Developing predictive thermal property models for liquids based on microscopic principles has been elusive. The difficulty is that liquids have gas-like and solid-like attributes that are at odds when considering the frameworks of microscopic models: Models for gases are simple due to randomness and low density, whereas models for crystalline solids rely on symmetry and long-range order for easier calculation. The short-range order in liquids does, however, provide structure to neighboring molecules similar to amorphous solids, and there have been recent advances indicating that collective vibrational modes store heat in liquids. Models combining Debye approximations from solid-state physics and Frenkel’s theory of liquids can accurately predict the heat capacity of liquids. Phonon-like dispersions in liquids have also been widely observed in neutron scattering experiments. These developments motivate us to propose a model where high-frequency vibrational modes, which travel at the speed of sound and have a mean free path on the order of the average intermolecular distance, conduct heat in liquids. We use this liquid phonon gas model to calculate the thermal conductivity of liquids with varying intermolecular interaction energies from strongest to weakest—Coulomb, hydrogen-bonding, Keesom, and London dispersion energy. Generally, the model is more accurate as the intermolecular interaction energy and density of liquids increase. The calculated thermal conductivity of Coulombic-bound molten sodium nitrate and hydrogen-bonded water is within 1.46% and 2.98% of the experimentally measured values, respectively, across their entire temperature ranges. Further modal analysis of the velocity and the mean free path of collective vibrations could establish the liquid phonon gas model as an accurate model for weakly interacting liquids as well.

1.
V. M. B.
Nunes
,
C. S.
Queirós
,
M. J. V.
Lourenço
,
F. J. V.
Santos
,
N.
de Castro
, and
C.
A
, “
Molten salts as engineering fluids—A review: Part I. Molten alkali nitrates
,”
Appl. Energy
183
,
603
611
(
2016
).
2.
C. K.
Ho
, “
Advances in central receivers for concentrating solar applications
,”
Sol. Energy
152
,
38
56
(
2017
).
3.
M.
Mehos
,
C.
Turchi
,
J.
Vidal
,
M.
Wagner
,
Z.
Ma
,
C.
Ho
,
W.
Kolb
,
C.
Andraka
, and
A.
Kruizenga
, “Concentrating Solar Power Gen3 Demonstration Roadmap,”
NREL Technical Report
, 2017.
4.
C. W.
Forsberg
,
P. F.
Peterson
, and
P. S.
Pickard
, “
Molten-salt-cooled advanced high-temperature reactor for production of hydrogen and electricity
,”
Nucl. Technol.
144
(
3
),
289
302
(
2003
).
5.
J.
Serp
,
M.
Allibert
,
O.
Beneš
,
S.
Delpech
,
O.
Feynberg
,
V.
Ghetta
,
D.
Heuer
,
D.
Holcomb
,
V.
Ignatiev
,
J. L.
Kloosterman
,
L.
Luzzi
,
E.
Merle-Lucotte
,
J.
Uhlíř
,
R.
Yoshioka
, and
D.
Zhimin
, “
The molten salt reactor (MSR) in generation IV: Overview and perspectives
,”
Prog. Nucl. Energy
77
,
308
319
(
2014
).
6.
M. L.
Huber
,
R. A.
Perkins
,
D. G.
Friend
,
J. V.
Sengers
,
M. J.
Assael
,
I. N.
Metaxa
,
K.
Miyagawa
,
R.
Hellmann
, and
E.
Vogel
, “
New international formulation for the thermal conductivity of H2O
,”
J. Phys. Chem. Ref. Data
41
(
3
),
033102
(
2012
).
7.
A. Z.
Zhao
,
M. C.
Wingert
, and
J. E.
Garay
, “
Frequency-domain hot-wire measurements of molten nitrate salt thermal conductivity
,”
J. Chem. Eng. Data
66
(
1
),
262
270
(
2021
).
8.
Ch. D.
Chliatzou
,
M. J.
Assael
,
K. D.
Antoniadis
,
M. L.
Huber
, and
W. A.
Wakeham
, “
Reference correlations for the thermal conductivity of 13 inorganic molten salts
,”
J. Phys. Chem. Ref. Data
47
,
033104
(
2018
).
9.
E.
Mclaughlin
, “
The thermal conductivity of liquids and dense gases
,”
Chem. Rev.
64
(
4
),
389
428
(
1964
).
10.
W.
Lv
and
A.
Henry
, “
Examining the validity of the phonon Gas model in amorphous materials
,”
Sci. Rep.
6
(
1
),
37675
(
2016
).
11.
E. S.
Toberer
,
L. L.
Baranowski
, and
C.
Dames
, “
Advances in thermal conductivity
,”
Annu. Rev. Mater. Res.
42
,
179
209
(
2012
).
12.
P. B.
Allen
,
J. L.
Feldman
,
J.
Fabian
, and
F.
Wooten
, “
Diffusons, locons and propagons: Character of atomic vibrations in amorphous Si
,”
Philos. Mag. B
79
(
11–12
),
1715
1731
(
1999
).
13.
F.
DeAngelis
,
M. G.
Muraleedharan
,
J.
Moon
,
H. R.
Seyf
,
A. J.
Minnich
,
A. J. H.
McGaughey
, and
A.
Henry
, “
Thermal transport in disordered materials
,”
Nanoscale Microscale Thermophys. Eng.
23
,
1
36
(
2018
).
14.
M. C.
Wingert
,
J.
Zheng
,
S.
Kwon
, and
R.
Chen
, “
Thermal transport in amorphous materials: A review
,”
Semicond. Sci. Technol.
31
(
11
),
113003
(
2016
).
15.
D. G.
Cahill
,
S. K.
Watson
, and
R. O.
Pohl
, “
Lower limit to the thermal conductivity of disordered crystals
,”
Phys. Rev. B
46
(
10
),
6131
6140
(
1992
).
16.
M. T.
Agne
,
R.
Hanus
, and
G. J.
Snyder
, “
Minimum thermal conductivity in the context of diffuson -mediated thermal transport
,”
Energy Environ. Sci.
11
(
3
),
609
616
(
2018
).
17.
Q.
Xi
,
J.
Zhong
,
J.
He
,
X.
Xu
,
T.
Nakayama
,
Y.
Wang
,
J.
Liu
,
J.
Zhou
, and
B.
Li
, “
A ubiquitous thermal conductivity formula for liquids, polymer glass, and amorphous solids
,”
Chin. Phys. Lett.
37
(
10
),
1
6
(
2020
).
18.
P. W.
Bridgman
, “
The thermal conductivity of liquids
,”
Proc. Natl. Acad. Sci. U.S.A.
9
(
10
),
341
345
(
1923
).
19.
P. W.
Bridgman
, “
The thermal conductivity of liquids under pressure
,”
Proc. Am. Acad. Arts Sci.
59
(
7
),
141
169
(
1923
).
20.
D.
Enskog
, “
Kungliga Svenska Vatenskapsakademiens Handlingar for English translation
,” in
Kinetic Theory
, edited by
S. G.
Brush
(
Pergamon Press
,
Oxford
,
1972
), Vol. 3 Ny Föld 1922, 63 (2).
21.
S.
Chapman
and
T. G.
Cowling
,
The Mathematical Theory of Non-Uniform Gases
(
Cambridge University Press London
,
1953
).
22.
M. G.
Velarde
, “
On the Enskog hard-sphere kinetic equation and the transport phenomena of dense simple gases
,” in
Transport Phenomena
, edited by
G.
Kirczenow
and
J.
Marro
(
Springer-Verlag
,
Berlin
,
1974
), pp.
289
336
.
23.
G.
Chen
,
Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons
(
Oxford University Press
,
New York
,
2005
).
24.
E. A.
Sykioti
,
M. J.
Assael
,
M. L.
Huber
, and
R. A.
Perkins
, “
Reference correlation of the thermal conductivity of methanol from the triple point to 660 K and up to 245 MPa
,”
J. Phys. Chem. Ref. Data
42
,
043101
(
2013
).
25.
M. J.
Assael
,
E. A.
Sykioti
,
M. L.
Huber
, and
R. A.
Perkins
, “
Reference correlation of the thermal conductivity of ethanol from the triple point to 600 K and up to 245 MPa
,”
J. Phys. Chem. Ref. Data
42
,
023102
(
2013
).
26.
M. J.
Assael
,
S. K.
Mylona
,
M. L.
Huber
, and
R. A.
Perkins
, “
Reference correlation of the thermal conductivity of toluene from the triple point to 1000 K and up to 1000 MPa
,”
J. Phys. Chem. Ref. Data
41
,
023101
(
2012
).
27.
M. J.
Assael
,
E. K.
Mihailidou
,
M. L.
Huber
, and
R. A.
Perkins
, “
Reference correlation of the thermal conductivity of benzene from the triple point to 725 K and up to 500 MPa
,”
J. Phys. Chem. Ref. Data
41
,
043102
(
2012
).
28.
K.
Trachenko
, “
Heat capacity of liquids: An approach from the solid phase
,”
Phys. Rev. B
78
(
10
),
104201
(
2008
).
29.
D.
Bolmatov
,
V. V.
Brazhkin
, and
K.
Trachenko
, “
The phonon theory of liquid thermodynamics
,”
Sci. Rep.
2
,
421
(
2012
).
30.
V. V.
Brazhkin
and
K.
Trachenko
, “
Collective excitations and thermodynamics of disordered state: New insights into an old problem
,”
J. Phys. Chem. B
118
,
11417
11427
(
2014
).
31.
V. V.
Brazhkin
and
K.
Trachenko
, “
Between glass and gas: Thermodynamics of liquid matter
,”
J. Non-Cryst. Solids
407
,
149
153
(
2015
).
32.
K.
Trachenko
and
V. V.
Brazhkin
, “
Collective modes and thermodynamics of the liquid state
,”
Rep. Prog. Phys.
79
(
1
),
016502
(
2016
).
33.
P.
Debye
, “
Zur Theorie Der Spezifischen Wärmen
,”
Ann. Phys.
344
(
14
),
789
839
(
1912
).
34.
C.
Kittel
,
Introduction to Solid State Physics
, 8th ed. (Wiley,
1996
).
35.
J.
Frenkel
,
Kinetic Theory of Liquids
(
Clarendon Press
,
Oxford
,
1946
).
36.
E.
Burkel
, “
Phonon spectroscopy by inelastic x-ray scattering
,”
Rep. Prog. Phys.
63
,
171
232
(
2000
).
37.
V. M.
Giordano
and
G.
Monaco
, “
Inelastic x-ray scattering study of liquid Ga: Implications for the short-range order
,”
Phys. Rev. B
84
(
4
),
52201
52202
(
2011
).
38.
S.
Hosokawa
,
S.
Munejiri
,
M.
Inui
,
Y.
Kajihara
,
W. C.
Pilgrim
,
Y.
Ohmasa
,
S.
Tsutsui
,
A. Q. R.
Baron
,
F.
Shimojo
, and
K.
Hoshino
, “
Transverse excitations in liquid Sn
,”
J. Phys.: Condens. Matter
25
(
11
),
112101
(
2013
).
39.
S.
Hosokawa
,
M.
Inui
,
Y.
Kajihara
,
S.
Tsutsui
, and
A. Q. R.
Baron
, “
Transverse excitations in liquid Fe, Cu and Zn
,”
J. Phys.: Condens. Matter
27
,
194104
(
2015
).
40.
A.
Cunsolo
, “
The terahertz dynamics of simplest fluids probed by inelastic x-ray scattering
,”
Int. Rev. Phys. Chem.
36
(
3
),
433
539
(
2017
).
41.
D. C.
Elton
and
M.
Fernández-Serra
, “
The hydrogen-bond network of water supports propagating optical phonon-like modes
,”
Nat. Commun.
7
(
1
),
10193
(
2016
).
42.
M. E.
Caplan
,
A.
Giri
, and
P. E.
Hopkins
, “
Analytical model for the effects of wetting on thermal boundary conductance across solid/classical liquid interfaces
,”
J. Chem. Phys.
140
(
15
),
154701
(
2014
).
43.
A. S.
Henry
and
G.
Chen
, “
Spectral phonon transport properties of silicon based on molecular dynamics simulations and lattice dynamics
,”
J. Comput. Theor. Nanosci.
5
(
7
),
1
12
(
2008
).
44.
H. R.
Seyf
,
K.
Gordiz
,
F.
Deangelis
, and
A.
Henry
, “
Using Green-Kubo modal analysis (GKMA) and interface conductance modal analysis (ICMA) to study phonon transport with molecular dynamics
,”
J. Appl. Phys.
125
(
8
),
081101
(
2019
).
45.
A.
Rahman
, “
Correlations in the motion of atoms in liquid argon
,”
Phys. Rev.
136
(
2A
),
A405
A411
(
1964
).
46.
K.
Skold
and
K. E.
Larsson
, “
Atomic motion in liquid argon
,”
Phys. Rev.
161
(
1
),
102
116
(
1967
).
47.
K. S.
Singwi
,
K.
Sköld
, and
M. P.
Tosi
, “
Zero sound in classical liquids
,”
Phys. Rev. Lett.
21
(
13
),
881
884
(
1968
).
48.
J. R. D.
Copley
and
S. W.
Lovesey
, “
The dynamic properties of monatomic liquids
,”
Rep. Prog. Phys.
38
(
4
),
461
563
(
1975
).
49.
I.
Ohmine
, “
Liquid water dynamics: Collective motions, fluctuation, and relaxation
,”
J. Phys. Chem.
99
(
18
),
6767
6776
(
1995
).
50.
T.
Scopigno
,
U.
Balucani
,
G.
Ruocco
, and
F.
Sette
, “
Evidence of two viscous relaxation processes in the collective dynamics of liquid lithium
,”
Phys. Rev. Lett.
85
(
19
),
4076
4079
(
2000
).
51.
H. J.
Bakker
and
J. L.
Skinner
, “
Vibrational spectroscopy as a probe of structure and dynamics in liquid water
,”
Chem. Rev.
110
(
3
),
1498
1517
(
2010
).
52.
K.
Yoshida
,
T.
Yamaguchi
,
T.
Yokoo
, and
S.
Itoh
, “
Collective dynamics measurement of liquid methanol by inelastic neutron scattering
,”
J. Mol. Liq.
222
,
395
397
(
2016
).
53.
K.
Amann-Winkel
,
M.-C.
Bellissent-Funel
,
L. E.
Bove
,
T.
Loerting
,
A.
Nilsson
,
A.
Paciaroni
,
D.
Schlesinger
, and
L.
Skinner
, “
X-ray and neutron scattering of water
,”
Chem. Rev.
116
,
7570
7589
(
2016
).
54.
E. W.
Lemmon
,
M. O.
McLinden
, and
D. G.
Friend
, “
Thermophysical properties of fluid systems
,” in
NIST Chemistry WebBook, NIST Standard Reference Database Number 69
, edited by
P. J.
Linstrom
and
W. G.
Mallard
(
National Institute of Standards and Technology
,
Gaithersburg
,
MD
, 2020), p.
20899
.
55.
G.
Ruocco
and
F.
Sette
, “
The high-frequency dynamics of liquid water
,”
J. Phys.: Condens. Matter
11
,
R259
R293
(
1999
).
56.
J. M.
Larkin
and
A. J. H.
McGaughey
, “
Thermal conductivity accumulation in amorphous silica and amorphous silicon
,”
Phys. Rev. B
89
(
14
),
144303
(
2014
).
57.
G. A.
Slack
, “
The thermal conductivity of nonmetallic crystals
,”
Solid State Phys.
34
,
1
71
(
1979
).
58.
B.
Jip Yoon
,
M.
Shik Jhon
, and
H.
Eyring
, “
Radial distribution function of liquid argon according to significant structure theory
,”
Proc. Natl. Acad. Sci. U.S.A.
78
(
11
),
6588
6591
(
1981
).
59.
A. K.
Soper
, “
The radial distribution functions of water and ice from 220 to 673 K and at pressures up to 400 MPa
,”
Chem. Phys.
258
,
121
137
(
2000
).
60.
M. C.
Wilding
,
M.
Wilson
,
M. C.
Ribeiro
,
C. J.
Benmore
,
J. K. R.
Weber
,
O. L. G.
Alderman
,
A.
Tamalonis
, and
J. B.
Parise
, “
The structure of liquid alkali nitrates and nitrites
,”
Phys. Chem. Chem. Phys
19
,
21625
(
2017
).
61.
R. D.
Shannon
, “
Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides
,”
Acta Crystallogr. Sect. A
32
(
5
),
751
767
(
1976
).
62.
A. E.
Gheribi
,
J. A.
Torres
, and
P.
Chartrand
, “
Recommended values for the thermal conductivity of molten salts between the melting and boiling points
,”
Sol. Energy Mater. Sol. Cells
126
,
11
25
(
2014
).
63.
J. N.
Israelachvili
,
Intermolecular and Surface Forces
(
Academic Press
,
2011
).
64.
M. C.
Wingert
,
A. Z.
Zhao
,
Y.
Kodera
,
S. J.
Obrey
, and
J. E.
Garay
, “
Frequency-domain hot-wire sensor and 3D model for thermal conductivity measurements of reactive and corrosive materials at high temperatures
,”
Rev. Sci. Instrum.
91
(
5
),
054904
(
2020
).
65.
Y.
Takahashi
,
R.
Sakamoto
, and
M.
Kamimoto
, “
Heat capacities and latent heats of LiNO3, NaNO3, and KNO3
,”
Int. J. Thermophys.
9
,
1081
1090
(
1988
).
66.
G. J.
Janz
,
C. B.
Allen
,
N. P.
Bansal
,
R. M.
Murphy
, and
R. P. T.
Tomkins
,
Physical Properties Data Compilations Relevant to Energy Storage. II. Molten Salts: Data on Single and Multi-Component Salt Systems
(United States Government Printing Office,
Washington
,
D.C.
,
1979
).
67.
K. Α
Tasidou
,
Ch. D.
Chliatzou
,
M. J.
Assael
,
K. D.
Antoniadis
,
S. K.
Mylona
,
M. L.
Huber
, and
W. A.
Wakeham
, “
Reference correlations for the viscosity of 13 inorganic molten salts
,”
J. Phys. Chem. Ref. Data
48
(
1
),
013101
(
2019
).
68.
R. W.
Higgs
and
T. A.
Litovitz
, “
Ultrasonic absorption and velocity in molten salts
,”
J. Acoust. Soc. Am.
32
,
1108
(
1960
).
69.
J. R.
Rumble
,
D. R.
Lide
and
T. J.
Bruno
, “
Dipole moments
,” in
CRC Handbook of Chemistry and Physics
, 100th ed., edited by
J. R.
Rumble
(
CRC Press
,
Boca Raton
,
FL
, 2019).
70.
J. R.
Lane
, “
CCSDTQ optimized geometry of water dimer
,”
J. Chem. Theory Comput.
9
(
1
),
316
323
(
2013
).
71.
A.
Das
,
P. K.
Mandal
,
F. J.
Lovas
,
C.
Medcraft
,
N. R.
Walker
, and
E.
Arunan
, “
The H2S dimer is hydrogen-bonded: Direct confirmation from microwave spectroscopy
,”
Angew. Chem. Int. Ed.
57
(
46
),
15199
15203
(
2018
).
72.
M. A.
Perkins
,
K. R.
Barlow
,
K. M.
Dreux
, and
G. S.
Tschumper
, “
Anchoring the hydrogen sulfide dimer potential energy surface to juxtapose (H2S)2 with (H2O)2
,”
J. Chem. Phys.
152
(
21
),
214306
(
2020
).
73.
C. M.
Lousada
and
P. A.
Korzhavyi
, “
The H2S dimer revisited—Insights from wave-function and density functional theory methods. Ab initio molecular dynamics simulations of liquid H2S
,”
Comput. Theor. Chem.
1180
,
112821
(
2020
).
74.
M. J.
Pedersen
,
W. B.
Kay
, and
H. C.
Hershey
, “
Excess enthalpies, heat capacities, and excess heat capacities as a function of temperature in liquid mixtures of ethanol + toluene, ethanol + hexamethyldisiloxane, and hexamethyldisiloxane + toluene
,”
J. Chem. Thermodyn.
7
(
12
),
1107
1118
(
1975
).
75.
J. A.
Schroeder
,
S. G.
Penoncello
, and
J. S.
Schroeder
, “
A fundamental equation of state for ethanol
,”
J. Phys. Chem. Ref. Data
43
,
043102
(
2014
).
76.
T. S.
Khasanshin
and
A. A.
Aleksandrov
, “
Thermodynamic properties of ethanol at atmospheric pressure
,”
J. Eng. Phys.
47
(
3
),
1046
1052
(
1984
).
77.
D.
Torii
,
T.
Nakano
, and
T.
Ohara
, “
Contribution of inter- and intramolecular energy transfers to heat conduction in liquids
,”
J. Chem. Phys.
128
,
044504
(
2008
).
78.
T.
Ohara
,
T. C.
Yuan
,
D.
Torii
,
G.
Kikugawa
, and
N.
Kosugi
, “
Heat conduction in chain polymer liquids: Molecular dynamics study on the contributions of inter-and intramolecular energy transfer
,”
J. Chem. Phys
135
,
034507
(
2011
).
79.
H.
Matsubara
,
G.
Kikugawa
,
T.
Bessho
,
S.
Yamashita
, and
T.
Ohara
, “
Effects of molecular structure on microscopic heat transport in chain polymer liquids
,”
J. Chem. Phys.
142
,
164509
(
2015
).
80.
H.
Matsubara
,
G.
Kikugawa
,
T.
Bessho
,
S.
Yamashita
, and
T.
Ohara
, “
Understanding the chain length dependence of thermal conductivity of liquid alcohols at 298 K on the basis of molecular-scale energy transfer
,”
Fluid Phase Equilib.
441
,
24
32
(
2017
).
81.
L.
Wang
,
C.
Yang
,
M. T.
Dove
,
A. V.
Mokshin
,
V. V.
Brazhkin
, and
K.
Trachenko
, “
The nature of collective excitations and their crossover at extreme supercritical conditions
,”
Sci. Rep.
9
(
1
),
1
9
(
2019
).
82.
R. E.
Powell
,
W. E.
Roseveare
, and
H.
Eyring
, “
Diffusion, thermal conductivity, and viscous flow of liquids
,”
Ind. Eng. Chem.
33
(
4
),
430
435
(
1941
).
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