It is shown that the residual entropy (entropy minus that of the ideal gas at the same temperature and density) is mostly synonymous with the independent variable of density scaling, identifying a direct link between these two approaches. The residual entropy and the effective hardness of interaction (itself a derivative at constant residual entropy) are studied for the Lennard-Jones monomer and dimer as well as a range of rigid molecular models for carbon dioxide. It is observed that the density scaling exponent appears to be related to the two-body interactions in the dilute-gas limit.

1.
J. C.
Dyre
, “
Perspective: Excess-entropy scaling
,”
J. Chem. Phys.
149
,
210901
(
2018
).
2.
D.
Fragiadakis
and
C. M.
Roland
, “
Intermolecular distance and density scaling of dynamics in molecular liquids
,”
J. Chem. Phys.
150
,
204501
(
2019
).
3.
I. H.
Bell
, “
Probing the link between residual entropy and viscosity of molecular fluids and model potentials
,”
Proc. Natl. Acad. Sci. U. S. A.
116
,
4070
4079
(
2019
).
4.
D.
Fragiadakis
and
C. M.
Roland
, “
Connection between dynamics and thermodynamics of liquids on the melting line
,”
Phys. Rev. E
83
,
031504
(
2011
).
5.
Y.
Rosenfeld
, “
Relation between the transport coefficients and the internal entropy of simple systems
,”
Phys. Rev. A
15
,
2545
2549
(
1977
).
6.
Y.
Rosenfeld
, “
A quasi-universal scaling law for atomic transport in simple fluids
,”
J. Phys.: Condens. Matter
11
,
5415
5427
(
1999
).
7.
M.
Dzugutov
, “
A universal scaling law for atomic diffusion in condensed matter
,”
Nature
381
,
137
139
(
1996
).
8.
I. H.
Bell
,
R.
Hellmann
, and
A. H.
Harvey
, “
Zero-density limit of the residual entropy scaling of transport properties
,”
J. Chem. Eng. Data
65
,
1038
1050
(
2019
).
9.
I. H.
Bell
,
R.
Messerly
,
M.
Thol
,
L.
Costigliola
, and
J. C.
Dyre
, “
Modified entropy scaling of the transport properties of the Lennard-Jones fluid
,”
J. Phys. Chem. B
123
,
6345
6363
(
2019
).
10.
X.
Yang
,
D.
Kim
,
E. F.
May
, and
I. H.
Bell
, “
Entropy scaling of thermal conductivity: Application to refrigerants and their mixtures
,”
Ind. Eng. Chem. Res.
60
,
13052
13070
(
2021
).
11.
X.
Yang
,
X.
Xiao
,
E. F.
May
, and
I. H.
Bell
, “
Entropy scaling of viscosity—III: Application to refrigerants and their mixtures
,”
J. Chem. Eng. Data
66
,
1385
1398
(
2021
).
12.
I. H.
Bell
, “
Entropy scaling of viscosity—II: Predictive scheme for normal alkanes
,”
J. Chem. Eng. Data
65
,
5606
5616
(
2020
).
13.
I. H.
Bell
, “
Entropy scaling of viscosity—I: A case study of propane
,”
J. Chem. Eng. Data
65
,
3203
3215
(
2020
).
14.
R.
Casalini
and
T. C.
Ransom
, “
On the experimental determination of the repulsive component of the potential from high pressure measurements: What is special about twelve?
,”
J. Chem. Phys.
151
,
194504
(
2019
).
15.
F.
Hummel
,
G.
Kresse
,
J. C.
Dyre
, and
U. R.
Pedersen
, “
Hidden scale invariance of metals
,”
Phys. Rev. B
92
,
174116
(
2015
).
16.
G.
Galliero
,
C.
Boned
, and
J.
Fernández
, “
Scaling of the viscosity of the Lennard-Jones chain fluid model, argon, and some normal alkanes
,”
J. Chem. Phys.
134
,
064505
(
2011
).
17.
C.
Alba-Simionesco
,
D.
Kivelson
, and
G.
Tarjus
, “
Temperature, density, and pressure dependence of relaxation times in supercooled liquids
,”
J. Chem. Phys.
116
,
5033
5038
(
2002
).
18.
L.
Bøhling
,
T. S.
Ingebrigtsen
,
A.
Grzybowski
,
M.
Paluch
,
J. C.
Dyre
, and
T. B.
Schrøder
, “
Scaling of viscous dynamics in simple liquids: Theory, simulation and experiment
,”
New J. Phys.
14
,
113035
(
2012
).
19.
A.
Sanz
,
T.
Hecksher
,
H. W.
Hansen
,
J. C.
Dyre
,
K.
Niss
, and
U. R.
Pedersen
, “
Experimental evidence for a state-point-dependent density-scaling exponent of liquid dynamics
,”
Phys. Rev. Lett.
122
,
055501
(
2019
).
20.
T. C.
Ransom
,
R.
Casalini
,
D.
Fragiadakis
, and
C. M.
Roland
, “
The complex behavior of the ‘simplest’ liquid: Breakdown of density scaling in tetramethyl tetraphenyl trisiloxane
,”
J. Chem. Phys.
151
,
174501
(
2019
).
21.
J. C.
Dyre
, “
Simple liquids’ quasiuniversality and the hard-sphere paradigm
,”
J. Phys.: Condens. Matter
28
,
323001
(
2016
).
22.
N.
Gnan
,
T. B.
Schrøder
,
U. R.
Pedersen
,
N. P.
Bailey
, and
J. C.
Dyre
, “
Pressure-energy correlations in liquids. IV. ‘Isomorphs’ in liquid phase diagrams
,”
J. Chem. Phys.
131
,
234504
(
2009
).
23.
N. P.
Bailey
,
U. R.
Pedersen
,
N.
Gnan
,
T. B.
Schrøder
, and
J. C.
Dyre
, “
Pressure-energy correlations in liquids. I. Results from computer simulations
,”
J. Chem. Phys.
129
,
184507
(
2008
).
24.
U. R.
Pedersen
,
N. P.
Bailey
,
T. B.
Schrøder
, and
J. C.
Dyre
, “
Strong pressure-energy correlations in van der Waals liquids
,”
Phys. Rev. Lett.
100
,
015701
(
2008
).
25.
T. S.
Ingebrigtsen
,
T. B.
Schrøder
, and
J. C.
Dyre
, “
What is a simple liquid?
,”
Phys. Rev. X
2
,
011011
(
2012
).
26.
H. W.
Hansen
,
A.
Sanz
,
K.
Adrjanowicz
,
B.
Frick
, and
K.
Niss
, “
Evidence of a one-dimensional thermodynamic phase diagram for simple glass-formers
,”
Nat. Commun.
9
,
518
(
2018
).
27.

cvT(∂s/∂T)ρ, so we may write cvrTsr/Tρ or cvr/RTs+/Tρ. A similar starting identity of (∂s/∂v)T = (∂p/∂T)v with v = 1/ρ yields the transformation of the numerator.

28.
L.
Costigliola
, “
Isomorph theory and extensions
,” Ph.D. thesis,
Roskilde University
,
Denmark
,
2016
.
29.
R.
Lustig
, “
Statistical analogues for fundamental equation of state derivatives
,”
Mol. Phys.
110
,
3041
3052
(
2012
).
30.
I. H.
Bell
, “
Effective hardness of interaction from thermodynamics and viscosity in dilute gases
,”
J. Chem. Phys.
152
,
164508
(
2020
).
31.
T.
Maimbourg
,
J. C.
Dyre
, and
L.
Costigliola
, “
Density scaling of generalized Lennard-Jones fluids in different dimensions
,”
SciPost Phys.
9
,
90
(
2020
).
32.
N.
Bailey
,
T.
Ingebrigtsen
,
J. S.
Hansen
,
A.
Veldhorst
,
L.
Bøhling
,
C.
Lemarchand
,
A.
Olsen
,
A.
Bacher
,
L.
Costigliola
,
U.
Pedersen
,
H.
Larsen
,
J.
Dyre
, and
T.
Schrøder
, “
RUMD: A general purpose molecular dynamics package optimized to utilize GPU hardware down to a few thousand particles
,”
SciPost Phys.
3
,
038
(
2017
).
33.
S.
Deublein
,
B.
Eckl
,
J.
Stoll
,
S. V.
Lishchuk
,
G.
Guevara-Carrion
,
C. W.
Glass
,
T.
Merker
,
M.
Bernreuther
,
H.
Hasse
, and
J.
Vrabec
, “
ms2: A molecular simulation tool for thermodynamic properties
,”
Comput. Phys. Commun.
182
,
2350
2367
(
2011
).
34.
C. W.
Glass
,
S.
Reiser
,
G.
Rutkai
,
S.
Deublein
,
A.
Köster
,
G.
Guevara-Carrion
,
A.
Wafai
,
M.
Horsch
,
M.
Bernreuther
,
T.
Windmann
,
H.
Hasse
, and
J.
Vrabec
, “
ms2: A molecular simulation tool for thermodynamic properties, new version release
,”
Comput. Phys. Commun.
185
,
3302
3306
(
2014
).
35.
G.
Rutkai
,
A.
Köster
,
G.
Guevara-Carrion
,
T.
Janzen
,
M.
Schappals
,
C. W.
Glass
,
M.
Bernreuther
,
A.
Wafai
,
S.
Stephan
,
M.
Kohns
,
S.
Reiser
,
S.
Deublein
,
M.
Horsch
,
H.
Hasse
, and
J.
Vrabec
, “
ms2: A molecular simulation tool for thermodynamic properties, release 3.0
,”
Comput. Phys. Commun.
221
,
343
351
(
2017
).
36.
R.
Fingerhut
,
G.
Guevara-Carrion
,
I.
Nitzke
,
D.
Saric
,
J.
Marx
,
K.
Langenbach
,
S.
Prokopev
,
D.
Celný
,
M.
Bernreuther
,
S.
Stephan
,
M.
Kohns
,
H.
Hasse
, and
J.
Vrabec
, “
ms2: A molecular simulation tool for thermodynamic properties, release 4.0
,”
Comput. Phys. Commun.
262
,
107860
(
2021
).
37.
B.
Widom
, “
Some topics in the theory of fluids
,”
J. Chem. Phys.
39
,
2808
2812
(
1963
).
38.
M. S.
Green
, “
Markoff random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids
,”
J. Chem. Phys.
22
,
398
413
(
1954
).
39.
R.
Kubo
, “
Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems
,”
J. Phys. Soc. Jpn.
12
,
570
586
(
1957
).
40.
Z.
Zhang
and
Z.
Duan
, “
An optimized molecular potential for carbon dioxide
,”
J. Chem. Phys.
122
,
214507
(
2005
).
41.
J. G.
Harris
and
K. H.
Yung
, “
Carbon dioxide’s liquid-vapor coexistence curve and critical properties as predicted by a simple molecular model
,”
J. Phys. Chem.
99
,
12021
12024
(
1995
).
42.
J.
Vrabec
,
J.
Stoll
, and
H.
Hasse
, “
A set of molecular models for symmetric quadrupolar fluids
,”
J. Phys. Chem. B
105
,
12126
12133
(
2001
).
43.
T.
Merker
,
C.
Engin
,
J.
Vrabec
, and
H.
Hasse
, “
Molecular model for carbon dioxide optimized to vapor-liquid equilibria
,”
J. Chem. Phys.
132
,
234512
(
2010
).
44.
J.
Errington
, “
The development of novel simulation methodologies and intermolecular potential models for real fluids
,” Ph.D. thesis,
Cornell University
,
1999
.
45.
R.
Hellmann
, “
Ab initio potential energy surface for the carbon dioxide molecule pair and thermophysical properties of dilute carbon dioxide gas
,”
Chem. Phys. Lett.
613
,
133
138
(
2014
).
46.
P.
Mausbach
,
A.
Köster
, and
J.
Vrabec
, “
Liquid state isomorphism, Rosenfeld-Tarazona temperature scaling, and Riemannian thermodynamic geometry
,”
Phys. Rev. E
97
,
052149
(
2018
).
47.
C. S.
Murthy
,
K.
Singer
, and
I. R.
McDonald
, “
Interaction site models for carbon dioxide
,”
Mol. Phys.
44
,
135
143
(
1981
).
48.
J. J.
Potoff
and
J. I.
Siepmann
, “
Vapor–liquid equilibria of mixtures containing alkanes, carbon dioxide, and nitrogen
,”
AIChE J.
47
,
1676
1682
(
2001
).
49.
D.
Möller
and
J.
Fischer
, “
Determination of an effective intermolecular potential for carbon dioxide using vapour-liquid phase equilibria from NpT + test particle simulations
,”
Fluid Phase Equilib.
100
,
35
61
(
1994
).
50.
J. J.
Potoff
,
J. R.
Errington
, and
A. Z.
Panagiotopoulos
, “
Molecular simulation of phase equilibria for mixtures of polar and non-polar components
,”
Mol. Phys.
97
,
1073
1083
(
1999
).
51.
N.
Chetty
and
V. W.
Couling
, “
Measurement of the electric quadrupole moments of CO2 and OCS
,”
Mol. Phys.
109
,
655
666
(
2011
).
52.
R. L.
Beil
and
R. J.
Hinde
, “
Ab initio electrical properties of CO2: Polarizabilities, hyperpolarizabilities, and multipole moments
,”
Theor. Chem. Acc.
140
,
120
(
2021
).
53.
I.
Bell
,
Archival version of potter
,
2022
.
54.
I. H.
Bell
,
J. C.
Dyre
, and
T. S.
Ingebrigtsen
, “
Excess-entropy scaling in supercooled binary mixtures
,”
Nat. Commun.
11
,
4300
(
2020
).
55.
M.
Thol
,
G.
Rutkai
,
A.
Köster
,
R.
Lustig
,
R.
Span
, and
J.
Vrabec
, “
Equation of state for the Lennard-Jones fluid
,”
J. Phys. Chem. Ref. Data
45
,
023101
(
2016
).
56.
R.
Span
and
W.
Wagner
, “
A new equation of state for carbon dioxide covering the fluid region from the triple point temperature to 1100 K at pressures up to 800 MPa
,”
J. Phys. Chem. Ref. Data
25
,
1509
1596
(
1996
).
57.
E. W.
Lemmon
,
I. H.
Bell
,
M. L.
Huber
, and
M. O.
McLinden
, NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 10.0, National Institute of Standards and Technology, http://www.nist.gov/srd/nist23.cfm,
2018
.
58.
T. L.
Hill
,
An Introduction to Statistical Thermodynamics
(
Dover Publications, Inc.
,
New York
,
1986
).
59.
N. P.
Bailey
,
U. R.
Pedersen
,
N.
Gnan
,
T. B.
Schrøder
, and
J. C.
Dyre
, “
Pressure-energy correlations in liquids. II. Analysis and consequences
,”
J. Chem. Phys.
129
,
184508
(
2008
).
60.
R. J.
Sadus
, “
Second virial coefficient properties of the n-m Lennard-Jones/Mie potential
,”
J. Chem. Phys.
149
,
074504
(
2018
).
61.
R. J.
Sadus
, “
Erratum: ‘Second virial coefficient properties of the n-m Lennard-Jones/Mie potential’ [J. Chem. Phys. 149, 074504 (2018)]
,”
J. Chem. Phys.
150
,
079902
(
2019
).
62.
S.
Polychroniadou
,
K. D.
Antoniadis
,
M. J.
Assael
, and
I. H.
Bell
, “
A reference correlation for the viscosity of krypton from entropy scaling
,”
Int. J. Thermophys.
43
,
6
(
2021
).

Supplementary Material

You do not currently have access to this content.