a, and R-125. Absolute deviations are given by $\left|\mathrm{\Delta }\eta \right|=|\left({\eta }_{\mathrm{fi}\mathrm{t}}/{\eta }_{\exp }-1\right)\times 100|$. The colored rectangle is the approximate range of validity of this method, and the dashed line indicates 10%.

The same exercise was made for all of the molecular fluids that (i) have a Helmholtz-energy explicit EOS available in the NIST REFPROP thermophysical property library (45) and (ii) have experimental liquid viscosity data available in the NIST ThermoData Engine version 10.1 (94). The fluids included in this suite include hydrocarbons, refrigerants, siloxanes, noble gases, fatty-acid methyl esters, etc. In total, 120 fluids were included in the analysis, with molar masses ranging from 2 g$\cdot$mol⁻¹ (hydrogen) to 459 g$\cdot$mol⁻¹ (${\mathrm{M}\mathrm{D}}_4$M).

The fitted values for σ are shown in Fig. 7 as a function of three characteristic volumes: those at the critical point, the liquid at the triple point, and the saturated liquid at $0.8T_{\mathrm{c}}$. The fact that ${\sigma }^3$ should be proportional to the critical volume was originally proposed in corresponding-states theory (28, 95), and these data confirm this proposition. A similar linear relationship between the cube root of the critical volume and the length scale was seen by Liu et al. (28) with diffusivity data. It is remarkable that this behavior holds even for fluids that are associating (ethanol, water, etc.), for which this relationship is not expected to be followed. The proportionality constant of the critical point volume scaling is approximately 0.7, which is quite different from the value given by the Chung et al. (96, 97) model for extended corresponding states of 0.958. An even more remarkable relationship is found when the length-scaling parameter is plotted against the cube root of the volume of the liquid at the triple point; the length-scaling parameter is approximately equal to $v_{\mathrm{N},\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}}^{1/3}$. We currently have no theoretical explanation for this behavior. A third length scale based on the cube root of the volume of the saturated liquid at $0.8T_{\mathrm{c}}$ also results in a nearly linear functional dependence; this is a more meaningful liquid corresponding-states point than the triple point because the latter depends on solid-phase properties. In SI Appendix, section 4.A additional candidate length scales are further described, including the length scale obtained from Noro–Frenkel universalism (98) and the length scales obtained for $\epsilon /k_{\mathrm{B}}=T_{\mathrm{c}}$ and $\epsilon /k_{\mathrm{B}}=T_{\mathrm{c}}/0.7$ for each reference potential.

Fig. 7.

Optimized values of $\sigma$ for each fluid for the mapping to the repulsive WCA reference potential for $\epsilon /k_{\mathrm{B}}=T_{\mathrm{c}}/1.32$. The light-colored symbols correspond to the full set of fluids from NIST REFPROP and to experimental viscosity data from NIST ThermoData Engine no. 103b version 10.1, and the dark-colored symbols correspond to the fluids selected in Molecular Fluids. The dashed line for the critical point is given by $\sigma =0.6786{\left(v_{\mathrm{N},\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}\right)}^{1/3}$, that for the triple point is given by $\sigma ={\left(v_{\mathrm{N},\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}}\right)}^{1/3}$, and that for the saturated liquid at $0.8T_{\mathrm{c}}$ is given by $\sigma =0.8984{\left(v_{\mathrm{N},\mathrm{0.8}{\mathrm{T}}_{\mathrm{c}}}\right)}^{1/3}$.

While the mapping between experimental data and model potentials via the residual entropy is fruitful, one challenge is that the repulsive model potentials reach their respective solid–liquid-equilibrium curve at smaller values of $-s^{\mathrm{r}}/R$ than the experimental data scaled into simulation units. The largest value of ${\eta }^{*}/\left[{\left({\rho }^{*}\right)}^{2/3}\sqrt{T^{*}}\right]$ available for the repulsive WCA potential is 4.3 for the highest-density simulation run of ref. 99. Therefore, the mapped data for ${\eta }^{*}/\left[{\left({\rho }^{*}\right)}^{2/3}\sqrt{T^{*}}\right]>4.3$ represent metastable extrapolation of the repulsive WCA results into the solid phase that should be considered with caution.

Finally, Fig. 8 presents a set of violin plots for all of the experimental data for the full set of fluids in NIST REFPROP with experimental viscosity data in NIST ThermoData Engine no. 103b version 10.1. Nearly 50,000 experimental data points are included in this collection. The optimized value of $\sigma$ for each fluid is used. In the recommended range of validity of the WCA potential ($0.5\lesssim -s^{\mathrm{r}}/R\lesssim 3.5$), 95% of the data points are predicted within 18.1%, and the worst median error is 4.2% for the bin at the largest value of $-s^{\mathrm{r}}/R$. The fully predictive mode, where $\sigma$ is taken from the correlation based upon the volume of the saturated liquid at $0.8T_{\mathrm{c}}$, as described in Fig. 7, results in a poorer representation of the experimental viscosity, as shown in SI Appendix. In predictive mode and in the same range of validity, 95% of the data points are predicted within 46.5%.

Fig. 8.

Violin plots of deviations in the prediction of viscosity with the optimized values of $\sigma$ for each fluid for the mapping to the repulsive WCA reference potential for nonassociating fluids. The range of $-s^{\mathrm{r}}/R$ between 0.5 and 3.5 was split into bins of width 0.5. A violin distribution was constructed [by matplotlib (100)] for the results in each bin. The 97.5% and 2.5% percentiles are indicated with horizontal lines and the solid circle is the median value. Experimental data points for $-s^{\mathrm{r}}/R$ greater than 3.5 or less than 0.5 are not shown and in general correspond to much larger deviations.

Conclusions and Outlook

This work has demonstrated that the Rosenfeld scaling of viscosity allows the viscosity of pure fluids and model potentials to collapse to nearly monovariate functions of the residual entropy. This monovariability allows for a mapping from molecular fluid properties onto the properties of the model potentials for nonassociating molecular fluids. Thus, a theoretically grounded approach is demonstrated that connects model potentials and molecular fluids through the residual entropy. The scaling parameter $\sigma$ is shown to be nearly proportional to measurable length scales of the molecular fluids.

It is not conclusively shown that the WCA potential is the best possible model potential for the residual entropy corresponding states in viscosity; further study should consider whether other model potentials would be more suitable. For instance, it is seen that the scaled viscosity data in Fig. 5 do not have the same curvature as the repulsive WCA potential. A better potential would more faithfully represent the shape of the viscosity data in these scaled coordinates.

Ultimately, the connection between residual entropy and viscosity stems from the fact that viscosity is primarily governed by the repulsive interactions between molecules. The structure in the dense fluid is driven by the repulsive interactions (3, 77, 101), so if structure is the determinant of viscosity, and if structure can be quantified by the residual entropy, then it follows that the viscosity should be closely related to the residual entropy.

There are many molecular fluids for which no experimental viscosity data exist. This universal scaling approach, along with the scaling parameters of Fig. 7, can yield a reasonable estimate for viscosities of heretofore unmeasured fluids, as long as they are not associating. Or, if a small number of viscosity measurements are available, $\sigma$ could be fitted to those data points and the entire liquid viscosity surface accurately predicted within perhaps 20%. The mapping of associating fluids onto model potentials remains a challenging endeavor and is worth continued research effort.

Materials and Methods

SI Appendix includes detailed information on the literature data sources for each molecular fluid and model potential, mathematical derivations that complement the analysis in this paper, and additional figures and tables for completeness.