verbose *SI

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

Ian H. Bell

## 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, 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, CO2, SF6, 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 non-associating 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 measureable 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

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

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

$\eta^{\rm R}=\rho^{2/3}_{\rm N}\sqrt{mk_{\rm B}T},$ (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: $m$, time: $\rho_{\rm
N}^{-1/3}\sqrt{m/(k_{\rm B}T)}$, length: $\rho_{\rm N}^{-1/3}$ )[2, 3]. The
parameter $\rho_{\rm N}$ is the number density, not to be confused with the
molar density $\rho$, $m$ is the mass of one particle or molecule in kg,
$k_{\rm B}$ is the Boltzmann constant in J mol-1, $T$ is the temperature in
kelvins, and $-s^{\rm r}/R$ is the reduced residual entropy. For more on the
selected unit system and nomenclature, see the supporting information (SI) in
sec:units.

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

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

Over the last few years, a theoretical basis for the scaling effects that
Rosenfeld saw four decades ago has been developed with isomorph theory[5, 6,
7, 2, 8, 3]. 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 an
R-simple (Roskilde simple) fluid [3]. No molecular fluids are truly perfectly
correlating in the R-simple sense, and furthermore, this R-simple scaling may
only apply in part of the liquid domain, but this is a powerful theoretical
tool to understand the dynamic behavior of molecular fluids.

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 (see for instance [9, 10, 11, 12, 13, 14]) 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 [15, 13, 16, 17]. As is
highlighted by [15, 13], 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[18, 19, 20, 21, 22, 23] 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 [24,
25, 26, 27, 28, 29, 30].

This work investigates the hypothesis that the Rosenfeld-scaled viscosity
should in general be invariant along lines of residual entropy, as is proposed
by isomorph theory. The most comprehensive study to date of this hypothesis
based upon viscosities obtained from experimental measurements of molecular
fluids and molecular simulation of model potentials has been carried out. The
nearly monovariate relationship 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
result in a non-monovariate 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 are collapsed by a residual entropy corresponding states approach.
This residual entropy corresponding states is unlike the classical
corresponding states[31] in which the thermodynamic states, relative to the
respective critical points, are equated. 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. That is to say: residual
entropy is the scaling parameter that connects the thermodynamic and transport
properties of dense molecular fluids.

##  1 Molecular fluids

The term “molecular fluid” is used in this work to differentiate from model
intermolecular potentials that 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.

###  1.1 Fluid selection

Molecular fluids for this study were selected according to the availability
of:

  1. 1.

a significant body of high-quality experimental viscosity data covering most
of the liquid, gas, and supercritical states, and

  2. 2.

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:

  * •

monatomic gas (argon),

  * •

predominantly repulsive molecules (methane, carbon dioxide, and sulfur
hexafluoride),

  * •

halogenated refrigerants (R-134a and R-125) with electrostatic interactions
due to polarity,

  * •

strongly associating fluids (methanol and water)

Table 1 lists the equations of state that were employed in this work. All of
the equations of state are multiparameter reference equations. The NIST
REFPROP thermophysical property library [32] was used to carry out all 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 |  $T_{\rm max}$ (K) |  $p_{\rm max}$ (MPa)
---|---|---|---
Argon | [33] | 2000 | 1000
Methane | [34] | 625 | 1000
SF6 | [35] | 625 | 150
CO2 |  [36]a | 2000 | 800
R-134a | [37] | 455 | 70
R-125 | [38] | 500 | 60
Methanol | [39] | 620 | 800
Water | [40] | 2000 | 1000

a: The equation of state of Giordano et al. [41] is used above 800 MPa instead
of that of Span and Wagner[36].

If temperature and pressure are known for the experimental state point, the
density is iteratively obtained from the equation of state. With the exception
of carbon dioxide, for which the equation of state of [41] (see the SI Section
sec:Giordano for the use of this EOS) was used above the maximum pressure of
the equation of state of [36], any measurements carried out at pressures above
the stated maximum pressure of the EOS were excluded to avoid errors
associated with extrapolation.

In the SI, Fig. fig:coverage_overview shows the coverage of the experimental
viscosity data available for the studied fluids, and the limits of the
equations of state for these fluids. This figure demonstrates that there is
significant disparity in data coverage, even among the best-studied fluids.

###  1.2 Evaluation of $s^{\rm{r}}$

The state-of-the-art equations of state 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 $a^{\rm r}=RT\alpha^{\rm r}$
contributions, given as

$\alpha(\tau,\delta)=\frac{a}{RT}=\alpha^{0}(\tau,\delta)+\alpha^{\rm{r}}(\tau,\delta),$
(4)

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

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

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

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

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

For additional details of the the use of multiparameter EOS, the reader is
directed to the literature [42, 43, 44].

The residual entropy should not be confused with the term “excess” entropy
[45, 15], 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 non-interacting ideal gas[46]. To illustrate this property,
Fig. 1 shows contours of the reduced residual entropy for ordinary water,
where $-s^{\rm{r}}/R$ is evaluated from the equation of state of Wagner and
Pruß[40]. In the zero-density limit, $-s^{\rm{r}}/R$ is zero (no increase in
structure caused by molecular interactions), and as the density increases, so
does $-s^{\rm{r}}/R$. The maximum value for $-s^{\rm{r}}/R$ is found along the
melting line at the maximum pressure of the equation of state; this can be
intuitively understood as the state within the fluid domain where the fluid is
most structured.

Figure 1: Contours of the residual entropy $s^{\rm{r}}/R$ for water from the
equation of state of Wagner and Pruß[40]. The dashed curve is the line of
maximum pressure of the equation of state, the solid red curve is the melting
curve, and the solid black curve is the vapor-liquid co-existence curve (the
binodal).

###  1.3 Data Analysis

Figure 2: Overview of relationship between reduced viscosity and residual
entropy for the molecular fluids from a total of 12987 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.

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, either taken directly from the measurement or from an iterative
thermodynamic calculation of the equation of state. The residual entropy was
then evaluated at the specified molar density and temperature as described in
Section 1.2.

Figure 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 has a
characteristic, and roughly similar, shape. The data for other molecular
fluids (investigated but not discussed in this paper) also have the same
shape.

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 ($-s^{\rm{r}}/R\gtrsim 1$) at densities that are not “too high”. As
the intermolecular interactions qualitatively increase in intensity, (i.e.,
for the associating fluids), the Rosenfeld “universal” relationship does not
agree either qualitatively or quantitatively in the liquid-like phase.

Costigliola et al. [47] and others [46, 48, 49] 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 into a monovariate
dependency on the reduced residual entropy, with the exception of states
approaching the melting line, where the analysis of Ruppeiner and co-authors
(see Section 1.3.1) suggests a means of identifying the presence of hydrogen-
bonding networks from a high accuracy equation of state.

####  1.3.1 Liquids

For liquid-like states ($-s^{\rm{r}}/R\gtrsim 1$), the experimental data for
each non-associating molecular fluid (aside from some scatter in the
experimental measurements) collapse onto master curves – a monovariate
functional dependence. The curvature in semi-log coordinates differs depending
on the intermolecular interactions. In the case of argon, methane, SF6, CO2,
and the refrigerants R-134a and R-125, the liquid-like scaling is roughly
linear in semi-log coordinates. The experimental data for these fluids
(particularly for SF6 and CO2 and less so for R-134a and R-125) extend to the
melting line (see Fig. fig:coverage_overview in the SI).

For the associating fluids methanol and water, a more complicated functional
dependence is seen, particularly at large values of $-s^{\rm{r}}/R$. 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
$-s^{\rm{r}}/R$. 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[13] and viscosity[22] of water. The thermodynamic states where
these networks are present can be identified by states with positive
Riemannian curvature [50, 51, 52, 53].

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

$\ln\left[\frac{\eta}{\rho_{\rm N}^{2/3}\sqrt{mk_{\rm
B}T}}\cdot\left(-\frac{s^{\rm
r}}{R}\right)^{2/3}\right]=\sum_{i}c_{i}\left(-\frac{s^{\rm
r}}{R}\right)^{i}.$ (7)

Multiplication of the reduced viscosity by $(-s^{\rm r}/R)^{2/3}$ was used to
remove the divergence in the dilute-gas limit. The coefficients of the
polynomial fits are in the SI (Table tab:srfit_fits). Figure 3 shows the
deviations between the experimental data points and the fits obtained 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 ($-s^{\rm{r}}/R>0.5$). This shows
that the relationship between reduced viscosity and residual entropy is indeed
approximately monovariate (except for the associating fluids).

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 Fig. 2 and Fig. 3 indicate that this
monovariate relationship is present, even if the relationship might be
different for each fluid. As shown in Section 3, the mapping onto the results
of model intermolecular potentials can reduce the data so that non-associating
fluids can all be collapsed onto a single curve with one adjustable parameter.

Figure 3: Deviations from monovariability between data and fits from Eq. 7.
Deviation term is given by $\Delta\eta=100\times(\eta_{\rm fit}/\eta_{\rm
exp}-1)$ and dashed lines represent $\pm 10$%. The correlation is fit for
$-s^{\rm r}/R>0.5$, values below the range of the fit are impacted by the
extrapolation behavior of the fit and non-monovariate scaling.

####  1.3.2 Gas

In the gaseous domain where $-s^{\rm{r}}/R\lesssim 1$, 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 the SI (Fig.
fig:scaled_data_12987pts_gas_zoomed). This figure demonstrates two
deficiencies in the scaling proposed by Rosenfeld:

  * •

The scaling diverges at zero density (where $-s^{\rm{r}}/R=0$)

  * •

The gaseous region does not reduce to a monovariate dependence of reduced
viscosity with $-s^{\rm{r}}/R$

Other authors [24, 25, 26, 27, 28, 54, 29, 30] 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 non-
associating molecules and in the gaseous phase for associating molecules. For
that reason the alternative entropy scalings are not discussed in this work;
further work will investigate potential means of reconciling these different
scaling approaches.

The isomorph theory describes why the reduced viscosity should not be a
monovariate function of the residual entropy in the gas phase. In the gaseous
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.

##  2 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 $\eta^{*}=\eta\sigma^{2}/\sqrt{m\varepsilon}$ [55], in terms
of the reduced temperature $T^{*}=T/(\varepsilon/k_{\rm B})$ and reduced
density $\rho^{*}=\rho_{\rm ND}\sigma^{3}$, where $\varepsilon$ is an energy
scale, $\sigma$ is a length scale, and $\rho_{\rm ND}$ is the number density
in m-3; for more information on working in molecular simulation units, see
[56].

###  2.1 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 each 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. In the SI (Fig. fig:hard_sphere)
is a graphical representation of the scaling for the hard sphere, and the
curve of the reduced viscosity versus the residual entropy is also shown in
Fig. 4. The shape of the viscosity versus $-s^{\rm r}$ curve in scaled
coordinates bears significant but imperfect resemblance to that of the argon
data in Fig. 2.

###  2.2 Inverse-power pair potential

Real molecules are not rigid; they are more rubber ball than billiard ball. As
a result, it is more reasonable to treat molecules as soft spheres than hard
spheres. The inverse-power pair potential (IPP) is a repulsive potential
commonly used to model fluids with soft repulsive interactions given by
$U=\varepsilon\left(\sigma/r\right)^{n}$. The density and temperature are not
independent for the IPP potential [57, 58, 6]; they are linked via the scaling
variable $\gamma=\rho_{\rm N}\sigma^{3}(T^{*})^{-3/n}$.

The ratio of viscosity $\eta$ to $\rho_{\rm N}^{2/3}\sqrt{mk_{\rm B}T}$ is
then (see the SI)

$\frac{\eta}{\rho_{\rm N}^{2/3}\sqrt{mk_{\rm
B}T}}=\dfrac{\eta^{*}/(T^{*})^{n^{\prime}}}{\gamma^{2/3}},$ (8)

in which $n^{\prime}=(2/n)+(1/2)$. The simulation for the IPP potential is
carried out at specified pairs of $\rho^{*}=\rho_{\rm N}\sigma^{3}$ and
$T^{*}=Tk_{\rm B}/\varepsilon$, for which the simulation data are expressed in
terms of $\eta^{*}/(T^{*})^{n^{\prime}}$ as a function of the scaling variable
$\gamma$ (see [59] and Section sec:IPPdense). For the $n=12$ IPP potential,
the residual entropy is obtained from integration of the convergent virial
expansion given by [57]. For other values of $n$, the asymptotically
convergent approximation of [58] is used (see the SI Section sec:IPPsr for
further description of this method).

###  2.3 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

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

A number of researchers have carried out molecular simulation on the Lennard-
Jones 12-6 potential, and through application of the Green-Kubo formalism,
evaluated viscosities [60]. The coverage of the simulation results for the
Lennard-Jones fluid is shown in the SI Fig. fig:coverage_LJ. The most accurate
equation of state for the Lennard-Jones 12-6 potential is the one recently
developed by Thol et al. [61], which is valid up to $T^{*}=9$ and $p^{*}=65$,
where $p^{*}=p\sigma^{3}/\varepsilon$. Due to the availability of the
multiparameter EOS for the Lennard-Jones 12-6 potential[61], the same
methodology for the Lennard-Jones 12-6 potential is applied as with the
molecular fluids in Section 1.2 – for a given set of $T^{*}$, $\rho^{*}$,
$\eta^{*}$ from one simulation, the residual entropy is evaluated from the
equation of state as described in Eq. 6.

###  2.4 WCA

Weeks, Chandler, and Andersen (WCA) [62] 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
upwards by $\varepsilon$, or

$U=\left\\{\begin{array}[]{cc}4\varepsilon\left[\left(\dfrac{\sigma}{r}\right)^{12}-\left(\dfrac{\sigma}{r}\right)^{6}\right]+\varepsilon,&r\leq
2^{1/6}\sigma\\\ 0,&r>2^{1/6}\sigma.\end{array}\right.$ (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 between
molecules 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 $r\leq 2^{1/6}\sigma$. 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 equation of state reduces to a quasi-monovariate function of the
effective packing fraction, without a liquid phase or a critical point [63].

Analogously to the soft-sphere potential, an effective packing fraction (with
implicit temperature dependence) is defined by Heyes and Okumara [64] by
$\zeta_{\rm e}=\pi\rho^{*}\left(\sigma_{\rm e}/\sigma\right)^{3}/6$, with the
effective particle diameter given by $\sigma_{\rm
e}/\sigma=[2/(1+\sqrt{T^{*}})]^{1/6}$. Alternative effective particle diameter
models are described in the literature [65, 66, 67, 68]. The residual entropy
of the repulsive WCA potential is obtained by integration of the empirical
compressibility factor model proposed by Heyes and Okumara [64] (see
description in the SI, Section sec:WCA). 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
the SI. Of particular interest are the new simulation results from Krekelberg
provided in the SI with permission, Section sec:BillWCA..

###  2.5 Overview

Figure 4 presents the simulation results for all 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 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 data set
comprising four data points at zero shear rate from Ashurst and Hoover [69,
Table VI]; the present data coverage of results on the Lennard-Jones fluid
here 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.

Figure 4: Overlaid data for each of the model potentials studied in this work
(blue $\triangle$: Lennard-Jones 12-6 potential[70, 71, 72, 55, 73, 74, 75];
black $\lhd$: IPP with $n=12$ [59]; red $\bigcirc$: repulsive WCA potential;
yellow $\triangle$: Lennard-Jones data from Ashurst and Hoover[69] considered
by Rosenfeld [1]; orange dashed curve: hard sphere (Enskog theory plus
correction of [76]); black dashed line: correlation from Rosenfeld[4]). A
larger version of this figure is available in the SI (Fig.
fig:overlay_simulations_big).

##  3 Residual Entropy Corresponding States

Figures 2 and 4 demonstrate a remarkable similarity for the non-associating
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 $T^{*}$,
$\rho^{*}$, etc. by adjusting the parameters $\varepsilon/k_{\rm B}$ and
$\sigma$. Carrying out the appropriate cancellation results in

$\frac{\eta}{\rho_{\rm N}^{2/3}\sqrt{mk_{\rm
B}T}}=\frac{\eta^{*}}{\left(\rho^{*}\right)^{2/3}\sqrt{T^{*}}}$ (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 $\varepsilon/k_{\rm
B}$ and $\sigma$ adjust the residual entropy.

At this point, it is necessary to determine:

  * •

the most appropriate reference potential

  * •

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

  * •

the Lennard-Jones 12-6 potential behaves like a molecular fluid and has a
liquid phase, and also no convenient scaling variable such as the $\gamma$ of
the IPP potential,

  * •

the equation of state for the Lennard-Jones 12-6 potential has areas in the
unstable region between the spinodals where non-physical residual entropies
are obtained (see the SI),

  * •

no highly accurate viscosity correlation for the Lennard-Jones 12-6 potential
exists, though several empirical viscosity correlations of poorer accuracy are
available in the literature [77, 78].

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 $\zeta_{\rm e}$, and the
simulation results for the viscosity of the repulsive WCA potential lie within
the range of results from the Lennard-Jones simulations (see Fig. 4). Mapping
the properties onto the $n=12$ IPP potential was slightly less successful, as
described in the SI (Fig. fig:simscaled_non_hydrogen). The mapping onto the
hard-sphere potential was also carried out with the same methodology. The
hard-sphere mapping was not successful, as is shown in the SI (Fig.
fig:simscaled_non_hydrogen_HS).

The value of $\varepsilon/k_{\rm B}$ was set equal to the critical temperature
of the molecular fluid divided by 1.32 ($T_{\rm c}^{*}=1.32$ for the Lennard-
Jones equation of state[61]; 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 $\varepsilon/k_{\rm B}=T_{\rm{c}}$ and $\varepsilon/k_{\rm
B}=T_{\rm{c}}/0.7$ were also considered, as described in the SI; there is a
very weak dependence of $\sigma$ on $\varepsilon/k_{\rm B}$.

Table 2: Optimized values of $\sigma$ for the eight molecular fluids included in this study. Units of all variables are $10^{-10}$ m (Å) fluid | $v_{\rm N,triple}^{1/3}$ | $v_{\rm N,0.8T_{\rm{c}}}^{1/3}$ | $v_{\rm N,crit}^{1/3}$ | $\sigma_{\rm IPP}$ | $\sigma_{\rm WCA}$
---|---|---|---|---|---
Water | 3.104 | 3.334 | 4.529 | 3.084 | 2.973
Argon | 3.615 | 3.855 | 4.985 | 3.676 | 3.476
CO2 | 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
SF6 | 5.094 | 5.250 | 6.888 | 5.244 | 4.995
R-125 | 4.909 | 5.314 | 7.030 | 5.265 | 4.992

Each fluid was mapped onto the residual entropy of the reference potential. In
order to do this, a one-dimensional optimization of $\sigma$ was carried out
to minimize the difference between the Rosenfeld scalings at liquid-like
states. The approach is as follows:

  1. 1.

For a given molecular fluid experimental datapoint for which $-s^{\rm r}/R>1$,
calculate the reduced quantity $\eta/[\rho_{\rm N}^{2/3}\sqrt{mk_{\rm B}T}]$.

  2. 2.

At the same value of reduced viscosity for the repulsive WCA correlation,
calculate the corresponding value of $\zeta_{\rm e}$ for the repulsive WCA
potential; the correlation is monotonic.

  3. 3.

From $\zeta_{\rm e}$, calculate $\rho^{*}$ for the given $\sigma_{\rm
e}/\sigma$ from $\rho^{*}=6\zeta_{\rm e}(\sigma_{\rm e}/\sigma)^{3}/\pi$, and
then obtain $\sigma=(\rho^{*}/\rho_{\rm N})^{1/3}$.

The median value of $\sigma$ among all the experimental data points for which
a value of $\sigma$ is successfully obtained is retained; the median $\sigma$
was used in order to avoid the influence of outliers. It may not be possible
to obtain the value for $\sigma$ if $\eta/[\rho_{\rm N}^{2/3}\sqrt{mk_{\rm
B}T}]$ is below the minimum value of $\eta/[\rho_{\rm N}^{2/3}\sqrt{mk_{\rm
B}T}]\approx 0.57$ 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 the experimental data into the “simulation”
units of $T^{*}$, $\rho^{*}$, $\eta^{*}$.

Figure 5: Scaled experimental data mapped onto the repulsive WCA potential for
the non-associating fluids argon, methane, CO2, SF6, 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 $\pm 20\%$.

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. Figure 5
shows the scaled experimental data for the non-associating fluids; the results
for assocating fluids are shown in Fig. 6. In the case of the non-associating
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.

Figure 6: Scaled experimental data mapped onto the repulsive WCA potential for
the hydrogen-bonding fluids methanol and water. 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 $\pm 20\%$. The
blue markers are for methanol, and the green ones for water.

Figure 7 shows the deviations between the experimental viscosities and the
viscosities calculated by the fitted values of $\sigma$ for each of the non-
associating fluids described in Fig. 5. Within the recommended range of
validity ($0.5\lesssim-s^{\rm r}/R\lesssim 3.5$) of the repulsive WCA
potential, the deviations are in general less than 10% within the bulk of the
range, except at larger values of $-s^{\rm r}/R$, where the curvature of the
repulsive WCA potential results begins to move the correlation away from the
experimental data (see Fig. 5). Within the center of the region of validity,
the absolute deviations are in general less than 5%.

The same exercise was made for all the molecular fluids that a) have a
Helmholtz-energy-explicit equation of state available in the NIST REFPROP
thermophysical property library [32] and b) have experimental liquid viscosity
data available in the NIST ThermoData Engine #103b version 10.1 [79]. 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 kg kmol-1 (hydrogen) to 459
kg kmol-1 (MD4M).

The fitted values for $\sigma$ are shown in Fig. 8 as a function of three
characteristic volumes, those at the critical point, the liquid at the triple
point, and the saturated liquid at $0.8T_{\rm{c}}$. The fact that $\sigma^{3}$
should be proportional to the critical volume was originally proposed in
corresponding states theory [31, 16], and these data confirm this proposition.
A similar linear relationship between cube root of critical volume and length
scale was seen by Liu et al. [16] 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. In this
case the proportionality constant of the critical point volume scaling is
approximately 0.7, which is quite different than the value given by the Chung
model for extended corresponding states [80, 81] 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 $v^{1/3}_{\rm N,triple}$.
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
$0.8T_{\rm{c}}$ 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 the SI (Section
sec:altlengthscales) additional candidate length scales are further described,
including the length scale obtained from Noro-Frenkel universalism [82] and
the length scales obtained for $\varepsilon/k_{\rm B}=T_{\rm{c}}$ and
$\varepsilon/k_{\rm B}=T_{\rm{c}}/0.7$ for each reference potential.

Figure 7: Deviations between Rosenfeld-scaled experimental data mapped onto
the repulsive WCA potential and experimental data for the non-associating
fluids argon, methane, CO2, SF6, R-134a, and R-125. Absolute deviation is
given by $|\Delta\eta|=|(\eta_{\rm fit}/\eta_{\rm exp}-1)\times 100|$. The
colored rectangle is the approximate range of validity of this method, and the
dashed line indicates 10% Figure 8: Optimized values of $\sigma$ for each
fluid for the mapping to the repulsive WCA reference potential for
$\varepsilon/k_{\rm B}=T_{\rm{c}}/1.32$. The slightly transparent markers
correspond to the full set of fluids from NIST REFPROP and with experimental
viscosity data from NIST ThermoData Engine #103b version 10.1, and the solid
markers correspond to the fluids selected in Section 1. Dashed line for the
critical point is given by $\sigma=0.6786(v_{\rm N,crit})^{1/3}$, for the
triple point is given by $\sigma=(v_{\rm N,triple})^{1/3}$, and for the
saturated liquid at $0.8T_{\rm{c}}$ is given by by $\sigma=0.8984(v_{\rm
N,0.8T_{\rm{c}}})^{1/3}$.

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 $-s^{\rm r}/R$ than the experimental data scaled into simulation
units. The largest value of $\eta^{*}/[(\rho^{*})^{2/3}\sqrt{T^{*}}]$
available for the repulsive WCA potential is 4.3 for the highest density
simulation run of [83]. Therefore, the mapped data for
$\eta^{*}/[(\rho^{*})^{2/3}\sqrt{T^{*}}]>4.3$ represent a metastable
extrapolation of the repulsive WCA results into the solid phase that should be
considered with suspicion.

Finally, Fig. 9 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 #103b version 10.1. Nearly 50,000 experimental
data points are included in this collection. For each fluid, the optimized
value of $\sigma$ for each fluid is used. In the recommended range of validity
of the WCA potential ($0.5\lesssim-s^{\rm r}/R\lesssim 3.5$), 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 $-s^{\rm r}/R$. The fully predictive mode, where
$\sigma$ is taken from the correlation based upon the volume of the saturated
liquid at $0.8T_{\rm{c}}$, as described in Fig. 8, results in a poorer
representation of the experimental viscosity, as shown in the SI. In
predictive mode, in the same range of validity, 95% of the data points are
predicted within 46.5%.

Figure 9: Violin plots of the deviations in prediction of viscosity with the
optimized values of $\sigma$ for each fluid for the mapping to the repulsive
WCA reference potential for non-associating fluids. The range of $-s^{\rm
r}/R$ between 0.5 and 3.5 was split into bins of width 0.5. A violin
distribution was constructed (by matplotlib[84]) for the results in each bin.
The 97.5% and 2.5% percentiles are indicated with horizontal lines and the
marker is the median value. Experimental data points for $-s^{\rm r}/R$
greater than 3.5 or less than 0.5 are not shown, and in general correspond to
much larger deviations.

##  4 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 non-associating 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 also
shown to be nearly proportional to measureable 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 does 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 fluid is driven by the repulsive
interactions [62, 85, 3], 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. 8, 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 fit 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 worth continued research effort.

##  5 Supplemental Material

  * •

Detailed information on the data sources for each molecular fluid and model
potential

  * •

Derivations and description of the analysis required for the model potentials

  * •

Additional figures for completeness

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